Nicholas Ayache

Thomas

Rocio Sophie

Krissy

Bishesh, Vikash, Flo, Florent, Mike, Loïc C Hakim Jan

Pawel Aziz

Mika Claire

Lucas

Julia Kris

Ben Then

Matteo Damien Louis

Montana Jan

Medical Imaging community. He showed me how mathematical tools -that I had learned in Theoretical Physics -can be used to address cutting-edge problems in this exciting eld.

This thesis develops Geometric Statistics for the quantitative analysis of organ shapes in Computational Anatomy. Statistics is a science that studies methods of inference from data which often belong to vector spaces, i.e. linear spaces. In contrast, Geometric statistics generalizes statistics for data belonging to manifolds, i.e. non-linear spaces. Non-linear data spaces emerge naturally in Computational Anatomy, where statistics aims to uncover the normal and pathological variability of the human anatomy. First, organ shapes -seen in medical images -can be modeled as deformations of the same template shape: they are elements of a Lie group of deformations, which is a manifold with an additional group structure. Second, organ shapes can be modeled as the equivalence classes of their 3D congurations under the action of a Lie group of transformations, e.g. of rotations and translations: they belong to a quotient space, which is a manifold with an additional stratication. Third, the medical images themselves can be represented as manifolds with an additional structure of horizontal distribution, when considering both the image's intensity and its gradient. Dening a metric or a distance to perform statistics on these manifolds might not be enough, as the metric or distance might not consistently account for the aforementioned additional structures. Theories of statistics on Riemannian manifolds and metric spaces need to be extended. This thesis advocates that Geometric statistics is a very natural extension that applies to manifolds with additional geometric structures in Computational Anatomy.

Our rst research axis tackles the denition of Geometric Statistics. The model of organ shapes as deformations of a template leads to the challenging problem of dening statistics on Lie groups. We provide an algorithm that constructs a (pseudo-)Riemannian metric compatible with the group structure, i.e. bi-invariant, when it exists. We nd that some Lie groups do not admit any such (pseudo-)metric. This result thus advocates for non-metric statistics on Lie groups. In our second axis, we use Geometric Statistics for the analysis of existing algorithms in Computational Anatomy. We reformulate the widespread algorithm of template shape computation with organ shapes modeled as elements of a quotient space. This geometric approach enables us to show its asymptotic bias. We illustrate our results on brain templates and suggest an improved algorithm. We then study algorithms that have a pre-processing registration step before a statistical analysis. We show again their bias and provide a geometric correction, by taking into account the quotient space stratication in addition to the Riemannian framework. Our third axis focuses on the applications of Geometric Statistics to medical image processing, like in-painting or anisotropic smoothing. Considering the image intensity with its gradient leads to statistics on manifolds with a horizontal distribution. We provide the mathematical basis to extend sub-Riemannian structures, currently used for 2D processing, to our 3D images. In all these aspects, this work develops Geometric Statistics well beyond Riemannian geometry.

Résumé : Les statistiques géométriques pour l'anatomie numérique

Cette thèse développe les statistiques géométriques pour l'analyse quantitative des formes d'organes en anatomie numérique. Les statistiques étudient les méthodes d'inférence à partir de données, qui appartiennent le plus souvent à des espaces vectoriels, i.e. des espaces linéaires. Les statistiques géométriques, elles, généralisent les statistiques pour des données appartenant à des variétés diérentielles, i.e. des espaces non-linéaires. Les espaces de données non-linéaires apparaissent naturellement en anatomie numérique, où l'on cherche à dévoiler la variabilité normale et pathologique de l'anatomie humaine. Premièrement, les formes d'un organevues dans les images médicales -peuvent être représentées par des déformations du même organe de référence: donc par des éléments du groupe de Lie des déformations, qui est une variété diérentielle avec une structure additionnelle de groupe. Deuxièmement, les formes d'un organe peuvent être représentées comme les classes d'équivalence de leur conguration tridimensionnelle sous l'action d'un groupe de Lie de transformations, par exemple de rotations et de translations: elles appartiennent donc à un espace quotient, qui est une variété diérentielle avec une structure additionnelle de stratication. Troisièmement, les images médicales elles-mêmes peuvent être représentées comme des variétés diérentielles avec une structure additionnelle de distribution horizontale, lorsque l'on considère à la fois l'intensité de l'image et son gradient. Dénir une métrique ou une distance pour faire des Statistiques sur ces variétés diérentielles n'est probablement pas susant : la métrique ou distance ne tient pas toujours compte des structures additionnelles mentionnées ci-dessus. Les théories de statistiques sur les variétés diérentielles Riemanniennes et les espaces métriques doivent donc être généralisées. Cette thèse défend les statistiques géométriques comme une extension très naturelle qui s'applique aux variétés différentielles avec des structures géométriques additionnelles en anatomie numérique.

Premièrement, nous dénissons les statistiques géométriques sur les groupes de Lie. Nous proposons une construction algorithmique de (pseudo-)métrique Riemannienne, compatible avec la structure de groupe, lorsqu'elle existe. Nous trouvons que certains groupes n'admettent pas de telle (pseudo-)métrique et défendons l'idée de statistiques non-métriques sur les groupes de Lie. Ensuite, nous utilisons les statistiques géométriques pour analyser l'algorithme de calcul d'organe de référence, reformulé avec des espaces quotient. Nous montrons son biais et suggérons un algorithme amélioré. Enn, nous appliquons les statistiques géométriques au traitement d'images, en généralisant les structures sous-Riemanniennes, utilisées en 2D, au 3D. A travers tous ces aspects, notre travail développe donc les Statistiques Géométriques bien au delà de la Géométrie Riemannienne.

Chapter 1

Introduction

Fundamental research in anatomy seeks to understand both healthy and pathological structure and functioning of our organs, in order to help the diagnosis and treatment of diseases. The better our anatomical knowledge, the better our medicine. This introduction presents an historical overview of research in anatomy, and shows how successive methods or technologies used for anatomical investigations -dissections, medical imaging, computer science -have enabled progress in medicine. We then turn to Computational Anatomy -the main purpose of this thesis -and describe the geometric nature of the data spaces involved in this eld. We argue for the need of a statistical theory beyond statistics on metric spaces and Riemannian manifolds, in order to handle them. Ultimately, we present the overview of our manuscript.

A brief history of medicine and anatomy

Medicine is the science and practice of the diagnosis, treatment, healing, and prevention of diseases. In the ancient world already, the Greek mythology had its own god of medicine: Asclepius, represented on Figure 1.1(left). His rod, a snake-entwined sta, was supposed to heal any disease and remains a symbol of medicine in today's western civilization, see Figure 1.1(middle). is often considered the "father of Western medicine", because he was the rst to argue that diseases were not caused by any action of the gods but rather by bad environmental factors, diet, and living habits:

It is thus with regard divine nor more sacred than other diseases, but has a natural cause from the originates like other aections. Men regard its nature and cause as divine from ignorance and wonder..." -Hippocrates, On the Sacred Disease.

His argument implied that humans could understand and act on their own health.

Hippocrates thus created Medicine as a science. However, the Greeks knew almost nothing about human anatomy because of a taboo forbidding human dissections. Their anatomical knowledge essentially came from the work of , who postulated theories by dissecting monkeys and pigs, but never human bodies. Some of these theories are now known to be incorrect, like Humorism, which is the belief that the balance four bodily uids , known as humors, inuences one's health. Even if incorrect, they remained uncontested for almost 1,300 years, until the Renaissance.

Medicine and Anatomy in the Medieval Age and Renaissance Anatomical knowledge was recognized as pivotal for medicine only during the Renaissance. Anatomy experienced a revolution led by Andreas Vesalius, see Figure 1.1 (right), who advocated that direct observation through human dissections was the only reliable resource, in a huge break with ancient taboos. He published a seven-volume De humani corporis fabrica (On the Fabric of the Human Body) showing printed descriptions and illustrations of human dissections.

Moreover, Vesalius defended the idea that humans were sharing a common inner structure, like a prototype anatomy. His illustrations of the human anatomy were meant to show this prototype, by a mental averaging of the anatomies he had seen during his dissections. Such a prototypical anatomy is still a central aspect of anatomy today, and especially in Computational Anatomy where it is called a template anatomy.

Human dissection can be seen as the rst revolution in the history of anatomical research. It represented a fundamentally new tool for understanding the human body's interior. Medical imaging can be seen as the second revolution.

Relying on medical imaging

Medical imaging and computer science Medical imaging was invented in the end of the 19th century, with the discovery of X-rays by Rontgen [START_REF] Roentgen | On a New Kind of Rays[END_REF]], Physics Nobel Prize in 1901. Today, medical imaging equipments are omnipresent in hospitals around the world, see Figure 1.2. Their number even increases: the number of MRIs in French hospitals has increased of 43% since 2010 and the number of PET scans has increased by 49,4% in four years from 2010 to 2014, see Figure 1.3 [START_REF] Chasseing | 'enquete de la Cour des comptes sur l'adaptation aux besoins des moyens materiels et humains consacres a l'imagerie medicale[END_REF]]. These devices produce medical images in several modalities: X-ray radiography or X-ray CT scans, magnetic resonance imaging (MRI), medical ultrasonography or ultrasound or echography, endoscopy, elastography, tactile imaging, thermography, medical photography and nuclear medicine functional imaging techniques such as positron emission tomography (PET) and Single-photon emission computed tomography (SPECT). Medicine heavily relies on these images, as shown on Figure 1.4: for the countries members of the OECD (Organization for Economic Cooperation and Development

These devices produce medical images in several modalities: X-ray radiography or X-ray CT scans, magnetic resonance imaging (MRI), medical ultrasonography or ultrasound or echography, endoscopy, elastography, tactile imaging, thermography, medical photography and nuclear medicine functional imaging techniques such as positron emission tomography (PET) and Single-photon emission computed tomography (SPECT). Medicine heavily relies on these images, as shown on Figure 1.4: for the countries members of the OECD (Organization for Economic Cooperation and Development) for instance, the number of MRIs exams is 52 for 1000 inhabitants on average and the number of scan exams is 120 per 1000 inhabitants on average in 2013. Medical imaging noninvasively produce databases of thousands of tridimensional visual representations of the human organs, bones and tissues (anatomy) as well as visual representations of the function of some organs or tissues (physiology). For example, the UK Biobank [Sudlow 2015] collects medical images of multiple organs at a population level, together with information such as lifestyle, demographic and genomic data. There are currently around 10,000 participants in the study. Each image data of such database already contains an abundance of information, which is illustrated in the size it takes in a computer's memory. For example, tridimensional X-ray CT videos at a rate of 2-3 images per second for 40 seconds already represent 200 GB: only 5 such sequences can be stored on a 1 terabyte hard disk.

Clinicians often rely on these images to decide on diagnosis, treatment and medical intervention. But their eyes and memory may not be enough to fully analyze the complexity of information shown in the images. For example the intensity in X-Ray images contains quantied information, the amount of radiation absorbed by a given tissue, which the eye cannot quantitatively processed.

There is a need of automatic tools designed to assist the clinicians in the analysis of medical images. The invention of the computer in the beginning of the 20th century and the explosion of computer science thereafter provides the technology to do so. For example in the US, the resources allocated specically for medical image computing are growing exponentially: as an example, the cloud computing market allowed specically for medical images computing shows a growth of 27% in less than 10 years, see Figure 1.5. Today's increasing availability of high-performance computing -for example GPU (Graphics Processing Unit) computing -makes it possible to extract clinically relevant information from the medical images. 1.2 Why Geometric Statistics for Computational Anatomy 1.2.1 What is Computational Anatomy?

Computational Anatomy aims to uncover the normal and pathological variability of the human anatomy, through the processing and analysis organ shapes shown on medical images [Miller 1997, Grenander 1993]. The eld is multidisciplinary as it ranges from mathematics to statistics, to computer science, to anatomy and medicine. In particular, Computational Anatomy considers questions such as: how dierent is the shape of the brain for patients with a risk for depression? or: can schizophrenia aect the visual aspect of the cerebral cortex? Computational Anatomy is one pillar of Computational Medicine, together with Computational Physiology, Computational Molecular Medicine and Computational Healthcare. As such, its direct application is the development of tools helping the daily medical practice. In fact, organ shapes often represent a statistically signicant predictors for various clinical parameters: the brain shape is corelated to neurodegenerative diseases like Alzheimer [Gerber 2010, Chou 2008, Baron 2001[START_REF] Lorenzi | [END_REF], Apostolova 2007] but also to other neurological and neuropsychiatric illnesses, including epilepsy [Eriksson 2001] and schizophrenia [Kubicki 2007, Brignell 2010] for example.

Therefore, a computational representation of the anatomy can serve the diagnosis of diseases from medical images. We may characterize the brain morphology associated with a particular pathology such as Alzheimer's disease and learn how to discriminate between groups of healthy and diseased subjects [START_REF] Lorenzi | [END_REF]]. Furthermore, Computational Anatomy contributes to the transition to a personalized medicine: for example, we can model the evolution of a brain shape through time [START_REF] Lorenzi | [END_REF], Singh 2013], and especially the evolution of a diseases like Alzheimer's for example [Hadj-Hamou 2016[START_REF] Khanal | A biophysical model of brain deformation to simulate and analyze longitudinal {MRIs} of patients with Alzheimer's disease[END_REF]].

Modeling the data in Computational Anatomy

Computational Anatomy relies on the statistical analysis of organ shapes shown in medical images. Organ shapes are complicated data which are usually modeled as elements of non-linear data spaces. More precisely, these data spaces are manifolds with additional geometric properties which we introduce here.

Manifolds with Lie group actions We rst consider the raw data, which are the medical images. They are intensity functions I over a tridimensional domain and as such: elements of a vector space, which is a linear manifold. The vector space is equipped with an additional structure of Lie group action as soon as we consider registration of images. The registration of two images is the problem of nding correspondences between the anatomies they show, i.e. to look for an optimal way of geometrically deforming one image into the other [Brown 1992]. The geometric deformation belongs to a Lie groups and is said to act on the image.

We can also put landmarks on the organ shown in the medical image [START_REF] Darmanté | Analyse biométrique de lánneau pelvien en 3 dimensions â à propos de 100 scanners[END_REF]], or consider the organ's surface, eventually represented with the framework of currents [START_REF] Durrleman | Modèles statistiques de courants pour mesurer la variabilité anatomique de courbes, de surfaces et de leur évolution[END_REF]]. The space of landmarks or the space of currents are vector spaces too. Likewise, geometric deformations can be used to register one set of landmarks to another [Joshi 2000], or one surface to another [Vaillant 2005]. In all these cases, the anatomical data are modeled as elements of a manifold with a Lie group action.

Lie groups We can deform one anatomy onto another through the process of registration. If we x one anatomical image as a reference -called the template -we can represent all other images as deformations of this template [Amit 1991], i.e. as an element of a Lie group [Thompson 1992]. The same works with sets of landmarks or surfaces. In a longitudinal study, the template may be the rst image of the time line and all other images are represented as elements of a Lie group.

The Lie groups may be nite-dimensional, like rigid or ane transformations [Ashburner 2003, Boisvert 2008] or innite-dimensional, like Lie groups of dieomorphisms [START_REF] Trouvé | Dieomorphisms Groups and Pattern Matching in Image Analysis[END_REF][START_REF] Dupuis | [END_REF], Beg 2005].

Quotient spaces We consider the raw data: the medical image, or alternatively sets of landmarks or surfaces. Some of the Lie group acting on these data represent information that have no impact for the analysis of the organ's shapes. For example rotations or translations of the anatomical shape may only denote the fact that patients were imaged in dierent positions. In fact, the shape of a set of landmarks, the shape of a surface, or the shapes in an image can be dened as the remainder after we have ltered out the position and the orientation of the object [Kendall 1984, Small 1989, Bookstein 1997, Dryden 1998]. The resulting data space of (anatomical) shapes is thus a quotient space, called the shape space [START_REF] S Huckemann | Intrinsic shape analysis: Geodesic principal component analysis for Riemannian manifolds modulo Lie group actions[END_REF]].

An important literature has been developed for shape spaces of landmarks, especially in 2D and 3D [I. L. Dryden 1991[START_REF] Dryden | [END_REF], Goodall 1993[START_REF] Weinberger | [END_REF]. In this case, the landmarks congurations can be represented as matrices and linear algebra operations are performed to handle these data: this line of work is known as Procrustean analysis [Goodall 1991, Dryden 1998, Gower 2004].

Horizontal distributions Lastly, we move away from models of shapes applied to the analysis of medical images: we consider models involved in the processing of images. We consider one image, and the data are now the voxels. By default, the voxels belong to the tridimensional domain of the image, which is included in the vector space R 3 : the voxels are elements of a linear space. Recent studies oer to embed this space into a larger space which is a manifold [START_REF] Citti | [END_REF], Petitot 2013]. Instead of considering only the position of the voxel in R 3 , we also consider the orientation of the intensity gradient at this position. The domain of the image is now a lifted space of positions and directions, i.e. it has a structure of horizontal distribution. Considering this extended domain may lead to powerful image processing, like in-painting [Boscain 2014] or anisotropic smoothing [Duits 2010a].

Analyzing the data in Computational Anatomy

Data in Computational Anatomy are often modeled as elements of manifolds. Eorts have thus being engaged to generalize usual statistics on linear spaces to a theory of statistics on manifolds [Bhattacharya 2012, Patrangenaru 2015].

Theories of statistics on manifolds A usual procedure -referred to as extrinsic -is to embed the manifold in a large enough vector space of parameters.

Then, the Euclidean distance dened on the embedding space induces a distance on the embedded manifold which can be used in evaluating estimators on a manifold [Hendriks 1991]. This approach is often chosen in Directional Statistics, for example statistics on rotations in R n [START_REF] Jupp | [END_REF], Mardia 1999], where the manifolds considered here are spheres and projective spaces which are easily embedded.

In contrast, the intrinsic approach does not rely on an embedding. An intrinsic distance may be determined by the experimental setting and denes a metric structure on the manifold [Fréchet 1948]. One can generalize denitions from statistics on Euclidean space to statistics on metric spaces, by replacing the Euclidean distance by the intrinsic distance: for example, the mean of a dataset may be dened as the global minimum of the sum of the squared intrinsic distances to the data, a Chapter 1. Introduction 8 denition known as the Fréchet mean [Fréchet 1948]. Several means may be dened as the local minima of the same sum, a denition known as the Karcher means [Karcher 1977]. Statistical asymptotic properties of such means have been studied in [Bhattacharya 2003, Bhattacharya 2005]. Statistics on metric spaces have been employed successfully, for example in Phylogenetics and its tree spaces [Billera 2001].

The intrinsic distance may also be induced by a stronger structure on the manifold: a Riemannian structure, where one has an inner product dened at each tangent space of the manifold [Postnikov 2001, Pennec 2006]. First, one may use the denitions of the statistical theory on the induced metric spaces. But one may also rely on the additional structures provided by the Riemannian framework. One may use the Levi-Civita connection associated to the Riemannian metric and dene the mean as an Exponential Barycenter [START_REF] Émery | [END_REF][Émery , Corcuera 1998]. The mean can be seen as the 0-dimensional space which best represents the data set. One may consider higher dimensional subspaces that describe the dataset, and perform Principal Geodesic Analysis proposed in [Fletcher 2004] and implemented in [Sommer 2014], Geodesic Principal Component Analysis [START_REF] S Huckemann | Intrinsic shape analysis: Geodesic principal component analysis for Riemannian manifolds modulo Lie group actions[END_REF] or Barycentric Subspaces [Pennec 2015]. Regression techniques may also be generalized to Riemannian manifolds, for example on manifolds dening brain shapes [Davis 2010].

Towards Geometric Statistics Shall we use the theories of statistics on metric spaces or on Riemannian manifolds for the data spaces of Computational Anatomy detailed in the previous subsection? These data spaces possess additional structures, like the action of a Lie group, a Lie group structure, an horizontal distribution. We shall acknowledge this additional structures to build, and use, a consistent new statistical theory.

We take one example to illustrate the need of enforcing the compatibility between the statistical theory and the structures on the manifolds. We consider the denition of the mean on Lie groups. We could construct a distance or a Riemannian metric on the group and apply statistics on metric spaces or Riemannian manifolds, using for example the Fréchet mean. To respect the Lie group structure, it would be judicious to have a notion of mean that is stable by the group composition, by the left and by the right: so that the mean of a translated dataset is the translation of the original dataset's mean. This property is however not enforced in general [Pennec 2012]. The Ph.D thesis describes these situations where the consistency between statistics and geometry needs to be studied.

What is Geometric Statistics?

Geometric Statistics is a theory of statistics on manifolds, that generalizes statistics on metric spaces and statistics on Riemannian spaces to take into account other additional geometric structures on manifolds. "Geometric Statistics" is an appellation towards which the community seems to converge to design these research eorts. This Ph.D thesis will present Geometric Statistics for Computational Anatomy. So let rst present other theories that merge Statistics and Dierential Geometry but still are dierent from Geometric Statistics, i.e. let rst present what Geometric Statistics is not.

Geometric Statistics is dierent from Information geometry [START_REF] Amari | Dierential-geometrical methods in statistics[END_REF]], which applies techniques of dierential geometry to probability theory. A probability distribution is modeled as a point of a Riemannian manifold, called a statistical manifold.

In contrast in Geometric Statistics, Dierential Geometry is not used to represent a space of probability distributions of some data but rather to represent the space of the data themselves.

Geometric Statistics is also dierent from Manifold Learning [Lee 2007], which is a subeld of Machine Learning focusing on non-linear dimensionality reduction. Manifold Learning algorithms are based on the idea that the dimensionality of many data sets is only articially high. They seek to learn lower dimensional manifolds of the high dimensional vector space of the data. The Dierential Geometry involved is thus embedded in the Euclidean geometry. In contrast, Geometric Statistics model the data space as non-Euclidean from the very start.

Geometric Statistics is dierent from Stochastic Geometry, which studies random geometric patterns and especially the study of random point patterns [Matheron 1975]. And Geometric statistics is dierent from Geometric probability which studies probability functions dened on convex, or one studies the densities of lines in 2D or 3D that verify certain conditions [START_REF] Santalo | [END_REF]]. In fact, Geometric Statistics may encompass Stochastic Geometry and Geometric Probability, by building a statistical framework on the manifold of point patterns or of lines.

Manuscript overview

This Ph.D thesis advocates for Geometric Statistics in Computational Anatomy. Geometric Statistics naturally emerges in the various subelds of Computational Anatomy and has three objectives. First, it denes a statistical framework for creating new algorithms analyzing anatomical shapes. Second, it provides a formal setting to analyze the statistical properties of computational methods already employed for many years in the eld, but whose statistical consistency could not be evaluated. Third, it provides new tools for (medical) image processing.

This manuscript is divided into two parts. In the rst part, we introduce elements from Statistics -especially Estimation theory -and Geometry independently -with the mathematical denitions of the geometric nature of our data spaces. In the second part, we show the applications of Geometric Statistics on dierent essential problems of Computational Anatomy varying both the geometric structures and the statistical reasoning to give a wide view of the eld.

Our contributions are detailed in six core Chapters, which essentially correspond to the journal articles we have published so far, except for the last Chapter, which contains promising preliminary results that have been presented at a workshop of MICCAI'2015.

Part I: Foundations of Geometric Statistics

• Chapter 2 introduces Estimation theory, as it is well-known in the context of linear spaces. We oer a condensed illustration of the statistical problems in estimation procedures, exemplied with the computation of the sample mean. Computing the sample mean, in dierent geometric spaces, will indeed be one leading interest in this thesis.

• Chapter 3 presents the mathematical framework for the geometric spaces that will be studied in the thesis: Riemannian (pseudo-)metrics, ane connection spaces, Lie groups, Lie group actions, quotient spaces, ber bundles, etc. We also emphasize the links and interactions between these structures.

Part II: Geometric Statistics applied to Computational Anatomy

• Chapter 4 turns to Lie groups and the denition of a consistent statistical framework on these. We investigate the relationship between the Riemannian and pseudo-Riemannian framework with the group structure. We derive an algorithm for deciding if one can construct a (pseudo-)Riemannian metric that is compatible (i.e. "bi-invariant") with the Lie group given as input. The algorithmic procedure gives such a metric as output, in case of existence. Our procedure shows that a bi-invariant (pseudo-) metric fails to exists on general Lie groups. As a consequence, the denition of a consistent mean -and more generally of statistics-on Lie groups cannot rely on the (pseudo-)Riemannian framework, that is so widespread in our community. We advocate for other methods to dene consistent statistics on Lie groups, like for example ane connection spaces.

• Chapter 5 considers Lie group actions and quotient spaces. We introduce a new approach to analyze the statistical properties of well-known algorithms of Computational anatomy, like the algorithm of template anatomy estimation.

The approach relies on Geometric Statistics, as we reformulate the algorithm through the lenses of statistics on quotient spaces. The algorithm then boils down to the computation of the mean, on a stratied space which is a quotient space. Geometric Statistics thus enable us to demonstrate its asymptotic bias: even with an innite number of data, the algorithm does not converge to the anatomy it is designed to estimate. Our result is the rst proof of this eect that had been already observed experimentally. Our study shows that Geometric Statistics not only provides an elegant framework to analyze algorithms, it is also needed to avoid statistical caveats. The intrinsic geometry of a computational procedure in the data space needs to be taken into account, as it can create unexpected pathological behaviors in its daily use.

• Chapter 6 exemplies the previous results for neuroimaging studies, and more precisely for the computation of a brain template from MRIs. Computing the brain template is the rst step in almost all neuroimaging pipelines. We illustrate the consequences of the results of Chapter 5 and 6 by quantifying spatially the systematic bias on the brain template. We also suggest a new algorithm that aims to control the bias by a topological method.

• Chapter 7 applies tools of Geometric Statistics to other statistical procedures, beyond the computation of the template anatomical shape of Chapter 5. We consider unsupervised learning algorithms on shapes that are performed after a pre-processing step of data (images, meshes...) registration. We show that the registration step leads to bias in these algorithms, just as it led to bias for the computation of the template anatomy. We suggest a geometric correction of this bias, taking into account the stratied geometry of the data space. Again, one needs to take into account the additional structure of the manifold, in order to understand the algorithms and to adapt them when they fail.

• Chapter 8 considers statistics on sub-Riemannian manifolds and their use for medical image processing. The sub-Riemannian framework enables to consider both the intensity on the image and the direction of the intensity gradient. This is very useful for anisotropic smoothing or in-painting techniques. Sub-Riemannian geometry has been successfully used in 2D computer vision. Our contribution is to give the mathematical basis for using the structures for 3D images, which is absolutely not straightforward as new geometric structures appear while going from the 2D to the 3D. We provide a mathematical toolbox that make possible the use of sub-Riemannian geometry for 3D medical images processing.

Publications

The publications and awards of the author are listed below.

Peer-reviewed journal papers 1. Miolane, N., Holmes, S., Pennec, X.: Topologically constrained template estimation via Morse-Smale complexes allows to control its statistical consistency.

SIAM Journal on Applied Algebra and Geometry (2016). To be submitted. Invited talks

Introduction

The science of statistics is dened as the collection of data, their analysis and interpretation.

Historical landmarks of statistics The work of Al-Kindi in the 9th century is one of the rst occurrence of statistics in the literature. One can read in his book: "One way to solve an encrypted message, if we know its language, is to nd a dierent plain text of the same language [...] and then we count the occurrences of each letter. We call the most frequently occurring letter the 'rst', the next most occurring letter the 'second', the following most occurring the 'third', and so on [...].

Then we look at the cipher text we want to solve and we also classify its symbols. We nd the most occurring symbol and change it to the form of the 'rst' letter of the plain text sample, the next most common symbol is changed to the form of the 'second' letter, and so on...". In other words, Al-Kindi presented the rst code breaking algorithm based on frequency analysis. Nevertheless, statistics in their modern form -and in particular Estimation theory -are said to start in the 17th century. The mathematical foundations of Probability theory needed for its development were laid by Cardano, Pascal and Fermat. In the 18th century, Gauss and Legendre developed simultaneously the least squares method [START_REF] Stigler | [END_REF]], today's widespread procedure for tting an equation or a curve to data. These theories deal with data and parameters that are elements of Euclidean spaces. We can also cite Galton, Pearson, Gosset and Fisher among other pioneers of the eld. They all contributed to make statistics a rigorous mathematical discipline used for analysis across many elds and not only in science. In the late 20th century and 21st century, statistical methods extensively use modern computers that have made practicable new methods.

Estimation theory Within statistics, Estimation theory gives a mathematical framework to answer questions like: what is the typical blood pressure of an individual in the population? how many individuals will develop this pathology? what is the healthy anatomy of the human brain? The goal of Estimation theory in Computational Medicine is to extract information from the data, in order to answer these questions.

Exact answers, of course, are almost impossible to obtain. But approximate answers may be accessible and Estimation theory provides a set of rules for obtaining them. Above all, the theory makes precise such assertions as "this approximated answer is likely to be close to the exact answer" or "this answer is better than that one". Asserting the reliability of an answer is crucial with respect to the applications of Computational Medicine, that is: patients' health.

As an introduction to some problems involved in Estimation theory, we present here a toy example. We will use this example throughout this chapter. We consider estimating the human typical blood pressure, which we call µ. We have at our disposal a collection of blood pressures x i 's among n subjects. An intuitive estimate μ of µ is to compute the average -i.e. the sample mean -of the n observed blood pressures: μ = 1 n n i=1 x i . Why is it reasonable to consider the sample mean μ as an estimate of the human blood pressure µ? And how well does μ evaluate the real typical blood pressure µ, how does it depend on the number of observed data n? Estimation theory brings quantied answers to these interrogations.

Outline of this Chapter This Chapter introduces elements of Estimation theory

needed in this Ph.D thesis. Section 1 presents denitions on estimators and measures of their accuracy. Section 2 shows how to compute an estimator's measures of accuracy in practice. Section 3 provides methods to improve the accuracy of an estimator. The notions are illustrated with the example of the blood pressures given above. This Chapter is useful to understand the statistical issues arising in the usual estimators in Computational Medicine in the later chapters.

Theory on estimators

We refer to [START_REF] Lehmann | [END_REF]] for additional details on this section.

First denitions

A model describing the observations We consider the toy example given in the introduction. The goal is to answer the following question: what is the human typical blood pressure? This question implicitly relies on the assumption that there exists a unique blood pressure -which we call µ-that is shared across (healthy) humans. The fact that we measure n dierent blood pressures x i 's for the n dierent subjects is explained by measurement errors, for example from imperfect measurement devices. In other words, the observed data x i 's are generated following a model, described by the parameter µ and additional parameters representing the measurement error.

Specically, one assumes that the data x i 's are observations of a random variable

x, which takes values in the sample space X . In our example, the observations x i 's are real numbers x i ∈ R and the sample space is the real line X = R. On Figure 2.1, the data x i 's are represented as white dots on the real horizontal line X = R. In contrast, the next Chapter introduces data that belong to sample spaces X that are not necessarily vector spaces.

One further assumes that the random variable x follows some probability distribution P , dened over the sample space X , and that P belongs to a class P. The elements of P are usually indexed by a parameter denoted θ:

P = {P θ , θ ∈ Θ} (2.1)
For example, one may assume that the blood pressures x i 's are generated from a Gaussian distribution of parameters θ = (µ, σ), where µ is the mean and σ the standard deviation, see Figure 2.1 where the probability distribution P θ is represented in green. A statistical model is dened as a set of assumptions concerning the generation of observed data x i 's and usually refers to the class P.

The notation θ is standard for referring to the parameter characterizing a given P . The parameter θ can be a vector belonging to a nite-dimensional vector space, like in parametric and semi-parametric models, or to an innite-dimensional, like in non-parametric models where θ can represent a function. In contrast, the next Chapter introduces parameters θ that do not necessarily belong to vector spaces.

X P θ

x 1

x n µ Denitions: estimator, estimate, estimand The goal is to estimate the typical human blood pressure µ from the information available, that is the observed pressures x i 's. In general, an estimation procedure looks for the parameter θ of the generative model relying on the data x i 's. It denes a function θ over the sample space X , which is called an estimator. Note that θ is the function and θ(x) is a random variable, whose distribution is represented in orange in Figure 2.2 (b), the estimand being in green.

The parameter we want to estimate θ is called the estimand. θ(x) is called the estimate. θ(x) represents our best guess for θ given the data x and is illustrated on Figure 2.2 (a). Both θ and θ(x) belong to X . All in all, an estimator θ is a rule for calculating an estimate θ(x) of an estimand θ based on observed data x. We note that the terms "estimator" and "estimate" are often used interchangeably in the literature. The goal of estimation theory is to construct and compare estimators.

In our example, the pressures x i 's are generated by a Gaussian distribution of mean µ and standard deviation σ. We seek to estimate the parameter µ, which is the estimand. To this aim, we can compute the estimate μ(x) = 1 n n i=1 x i . The estimator is the function μ, which is called the sample mean.

Example of estimators: Maximum-Likelihood The denition of an estimator gives no restrictions on which procedure one shall use to estimate θ. Statisticians have developed paradigms to construct estimators. One of them is Maximum likelihood estimation (MLE).

The paradigm of Maximum Likelihood selects for θ(x) the set of values of the model parameters θ that maximizes the likelihood function:

L(θ) = p(x 1 , ..., x n |θ), (2.2)
which is the probability that this parameter θ has generated the data x i 's, i = 1...n.

The Maximum Likelihood method seeks to maximize the "agreement" of the model parameterized by θ with the observed data.

In our example, the estimator μ chosen is the sample mean. In fact, the sample mean is the Maximum Likelihood estimator of the mean of a Gaussian distribution. So we are using the Maximum Likelihood paradigm assuming that the blood pressures are generated with our Gaussian distribution.

Estimators' properties

We assume that we have available dierent rules to construct dierent estimators of the same parameter θ. How can we compare them? What is the "best" estimator to use in a given situation? And does "best estimator" even make sense? This leads

( θ) = E[( θ -θ) 2 ] Std deviation/error d = E[( θ -E[ θ]) 2 ] Variance var( θ) = E[( θ -E[ θ]) 2 ] Bias B( θ) = E[ θ] -θ Consistency θn P → n→∞ θ Asymptotic Normality √ n( θn -θ) D -→ n→+∞ N (0, V ), for some variance V .
to the question of the accuracy of an estimator, with respect to the parameter it is designed to estimate.

To evaluate the estimator θ, we consider the quantity θ(x) where x is the random variable that generates the observations x i , i = 1...n. Thus θ(x) is also a random variable and can consider its probability distribution function (p.d.f.). The properties of its p.d.f., with respect to the parameter θ, give indications on the accuracy of the estimator θ. The usual properties of estimators are summarized in Table 2.1 and we detail them below.

Evaluation criterion: Error and Mean square error of the estimator θ We want to compare θ to the parameter θ it is designed to estimate. The most intuitive idea is to consider their dierence: e = θ -θ, or e(x) = θ(x) -θ for a given set of observations x i , i = 1...n. This is the error e(x). However, this depends on the specic set of observations x and we would rather have a quantity that is independent of a sample. Therefore, we consider the random variable θ(x) -θ and, to get rid of the sign of the error, its squared version: (θ(x) -θ) 2 . Ultimately, as we seek one real number that can quantify the error of the estimator, we get to the denition of the Mean Square Error as:

MSE( θ) = E[( θ(X) -θ) 2 ].
The MSE indicates how far, on average, the estimates are from the single parameter being estimated. In our example of estimating the blood pressure µ, it indicates how far, on average, the sample means µ(x) computed from dierent samples x's are from the actual parameter µ.

We remark that the MSE is just a criterion indicating that θ(x) is close to θ.

Such a criterion is called the risk or the loss function associated with the estimator θ.

We could as well have want another criterion, for example the following probability to be large: P (| θ -θ| < c), or the following expectation to be small: E(| θ -θ|) p or, for a positive estimator and estimand, the following to be small: E(| θ θ -1| p ). But the MSE has good computational properties and is widely used in the literature.

Bias and variance of the estimator θ We can consider other measures of accuracy of the estimator θ. The bias of θ is the expected value of the error:

B( θ) = E( θ) -θ.
But a large error for one estimate does not mean the estimator is biased. Moreover, we can have a null error for a given estimate of a biased estimator just by chance. The sample mean μ is an unbiased estimator of the mean µ of a Gaussian distribution. In our blood pressure example, our estimator is thus unbiased.

The standard error of θ is:

d = E[( θ -E[ θ]) 2 ].
Its square is the variance of θ. Both quantify the amount of dispersion of the estimates around the true value of the parameter θ. For our example of the blood pressures, the standard error is equal to σ/n, where n is the sample size. Bias and variance are linked through the denition of the MSE. The MSE indeed decomposes as:

MSE( θ) = E[( θ -θ) 2 ] = E( θ -θ) 2 + E ( θ -E( θ)) 2 (2.3)
where the rst term is the squared bias of θ and the second term is the squared standard error of θ. We note that the variance is the MSE when the estimator θ is The ideal situation is to have an unbiased estimator with low variance. Additionally, we want to limit the number of outliers, i.e. samples where the error is extreme. But unbiasedness is not essential. If one allows for a little bias, one may nd an estimator with lower MSE and/or fewer outlier sample estimates. A ruleof-thumb states that the bias can be neglected as long at it is less that 0.25 of the variance [Efron 1993].

Skewness and kurtosis

We can borrow denitions associated to p.d.f's to further characterize the estimator θ. We consider the third and fourth moments of the p.d.f. Chapter 2. Elements of Statistics: Estimation theory. 22 of θ, just as the bias and variance referred to rst and second moments.

The skewness of θ is the third standardized moment:

Skew[ θ] = E ( θ -E[ θ]) 3 E ( θ -E[ θ]) 2 3 2 (2.4)
The skewness measures the asymmetry of the probability distribution of θ about its mean E( θ).

The kurtosis of θ is the fourth standardized moment:

Kurt[ θ] = E ( θ -E[ θ]) 4 E ( θ -E[ θ]) 2 2 , (2.5)
where σ is the standard deviation. The kurtosis is a measure of the "tailedness" of the probability distribution of a real-valued random variable, and in particular of a real-valued estimator.

Asymptotic behaviors: consistency and asymptotic normality The properties above have been dened for a given size n of the data set and corresponding estimator θn . We consider now the behavior of the sequence ( θn ) n for n → +∞, i.e. we tackle the question: how does the estimator's accuracy depend on the number of observed data n?

A consistent estimator θ is an estimator converging in distribution to a Dirac delta function centered at θ. The probability distributions of the θn 's become more and more concentrated, near the true value θ. In other words, the estimator nds the real parameter θ when there is an innite number of data.The sample mean is a consistent estimator of the mean on a vector space, such that, for the blood pressure example, we can nd the exact value of the human blood pressure if we measure the blood pressure of an innite number of subjects. We remark that bias is related to consistency as follows: a sequence of estimators is consistent if and only if it converges to a value and the bias converges to zero. An estimator can be unbiased but not consistent, or biased but consistent.

An asymptotically normal estimator θ is a consistent estimator whose distribution around the true parameter θ approaches a normal distribution with standard deviation shrinking in proportion to 1/ √ n as the sample size n grows. For example, the central limit theorem implies asymptotic normality of the sample mean μ as an estimator of the true mean µ. For our example, this means that with a lot of subjects, the distribution of the estimates will look normal, centered around the human blood pressure. More generally, maximum likelihood estimators are asymptotically normal under some regularity conditions.

"Best" estimators? We have dened some accuracy measures to evaluate the quality of an estimator. Given a statistical problem and a parameter we wish to estimate, can we answer the question: what is the best estimator? The denition of eciency provides answers to this.

An ecient estimator θ is an estimator that estimates the parameter θ in some "best possible" manner. This depends on the criterion chosen to evaluate the estimator, so usually the mean squared error MSE. If θ1 and θ2 are both estimators for the parameter θ, then θ1 is said to dominate θ2 if:

MSE( θ1 ) ≤ MSE( θ2 ) (2.6)
holds for all θ, with strict inequality holding somewhere. One usually says that a given estimator is "inecient" if there exists another estimator that dominates it.

Let us consider only unbiased estimators. A uniformly minimum-variance unbiased estimator or minimum-variance unbiased estimator (UMVUE or MVUE) is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of θ. The MVUE is thus ecient among unbiased estimators. If an unbiased estimator of θ exists, then one can prove there is an essentially unique MVUE. For example, the sample mean is an ecient estimator for the mean of the Gaussian distribution. The sample mean is also a MVUE. As a consequence, it represents a notion of "best" possible estimator for the mean µ.

Estimating an estimator's properties

The measures of accuracy of the estimator θ use the parameter θ in their denition. However, θ is not known since our goal is to estimate it. For example, we do not actually know the value of the parameter µ of the Gaussian distribution. So one cannot apply the above denitions of measure of accuracy directly. This section shows how to compute these measures in practice, and more precisely how to estimate these measures.

Using closed forms

Some estimators, like the sample mean θ = μ for estimating the mean µ, are widespread in the literature. The computation of their measures of accuracy in practice can be done using closed forms.

The sample mean μ is known to be unbiased. Its standard error is computed as follows:

d(x) = 1 n -1 n i=1 (x i -μ) 2
where {x 1 , x 2 , . . . , x n } is the sample. This is actually a biased estimate of the standard deviation. Bessel's correction is the use of n -1 instead of n in the formula:

d(x) = 1 n -1 n i=1 (x i -x) 2
This corrects the bias in the estimation of the standard error on the sample mean.

Resampling methods: bootstrap

But often, there is no closed form formula for computing -or estimating -the measures of accuracy of the estimator θ. We need an "universal" method that can estimate the accuracy of θ, that works for any θ. The "resampling methods" like the bootstrap [Efron 1993], which we introduce in this subsection. They are called "resampling methods" because we sample, often with replacement, from the original sample of the x i 's.

Plug-in principle: P θ and Pθ The bootstrap is an application of the plug-in principle, which we present here. We recall that P θ is the probability distribution that generates the data x i 's:

P θ → x = (x 1 , ..., x n ) (2.7)
The empirical distribution Pθ is dened to be the discrete distribution which puts probability 1/n on each observed value x i . It is a simple estimate of the distribution P . Alternatively, one can consider P θ, which is the probability distribution having θ as parameter and is also a simple estimate of P .

If one wants to estimate some aspects of P θ , like its mean or the parameter θ, one can consider the corresponding aspect of P θ or P θ. This is called the plug-in principle. For example, let us denote t the operation that extracts the parameter θ from the probability distribution P θ . Then, the plug-in estimate of θ = t(P θ ) is naturally θ = t( Pθ ). We consider our running example of estimating the mean of the Gaussian. The plug-in estimate of the mean θ = µ of P θ is the mean of Pθ , i.e. the sample mean.

Non-parametric and parametric bootstrap: sampling from Pθ and P θ A non-parametric bootstrap sample is dened to be a sample generated by the empirical distribution Pθ , so that:

Pθ → x * = (x * 1 , ..., x * n ) (2.8)
It is common to use the star-notation to refer to the bootstrap sample. We note that creating a bootstrap sample from Pθ can be done by drawing n times from the n data in x with replacements. The bootstrap sample has elements of the original sample, some appearing several times and some not appearing at all.

In contrast, a parametric bootstrap sample is dened to be a sample generated by the distribution P θ, so that:

P θ → x * = (x * 1 , ..., x * n ) (2.9)
In this case, one generates n data from P θ.

Using bootstrap samples to estimate the accuracy of θ We use the (parametric or non-parametric) bootstrap samples to estimate the measures of accuracy as follows. The estimate θ(x) was computed from the data set x as: θ(x) = s(x),

i.e. using a function s that takes the sample as input. A bootstrap replication of θ is dened as: θ * = s(x * ). In other words, we apply the same function s we had use on the original data set, but we apply it to the bootstrap data set. We can perform B resampling, leading to B bootstrap samples, each of size n. Each time, we compute a bootstrap replication θ * of θ. The bootstrap procedure writes:

• For b = 1...B:
Generate b-th bootstrap sample:

Pθ → x b * = (x b * 1 , ..., x b * n ) (non-parametric) or P θ → x b * = (x b * 1 , ..., x b * n ) (parametric) Compute b-th bootstrap replication: θ * b = s(x b * )
• From the empirical distribution of θ * 's, compute the bootstrapped desired property: bias, standard error, etc.

As an illustration, we estimate the properties of the sample mean as an estimator of the real mean parameter. To this aim, we resample B times, with replacement, from the original data set x i , i = 1...n. This gives us B data sets each of size n, from which we compute the sample mean. We then use the distribution of the sample mean as an approximation of the distribution of the corresponding estimator.

If we repeat the bootstrap sampling an innite number of times, i.e. for B → ∞ in the above procedure, we get the plug-in estimate of the distribution of the estimator θ and the plug-in estimates of various measures of accuracy that we could compute from it. But in practice, we do not repeat an innite number of times the experiment so that B is nite. So we get an approximation of the plug-in estimate of the distribution of θ and approximation of the plug-in estimates of the measures of accuracy. These are called bootstrap estimates for a number B of replications. The convergence of the bootstrap estimates to the ideal bootstrap estimate are linked to Monte-Carlo methods. In practice, the number of replications B needed will depend on the property of θ we want to estimate. We give examples below.

Bootstrap estimation of MSE, standard error, and bias We have the bootstrap estimate, for a number B of replications, of the distribution of θ. We can compute the moments of this distribution, and for example its mean and its variance which lead to bootstrap estimates of the bias and standard error of θ. We note the subscript B that illustrates the fact that they are bootstrap estimates. We provide rules of thumb regarding the recommended number B of replications for each case, which comes from the convergence properties of the Monte-Carlo [Efron 1993].

The bootstrap estimate of the standard error of θ is:

dB ( θ) = B b=1 ( θ * b -mean( θ * )) 2 B -1 1/2
(2.10)

Chapter 2. Elements of Statistics: Estimation theory.

26

The recommended number of replications is around B = 50. Then, the bootstrap estimate for the bias is:

b ias B = 1 B B b=1 θ * b -θ (2.11)
which takes around B = 3000 to converge. One can also use:

bias B = 1 B B b=1 θ * b -θ * (2.12)
where θ * is the estimator computed from the P * * , the average of the bootstrap distributions. The recommended number of replications is B = 400. Ultimately, the bootstrap estimate of the MSE of θ is: : one can compute the properties of this estimator.

M SE B ( θ) = 1 B B b=1 ( θ * b -θ) 2 (2.

Improving an estimator

We have seen how to compute measures of accuracy of an estimator. Now we present computational methods to improve the estimator by improving the corresponding measures of accuracy. In particular, we show how to reduce the bias and the variance of an estimator.

Bias correction

Assume we have an estimator known to be biased. We present one methods for correcting the bias of θ [Efron 1987], which is a direct application of the bootstrap procedures above.

We estimate the bias with a parametric or non-parametric procedure. Then, we modify the estimator by subtracting the estimated bias from the original estimate. If in addition, we believe that the estimate of the bias is itself biased, we can perform a nested bias-correction. We rst correct the bias in the bias' estimate and then correct the estimate θ with the bias-corrected bias' estimate. However, while it will reduce the bias of θ it will also increase the variance. So for it to be useful the improvement in bias must be large relative to the loss in the variance.

Bagging: variance reduction

Bootstrap aggregating, also called bagging, is a procedure meant to improve the variance of an estimator [Breiman 1996]. It is usually performed to avoid overtting for high variance low bias models like decisions trees or neural networks.

The procedure, very close to the bootstrap, is:

• For b = 1...B:
Generate b-th bootstrap sample:

Pθ → x b * = (x b * 1 , ..., x b * n ) (non-parametric) or P θ → x b * = (x b * 1 , ..., x b * n ) (parametric) Compute b-th bootstrap replication: θ * b = s(x b * ) • Aggregate the estimates to get θ = 1 B B b=1 θ * b .
Bagging averages B estimators θ * b 's computed from B dierent bootstrap samples. This average has smaller variance than just one sampling from the same distribution. To see it, we can consider a the standard Gaussian with 0 mean and σ = 1. One sample x has mean 0 and variance 1. When we sample B times and average the results, the mean is still be 0, but the variance becomes 1/B.

Conclusion

In this chapter, we have presented an overview of Estimation theory. We have introduced the concepts in the case of data and parameters belonging to a Euclidean space. This chapter represents the basis of the vocabulary that will be used in this thesis. We will use it to evaluate the estimations procedures that are used in Computational Medicine and that show non-Euclidean behaviors.

We have especially emphasized examples linked to estimating the mean of a sample. We have detailed procedures to estimate the mean of a data set of blood pressures x i ∈ R and introduced the generalization needed to estimate the mean of a data set of brain anatomies. Estimating the mean -which represents the easiest Chapter 3

Elements of Geometry: manifolds with additional structures

This Chapter introduces elements of Dierential Geometry used in this Ph.D thesis. They describe some non-linear data spaces of Computational Anatomy. The Chapter can be read independently from Chapter 2. Chapter 2 dened the basics of the "Statistics" aspect of "Geometric Statistics". This Chapter now introduces the "Geometric" aspect.

While reading this Chapter, we invite the reader to think about the generalization of Estimation theory of Chapter 2 to non-linear data spaces. One can consider data x i 's belonging to a non-linear space X . One can also take the model's parameter θ belonging to a non-linear space. The remaining of the Ph.D thesis will apply the statistical principles of Estimation theory to these non-linear data spaces, in order to investigate open issues in Computational Anatomy.

Introduction

Historical landmarks of Dierential Geometry The theory emerged from the study of curves and surfaces in 2D and 3D Euclidean spaces in the 18th and 19th centuries. Since the late 19th century Dierential Geometry has grown into a more abstract elds, concerned with structures on dierentiable manifolds and among them: (pseudo-) Riemannian geometry, Finsler geometry, Symplectic geometry, Contact geometry, Complex and Káhler geometry, CR geometry, Lie groups, Bundles and connections, etc.

Dierential Geometry has been extremely successful in describing phenomena in natural sciences, and for example in Physics. Riemannian geometry for example represents the mathematical foundations of Einstein's General Relativity in the 20th century. Symplectic geometry is used to describe Hamiltonian systems and Contact geometry has found applications in Thermodynamics. One may wonder which branches of Dierential Geometry are the most suited to Computational Medicine and Computational Anatomy.

Dierential Geometry for Computational Anatomy Dierential Geometry appears naturally in Computational Anatomy where the data spaces can be described as manifolds. The data are organ shapes -as seen in medical images -which may be modeled in several ways: (i) as deformations of a reference shape i.e. as elements of a Lie group, which is a manifold with a group structure, and (ii) as the equivalence class of their conguration in the 3D space under the action of rotations, translations i.e. as elements of a quotient space. Furthermore, the medical images themselves can be embedded in a sub-Riemannian manifold for better processing.

What do these data spaces have in common: the Lie groups, the manifolds with Lie group actions, and the sub-Riemannian spaces? They are all manifolds, i.e. non-linear spaces that locally look like vector spaces. Dening a theory of statistics on manifolds is still considered open research. A widespread approach is to dene a Riemannian metric on the manifold and then dene statistics on Riemannian manifolds [Pennec 2006]. Indeed, the Riemannian metric enables to dene a Riemannian distance between the points on the manifold. The distance can be used to generalize denitions of Statistics to the manifold.

How do these data spaces dier? While being all manifolds, they have structural specications of their own. They may be equipped with: a Lie group action, a stratication, an horizontal distribution, etc.

Outline of this Chapter This Chapter introduces elements of Dierential Geometry needed in this Ph.D thesis. The rst part of this chapter -Section 3.2thus focuses on the denition of manifold and of the (pseudo-) Riemannian metrics that we may consider on them. Then, Sections 3.3-3.7 dene some additional structures on manifolds and their relations. For each structure, we give a formal and intuitive denition and we show the compatibility conditions arising while dening a Riemannian pseudo-metric on it.

Manifolds and Riemannian manifolds

Vector spaces, high-dimensional smooth surfaces and the data spaces encountered in Computational Anatomy are all "manifolds". This section gives some mathematical denitions about manifolds. We refer to [Nakahara 2003] and [Postnikov 2001] for formal details and the Einstein notation on indices.

Dierentiable structure of manifolds

A manifold M of dimension m is a generalization of a vector space of dimension m: M is a space that looks like a m-dimensional vector space in the neighborhood of every point. A dierentiable manifold M of dimension m is a type of m-dimensional manifold that is locally similar enough to a vector space to allow one to do calculus. Such calculus is performed through a chart, which is an invertible map φ between a subset of the manifold M and R m , such that both the map and its inverse preserve the desired structure. This type of calculus is called "Dierential Calculus" and represents the basis of "Dierential Geometry". The next paragraph briey explains why one uses the term "Dierential".

We can dene the space C ∞ (M ) of smooth functions f on the manifold M , which associate a real number to a given point X ∈ M . But how do we dene the dierentiation of such a function? There may be several coordinate systems on the manifold: how do we give a coordinate-independent denition of derivative? The tangent vectors of M , dened below, allow us to do exactly this.

We consider a chart on M , φ : U → R m where U is an open subset of M containing a point X. We take two curves

C 1 : [-1, 1] → M and C 2 : [-1, 1] → M that coincide at 0: C 1 (0) = C 2 (0) = X. Moreover φ • C 1 and φ • C 2
are assumed to be dierentiable at 0. The curves are called equivalent at X if the ordinary derivatives of φ • C 1 and φ • C 2 at 0 coincide. One can show that this denes an equivalence relation on such curves. The equivalence classes are known as the tangent vectors of M at X. The space of tangent vectors at X is called the tangent space of M at X and is denoted T X M .

More generally, T X M forms the algebra of derivations at X of smooth functions f ∈ C ∞ (M ). Indeed, we have the fundamental isomorphism of Dierential Geometry at each point X ∈ M :

D X (M ) T X M (3.1)
where D X (M ) is the vector space of directional derivatives at X [Nakahara 2003]. This is the reason of the term "Dierential" in "Dierential Geometry".

The set of all tangent spaces of M , at any point of M , is called the tangent bundle T M of M . Then, we denote Γ(M ) the space of vector elds on M . In a local coordinate system, the action of a vector eld V ∈ Γ(M ) at X on f is :

V [f ] X = ∂ X f X = d dλ (f (X + λV | X )) λ=0 . (3.2) When composing the derivatives, V [U [f ]]| X = ∂ V ∂ U f | X involves
second order derivations and does not behave like a vector (or a directional derivation). We usually remove the second order terms by subtracting ∂ V ∂ U to dene the Lie Bracket of vector elds:

[V, U ][f ] = V [U [f ]] -U [V [f ]] (3.3)
which is now a derivation [Nakahara 2003].

The above structures on tangent vectors and tangent spaces are dened on any dierentiable manifold. In particular they are dened on all Lie groups, manifolds with group actions, or bundles that we will encounter in the next sections.

Riemannian and pseudo-Riemannian structure on manifolds

So far, the manifold M is only equipped with a dierentiable structure. Distance between points are not yet dened, and neither are angles between tangent vectors. One can imagine M as a soft and elastic high-dimensional smooth surface. Dening a (Riemannian) metric on M provides a way to dene distances and angles. As such, the metric gives the manifold M some rigidity: the high-dimensional surface chooses a shape.

A Riemannian metric is a smooth collection of positive denite bilinear 2-forms on the tangent spaces of the manifold M , i.e. it denes the dot product of tangent vectors. This structure is the most commonly used in Computational Anatomy.

Then, by relaxing the positivity assumption on the metric, one denes a Riemannian pseudo-metric. A Riemannian pseudo-metric is a smooth collection of denite bilinear 2-forms on tangent spaces of the manifold M , i.e. non necessarily positive denite. We denote (p, q) its signature, p + q = dim M , where p is the number of strictly positive eigenvalues, and q the number of strictly negative ones. We remark that a Riemannian metric is a Riemannian pseudo-metric of signature (dim M, 0).

A Riemannian metric allows us to determine angle between tangent vectors (of the same tangent space) and their lengths. Moreover, the metric induces a distance along a curve C of the manifold M through integration of its tangent vector in ane parameterization s:

I[γ] = γ ds < γ(x), γ(s) > | γ(s) (i.e. the length).
Then, a Riemannian geodesic γ of M is a curve minimizing the functional

I[γ]
. Going now to pseudo-metrics, one can equivalenly minimize the functional [Nakahara 2003]:

K[γ] = 1 2 γ ds < γ(x), γ(s) > | γ(s) (i.e.

the kinetic energy)

The Riemannian (pseudo-)metric gives an intrinsic way of measuring length between two points on a manifold: we compute the length of the geodesic linking the two points, provided that it exists and is unique.

Lie groups

In this section, we turn to a rst additional structure one may consider on a manifold.

We add an algebraic structure on M , the structure of a group. This leads us to the denition of Lie groups.

Lie groups are extensively used in Physics where they describe symmetries in physical systems. For example, they express the relativity principle of General Relativity: this is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference. Going from one frame to another is expressed through a Lie group element. Lie groups are also used to represent elementary particle symmetries, where two dierent Lie groups describe quark elds or electron elds.

More recently, Lie groups have also been used to represent organ transformations or deformations in Computational Anatomy. For example, any rotation or translation of an imaged organ, due to the dierences in imaging procedures, is an element of a Lie group. Smoothly deforming an organ shape to have it match another also requires an element of a Lie group. The structure of Lie groups will be used in particular in Chapters 4 and 5 of this Ph.D thesis.

Lie groups and its translations

A Lie Group G is a smooth manifold with a compatible group structure. It is provided with an identity element e, a smooth composition law * : (g, h) → g * h ∈ G and a smooth inversion law Inv : g → g (-1) ∈ G. Its tangent space at g is written T g G.

There is a canonical way to dene two dieo/auto-morphisms families of G, called the left and right translations: L g : f → g * f and R g : f → f * g in addition to a third one called the conjugations

C g = f → g * f * g (-1
) [Nakahara 2003].

Adding a metric structure

We consider a Riemannian pseudo-metric g G dened on the Lie group G. As such pseudo-metric can be dened on any manifold, it can also be dened on G. We investigate the compatibility of g G with the Lie group structure.

The pseudo-metric g G is said left-invariant pseudo-metric if for all U, V ∈ T g G and for all g, h ∈ G we have:

< DL h (g)U, DL h (g)V > | L h g =< U, V > | g (3.4)
where L h is the left translation by h, i.e. if the left translations are isometries for this pseudo-metric. Similarly, g G could be right-invariant and bi-invariant. Bi-invariant pseudo-metrics show the most compatibility with the group structure. Chapter 4 studies the bi-invariant property of pseudo-metrics and its importance for Statistics on Lie groups.

Quotient spaces

In this section, we turn to another additional structure on a manifold M : the structure of Lie group action. The manifold M is not the whole Lie group G like in the previous section, but G acts on M . Furthermore, we consider the quotient of the Lie group action which leads us to the structure of quotient space. Such structures intuitively describe several situations in Computational Anatomy. For example rotating and translating an organ's surface involves the action of a rotation and a translation on the surface. Then, the shape of the organ's surface may be dened by the geometry that is preserved while rotating and translating the surface. The quotient of the space of surfaces by the action of rotations and translation denes the space of the surfaces' shapes. The structures of Lie group actions and quotient spaces will be used in particular in Chapters 5 and 6 of this Ph.D thesis.

Lie group actions and quotient spaces

A Lie group (left) action of G on a manifold M is a map:

ρ : G × M → M, (g, X) → g • X (3.5)
such that for all X ∈ M, e • X = X, where e is the identity, and for all (g, g ) ∈ G 2 and for all X ∈ M, g • (g • X) = (g g) • X. Right group actions are dened similarly. We will consider proper and eective actions (for denitions, see [Huckemann 2010]).

The orbit of X under the action of G, written O X , is dened as all points of

M reachable the action of G on X, thus O X = {g • X | g ∈ G}. Each orbit O X is
a submanifold of M . The isotropy group of X or stabilizer of X, written G X , is dened as the elements G that leave X unmoved:

G X = {g ∈ G | g • X = X}. Each isotropy group G X is a Lie subgroup of G. The set of orbits {O X } X forms a partition of M . It is written Q = M/G
and is called the quotient space of M by the action of G.

Adding a metric structure

We consider a Riemannian metric g M on the manifold M . The compatibility condition between the metric and the Lie group action is the following. The action of

G on M is called isometric with respect to g M if dρ g : T X M → T g•X M leaves g M invariant.
Equivalently, g M is said to be G-invariant. It means that the distance between two data does not change if we act on both data in the same manner. This compatibility condition can always be satised in theory: given the action of G of M , we can always nd a g M that is G-invariant.

The compatibility condition with respect to the Lie group action is important in order to induce a Riemannian metric g Q on the quotient space Q = M/G. From now on, we assume that g M is G-invariant. We consider d M the distance associated to g M . The metric g M being G-invariant, d M is also G-invariant. Let π : M → M/G be the canonical projection to the quotient space with quotient topology. The Ginvariant distance d M on M induces a quasi-distance d Q on Q, interpreted as the quasi-distance between shapes. It is dened as:

d Q (O X 1 , O X 2 ) = inf g∈G d M (g • X 1 , X 2 ), (3.6)
for X 1 , X 2 ∈ M .

Fiber bundles

We now consider the structure of ber bundles over a manifold M . Particles elds and the corresponding mathematics of ber bundles are the core of Relativistic Quantum Field theory. They represent some of the most advanced model used to describe nature, that has been conrmed by experiments. In Relativistic Quantum Field theory, a eld of quarks can be roughly thought as follows: there is a (probability of nding a) particle at each point of space-time R 4 . The particles interact spatially, so there exists some correlation on nding a particle at two neighboring locations.

There are similarities between elds of elementary particles of Theoretical Physics and elds encountered in Medical Images. A tridimensional medical image can be seen as a eld over a domain of R 3 , or over a grid of voxels which is a discretized version of R 3 . One may consider that there is a "particle" at each point of the grid, which can be an intensity (MRIs...) or a tensor (DTI images), a p-value etc. These "particles" interact spatially in the sense that there is a correlation in the intensity of neighboring voxels. The structure of ber bundles will be useful in Chapter 8.

Fiber bundles

A ber bundle (P, M, π) of base M is a manifold P equipped with a dierentiable projection π onto the manifold M , such that the reciprocal images π -1 (X), for X ∈ M , are all dieomorphic. Such a F X = π -1 (X) is called the ber above X.

For example, the tangent bundle T M of M is a ber bundle of base M (see Figure 3.1, middle). In this case, the projection π is the operator which maps each tangent vector to its foot, i.e. each v ∈ T X M to X. Fiber bundles are generalizations of product spaces: ber bundles have no projection onto the ber F , only a projection onto the base M , whereas product spaces have both.

The projection π : P → M is often surjective and if not, we can restrict M to match π's image. However, π is not injective in general: the cardinal of the ber is dierent from 1 in general. For each X ∈ M , we can choose an element of the ber F X to build a right inverse for π. We call this element σ(X) and by construction we have: π • σ = Id. The map σ is called a section of the ber bundle. There are dierent choices possible for dening a section. In the example of the tangent bundle, a section denes a vector eld over M . There are dierent possible sections just as we have dierent vector elds on a given manifold M .

Examples We can consider the image's domain (in its continuous version) to be the base manifold M . Taking the ber to be R, we can build the trivial ber bundle: M × R which can be seen as the set of all possible images over the domain M . A section of this bundle gives an image. There are dierent sections possibles just as there are dierent images that can be dened on the same domain.

Alternatively we can choose the ber to be Sym 3 (R) i.e. the vector space of symmetric matrices over R in 3 dimensions. In this case, we can build the (trivial) ber bundle M × Sym 3 (R). A section of this bundle represents a image of DTI: at each point of the domain, we have a symmetric matrix. The ber bundle itself represents the set of all possible DTI images over this domain.

We can also consider the vector eld dened by the intensity gradient of the given image. This vector eld belongs to a vector bundle dened over the image's domain. In Physics, we can take the base manifold to be space-time R 4 and the ber to be C 3 . In this case, a section of the ber can be thought of as a simplied model of a quark eld.

These may seem an overly complicated representation of images. But this additional structure enables to describe operations on images in a compact way, as we shall see in the next sections.

Principal bundles

A principal bundle (P, M, π) of structure group G is a special type of ber bundle, where each ber looks like G: see Figure 3.1, right, where each ber looks like SO(2), i.e. S 1 . For example the bundle of all local frames of the manifold M is a principal bundle of structure group GL m . Thus, it can be thought about as a generalization of the direct product spaces M × G. Formally, a principal bundle is a ber bundle, equipped with a right action of the group G on P , which is transitive on the bers, and such that the bers are all homeomorphic to G.

Examples In Physics and more particularly in General Relativity, we can consider the principal bundle of all frames over space-time R 4 , i.e. F M . These frames are called external frames. In contrast, bundles of internal structure are dened by principal bundles that have nothing to do with the frame bundle F M . The external structure denes the properties of the eld in space-time, and the internal structure represents the intrinsic properties like particle symmetries properties of the eld.

We come back to the examples from Medical Imaging and take an image's domain as the base manifold M . Again, we can build several principal bundles with the same base manifold M . Taking the multiplicative group (R * + , .), we can form the trivial principal bundle M × R * + . Taking the additive group (R, +), we can form the trivial principal bundle M × R. We could also consider the general linear group in 3 dimensions GL 3 and create the trivial principal bundle M × GL 3 . These principal bundles do not dene new types of images. They rather dene objects that can interact with the previously dened images. They will dene the images' properties, just as they dened the particles' intrinsic symmetries in Physics. This is the aim of the associated ber bundles of the next section.

Figure 3.1: Left: a smooth curve C is an example of a 1-dim. manifold. Center: the tangent bundle of the curve C, whose bers are in blue. Right: a principal bundle with base C and structure group S 1 , the bers are in blue.

Connections

In this section we consider the structure of connections on the manifold M . We recall that, without any metric, the manifold M can be seen as a soft and elastic surface. The connection does not dene distances between points of M , but it denes some notion of parallelism. A manifold M with a connection is still allowed to deform, but in a restricted way that preserves the parallelism. The connection is a less restrictive structure than the metric as it does not completely x the shape of M . We use connections on Lie groups in Chapter 4.

Ane connection spaces

If one wants to compare data in the tangent space T X M at point X of the group, with data at another tangent space, say T Y M , one needs a specic mapping Π Y X between these two tangent spaces. This is the essence of parallel transport whose innitesimal version is the connection.

Let γ be a curve in M joining γ(0) to γ(1). A parallel transport along this curve is a collection of mappings Π(γ) t s : T γ(s) M → T γ(t) M such that:

• Π(γ) s s = Id (identity transformation of T γ(s) M ),
• Π(γ) t u • Π(γ) u s = Π(γ) t s i.e. consistency along the curve,

• The dependence of Π on γ, s and t is smooth.

The notion of (ane) connection is the innitesimal version of parallel transport. The connection hence represents how a vector transforms when it is parallel transported to an innitesimally close tangent space. Let x = γ(0) be the tangent vector at the initial point of the curve γ and U be a vector eld. Then the quantity:

lim t→0 Π(γ) 0 t U | γ(t) -U | γ(0) t = d dt Π(γ) 0 t U | γ(t) t=0 (3.7)
is independent of the curve γ [Postnikov 2001]. Hence we can forget about the previous expression of parallel transport and dene an ane connection without specifying a curve γ.

An ane connection ∇ is a bilinear map:

∇ : Γ(M ) × Γ(M ) -→ Γ(M ) U, V -→ ∇ U V
with the following properties:

• ∇ f U V = f ∇ U V , • ∇ U (f V ) = ∂ U f V + f ∇ U V i.e. Leibniz rule in the second variable for any U, V ∈ Γ(M ) and f ∈ C ∞ (M ).
The torsion induced by ∇ on the manifold M is dened as:

T ∇ (X, Y ) = ∇ X Y -∇ Y X -[X, Y ]. (3.8)
If the torsion vanishes, the connection is said to be torsion-free or symmetric.

Examples In DTI images, the connection denes what happens to the 3D reference frame when we travel from one tensor to a neighboring one in the 3D image.

For example, we can force the frame to be aligned with the principal axes of the tensors. In this case, we dene a connection that in fact represents the structure of the bers of the organ studied. The properties of this connection may give indication about the tissue that is being imaged. This directly links the Dierential Geometry of medical imaging to Dierential Geometry of continuous mechanics. The DTI imaging is a way to image the material connection, in this case the connection of the tissue.

Adding a metric structure

We consider a (pseudo-)metric g M on the manifold M . We investigate the compatibility condition with respect to the connection.The connection ∇ is compatible with g M if at any point X ∈ M we have:

∇ V g M = 0 ∀U ∈ T X M (3.9)
i.e. if the pseudo-metric is parallel transported along any curves, or equivalently, if we have the product rule:

U < V, W >=< ∇ U V, W > + < V, ∇ U W > . (3.10)
For a given pseudo-metric g M , there is a unique symmetric (i.e. torsion free) connection compatible with it. It is called the Levi-Civita connection and can be computed with the Koszul formula as follow:

< ∇ U V, W >= 1 2 (U < V, W > +V < U, W > -W < U, V > -< U, [V, W ] > -< V, [U, W ] > + < W, [U, V ] >)

Sub-Riemannian manifolds

This section considers another metric structure on a manifold M , the structure dened on M by a sub-Riemannian metric. An horizontal distribution on M is a subbundle HM of the tangent bundle of M. An horizontal distribution HM is called completely non-integrable if, for a given point X ∈ M , any tangent vector of T X M can be presented as a linear combination of vectors of the following types U (X), [U, V ](X), [U, [V, W ]](X), ... where the vector elds U, V, W, ... are themselves in H X M .

A sub-Riemannian manifold is a triple (M, H, g), where M is a dierentiable manifold, H is a completely non-integrable horizontal distribution and g is a smooth section of positive-denite quadratic forms on H. Any sub-Riemannian manifold carries a natural intrinsic metric, called the metric of Carnot-Caratheodory [Postnikov 2001].

Conclusion

This Chapter has introduced the basic denitions of the main structures of Differential Geometry that can be used in Computational Medicine to describe organ shapes. We have chosen an intuitive approach and given examples with medical images. For each structure, we have given its relation and compatibility with the introduction of a Riemannian (pseudo-)metric.

The next part of this Ph.D thesis generalizes the statistics of Chapter 2 to the geometric structures seen in this Chapter.

Introduction

Modeling with Lie groups Data can be modeled as elements of Lie groups in many dierent elds: Computational Anatomy, Robotics, Paleontology etc. Indeed, Lie groups are continuous groups of transformations and thus appear naturally whenever one deals with articulated objects or shapes.

Regarding articulated objects, one can take examples in Robotics or in Computational Anatomy. In Robotics rst, a spherical arm is obviously an articulated object. The positions of the arm can be modeled as the elements of the 3-dimensional Lie group of rotations SO(3). In Computational Anatomy then, the spine can be modeled as an articulated object. In this context, each vertebra is considered as a orthonormal frame that encodes the rigid body transformation from the previous vertebra. Thus, as the human spine has 24 vertebrae, a conguration of the spine can be modeled as an element of the Lie group SE(3) 23 , where SE(3) is the Lie group of rigid body transformations in 3D, i.e. the Lie group of rotations and translations in R 3 also called the Special Euclidean group.

Regarding shapes, the general model of d'Arcy Thompson suggests to represent shape data as the dieomorphic deformations of a reference shape [Thompson 1992], thus as elements of an innite dimensional Lie group of dieomorphisms. This framework can be applied as well in Paleontology than in Computational Medicine. In Paleontology rst, a monkey skull or a human skull can be modeled as the dieomorphic deformation of a reference skull. In Computational Medicine then, the shape of a patient's heart can be modeled as the dieomorphic deformation of a reference shape. Obviously, many more examples could be given, also in other elds.

Statistics on Lie groups Once data are represented as elements of a Lie group, we may want to perform statistical analysis on them for prediction or quantitative modeling. Thus we want to perform statistics on Lie groups. How can we dene an intrinsic statistical framework that is ecient on all Lie groups? How do we compute the mean or the principal modes of variation for a sample of Lie groups elements? In order to train our intuition, we consider nite dimensional Lie groups here.

To dene a statistical framework, it seems natural to start with the denition of a mean. The denition of mean on a Lie group exemplies the issues one can encounter while dening the whole statistical framework. We know that the usual denition of the mean is the weighted sum of the data elements of the sample. However, this denition is linear and Lie groups are not linear in general. Consequently, we cannot use this denition on Lie groups: we could get a mean of Lie group elements that is not a Lie group element. One can consider as an example the half sum of two rotations matrices that is not always a rotation matrix.

In fact, the denition of the mean on a Lie group should be consistent with the group structure. This consistence leads to several requirements of the mean, or properties. First, the mean of Lie group elements should be in the Lie group. Then, it seems natural to require that a left or right translation of the data set should translate its mean accordingly. The Figure (4.1) illustrates the case when this condition is fullled. Finally, the inversion of all data elements should lead to an inverted mean. A mean verifying all these properties is said bi-invariant. has a mean represented in green. We require that the mean of the (right or left) translated data set is the translation of the red mean, which is the case on this illustration: the blue mean is the left translation of the red mean, the green mean is the right translation of the red mean.

R h L h {g i } N i=1 {g i * h} N i=1 {h * g i } N i=1
A naturally bi-invariant candidate for the mean on Lie groups is the group exponential barycentre [Pennec 2012] dened as follows. A group exponential barycenter m of the data set {g i } i=1,..,N is a solution, if there are some, of the following group barycenter equation:

N i=1 Log(m (-1) * g i ) = 0 (4.1)
where Log is the group logarithm. As the group exponential barycenter is naturally bi-invariant, we call a group exponential barycenter a bi-invariant mean. The local existence and uniqueness of the bi-invariant mean have been proven if the dispersion of the data is small enough. "Local " means that the data are assumed to be in a suciently small normal convex neighborhood of some point of the Lie group.

Now we want to provide a computational framework for the bi-invariant mean that would set the foundations for computations on Lie groups statistics in general.

For that, we are interested in characterizing the global domains of existence and uniqueness of the bi-invariant mean. By "global domain", we mean for example a ball of maximal radius such that any probability measure with support included in it would have a unique bi-invariant mean. Note that there is a priori no problem having several means, which can be called several "modes", or no mean at all. Our aim is rather to characterize the dierent situations that may occur: no mean, one unique mean, several means.

Using Riemannian and pseudo-Riemannian structures for statistics on Lie groups To this aim, we are interested in additional geometric structures on Lie groups that could help, by providing computational tools. For example we are interested in a distance on a Lie group, that could enable to measure radii of balls. Such a distance could obviously help characterizing balls of maximal radius.

But a Lie group is a group that carries an additional manifold structure and one can dene a pseudo-metric on a manifold, making it a pseudo-Riemannian manifold. Thus, we can add a pseudo-metric on Lie groups which then induces a pseudo-distance. Could this additional pseudo-Riemannian structure help to dene the statistical framework on Lie groups in practice?

We consider rst the case of the Riemannian structure, i.e. when the pseudometric is in fact a metric (positive denite). Several denitions of the mean on Riemannian manifolds have been proposed in the literature: the Fréchet mean, the Karcher mean or the Riemannian exponential barycentre [Fréchet 1944, Fréchet 1948, Karcher 1977, Kendall 1990[START_REF] Émery | [END_REF], Corcuera 1998]. For example, the Riemannian exponential barycentres are dened as the critical points of the variance of the data, dened as:

σ 2 (y) = 1 N N i=1 dist(x i , y) 2 , where {x i } N
i=1 are the data and dist the distance induced by the Riemannian metric. The Riemannian framework provides theorems for the global existence and uniqueness domains of this mean [START_REF] Émery | [END_REF], Corcuera 1998, Huiling 2004, Yang 2010, Afsari 2011], ensuring the computability of statistics on Riemannian manifolds. These represent exactly the kind of results we would like to have for the bi-invariant mean on Lie groups. Thus, one may wonder if we can apply this computational framework for statistics on Lie groups, and more particularly for the bi-invariant mean, by adding a Riemannian metric on the Lie group.

Chapter 4. Geometric Statistics on Lie groups: analysis of anatomical deformations.

44

In fact, the notions of Riemannian mean and group exponential barycentre (or bi-invariant mean) coincide when the Riemannian metric is itself bi-invariant. In this case, the Riemannian geodesics coincide with the geodesics of the Cartan-Schouten connection [Sternberg 1964]. Thus, we can use the computational framework for Riemannian means only if we can add a bi-invariant metric on a Lie group.

But it is known that a Lie group does not have any bi-invariant Riemannian metric in general. The Lie group ST (n) of Scalings and Translations of R n , the Heisenberg group H, the Lie group U T (n) of Upper Triangular matrices of size n×n and the Lie group SE(n) of rotations and translations of R n do not have any biinvariant metric while they admit a locally unique bi-invariant mean [Pennec 2012]. Therefore, if we want to characterize the bi-invariant mean with an additional geometric structure on Lie groups, we have to consider a structure that is more general than the Riemannian one.

The pseudo-Riemannian framework is a generalization of the Riemannian framework. Thus, it represents a tempting alternative for the characterization of the biinvariant mean and for the denition of computational statistics on Lie groups in general. The pseudo-metric is not required to be positive denite anymore, only definite: the class of Lie groups that admit a bi-invariant pseudo-metric is larger than the class of those with a bi-invariant metric. Therefore, we could try to generalize the Riemannian statistical framework to a pseudo-Riemannian statistical framework and apply it for Lie groups. For instance, the mean on a pseudo-Riemannian manifold could still be dened as a critical point of the variance σ 2 (y) = 1 N N i=1 dist(x i , y) 2 , but dist would now be the pseudo-distance induced by the pseudo-metric. Of course, existence and uniqueness theorems would have to be re-established but we could get intuition from the Riemannian case.

In order to use the pseudo-Riemannian framework to characterize the biinvariant mean, the rst issue is: how many Lie groups do admit a bi-invariant pseudo-metric? Is it the case for the real Lie groups ST (n), H, U T (n) and SE(n), which have a locally unique bi-invariant mean?

Lie groups and Lie algebras with bi-invariant pseudo-metrics If G is a connected Lie group, it admits a bi-invariant non degenerate symmetric bilinear form if and only if its Lie algebra admits a nondegenerate symmetric bilinear inner product, also called a bi-invariant pseudo-metric. Lie algebras with biinvariant pseudo-metric were known to exist since the 1910's with the classication of simple Lie algebra [Cartan 1933] and the well-known Cartan-Killing form which is not degenerate in this case, but their specic study began in the 1950's with the works of [Tsou 1957, Tsou 1962]. Later, [Astrakhantsev 1978] started to study properties of these Lie algebras from their structural point of view and introduce the decomposability of indecomposability of these Lie algebras as a direct sum of ideals. However, the decomposition of [Astrakhantsev 1978] was not enough to characterize all Lie algebras with bi-invariant pseudo-metrics, as some authors [Medina 1985b, Keith 1984[START_REF] Guts | On the Foundations of Relativity Theory[END_REF]] remark that the so-called oscillators algebra arising in quantum mechanics carried a bi-invariant pseudo-metric without being decomposable in the sense of [Astrakhantsev 1978]. This leads, Medina and Revoy [Medina 1982, Medina 1985a] and Keith [Keith 1984] to build independently a classication of these Lie algebras, by showing that they all arise through direct sums and a structure called the double extension in [Medina 1982, Medina 1985a] and the bi-extension in [Keith 1984].

These results have been complemented by [Hofmann 1986], and then generalized by Bordemann to any non associative algebras with bi-invariant form through the T * -extension structure [Bordemann 1997]. They have been completely described for certain dimensions in specic cases. The classication of the nilpotent quadratic Lie algebras of dimensions ≤ 7 is obtained in [Favre 1987], of the real solvable quadratic Lie algebras of dimensions ≤ 6 in [START_REF] Campoamor-Stursberg | Quasi-Classical Lie Algebras and their Contractions[END_REF], the irreducible non solvable Lie algebras of dimensions ≤ 13 in [Benayadi 2014]. The specic cases of indecomposable quadratic Lie algebras with pseudo-metrics of different indices have been studied: bi-invariant pseudo-metrics of index 1 are described in [Medina 1985a, Hilgert 1985], of index 2 in [START_REF][END_REF]] and nally of general index in [Kath 2006]. The dimension of the space of bi-invariant pseudo-metrics has been studied in [Duong 2011] where bounds are provided.

Authors from other elds than pure algebra have also contributed to the study of bi-invariant pseudo-metrics. For example in Functionnal Analysis, Manin triple are a special type of Lie algebra with bi-invariant pseudo-metric that allow to interpret the solutions of the classical Yang-Baxter equation [Drinfeld 1987]. In this context, the Manin triple have been themselves classied for semi-simple Lie algebras in [Belavin 1998] and for complex reductive Lie algebra in [Delorme 2001].

Simultaneously, people started to gain interest in computational aspects on nite dimensional Lie algebras, implementing the identication of a Lie algebra from its structure constants given in any basis [Rand 1986, Rand 1988] or the Levi decomposition [Cohen 1997, Ronyai 1997]. The state-of-the-art regarding implementations on nite dimensional Lie algebra is summarized in [De Graaf 2000]. However, computations deal with the algebraic aspects of Lie algebras and, to the knowledge of the authors, do not consider metrics or pseudo-metrics.

Contributions and outline Our contribution is an algorithmic reformulation of a classication theorem for Lie algebras [Medina 1982, Medina 1985a] that answers these questions. More precisely, taking a Lie group G as input, the algorithm constructs a bi-invariant pseudo-metric on G in case of existence. Using this algorithm, we show that most Lie groups that have a locally unique bi-invariant mean do not possess a bi-invariant pseudo-metric. We conclude that, for the purpose of statistics on general real Lie groups -and more precisely for the computational framework of the bi-invariant mean-generalizing the Riemannian statistical framework to a pseudo-Riemannian framework may not be the optimal program.

The paper is organized as follows. In the rst section, we introduce notions on quadratic Lie groups that will be useful for the understanding of the paper. In the second section, we present the (tree-structured) algorithm that constructs biinvariant pseudo-metrics on a given Lie group, in case of existence. In the third section, we apply the algorithm on ST (n), H, U T (n) and SE(n) and show that most of them do not have any bi-invariant pseudo-metric.

Introduction to Lie groups with bi-invariant pseudometrics

Here, we dene the algebraic and geometric notions which will be used throughout the paper.

Quadratic Lie groups and Lie algebras

In the following, we consider nite dimensional connected simply connected Lie groups over the eld F, where F is R or C.

Lie groups A Lie Group G is a smooth manifold with a compatible group structure. It is provided with an identity element e, a smooth composition law

* : (g, h) → g * h ∈ G and a smooth inversion law Inv : f → f (-1) ∈ G. Its tangent space at g is written T g G. The map L h : G g → h * g ∈ G is the left translation by h and is a dif- feomorphism of G. Therefore, its dierential (at g), DL h (g) : T g G → T L h g G is an isomorphism that connects tangents spaces of G. Similarly, one can dene R h : G g → g * h ∈ G, the right translation by h. A vector eld X on G is left invariant if (dL h )(X(g)) = X(L h (g)) = X(h * g) for each g, h ∈ G.
Similarly, one could dene right invariant vector elds. The left invariant vector elds form a vector space which we denote Γ(T G) L and which is isomorphic to T e G. The Lie bracket of two left invariant vector elds is a leftinvariant vector elds [Postnikov 2001].

Lie algebras As Γ(T G) L is closed under the Lie bracket of vector elds, we can look at T e G as a Lie algebra. More precisely, we dene g the Lie algebra of G as T e G with the Lie bracket induced by its identication with Γ(T G) L . The Lie algebra essentially captures the local structure of the group. In the case of Lie algebras of matrices, the Lie bracket corresponds to the commutator. For a more complete presentation of Lie groups and Lie algebras, we refer the reader to [START_REF] Bourbaki | Lie groups and lie algebras: Chapters 1-3. Elements of mathematics[END_REF]].

Writing the expression of the Lie bracket [, ] g on a given basis B g = {e i } n i=1 of g, we dene the structure constants f ijk as:

[e i , e j ] g = f ijk e k (4.2)
The structure constants f ijk depend on the basis B g chosen. They are always skewsymmetric in the rst two indices but they may have additional symmetry properties if we write them in a well-chosen basis (see below). The structure constants f ijk Chapter 4. Geometric Statistics on Lie groups: analysis of anatomical deformations.

47 completely determine the algebraic structure of the Lie algebra. Therefore, the structure constants are often the starting point, or the input, of algorithms on Lie algebras [Rand 1986, Rand 1988, Cohen 1997, De Graaf 2000]. It will also be the case for the algorithm we present in this paper.

Pseudo-metrics A pseudo-metric <, > on G is dened as a smooth collection of denite inner products <, > | g on each tangent space T g G. Then, G becomes a pseudo-Riemannian manifold. A metric is dened as a pseudo-metric whose inner products are all positive denite. In this case, G is called a Riemannian manifold. The signature (p, q) of a pseudo-metric is the number (counted with multiplicity) of positive and negative eigenvalues of the real symmetric matrix representing the inner product <, > | g at a point g and with respect to a basis of T g G. The signature is independent on the choice of the point g and on the basis at T g G. By denition, a pseudo-metric is denite, thus there is no null eigenvalues and we have p + q = n, where n is the dimension of G. By denition, a metric is positive dene and thus its signature is (n, 0). Again, further details about such Dierential Geometry can be found in [Postnikov 2001].

Quadratic Lie groups and algebras A left-invariant pseudo-metric is a pseudometric <, > such that for all X, Y ∈ T g G and for all g, h ∈ G we have:

< DL h (g)X, DL h (g)Y > | L h g =< X, Y > | g (4.3)
where L h is the left translation by h. In other words, the left translations are isometries for this pseudo-metric. Similarly, we can dene right-invariant and biinvariant pseudo-metrics <, >. Note that any Lie group admits a left (or right) invariant pseudo-metric: we can dene an inner product on the Lie algebra g = T e G and propagate it on each tangent space T g G through DL g (e) (or DR g (e)). However, not any Lie group admits a bi-invariant pseudo-metric.

The Lie groups that admit a bi-invariant pseudo-metric are called quadratic Lie groups. The corresponding Lie algebras are called quadratic Lie algebras. Note that quadratic Lie groups or algebras are called dierently in the literature. We nd the appellation metrizable or metrized in [Tsou 1957, Tsou 1962, Astrakhantsev 1978], metric in [START_REF][END_REF], Kath 2006], quasi-classical in [START_REF] Campoamor-Stursberg | Quasi-Classical Lie Algebras and their Contractions[END_REF] and nally quadratic in [Favre 1987, Benayadi 2014].

Figure (4.
2) shows a summary of the structures we just introduced. We now recall that a non-degenerate bi-invariant inner product on a nite dimensional Lie algebra g gives rise to a bi-invariant pseudo-metric on every Lie group whose Lie algebra is g (see for example [Milnor 1976]). Therefore, we focus on Lie algebras from now on. We will still use the terms "pseudo-metric" or "metric" and the notation "<, >" in order to refer to the corresponding inner products on the Lie algebra g = T e G.

Characterization of quadratic Lie algebras

We give here dierent formulations of an equation characterizing a pair (g, <, >) as a quadratic Lie algebra. A Lie algebra g is quadratic if and only if it has a pseudo-metric <, > verifying:

∀x, y, t ∈ g, < [x, y] g , t > + < y, [x, t] g >= 0 (4.4)
A proof for this characterization is given in [Postnikov 2001] and [Sternberg 1964]. First, taking advantage of the linearity in x, y, t we can rewrite Equation (4.4) on basis vectors. Let B g = {e i } n i=1 be a basis of g, we consider: x = e i , y = e j and z = e k . Thus we can express the Lie bracket in terms of the structure constants and we get: ∀i, j, k ∈ {1, .., n} f ijl < e l , e k > +f jkl < e l , e j >= 0 (4.5)

In particular, we observe that the structure constants written in a basis orthonormal with respect to a bi-invariant metric are totally skew-symmetric. The structure constants written in a basis orthonormal with respect to a bi-invariant pseudo-metric will have additional symmetric properties as well.

Then, as we consider nite dimensional Lie groups, we can also rewrite Equation (4.4) in terms of matrices:

∀x ∈ g, A(x) T .Z + Z.A(x) = 0 (4.6) where A(x) is the matrix of the endomorphism denoted [x, •] dened as y → [x, y],
and Z a symmetric invertible (non necessarily positive) matrix representing <, > on B g , the basis of g. Note that: x → A(x) is itself linear. Finally, taking advantage of the linearity again, and writing: A(e i ) = A i , we can again reformulate Equation (4.4) and we get:

∀i ∈ {1, .., n}, A T i .Z + Z.A i = 0 (4.7)
which is now a linear system of n matrix equations. Note that Equation (4.5) corresponds to Equation (4.7) written in coordinates.

Chapter 4. Geometric Statistics on Lie groups: analysis of anatomical deformations.

49 How to compute bi-invariant pseudo-metrics? Given a Lie algebra g as input, we see now that the computation of bi-invariant pseudo-metrics on g amounts to the resolution of the linear system of equations (4.7) for Z. The solutions of the linear system (4.7) form a vector space, which is called the quadratic space Q(g) [Duong 2011]:

Q(g) = {Z ∈ Sym(n) | ∀i ∈ {1, .., n}, A T i .Z + Z.A i = 0} (4.8)
Obviously, the vector space Q(g) contains invertible and non invertible solutions.

Recalling the denition of a pseudo-metric, we emphasize that we will be interested in invertible solutions only.

In order to solve the system (4.7) for Z, i.e. to compute the quadratic space Q(g), we could adopt an analytic point of view. At i xed, a single equation of the system (4.7) is a particular case of a Lyapunov equation that is studied in the context of control theory [Bartels 1972]. Thus, computational methods exist for studying one of our linear matrix equations [Kitagawa 1977]. For our purpose however, we want to understand the structure of a quadratic Lie group, in order to get an intuition for the generalization to innite dimensional Lie groups of dieomorphisms. Thus, we do not rely on an analytic point of view to solve the system (4.7).

We rather consider the whole system of equations (4.7) from an algebraic point of view. The pure algebraic point of view enables to solve the system (4.7) completely in most cases, like in the examples provided at the end of the paper. In the other cases, it leads to a smaller system of equations that can be solved analytically or computationally. Thus, the algebraic point of view provides not only a theoretical understanding of quadratic Lie groups, it also either solves the problem, or reduces the problem in order for the analytic point of view to solve it.

Therefore, we present in the next subsection the algebraic and geometric notions needed to set up, and later implement, the algebraic point of view.

Lie algebra representations

How can we understand the structure of a Lie algebra? An idea is to represent the Lie algebra elements as matrices acting on vectors. Then, the study of the behavior of these matrices helps to understand the Lie algebra as a whole. This is the purpose of the theory of Lie algebra representations, which we present briey relying on [Cartan 1933, Medina 1985a[START_REF] Bourbaki | Lie groups and lie algebras: Chapters 1-3. Elements of mathematics[END_REF], De Graaf 2000] in all this subsection.

Lie algebras representations A g-representation on the vector space V is a Lie algebra homomorphism η : g → gl(V ), that represents the elements of g as matrices acting on the vector space V . The g-representations θ 1 and θ 2 are said to be isomorphic if there is an isomorphism of representations between them, i.e. an isomorphism of vector spaces l :

V 1 → V 2 that veries: θ 2 (x) • l = l • θ 1 (x). We denote Hom g (V 1 , V 2 ) the vector space of isomorphisms of representations between V 1 and V 2 .
Chapter 4. Geometric Statistics on Lie groups: analysis of anatomical deformations.

50

In order to understand the representations of a Lie algebra g, and thus the Lie algebra g itself, a strategy is to decompose the representations into smaller bricks, and then study those bricks. In this context, a g-subrepresentation of the grepresentation V is a subspace of V stable by the elements of η(g). An irreducible gsubrepresentation is a g-subrepresentation without proper g-subrepresentation. An indecomposable g-subrepresentation is a g-subrepresentation that can not be decomposed into g-subrepresentations.

Note that irreducibility implies indecomposability but the converse is false: a g-representation can have a g-subrepresentation which does not have a supplementary that is also a g-subrepresentation (it would be "only" a vector space). Thus, it is not always possible to decompose a g-representation into irreducible g-subrepresentations, but only into indecomposable ones. In this context, a grepresentation that can be decomposed into irreducible g-representations is called completely reducible.

Adjoint and co-adjoint representation We can choose the vector space V on which we represent g. Taking V = g, thus representing the Lie algebra on itself, we dene the so-called adjoint representation of g, ad :

g x → ad(x) = [x, •] g ∈ gl(g).
In its matrix version, we recognize the matrices A of the previous subsection. We see also that the set of matrices A i dening the adjoint representation is equivalent to the set of structure constants of g.

We can rewrite again the Equation (4.4), but now in terms of the adjoint representation. We get: ∀x, y, t ∈ g, < ad(x).y, t > + < y, ad(x).t >= 0 (4.9)

Thus, the statement that g is quadratic with bi-invariant pseudo-metric <, > is equivalent to the requirement that all endomorphisms ad(x) are skew-symmetric endomorphisms with respect to <, >. Recalling the matrix version of Equation (4.4), that is Equation (4.6), we see that solving for a bi-invariant Z amounts to nd a symmetric isomorphism of representations Z between the adjoint representation of g, written in its matrix form as x → A(x), and the representation written in its matrix form as x → -A(x) T . If we choose to represent the Lie algebra g on the dual vector space g * , i.e. we choose V = g * , we can dene the co-adjoint representation θ:

g x → θ(x) ∈ gl(g * ) where < θ(x).f, t >=< f, ad(x).t > for f ∈ g * , x
, y ∈ g and <, > the inner product used to dene the dual basis. If we write A(x) the matrix of the endomorphism ad(x), T (x) the matrix of the endomorphism θ(x) and Z the inner product dening the dual basis, the previous denition states that Z is in fact an isomorphism of representation between the co-adjoint representation x → T (x) and the representation: x → A(x) T . Now, if the inner product <, > used to dene the dual basis is bi-invariant, by identifying the vector spaces g and g * , we can again rewrite Equation (4.4) to get: ∀x, y, t ∈ g, < ad(x).y, t > + < θ(x).y, t >= 0 (4.10)

Chapter 4. Geometric Statistics on Lie groups: analysis of anatomical deformations.

51

We conclude that the bi-invariance of the inner product implies the following relation between the adjoint and co-adjoint representations: ad = -θ. As Z (that represents <, >) is an isomorphism of representations between the co-adjoint and the representation x → A(x) T , we recover that the statement of Z being a biinvariant pseudo-metric on g is equivalent to Z being a symmetric isomorphism of representations between x → A(x) and x → -A(x) T .

Some algebra vocabulary

The adjoint representation is related to the structure constants of g and thus completely characterizes g. Thus, it links the language of abstract algebras and the language of representations for g.

For the special case of the adjoint representation ad, g-subrepresentations are ideals of g, irreducible g-representations are minimal ideals of g, indecomposable g-representations are ideals of g that can not be decomposed into a direct sum of ideals of g. We will use the two languages of ideals or of representations.

If the adjoint representation is itself irreducible but not 1-dimensional, g is said to be simple. If the adjoint representation is completely reducible, g is said to be reductive. If the adjoint representation is completely reducible without 1-dimensional subrepresentations, g is semi-simple. If the adjoint representation is completely reducible with only 1-dimensional subrepresentations, g is abelian. A reductive Lie algebra is thus the sum (in the sense of subrepresentations) of a semi-simple Lie algebra and an abelian Lie algebra.

Some vocabulary of geometry An ideal I of a Lie algebra B is said to be isotropic with respect to a pseudo-metric given on B if I ∩ I ⊥ = {0}. The ideal I is said to be totally isotropic if I ⊂ I ⊥ . The intersection between I and I ⊥ represents the vectors that are orthogonal to themselves, and thus that have zero norm even if they are themselves non zero. Thus, isotropic ideals appear only in a case of a pseudo-metric that is not a metric. From the intuition provided by Theoretical Physics, we can interpret the vectors in I ∩I ⊥ as photons: they have zero mass even if they have non-zero velocity.

Constructions with Lie algebra representations

We have seen that we can study the structure of a given Lie algebra by looking at its representations and more particularly at its adjoint representation. Here, we study decompositions of the adjoint representation that will be pertinent for the characterization of quadratic Lie algebras: the direct sum decomposition and the double extension decomposition. We show how these decompositions can be implemented in a computational framework. In this subsection, we use the notation (B, [, ] B ) to denote the Lie algebra, because this is the notation that we will use in the core of our algorithm (see Section 4.4).

Denition of direct sum

B = B 1 ⊕ B B 2 is the direct sum of B 1 , B 2 if:
Chapter 4. Geometric Statistics on Lie groups: analysis of anatomical deformations.

52 This decomposition was rst studied by [Astrakhantsev 1978]. We illustrate it with the matrices

• B = B 1 ⊕ B 2 in terms of vector spaces, • [B, B 1 ] B ⊂ B 1 , and [B, B 2 ] B ⊂ B 2 ,
A representing the adjoint representation b → [b, •] B of B, i.e. the matrices denoted: b → A(b) = [b, •] B .
The direct sum of B is equivalent to the decomposition of the adjoint representation into the B-representations B 1 and B 2 i.e.:

A(b) =   A(b 1 ) 0 0 A(b 2 )   (4.11) on a basis respecting B = B 1 ⊕ B B 2 .
Note that we write ⊕ B to emphasize the fact that this direct sum decomposition is more than the direct sum decomposition into vector spaces.

Direct sum decomposition and bi-invariant pseudo-metrics We have the following property: B being quadratic is equivalent to B 1 and B 2 being quadratic. Medina 1982, Medina 1985a].

Indeed, if <, > B 1 , <, > B 2 are bi-invariant pseudo-metrics on B 1 , B 2 and represented by the matrices Z B 1 , Z B 2 then: Z B 1 ⊕ B B 2 =   Z B 1 0 0 Z B 2   (4.12) is bi-invariant on B. And conversely, if <, > B is bi-invariant on B, its restrictions <, > B | B 1 and <, > B | B 2 , are bi-invariant on B 1 , B 2 [
Computing the direct sum The direct sum decomposition of a Lie algebra B into indecomposable subrepresentations is unique, up to isomorphisms. In practice, writing

B B = {e k } dim(B) k=1
a basis of B and A k = A(e k ), computing the direct sum decomposition of B into indecomposable B i 's amounts to the simultaneous bloc diagonalization of the matrices A k .

Denition of double extension

B = W ⊕ S ⊕ S * is the double extension of W by a simple S if: • B = W ⊕ S ⊕ S * in terms of vector spaces, • (W , [, ] W ) is a Lie algebra and [S, W ] B ⊂ W makes W a S-representation, • (S, [, ] S ) is a simple Lie subalgebra of B: [s, s ] B = [s, s ] S ,
• S * is the dual space of S and [S, S * ] B ⊂ S * makes S * the co-adjoint representation,

Chapter 4. Geometric Statistics on Lie groups: analysis of anatomical deformations.

53

• ∀w, w ∈ W : [w, w ] B = [w, w ] W + β(w, w )
where β : Λ 2 W → S * is a (skew-symmetric) S-equivariant map, i.e. a map that commutes with the action of S.

This denition relies on the framework introduced in [Medina 1985a], or in [Keith 1984] under the appellation "bi-extension". Here, we can illustrate it with the matrices representing the adjoint representation b → [b, •] B of B, i.e. the matrices denoted: b → A(b). The double extension decomposition is equivalent to the following decomposition of the adjoint representation of B:

A(b) =      [w, •] W + [s, •] B [w, •] B 0 0 [s, •] S 0 β(w, •) [f, •] B [s, •] B      (4.13)
on a basis respecting B = W ⊕ S ⊕ S * and b = w + s + f . Note that, in the blocks of the matrix A(b), we have identied endomorphisms with their corresponding matrices.

The denition of double extension uses a number of dierent notations. First, we recognize ad(s) = [s, •] S and ad(w) = [w, •] W to be respectively the adjoint representation of S (on S) and the adjoint representation of W (on W ). But [s, •] B is a S-representation on W that has nothing to do with the adjoint (the adjoint is a representation of a Lie algebra on itself ).

Then, we should be careful with the structures that are manipulated. For example we can consider the vector space S * as an abelian Lie subalgebra of B. But we can not consider W as a subalgebra of B. The skew-symmetric map β represents precisely the corresponding obstruction.

Double extension decomposition and bi-invariant pseudo-metrics

We have the following property: B being quadratic is equivalent to W being quadratic. Indeed, if <, > W is bi-invariant on W , represented by Z W , then:

Z W ⊕S⊕S * =      Z W 0 0 0 0 I 0 I 0      (4.14) is bi-invariant on B.
Conversely, if B is quadratic and written as a double extension of W with S simple (or 1-dimensional), then the restriction [Medina 1982, Medina 1985a]. Note here that we can write the I-blocks because the basis of S and S * are chosen to be duals of each other. If two dierent basis were chosen, the corresponding bi-invariant pseudo-metric on B = W ⊕ S ⊕ S * would have the form:

<, > W =<, > B | W is bi-invariant
Z W ⊕S⊕S * =      Z W 0 0 0 0 L 0 L T 0      (4.15)
Chapter 4. Geometric Statistics on Lie groups: analysis of anatomical deformations.

54

with L an invertible matrix representing precisely the change of basis. More precisely, by computing Equation (4.6) on this last Z W ⊕S⊕S * while choosing s ∈ S, we show that L is necessarily an isomorphism of S-representations on S and I, i.e. L ∈ Hom S (S, S * ). This remark will be used in practice in the algorithm (see Section 4.4).

Computing double extensions Contrary to the direct sum decomposition, the decomposition of a quadratic Lie algebra B as a double extension is not necessary unique. For example, given a quadratic indecomposable non simple B, we can build a double extension decomposition from each minimal ideal of B [Medina 1985a]. It proceeds as follows. We take a minimal ideal I of B and consider I ⊥ its orthogonal with respect to a bi-invariant pseudo-metric <, > B . The decomposition:

B = W ⊕ S ⊕ S *
where:

W = I ⊥ /I, S = B/I ⊥ and S * = I
is a double extension of W with S simple (or 1-dimensional). Moreover, one can show that I and I ⊥ verify the following properties:

• I is abelian,

• I ⊥ is a maximal ideal,

• I ⊂ I ⊥ (total isotropy),

• [I, I ⊥ ] = 0 (commutativity),

• codim(I ⊥ ) = dim(I).

These necessary conditions are taken from [Medina 1982, Medina 1985a, Astrakhantsev 1978].

In practice in our algorithm, we will have to build a double extension from a B in order to compute a bi-invariant pseudo-metric on B, if it exists (see Section 4.4). Therefore, even if we know an abelian minimal ideal I of B, we will not have its orthogonal I ⊥ needed for the construction shown above: we do not know any bi-invariant pseudo-metric as we want to build one! Thus, given an abelian minimal ideal I, we shall test all ideals J that could be an I ⊥ for a bi-invariant pseudo-metric, i.e. all ideals J that verify the necessary conditions listed above.

We show here that the only plausible ideals that can play the role of

I ⊥ are, either J = C B (I) the centralizer of I in B in the case C B (I) = B, or the maximal ideals of codimension 1 containing I in the case C B (I) = B.
We have seen above that a rst necessary condition for a J to be an I ⊥ is its commutativity with I: [I, J] = 0. We recall that the centralizer C B (I) of I in B is dened as the set of elements that commute with I. Thus: J ⊂ C B (I).

Another necessary condition for a plausible J is to be a maximal ideal. As I is an ideal, C B (I) is also an ideal. Thus J is a maximal ideal included in the ideal C B (I): we have necessarily J = C B (I) in the case C B (I) = B. In this case, the Chapter 4. Geometric Statistics on Lie groups: analysis of anatomical deformations. 55 condition I ⊂ J is fullled as I is abelian. The last necessary condition to check is codim(C B (I)) = dim(I).

However if C B (I) = B, then we shall look for maximal ideals of B. But in this case, I commutes with all elements of B and therefore I is necessarily of dimension 1 as a minimal ideal. Therefore we shall look for maximal ideals J of codimension 1. Adding the last necessary condition, we conclude that in the case C B (I) = B, we shall consider only maximal ideals of codimension 1 containing I.

Structure of quadratic Lie groups

Here, we characterize the structure of quadratic Lie algebras, using the constructions dened in the previous section. We rst present a reformulation of a classication theorem of quadratic Lie algebras. Then, we emphasize which Lie algebras we add by asking for a bi-invariant pseudo-metric instead of a bi-invariant metric. We nally investigate how we can go from a bi-invariant pseudo-metric to a bi-invariant dual metric on a special class of Lie algebra with bi-invariant pseudo-metrics.

A classication theorem

To characterize the structure of a quadratic Lie algebra, we use a reformulation of a classication theorem than can be found in [Medina 1985a] or [Keith 1984].

Theorem 4.3.1 (Classication of quadratic Lie algebras) The Lie algebra g is quadratic if and only if its adjoint representation decomposes into indecomposable subrepresentations B that are of the following types:

• Type (1): B is simple (or 1-dimensional),

• Type (2): B = W ⊕ S ⊕ S * is a double extension of a quadratic W by S simple (or 1-dimensional).

This means that any quadratic Lie algebra writes g = B 1 ⊕ g ... ⊕ g B N where each B is of Type (1) or of Type (2). In particular, we can already conclude that any reductive (a fortiori, semi-simple or abelian) Lie algebra g is quadratic. Moreover, if g is quadratic but not reductive, then g has non-irreducible indecomposable subrepresentations, and these are necessarily double extensions of Type (2). We recall that the notions of representation decomposition come from a simultaneous diagonalization of matrices. Therefore, they depend on the base eld F: a Lie algebra reductive in R is reductive in C but the converse is false. And thus, being quadratic also depends on the eld we consider. A Lie algebra quadratic on R will be quadratic on C but the converse is false.

Elementary bi-invariant pseudo-metrics The previous characterization of quadratic Lie algebras in terms of their structure is useful in practice. It enables to construct a type of bi-invariant pseudo-metric <, > g that exists necessarily on a Chapter 4. Geometric Statistics on Lie groups: analysis of anatomical deformations. 56 quadratic g. We call this type of pseudo-metrics the elementary bi-invariant pseudometrics of g.

The elementary bi-invariant pseudo-metric <, > B of a 1-dimensional Lie algebra B is dened to be the multiplication. The elementary bi-invariant pseudo-metric <, > B of a simple Lie algebra B is dened to be the Killing form. Now let us dene recursively the elementary bi-invariant pseudo-metrics of a general quadratic g.

Let us be given a quadratic Lie algebra g on which we know an auxiliary biinvariant pseudo-metric <, > g (not necessarily of the elementary type). First, we decompose the adjoint representation of g into indecomposable subrepresentations B's:

g = B 1 ⊕ g ... ⊕ g B N .
Then, we study separately the two cases: the B's of Type (1) and the B's of Type (2).

On the B's of Type (1), we dene the elementary bi-invariant pseudo-metric <, > B as above: the multiplication if B is 1-dimensional or the Killing form if B is simple.

On the B's of Type (2), we build a double extension. To this aim, we consider a minimal ideal I and, using the auxiliary bi-invariant pseudo-metric <, > g of g,

we compute I ⊥ . We get the double extension B = W ⊕ S ⊕ S * with W = I ⊥ /I, S = B/I ⊥ and S * = I. We construct an elementary bi-invariant pseudo-metric <, > W on W recursively. We then dene an elementary bi-invariant pseudo-metric <, > B on the double extension B = W ⊕S ⊕S * to be of the form of Equation (4.14). Finally, we dene the elementary bi-invariant pseudo-metric <, > g on the direct sum decomposition g = B 1 ⊕ g ... ⊕ g B N to be of the form of Equation (4.12). This construction denes (and proves the existence of) elementary bi-invariant pseudometrics on a quadratic g.

Riemannian and pseudo-Riemannian quadratic Lie groups

The previous characterization of quadratic Lie algebras can be rened to distinguish between quadratic Lie algebras that admit bi-invariant metric with respect to quadratic Lie algebras with bi-invariant pseudo-metrics. In other words, it answers the questions: which Lie algebras do we add by removing the positivity of the metric?

Studying the signature We recall from Section 4.2 that a metric on g of dimension n has signature (n, 0). Now, we take a quadratic g that is decomposed into indecomposable pieces g = B 1 ⊕ g ... ⊕ g B N , where the B i are either simple (or 1-dimensional) or double extensions. The signature on the direct sum is the sum of the signatures on the B i [Postnikov 2001]:

sgn g = sgn B 1 + ... + sgn B N (4.16)
Therefore, asking for a positive denite signature on g is equivalent to asking for a positive denite signature on each of the B's.

If B is simple, it possess a bi-invariant metric if and only if it is compact. If B is a double extension, a bi-invariant pseudo-metric has necessary a non positive Chapter 4. Geometric Statistics on Lie groups: analysis of anatomical deformations. 57 denite signature of the form [Medina 1985a]:

sgn B = sgn W + (m, m) (4.17)
where m is the dimension of the minimal ideal I used to build the double extension .

We conclude that g admits a bi-invariant metric if and only if its indecomposable parts are simple compact or 1-dimensional, i.e. if and only if g is reductive with compact simple parts.

Comparison The trees of g Thus, going from Riemannian to pseudo-Riemannian enables to add the simple algebras that generalize the compact algebras and the double extension structures (in blue) with its recursive construction that is not present in the Riemannian case.

B g i B g 1 B g N 1-dimensional compact

From a bi-invariant pseudo-metric to a bi-invariant dual metric?

We investigate here a special case of Lie algebras we gain by going from Riemannian to pseudo-Riemannian: the double extension of W = {0} by a compact simple Lie algebra K, which is an example of Manin triple (see [Drinfeld 1987, Belavin 1998]).

Chapter 4. Geometric Statistics on Lie groups: analysis of anatomical deformations.

58

We will see in this subsection that we can view this case as a Riemannian case by changing the base eld F (which is R or C for us) to its dual algebra D. This development is a new contribution which is a justication and an extension of the dual quaternions for SE(3).

Dual numbers and vectors Given a eld F, the algebra D of dual numbers over this eld is dened as D = F + εF where ε 2 = 0 and ε = 0 denes the multiplication [Grunwald 1906]. We can dene a m-dimensional dual vector space D m = F m +εF m , whose elements are dual vectors. Note here that the term "vector" is abusive in the sense that a vector space is usually dened on a eld, not on an algebra. In the following, in order to study the properties of the dual vector space, we will use the dual map :

ψ : F m ⊕ F m → D m x 0 + x ε → x 0 + εx ε
using the same notation ψ for mapping either to dual numbers or to dual vectors.

From the double extension g = K ⊕ K * to its dual g = K + εK * Now we consider the double extension g = K ⊕ K * where K is compact simple and dim(K) = m so that dim(g) = 2m. We take the following elementary bi-invariant pseudo-metric on g:

Z K⊕K * =   I I I 0   (4.18)
As K and K * have same F-dimension m, we consider the dual space g = K + εK * , of D-dimension m. Its dual vectors write x = x 0 + ε.x ε where x 0 ∈ K and x ε ∈ K * .

Proposition 4.3.2 The dual map:

ψ : g = K ⊕ K * → g x 0 + x ε → x 0 + εx ε
is an isomorphism of Lie algebras that respects the sum K ⊕ K * . The canonical inner product on g is bi-invariant and corresponds to the bi-invariant pseudo-metric Z K⊕K * above.

This can been shown as follows. First consider the Lie bracket on g inherited from ψ. We have:

[ψ(x), ψ(x )] = [x 0 + εx ε , x 0 + εx ε ] = [x 0 , x 0 ] + ε([x 0 , x ε ] + [x ε , x 0 ]) (as ε 2 = 0) = ψ([x, x ]) (denition of double extension)
which proves the isomorphism of Lie algebras.

Chapter 4. Geometric Statistics on Lie groups: analysis of anatomical deformations.

59

We now show that the pseudo-metric Z K⊕K * on the Lie F-algebra g maps to the canonical metric Z = I on the Lie D-algebra g:

ψ(x) T .ψ(x ) = (x 0 + ε.x ε ) T .(x 0 + εx ε ) = x T 0 .x 0 + ε(x T ε .x 0 + x T 0 .x ε ) = ψ(x T .Z K⊕K * .x)
(using ψ for dual numbers)

In others words, the spaces g and g are isometric. But, again the term "isometric" is abusive, as we recall that g and g are not dened on the same eld -the latter being dened on an algebra.

Towards statistics on dual Riemannian manifolds We have shown that a double extension g = K ⊕ K * of W = {0} by a compact simple K, endowed with a bi-invariant pseudo-metric is isometrically isomorphic to a dual Lie algebra g with a bi-invariant metric. Thus, we could think of generalizing the theory of statistics on Riemannian manifolds to a theory of statistics on dual Riemannian manifolds. However, the fact that the space is dened on an algebra may cause some problems.

Generalization? One could wonder if we can use this construction for any general double extension. But we should note that this construction takes advantage of the fact that K * is totally isotropic and abelian. The element ε such that ε 2 = 0 enables to represent the commutativity of K * (Lie bracket is null) and the self orthogonality of K * (inner product is null) at the same time. A general Lie algebra with bi-invariant pseudo-metric, is not necessarily decomposable into two subspaces of same dimension such that one of them is abelian and isotropic. For example, take a Lie algebra of odd dimension.

An algorithm to compute bi-invariant pseudometrics on a given Lie group

We go back to the general case of any quadratic Lie algebra over the eld F (F = R or C). We present in this section an algorithm that computes bi-invariant pseudometrics on a Lie algebra given as input.

Then, we show how one could generalize the algorithm to compute all bi-invariant pseudo-metrics on g. Finally, we apply the algorithm to some Lie groups known to possess a unique bi-invariant mean: we nd that most of them are not quadratic.

The algorithm: computation of one bi-invariant pseudometric

For the computations, we will use matrix representations Z of pseudo-metrics <, >, where the basis will be specied. The input is B g = {e i } n i=1 , a basis of g and the structure constants f ijk on this basis. The output is a symmetric invertible matrix Chapter 4. Geometric Statistics on Lie groups: analysis of anatomical deformations. 60 Z g on the basis B g , representing an elementary bi-invariant pseudo-metric, or a message of error: "The Lie algebra g is not quadratic".

Core of the algorithm The core of the algorithm tests the structure of the Lie algebra given as input, to determine if it matches the characteristic tree-structure of quadratic Lie algebras described in the Section 4.3 (see Figure 4.3.2). Simultaneously with the progress through the tree, the algorithm tries to construct recursively an elementary bi-invariant pseudo-metric <, > g by testing all possible candidates. If it succeeds, we return the bi-invariant elementary pseudo-metric, proving that g is quadratic. If not, we conclude that g is not quadratic and we return the error message. More precisely, the algorithm is divided in four steps as follows.

Step 1: Direct sum decomposition In this step, we decompose the adjoint representation of g into indecomposable B's, in other words: we decompose g as a direct sum of B's.

g = B 1 ⊕ g ... ⊕ g B N (4.19)
An implementation of this step can be found in [Rand 1988].

From now on, we work on the basis B g that respects the direct sum: g = B 1 ⊕ g ... ⊕ g B N . The B's are indecomposable Lie algebras, thus we can take advantage of the classication theorem 4.3.1 of Section 4.3. In the following two steps, we test if each B is either of Type (1) (1-dimensional or simple) or of Type (2) (a double extension).

Step 2: Testing the Type (1) In this step, we test if the indecomposable B is of Type (1), i.e. if B is 1-dimensional or simple (see dichotomy of Theorem 4.3.1).

To test if B is 1-dimensional, we can obviously count the number of basis vectors of B in the basis B g . If B is found 1-dimensional, we return the multiplication, which is an elementary bi-invariant pseudo-metric on B.

To test if B is simple, we use a function that computes the radical of the Levi decomposition of B [START_REF] Levi | Sulla struttura dei gruppi niti e continui[END_REF]]. The indecomposable piece B is simple if and only if the radical is null. Such a function can be found in [Cohen 1997]. If B is found simple, we return the Killing form, which is an elementary bi-invariant pseudometric.

If B is neither 1-dimensional nor simple, we conclude that B is not of Type (1). We test in the following step if B is of Type (2).

Step 3: Testing the Type (2) In this step, we test if B is of Type (2), i.e. if B is a double extension of a quadratic W by a simple S (see dichotomy of Theorem 4.3.1). We recall that the double extension structure of B is not necessarily unique. Therefore it might seem that we need to test all possible candidates for a double extension structure of B, in order to answer if B is of Type (2). We proceed slightly dierently.

As B is indecomposable and not of Type (1) (see previous steps), B being of Type (2) is equivalent to B being quadratic. More precisely, at this step of the Chapter 4. Geometric Statistics on Lie groups: analysis of anatomical deformations. Thus we will consider only one minimal ideal I of B and try to construct a double extension out of it, of the form: B = W ⊕ S ⊕ I. Note that this step will need to call the algorithm recursively, to determine if the candidate for W in the double extension structure is quadratic or not. The details of this step are below.

Step 3.a First, we compute a minimal ideal I. More precisely, recalling the necessary conditions of the double extension structure of Section 4.2, we compute I an abelian minimal ideal, which is also a minimal abelian ideal. A function that nds a minimal abelian ideal of B can be derived from an algorithm of [START_REF][END_REF], that computes all abelian ideals of B: we can choose one of minimal dimension among those.

Step 3.b Then, we compute C B (I), the maximal ideals J's and the corresponding candidates for the double extension structure of B. The computation of C B (I) is implemented in [Motsak 2006].

If C B (I) = B, we take J = C B (I) and verify the condition codim(J) = dim(I). If the condition is not fullled, there is no double extension structure possible for B. Therefore, we conclude that B is not of Type(2).

If C B (I) = B, we compute the maximal ideals J of B of codimension 1 containing I (see Section 4.2). If no such ideals are found, there is no double extension structure possible for B. Again in this case, we conclude that B is not of Type(2).

If J's are found, we compute the corresponding double extension candidates of B, one per J, as: B = W ⊕ S ⊕ S * where: W = J/I, S = B/J and: S * = I.

(4.20)

We call the algorithm recursively on W , i.e. we determine recursively if W is quadratic. If there is no double extension candidate with a quadratic W , we conclude that B is not of Type (2). Otherwise, we keep the double extension candidates that have a quadratic W (with an elementary bi-invariant pseudo-metric Z W ).

Step 3.c Then, we try to compute an elementary pseudo-metric for all double extension candidates of the form: B = W ⊕ S ⊕ S * where W = J/I is quadratic with corresponding Z W , S = B/J and S * = I. Given a double extension candidate, we know from Section 4.2 that an elementary pseudo-metric on B has the form:

Z B=W ⊕S⊕I =      Z W 0 0 0 0 L 0 L T 0      (4.21)
Chapter 4. Geometric Statistics on Lie groups: analysis of anatomical deformations.

62

where L ∈ Hom S (S, I).

Therefore we need to compute Hom S (S, I). We recall that S is simple thus its adjoint representation is irreducible. As we are in the case of a nite dimensional irreducible representation, we can apply Schur's Lemma. Its general form states that Hom S (S, S) is a associative division algebra over F (= R or C), which is of nite degree because S is nite dimensional [Schur 1905]. When the base eld is F = C, we use the fact that a nite-dimensional division algebra over an algebraically closed eld is necessarily itself. Thus Hom S (S, S) = C and dim C (Hom S (S, S)) = 1. When the base eld is F = R, we use Frobenius theorem which asserts that the only real associative division algebras are R, C or H, the eld of quaternum numbers, [START_REF] Frobenius | Ueber lineare substitutionen und bilineare formen[END_REF]]. Thus Hom S (S, S) is R, C or H, and dim R (Hom S (S, S)) is 1, 2 or 4. Now if I and S are isomorphic, Hom S (S, I) is isomorphic to Hom S (S, S) and thus of maximal dimension 4 over F. Otherwise, if I and S are not isomorphic, we have Hom S (S, I) = {0}.

The computation of Hom S (S, I) is implemented in [Brooksbank 2008], more generally for any nite-dimensional modules of a nitely generated algebra.

Step 3.d To conclude Step 3, we determine if one of the possible elementary pseudo-metrics computed above is bi-invariant. To this aim, we plug the expression of Z B=W ⊕S⊕I in Equation (4.7) and solve it for L. Thus, the initial system of Equations 4.7 has been reduced to an equation in maximum 1 (complex case) or in 4 (real case) parameters.

We run this step for each double extension candidate. If a bi-invariant elementary pseudo-metric Z B is found on one of the candidates, we return Z B . Otherwise, we conclude B is not of Type (2).

Step 4: Construction of a bi-invariant pseudo-metric on the whole g In this step, we construct a bi-invariant (elementary) pseudo-metric on g, if it exists. If one B of the direct sum decomposition g = B 1 ⊕ g ... ⊕ g B N is neither of Type (1) nor of Type (2), we conclude from Theorem 4.3.1 that g is not quadratic. We return the error message. Otherwise, we glue together the elementary bi-invariant pseudo-metrics Z B 's that have been returned on the B's.

More precisely, we follow the construction of Section 4.2 to build the elementary bi-invariant pseudo-metric Z g on the basis B g of g that respects the direct sum decomposition:

Z g=B 1 ⊕g...⊕gB N =      Z B 1 0 0 0 . . . 0 0 0 Z B N      (4.22)
Finally, we perform a change of basis from B g to B g in order to return Z g , an elementary bi-invariant pseudo-metric on the basis of the Lie algebra given as input.

Tree structure of the algorithm The algorithm has a natural tree structure presented in Figure 4.5. The bi-invariant pseudo-metric Z g is computed in a postx manner. A tree level corresponds to a reduction of an adjoint representation: re-Chapter 4. Geometric Statistics on Lie groups: analysis of anatomical deformations.

63 duction of g into B's for the rst level, reductions of the W 's into B's for the others.

The arrows in dashes represent the cases we investigate to test if g is quadratic. If B is not in one of such cases, then B is not quadratic, so neither is g and we exit the algorithm. In pseudo-code, the algorithm is written as follows. This gives a bi-invariant pseudo-metric on the Lie algebra g. We can then make it a bi-invariant pseudo-metric on the Lie group G by propagating it through DL g (e) (or DR g (e)) on all tangent spaces T g G (see Section 4.2).

g B g 1 B g i B g N 1-dim. simple S ⊕ g S * W ⊕ g S ⊕ g S * EXIT ... ...
All in all, the algorithm allows to compute one bi-invariant pseudo-metric of g, i.e. one invertible element of the quadratic space Q(g). We can generalize the algorithm, in order to compute all bi-invariant pseudo-metrics of g, thus the whole quadratic space Q(g). This is the purpose of the next subsection.

Generalization of the algorithm: computation of all biinvariant pseudo-metrics

Here, we present how one should proceed in order to compute all bi-invariant pseudometrics of a given Lie algebra g, i.e. the whole quadratic space Q(g). Note that the dimension of Q(g) is unknown in the general case [Duong 2011]. However, the algorithmic procedure allows to compute the space anyway.

We follow the strategy of the previous algorithm: we decompose g into indecomposable B's, we compute the quadratic spaces Q(B) for each of them and then glue these spaces together to get Q(g).

Computing the quadratic space of indecomposable Lie algebras In this step, we compute the quadratic space for all indecomposable pieces B's of g, the simple (or 1-dimensional) and the double extensions.

The quadratic space of a 1-dimensional piece B is the weighted multiplication, so the whole base eld F:

Q(B) = {Z B = αI | ∀α ∈ F} = F (4.23)
Chapter 4. Geometric Statistics on Lie groups: analysis of anatomical deformations.

64 Algorithm 1 Computation of a bi-invariant pseudo-metric on g Input: B g = {e i } i basis of g, Structure constants f ijk on this basis. Initialization B = g. Core: Switch B:

• Case B is decomposable: decompose into B = B 1 ⊕ B ... ⊕ B B N , call algorithm recursively on the B i 's; return: Z B = Diag(Z B 1 , . . . , Z B N ). • Case B is 1-dimensional: return Z B = 1 . • Case B is simple: return Z B = Z Killing .
• Else:

Z B=W ⊕S⊕I =      Z W 0 0 0 0 L 0 L T 0      ,
If there is no solution: "EXIT". Else, return the corresponding Z B .

Output:

• if "EXIT": return the message "The Lie algebra g is not quadratic";

• Else: return the elementary bi-invariant pseudo metric on g.

Chapter 4. Geometric Statistics on Lie groups: analysis of anatomical deformations.

65

The quadratic space of a simple piece B is the vector space spanned by the Killing form.

Q(B) = {Z B = αZ Killing | ∀α ∈ F} (4.24)
The quadratic space of a double extension B = W ⊕ S ⊕ S * , where the basis of S and S * are chosen duals, is given by:

Q(B) =          Z B =      Z W M N M T αZ Killing βI N T βI (0)      ∀α, β ∈ F, ∀Z W ∈ Q(W ),
∀M, N sol. of eq. derived from (4.7)

         (4.25)
We leave to the reader the computations of the equations derived from Equation (4.7) that M and N are solving. Because of the dimension reduction, these equations can be solved in a lot of interesting cases. In our computations on selected Lie groups in the next subsection, N and N are vectors or scalars for example.

Computing the quadratic space of a direct sum The second step is the computation of the quadratic space of a direct sum g = B 1 ⊕ g ... ⊕ g B N , given the quadratic spaces of each of its indecomposable pieces B i . This gives:

Q(g) =          Z g ∈ Sym(n) | s.t. for i ∈ {1, ..N } (block index): Z g ii = Z B i ∈ Q(B i ) if i = j Z g ij = M ij if i < j          (4.26)
where M ij is a matrix that solves the following equation, derived from (4.7):

A(b i ) T .M ij + M ij .A(b j ) = 0 ∀b i ∈ B i , ∀b j ∈ B j (4.27)
In summary, the problem of computing all bi-invariant pseudo-metrics of a given g amounts to the resolution of a reduced number of algebraic equations of lower dimension.

Results of the algorithm on selected Lie groups

We run our algorithm manually to determine if a bi-invariant pseudo-metric exists on some real Lie groups for which there is a locally unique bi-invariant mean:

SE(n), ST (n), H and U T (n), for n ∈ N * [Pennec 2012].
We run the computations manually and illustrate them, for each example, with the corresponding progress through the tree of the algorithm. The results show that most of these Lie groups are not quadratic.

Chapter 4. Geometric Statistics on Lie groups: analysis of anatomical deformations. 66 Scalings and translations ST (n) The Lie group ST (n) comprises uniform scalings together with translations of R n . It is the semi-direct product R * + R n , its elements being written (λ, t). More precisely, ST (n) is dened by its action on R n : (λ, t).x = λ.x + t. The group law and the group inversion are written as follows:

(λ 1 , t 1 ) * (λ 2 , t 2 ) = (λ 1 .λ 2 , λ 1 * t 2 + t 1 ) and (λ, t) (-1) = (1/λ, -t/λ). The Lie algebra st(n) comprises the (µ, u) ∈ R ⊕ R n with Lie Bracket: [(µ 1 , u 1 ), (µ 2 , u 2 )] = (0, µ 2 .u 1 -µ 1 .u 2 ).
(4.28)

Input We choose the basis (D, {P a } n a=1 ) dened as: D = (1, 0) and P a = (0, e a ) with (e a ) n a=1 the canonical basis of R n . In this basis, the structure constants can be read in the following Lie brackets:

[P a , P b ] = 0, [D, P a ] = P a , [D, D] = 0.
Step 1 From the expression of the Lie brackets above, we can compute all ideals of st(n) manually and nd: Span(P 1 ), ..., Span(P n ) and their linear combinations. We remark that there is no ideal containing D. Thus st(n) cannot be written as the direct sum of ideals, i.e. st(n) is indecomposable.

Step 2 First, as n ∈ N * , we have dim(st(n)) > 1. Thus st(n) is not 1dimensional. Then, as Span(P 1 ) for example is an ideal, st(n) is not simple. We conclude that st(n) is not of Type (1).

Step 3 We take I = Span(P 1 ) which is obviously a minimal abelian ideal.

From the commutation relations given by the Lie brackets, we see that C st(n) (I) = Span({P a } n a=1 ) and we are in the case C st(n) (I) = st(n). Thus there is only one double extension candidate, with J = C st(n) (I). We dene S = st(n)/J = Span(D) and W = J/I = Span(P 2 , ..P n ). We call the algorithm recursively on W , which decomposes into 1-dimensional ideals on which we return the multiplication.

The S-representation on S is the null representation: [D, D] = 0. The Srepresentation on I is the trivial representation: [D, P 1 ] = P 1 . Hence, I and S are not isomorphic S-representations and Hom S (S, I) is zero. We conclude that st(n) is not of Type (2).

Output We have found that st(n) is indecomposable and neither of Type (1) nor of Type (2). Thus st(n) is not quadratic: there is no bi-invariant pseudo-metric <, > on st(n).

Heisenberg group H The Heisenberg group H comprises 3D upper triangular matrices M of the form:

M =      1 x z 0 1 y 0 0 1      .
Chapter 4. Geometric Statistics on Lie groups: analysis of anatomical deformations.

67 st(n) B 1 = st(n)
Exit: NO Figure 4.6: Schematical result for ST (n). We see on the top level that st(n) is indecomposable (it decomposes into itself). We see on the bottom level that st(n) is neither 1-dimensional, nor simple, nor a double extension and therefore we exit the algorithm: st(n) is not quadratic.

Thus, an element of this group can be written as (x, y, z) ∈ R 3 , with corresponding group law

(x 1 , y 1 , z 1 ) * (x 2 , y 2 , z 2 ) = (x 1 + x 2 , y 1 + y 2 , z 1 + z 2 + x 1 * y 2 ) and group inversion (x, y, z) (-1) = (-x, -y, -z + xy).
The Lie algebra h comprises the nilpotent matrices:

N =      0 p c 0 0 q 0 0 0      .
Input A basis for h is thus (P, Q, C) with clear notations. In this basis, the structure constants can be read in the following Lie brackets:

[C, P ] = 0, [C, Q] = 0, [P, Q] = C.
Step 1 From the expression of the Lie brackets above, we can compute all ideals of h manually and we nd: Span(C), Span(C, P ) and Span(C, Q). We remark that there is no ideal whose a supplementary is also an ideal. Thus h is indecomposable.

Step 2 h is obviously not 1-dimensional. Moreover, as Span(C) for example is an ideal, h is not simple. We conclude that h is not of Type (1).

Step 3 We take I = Span(C) which is a minimal abelian ideal of h. From the commutation relations given by the Lie brackets, we compute the commutator of I and we see that we are in the case C h (I) = h. Thus we consider all maximal ideals of h that are of codimension 1 and contain I. We get J = Span(C, P ) or J = Span(C, Q), thus we have two double extension candidates. By symmetry in P ↔ Q (see structures constants), we can consider J = Span(C, P ) only, without lost of generality. We dene S = h/J = Span(Q) and W = J/I = Span(P ). We call the algorithm recursively on W . As W is 1-dimensional, W is quadratic and we return Z W = 1 .

The S-representation on S is given by the bracket [Q, Q] = 0: it is the null representation. The S-representation on I is given by the bracket [Q, C] = 0: it is Chapter 4. Geometric Statistics on Lie groups: analysis of anatomical deformations. 68 also the null representation. The isomorphism of vector spaces L that maps C on Q is an isomorphism of representations whose matrix form is the identity in our basis. The dimension of Hom S (S, I) is obviously 1.

Thus we plug:

Z W ⊕S⊕I =      1 0 0 0 0 1 0 1 0      into Equation (4.6) to determine if it is bi-invariant.
Computations show that it is not. We conclude that h is not of Type (2).

Output We have found that h is indecomposable and neither of Type (1) nor of Type (2). Thus h is not quadratic: there is no bi-invariant pseudo-metric <, > on h.

We try the algorithm on the general Heisenberg algebra h 2m+1 which is dened abstractly by the basis C, {P i } m i=1 , {Q j } m j=1 and the Lie bracket:

[C, P i ] = 0, [C, Q j ] = 0, [P i , Q j ] = δ ij
where δ is the Kronecker symbol. We are in the same situation as with h, except that W is abelian (but not necessarily 1-dimensional). We thus decompose W into abelian 1-dimensional ideals and we return the following elementary bi-invariant pseudo-metric:

Z W =      1 0 0 0 . . . 0 0 0 1     
.

However, we exit the algorithm as previously. Thus the algorithm conrms that the general h 2m+1 has no bi-invariant pseudo-metric [Medina 1982].

The group of Scaled Upper Unitriangular Matrices U T (n) The group U T (n) comprises the upper triangular matrices M of the form: M = λ.Id + N where λ > 0 and N an upper triangular nilpotent matrix. The Lie algebra ut(n) comprises the matrices of the form X = µ.Id + Y where µ ∈ R and Y an upper triangular nilpotent matrix, the Lie bracket being the commutator of matrices. Now ut(n) is decomposable into the 1-dimensional Lie algebra generated by I and the Heisenberg algebra h. As h has no bi-invariant pseudo-metric, neither does ut(n).

Chapter 4. Geometric Statistics on Lie groups: analysis of anatomical deformations.

69 h. The bottom level for h indicates that h is neither 1-dimensional, nor simple, nor a double extension and therefore we exit the algorithm: h is not quadratic. The bottom level for ut(n) indicates that d is 1-dimensional and therefore quadratic but that h is not quadratic: ut(n) is not quadratic.

Exit: NO h B 1 = h Exit: NO ut(n) B 2 = h B 1 = d d=1-dim.

Rigid Body Transformations SE(n)

The group of isometries SE(n) comprises rotations together with translations of R n . It is the semi-direct product SO(n) R n , its elements being written (R, t). More precisely, SE(n) is dened by its action on R n as (R, t).x = R.x + t. The group law and the group inversion are (R

1 , t 1 ) * (R 2 , t 2 ) = (R 1 .R 2 , R 1 * t 2 + t 1 ) and (R, t) (-1) = (R (-1) , R (-1) .(-t)).
The Lie algebra se(n) comprises the (A, u) ∈ Skew(n) ⊕ R n with Lie Bracket:

[(A 1 , u 1 ), (A 2 , u 2 )] = (A 1 .A 2 -A 2 .A 1 , A 1 .u 2 -A 2 .u 1 ) (4.29)
Input We choose the basis: ({J ij } 1≤i<j≤n , {P a } n a=1 ) with J ij = e i .e T j -e j .e T i and {P a } n a=1 the canonical basis of R n . In this basis, the structure constants can be read in the following Lie brackets:

[J ij , J kl ] = δ ik .J jl -δ jk .J il + δ jl .J ik -δ il .J jk , [J ij , P a ] = δ aj .P i -δ ai .P j , [P a , P b ] = 0,
with δ the Kronecker symbol.

As preliminaries, we show that P = Span({P a } n a=1 ) is the only proper ideal of se(n). First, we see from the Lie brackets that P is a proper ideal of se(n). Suppose that se(n) has another proper ideal K. Then either K ∩ P is a proper ideal of se(n) included in P or K ⊂ so(n) is a proper ideal of se(n). P does not contain any proper ideal of se(n), because so(n) acts transitively on P with the Lie bracket. We can show that so(n) does not contain any proper ideal of se(n) (considering independently the case n = 4). Thus P is the only proper ideal of se(n).

Step 1 The Lie algebra se(n) has only one ideal P . Thus se(n) cannot be decomposed as a direct sum of ideals. We conclude that se(n) is indecomposable.

Step 2 If n = 1, se(1) is obviously 1-dimensional. We return the multiplication, which is a bi-invariant pseudo-metric on se(1). Otherwise dim(se(n)) > 1. As P is an ideal of se(n), se(n) is not simple. We conclude that se(1) is quadratic with Chapter 4. Geometric Statistics on Lie groups: analysis of anatomical deformations. 70 the multiplication as bi-invariant pseudo-metric and that se(n) with n > 1 is not of Type(1). We go on with n > 1.

Step 3 We take I = P and J = C se(n) (I) = P = I. The necessary condition codim(J) = dim(I) is veried only for n = 3. We conclude that se(n) is not of Type (2) if n = 3. We go on with n = 3. We compute S = se(3)/P ∼ so(3) and W = P/P = {0}.

In order to study the S-representations, we write the Lie bracket as:

[J m , J n ] = ε mnp .J p , [J m , P a ] = ε map .P p , [P a , P b ] = 0
where we dene J 1 = J 23 , J 2 = J 31 and J 3 = J 12 . The S-representation on S is the adjoint representation: [J m , J n ] = ε mnp .J p . The S-representation on I = P is given by: [J m , P a ] = ε map .P p . It is also the adjoint representation. The isomorphism of vector spaces L that maps each P a on J a is an isomorphism of representations whose matrix form is the identity in our basis.

Hence we write Z se(3) on the decomposition S ⊕ I = so(3) ⊕ P with basis ({J a } 3 a=1 , {P a } 3 a=1 ) and get:

Z se(3) =   0 I 3 I 3 0   . (4.30)
We plug it into Equation (4.7). Running the computation shows that the pseudometric Z se(3) is bi-invariant on se(3). Z se(3) is actually known as the Klein form [Karger 1985].

Output se(1) is quadratic, we return the multiplication which is a bi-invariant pseudo-metric on se(1). se(3) is quadratic, we return the Klein form which a biinvariant pseudo-metric on se(3). Otherwise, se(n) is indecomposable and neither of Type (1) nor of Type (2): it is not quadratic. not quadratic.

We can build the whole quadratic space of se(3). This gives the 2-dimensional vector space:

Q(se(3)) =      αZ Killing β.I β.I 0   ∀α, β ∈ F    (4.31)
Moreover, we have recognized in se(3) the special case of a double extension K ⊕ K * of W = {0} by a compact Lie algebra K = so(3). Therefore, the dual structure presented in Section 4.3 can be used in practice. We recall that we can represent the elements of SO(3) as unit quaternions. Thus, we can represent the elements of SE(3) as unit dual quaternions [Kenwright 2012]. A generalization of the theory of Riemannian statistics to a theory of dual Riemannian statistics would thus be useful for rigid body transformations, which are present in many dierent elds.

Chapter 4. Geometric Statistics on Lie groups: analysis of anatomical deformations.

71 se(1) = R B 1 = se(1) = R se(2) B 1 = se(2) Exit: NO se(3) B 1 = se(3) se(3) = Skew(3) ⊕ R 3 se(n > 3) B 1 = se(n > 3) Exit: NO se(1)=1-dim.

Conclusions

In this paper, we have presented an algorithmic method to compute a bi-invariant pseudo-metric on a Lie group, in the case of existence. The method allows to test simultaneously if the Lie group given as input is quadratic or not. We indicated how to compute all the bi-invariant pseudo-metrics on the given Lie group. First, the algorithm by itself represents a contribution to the eld of computational Lie algebra.

Then, regarding statistics on Lie groups which was our original motivation, we see two consequences of this article. First, it enables to distinguish, from a practical point of view, Lie groups on which a future pseudo-Riemannian theory of statistics could be used and implemented. This is the case of SE(3), the Lie group of rotations and translations of the 3D space which is found in various elds.

Second, this paper shows that a general Lie group with bi-invariant mean does not admit a bi-invariant metric. Therefore, if one wants to dene a general theory of statistics that works for all Lie groups, one needs to nd a geometric framework beyond the Riemannian and the pseudo-Riemannian ones.

Introduction

The shape of a set of points, the shape of a signal, the shape of a surface, or the shapes in an image can be dened as the remainder after we have ltered out the position and the orientation of the object [Kendall 1984]. Statistics on shapes appear in many elds. Paleontologists combine shape analysis of monkey skulls with ecological and biogeographic data to understand how the skull shapes have changed in space and time during evolution [Elewa 2012]. Molecular Biologists study how shapes of proteins are related to their function. Statistics on misfolding of proteins is used to understand diseases, like Parkinson's disease [Li 2008]. Orthopaedic surgeons analyze bones' shapes for surgical pre-planning [START_REF] Darmanté | Analyse biométrique de lánneau pelvien en 3 dimensions â à propos de 100 scanners[END_REF]]. In Signal processing, the shape of neural spike trains correlates with arm movement [Kurtek 2011]. In Computer Vision, classifying shapes of handwritten digits enables automatic reading of texts [Allassonnière 2015b]. In Medical Imaging and more precisely in Neuroimaging, studying brain shapes as they appear in the MRIs facilitates discoveries on diseases, like Alzheimer [START_REF] Lorenzi | [END_REF]].

What do these applications have in common? Position and orientation of the skulls, proteins, bones, neural spike trains, handwritten digits or brains do not Chapter 5. Geometric Statistics on quotient spaces: bias in anatomical template shape estimation. 73 matter for the studies' goal: only shapes matter. Mathematically, the study analyses the statistical distributions of the equivalence classes of the data under translations and rotations. They project the data in a quotient space, called the shape space.

The simplest -and most widely used -method for summarizing shapes is the computation of the mean shape. Almost all neuroimaging studies start with the computation of the mean brain shape [Evans 2012] for example. One refers to the mean shape with dierent terms depending on the eld: mean conguration, mean pattern, template, atlas, etc. The mean shape is an average of equivalence classes of the data: one computes the mean after projection of the data in the shape space. One may wonder if the projection biases the statistical procedure. This is a legitimate question as any bias introduced in this step would make the conclusions of the study less accurate. If the mean brain shape is biased, then neuroimaging's inferences on brain diseases will be too. This paper shows that a bias is indeed introduced in the mean shape estimation under certain conditions.

Related work

We review papers on the shape space's geometry as a quotient space, and existing results on the mean shape's bias.

Shapes of landmarks: Kendall analyses

The theory for shapes of landmarks was introduced by Kendall in the 1980's [START_REF] Kendall | [END_REF]]. He considered shapes of k labeled landmarks in R m . The size-and-shape space, written SΣ k m , takes also into account the overall size of the landmarks' set. The shape space, written Σ k m , quotients by the size as well. Both SΣ k m and Σ k m have a Riemannian geometry, whose metrics are given in [START_REF] Weinberger | [END_REF]]. These studies model the probability distribution of the data directly in the shape space Σ k m . They do not consider that the data are observed in the space of landmarks (R m ) k and projected in the shape space Σ k m . The question of bias is not raised.

We emphasize that there is a distinction between "form" and "shape" in the literature. "Form" relates to the quotient of the object by rotations and translations only. "Shape" denotes the quotient of the object by rotations, translations, and scalings. Kendall shape spaces refer to "shape": the scalings are quotiented by constraining the size of the landmarks' set to be 1.

Shapes of landmarks: Procrustean analyses Procrustean analysis is related to Kendall shape spaces but it also considers shapes of landmarks [Goodall 1991, Dryden 1998, Gower 2004]. Kendall analyses project the data in the shape space by explicitly computing their coordinates in Σ k m . In contrast, Procrustean analyses keep the coordinates in (R m ) k : they project the data in the shape space by "aligning" or "registering" them. Orthogonal Procrustes analysis "aligns" the sets of landmarks by rotating each set to minimize the Euclidean distance to the other sets. Procrustean analysis considers the fact that the data are observed in the space (R m ) k but does not consider the geometry of the shape space.

The mean "shape" was shown to be consistent for shapes of landmarks in 2D and 3D in [Lele 1993[START_REF] Le | On the Consistency of Procrustean Mean Shapes[END_REF]]. Such studies have a generative model with a scal-ing component α and a size constraint in the mean "shape" estimation procedure, which prevents the shapes from collapsing to 0 during registration. In constrast, the mean "form" -i.e. without considering scalings -is shown to be inconsistent in [Lele 1993] with an reducto ad absurdum proof. However, this proof does not provide any geometric intuition about how to control or correct the phenomenon. More recently, similar inconsistency eects have been observed in [START_REF] Du | [END_REF] showing that implementing ordinary Procrustes analysis without taking into account noise on the landmarks may compromise inference. The authors propose a conditional scoring method for matching congurations in order to guarantee consistency.

Shapes of curves Curve data are projected in their shape space by an alignment step [Joshi 2006], in the spirit of a Procrustean analysis. The bias of the mean shape is discussed in the literature. Unbiasedness was shown for shapes of signals in [Kurtek 2011] but under the simplifying assumption of no measurement error on the data. Some authors provide examples of bias when there is measurement error [START_REF] Allassonnière | Towards a Coherent Statistical Framework for Dense Deformable Template Estimation[END_REF]]. Their experiments show that the mean signal shape may converge to pure noise when the measurement error on simulated signals increases. The bias is proven in [Bigot 2011] for curves estimated from a nite number of points in the presence of error. But again, no geometric intuition nor correction strategy is given.

Abstract shape spaces [START_REF] S Huckemann | Intrinsic shape analysis: Geodesic principal component analysis for Riemannian manifolds modulo Lie group actions[END_REF]] studies statistics on abstract shape spaces: the shapes are dened as equivalence classes of objects in a manifold M under the isometric action of a Lie group G. This unies the theory for shapes of landmarks, of curves and of surfaces described above. [Huckemann 2010] introduces a generalization of Principal Component Analysis to such shape spaces and does not compute the mean shape as the 0-dimensional principal subspace. Therefore, the bias on the mean shape is not considered.

But in the same abstract setting, [Miolane 2015a] shows the bias of the mean shape, in the special case of a nite-dimensional at manifold M . The authors emphasize how the bias depends on the noise σ on the measured objects, more precisely on the ratio of σ with respect to the overall size of the objects. [START_REF] Stéphanie Allassonnière | Fréchet means top and quotient space may not be consistent[END_REF]] also presents a case study for an innite dimensional at manifold M quotiented by translations, where the noise σ is one of the crucial variables controlling the bias. However, the case of general curved manifolds M has not been investigated yet.

Contributions and outline

We are still missing a global geometric understanding of the bias. Which variables control its magnitude? Is it restricted to the mean shape or does it appear for other statistical analyses? How important is it in practice: do we even need to correct it? If so, how can we correct it? Our paper addresses these questions. We use a geometric framework that unies the cases of landmarks, curves, images etc.

Contributions We make three contributions. First, we show that statistics on shapes are biased when the data are measured with error. We explicitly compute Chapter 5. Geometric Statistics on quotient spaces: bias in anatomical template shape estimation. 75 the bias in the case of the mean shape. Formulated in the Procrustean terminology, our result is: the Generalized Procrustes Analysis (GPA) estimator of mean "form" is asymptotically biased, because we do not consider scalings. Second, we oer an interpretation of the bias through the geometry of the shape space. In applications, this aids in deciding when the bias can be neglected in contrast with situations when it must be corrected. Third, we leverage our understanding to suggest several correction approaches.

Outline The paper has four Sections. Section 1 introduces the geometric framework of shape spaces. Section 2 presents our rst two contributions: the proof and geometric interpretation of the bias. Section 3 describes the procedures to correct the bias. Section 4 validates and illustrates our results on synthetic and real data.

Geometrization of template shape estimation

Two running examples

We introduce two simple examples of shape spaces that we will use to provide intuition.

First, we consider two landmarks in the plane R 2 (Figure 5.1 (a)). The landmarks are parameterized each with 2 coordinates. For simplicity we consider that one landmark is xed at the origin on R 2 . Thus the system is now parameterized by the 2 coordinates of the second landmark only, e.g. in polar coordinates (r, θ). We are interested in the shape of the 2 landmarks, i.e. in their distance which is simply r.

Second, we consider two landmarks on the sphere S 2 (Figure 5.1 (b)). One of the landmark is xed at the north pole of S 2 . The system is now parameterized by the 2 coordinates of the second landmark only, i.e. (θ, φ), where θ is the latitude and φ the longitude. The shape of the two landmarks is the angle between them and is simply θ. The system is entirely represented by the coordinates X of the landmark in black.

X = (r, θ) X = (θ, φ) (a)
Chapter 5. Geometric Statistics on quotient spaces: bias in anatomical template shape estimation. 76 5.1.2 Dierential Geometry of shapes 5.1.2.1 The shape space is a quotient space

The data are objects {X i } n i=1 that are either sets of landmarks, curves, images, etc. We consider that each object X i is a point in a Riemannian manifold M . In this paper, we restrict ourselves to nite dimensional manifolds. We have M = R 2 in the plane example: a at manifold of dimension 2. We have M = S 2 in the sphere example: a manifold of constant positive curvature and of dimension 2.

By denition, the objects' shapes are their equivalence classes {[X i ]} n i=1 under the action of some nite dimensional Lie group G: G is a group of continuous transformations that models what does not change the shape. The action of G on M will be written with "•". In our examples, the rotations are the transformations that leave the shape of the systems invariant. Let us take g a rotation. The action of g on the landmark X is illustrated by a blue arrow in Figures 7.2 (a) for the plane and (d) for the sphere. We observe that the action does not change the shape of the systems: the distance between the two landmarks is preserved in (a), the angle between the two landmarks is preserved in (d). The equivalence class of X i is also called its orbit and written O X i . The equivalence class/orbit of X is illustrated with the blue dotted circle in Figure 7.2 (a) for the plane example and in Figure 7.2 (d) for the sphere example. The orbit of X in M is the submanifold of all objects in M that have the same shape as X. The curvature of the orbit as a submanifold of M is the key point of the results in Section 6.4.1.

The shape space is by denition the space of orbits. This is a quotient space denoted Q = M/G. One orbit in M , i.e. one circle in Figure 7.2 (b) or (e), corresponds to a point in Q. The shape space is Q = R + in the plane example. This is the space of all possible distances between the two landmarks, see Figure 7.2 (c). The shape space is Q = [0, π] in the sphere example. This is the space of all possible angles between the two landmarks, see Figure 7.2 (f).

The shape space is a metric space

We consider that the action of G on M is isometric with respect to the Riemannian metric of M . This implies that the distance d M between two objects in M does not change if we transform both objects in the same manner. In the plane example, rotating the landmark X 1 and another landmark X 2 with the same angle does not change the distance between them.

The distance in M induces a quasi-distance

d Q in Q: d Q (O X 1 , O X 2 ) = inf g∈G d M (g • X 1 , X 2 ) [Huckemann 2010].
The distance between the shapes of X 1 and X 2 is computed by rst registering/aligning X 1 onto X 2 by the minimizing g, and then using the distance in the ambient space M . In the plane example, the distance between two shapes is the dierence in distances between the landmarks. One can compute it by rst aligning the landmarks, say on the rst axis of R 2 , then one uses the distance in R 2 .

Chapter 5. Geometric Statistics on quotient spaces: bias in anatomical template shape estimation. 77 5.1.2.3 The shape space has a dense set of principal shapes

The isotropy group of X i is the subgroup of transformations of G that leave X i invariant. For the plane example, every X i = (0, 0) has isotropy group the identity and (0, 0) has isotropy group the whole group of 2D rotations. Objects on the same orbit, i.e. objects that have the same shape, have conjugate isotropy groups.

Principal orbits or principal shapes are orbits or shapes with smallest isotropy group conjugation class. In the plane example, R 2 \ (0, 0) is the set of objects with principal shapes. Indeed, every X in R 2 \ (0, 0) belongs to a non-degenerate circle centered at (0, 0) and has isotropy group the identity. The set of principal shapes corresponds to R * + in the shape space and is colored in blue on Figure 7.2 (c).

Singular orbits or singular shapes are orbits or shapes with larger isotropy group conjugation class. In the plane example, (0, 0) is the only object with singular shape. It corresponds to 0 in R + and is colored in red in Figure 7.2 (c).

Principal orbits form an open and dense subset of M , denoted M * . This means that there are objects with non-degenerated shapes almost everywhere. In the plane example, R 2 \ (0, 0) is dense in R 2 . In the sphere example, S 2 \ {(0, 0), (π, 0)} is dense in S 2 , where (0, 0) denotes the north pole and (π, 0) the south pole of S 2 . Likewise, principal shapes form an open and dense subset in Q, denoted Q * . In the plane example, R * + is dense in R + . In the sphere example, ]0, π[ is dense in [0, π]. The dense set M * makes the projection in the quotient space a Riemannian submersion [Huckemann 2010], which we use to embed Q * = M * /G in M * . In other words, regular shapes of Q * are embedded in the space of objects with regular shapes M * . Moreover, the computations in Section 6.4.1 will be carried out on the dense set M * of principal orbits. The curvature of these principal orbits -i.e. of the blue circles of Figures 7.2(b) and (e) -will be the main geometric parameter responsible for the asymptotic bias studied in this paper. We note that the curvature of principal orbits is closely related to the presence of singular orbits: principal orbits wrap around the singular orbits. In the plane example, any blue circle -i.e. any principal orbit -wraps around its center, the red dot (0, 0) -which is the singular orbit, see Figure 7.2(b).

We have focused on an intuitive introduction of the concepts. We refer to [Postnikov 2001, Alekseevsky 2003[START_REF] S Huckemann | Intrinsic shape analysis: Geodesic principal component analysis for Riemannian manifolds modulo Lie group actions[END_REF] for mathematical details. From now on, the mathematical setting is the following: we assume a proper, eective and isometric action of a nite dimensional Lie group G on a nite dimensional Riemannian manifold M .

Geometrization of generative models of shape data

We recall that the data are the {X i } n i=1 that are sets of landmarks, curves, images, etc. In the general case, one can interpret the data X i 's as random realizations of the generative model:

X i = Exp(g i • Y i , ε i ) i = 1...n, (5.1)
Chapter 5. Geometric Statistics on quotient spaces: bias in anatomical template shape estimation.

78 Chapter 5. Geometric Statistics on quotient spaces: bias in anatomical template shape estimation.

X g • X X g • X (a) (b) (c) (d) (e) (f)

79

where Exp(p, u) denotes the Riemannian exponential of u at point p. The Y i , g i , ε i are respectively i.i.d. realizations of random variables that are drawn independently.

In this paper as well as often in the literature [START_REF] Allassonnière | Towards a Coherent Statistical Framework for Dense Deformable Template Estimation[END_REF][START_REF] Stéphanie Allassonnière | Fréchet means top and quotient space may not be consistent[END_REF][START_REF] Bigot | [END_REF], Bigot 2011, Kurtek 2011], we consider mainly the following simpler generative model:

X i = Exp(g i • Y, ε i ) i = 1...n, (5.2)
where Y is a parameter which we call the template shape. The following three step formulation of the generative models 5.1 and (7.2) gives technical details and their interpretation in terms of shapes.

Step 1: Generate the shape

Y i ∈ M * /G ⊂ M *
In the full generative model (5.1), we assume that there is a probability density of shapes in the Riemannian manifold Q * = M * /G, with respect to the measure on Q * induced by the Riemannian measure of M * . The Y i 's are i.i.d. samples drawn from this distribution. For example, it can be a Gaussian -or one of its generalization to manifolds [Pennec 2006] -as illustrated in Figure 7.3 on the shape spaces for the plane and sphere examples. This is the variability that is meaningful for the statistical study, whether we are analyzing shapes of skulls, proteins, bones, neural spike trains, handwritten digits or brains.

We mainly assume in this paper the simpler generative model (7.2) with parameter: the template shape Y ∈ M * /G. In other words, we assume that the probability distribution is singular and more precisely that it is simply a Dirac at Y . This is the most common assumption within the model (5.1) [START_REF] Allassonnière | Towards a Coherent Statistical Framework for Dense Deformable Template Estimation[END_REF], Allassonnière 2015a, Kurtek 2011[START_REF] Bigot | [END_REF]]. We point out that Y is a point of the shape space M * /G, which is embedded in the object space by M * /G ⊂ M * , see previous subsection. The black curve illustrates the probability distribution function on shape space. This is a distribution on r ∈ R + for the plane example (a) and on θ ∈ [0, π] for the sphere example. The black square represents its expectation.

For the simpler generative model of Equation (7.2), the probability distribution boils down to a single point at Y i.e. at the black square.

Step 2: Generate its position/parameterization

g i ∈ G, to get g i • Y ∈ M *
We cannot observe shapes in Q = M/G. We rather observe objects in M , that Chapter 5. Geometric Statistics on quotient spaces: bias in anatomical template shape estimation. 80 are shapes posed or parameterized in a certain way. We assume that there is a probability distribution on the positions or parameterizations of G, or equivalently a probability distribution on principal orbits with respect to their intrinsic measure.

We assume that the distribution does not depend on the shape Y i that has been drawn. The g i 's are i.i.d. from this distribution. For example, it can be a Gaussian -or one of its generalization to manifolds [Pennec 2006] -as illustrated in Figure 7.4 on the shape spaces for the plane and sphere examples.

The drawn g i is used to pose/parameterize the shape Y i drawn in Step 1 (in the case of model of Equation (5.1)), where Y i = Y (in the case of model of Equation (7.2)). The shape is posed/parameterized through the isometric action of G on Step 3: Generate the noise

Q * ⊂ M * , to get the object g i • Y i ∈ M * , or the object g i • Y ∈ M * in
ε i ∈ T g i •Y i M
The observed X i 's are results of noisy measurements. We assume that there is a probability distribution function on T g i •Y i M representing the noise. We further assume that this is a Gaussian -or one of its generalization to manifolds [Pennec 2006] -centered at g i • Y i , the origin of the tangent space T g i •Y i M , and with standard deviation σ, see Figure 7.5. The parameter σ will be extremely important in the developments of Section 6.4.1, as we will compute Taylor expansions around σ = 0.

Other generative models may be considered in the literature. We nd in [Allassonnière 2015a] the model:

X i = g i • Exp(Y i , ε i ),
where the Riemannian exponential Exp is also performed in M through the embedding Y i ∈ Q ⊂ M . In [Kurtek 2011], we nd the model without noise:

X i = g i • Y i .
Chapter 5. Geometric Statistics on quotient spaces: bias in anatomical template shape estimation. 5.1.4 Learning the variability in shapes: estimating the template shape

Our goal is to unveil the variability of shapes in Q = M/G while we in fact observe the noisy objects X i 's in M . We focus on the case where the variability in the shape space is assumed to be a Dirac at Y . Our goal is thus to estimate the template shape Y , which is a parameter of the generative model.

Estimating the template shape with the Fréchet mean in the shape space We describe the procedure usually performed in the literature [Kurtek 2011[START_REF] Allassonnière | Towards a Coherent Statistical Framework for Dense Deformable Template Estimation[END_REF][START_REF] Stéphanie Allassonnière | Fréchet means top and quotient space may not be consistent[END_REF][START_REF] Bigot | [END_REF], Bigot 2011]. One initializes the estimate with Ŷ = X 1 . Then, one iterates the following two steps until convergence:

         (i) ĝi = argmin g∈G d M ( Ŷ, g • X i ), ∀i ∈ {1, ..., n}, (ii) Ŷ = argmin Y ∈M n i=1 d M (Y, ĝi • X i ) 2 .
(5.3)

This procedure has a very intuitive interpretation.

Step (i) is the projection of each object X i in the shape space Q, as illustrated in Figure 7.6 (a)-(i) and (b)-(i) with the blue arrows. We assume that each minimizer ĝi exists and is attained. In practice, this will often be the case and for example when the Lie group is compact, like the Lie group of rotations for example. We take X 1 , X 2 , X 3 three objects in R 2 in Figure 7.6 (a)-(i) and on S 2 in Figure 7.6 (b)-(i). One lters out the position/parameterization component, i.e. the coordinate on the orbit. One projects the objects X 1 , X 2 , X 3 in the shape space Q using the blue arrows.

Step (ii) is the computation of the mean of the registered data ĝi • X i , i.e. of the objects' shapes, as illustrated in Figure 7.6(a)-(i) and (b)-(i) where Ŷ is shown in orange. Again, we assume that the minimizer Ŷ exists and is attained. In practice, this will be the case as we will consider a low level of noise in Step 3 of the generative model. The registered data ĝi • X i will be concentrated on a small neighborhood Chapter 5. Geometric Statistics on quotient spaces: bias in anatomical template shape estimation. 82 of diameter of order σ. As a consequence, their Fréchet mean in the Riemannian manifold Q * is guaranteed to exist and be unique [Émery 1991].

The procedure of Equations (5.3) (i)-(ii) decreases at each step the following cost, which is bounded below by zero:

Cost(g 1 , ..., g n , Y ) = n i=1 d 2 M (Y, g • X i ).
(5.4)

Under the assumptions that both steps (i) and (ii) attained their minimizers, we are guaranteed convergence to a local minimum. We further assume that the procedure converges to the global minimum. The estimator computed with the procedure is then:

Ŷ = argmin Y ∈M n i=1 min g∈G d 2 M (Y, g • X i ).
(5.5)

The term min 6.1 is the distance in the shape space between the shapes of Y and X i . Thus, we recognize in Equation 6.1 the Fréchet mean on the shape space. The Fréchet mean is a denition of mean on manifolds [Pennec 2006]: it is the point that minimizes the squared distances to the data in the shape space. All in all, one projects the probability distribution function of the X i 's from M to M/G and computes its "expectation", in a sense made precise later.

g∈G d 2 M (Y, g • X i ) in Equation
(a) (b) X i ĝi • X i X i ĝi • X i Ŷ Ŷ (i) (ii) (i) (ii)
Figure 5.6: Steps (i) and (ii) of procedure of template shape estimation described in Equations (5.3) (i)-(ii) for the plane example (a) and the sphere example (b). The 3 black plus signs in R 2 (a) or S 2 (b) represent the 3 data. The 3 dotted blue curves are their orbits. In Step (i), the X i 's are registered, i.e. their projected in the shape space: 3 curved blue arrows represent their registration with the minimizers ĝi . The 3 black crosses in R + (positive x-axis) (a) or [0, π] (b) represent the registered data.

In Step (ii), the template shape estimate Ŷ is computed as the Fréchet mean of the registered data and is shown in orange.

We implemented the generative model and the estimation procedure on the plane and the sphere in shiny applications available online: https://nmiolane. shinyapps.io/shinyPlane and https://nmiolane.shinyapps.io/shinySphere. We invite the reader to look at the web pages and play with the dierent parameters of the generative model. Figure 5.7 shows screen shots of the applications.

Chapter 5. Geometric Statistics on quotient spaces: bias in anatomical template shape estimation. Simulated data X i 's (grey points), template shape Y (green), registered data ĝi • X i (black points), template shape estimate Ŷ (orange). Induced distributions on the shapes, template shape Y (green), template shape estimate Ŷ (orange).

Probabilistic interpretation of the procedure in Equations (5.3): an approximation of a Maximum-Likelihood Beside its intuitive interpretation, the procedure of template shape estimation of Equations ( 5.3) has a probabilistic interpretation. We have the generative model of the data X i 's: it is described in Equation (7.2) and Steps (1)-(3) of the previous subsections. Thus, one may consider the Maximum Likelihood (ML) estimate of Y , which is one of its parameters:

ŶML = argmax Y ∈Q L(Y ) = argmax Y ∈Q n i=1 log(P (X i |Y )) = argmax Y ∈Q n i=1 log g∈G P (X i |Y, g).P (g)dg .
In the above, P (X i |Y ) is the probability distribution of the data in M as a function of the parameter Y . P (g) is the probability distribution on the poses/parameterizations in G as described in Step 2 of the generative model given Chapter 5. Geometric Statistics on quotient spaces: bias in anatomical template shape estimation. 84 in the previous subsection. Then, P (X i |Y, g) is the probability distribution of the noise as described in Step 3. The g's are hidden variables in the model. The Expectation-Maximization (EM) algorithm is therefore the natural implementation for computing the ML estimator [START_REF] Allassonnière | Towards a Coherent Statistical Framework for Dense Deformable Template Estimation[END_REF]]. But the EM algorithm is computationally expensive, above all for tridimensional images. Thus, one can usually rely on an approximation of the EM, which is described in [START_REF] Allassonnière | Towards a Coherent Statistical Framework for Dense Deformable Template Estimation[END_REF]] as the "modal approximation" and used in [Allassonnière 2015a, Kurtek 2011[START_REF] Bigot | [END_REF].

We can check that this approximation is the procedure described in Equations (5.3). Step (i) is an estimation of the hidden observations g i and an approximation of the E-step of the EM algorithm. Step (ii) is the M-step of the EM algorithm: the maximization of the surrogate in the M-step amounts to the maximization of the variance of the projected data. This is exactly the minimization of the squared distances to the data of (ii). We refer to [START_REF] Allassonnière | Towards a Coherent Statistical Framework for Dense Deformable Template Estimation[END_REF]] for details.

Purpose of this paper reformulated with the geometrization Our main result is to show that the procedure presented in Equations (5.3) (and illustrated on Figure 7.6) gives an asymptotically biased estimate Ŷ for the template shape Y of the generative model presented in Equation (7.2) (and illustrated in Figures 7.3 that even for an innite number of grey points, the orange estimate will be dierent from the green parameter. We say that Ŷ has an asymptotic bias with respect to the parameter Y .

Where does this asymptotic bias come from and why doesn't Ŷ converge to Y ? In a nutshell, the bias comes from the external curvature of the template's orbit and we explain and summarize this in Figure 6.7 and its caption. The full geometric answer with its technical details is provided in the next section.

Quantication and correction of the asymptotic bias

This section explains, quanties and corrects the asymptotic bias of the template shape estimate Ŷ with respect to the parameter Y . We start from the denition of the asymptotic bias of an estimator with respect to the parameter it is designed to estimate. More precisely we start from a generalization of this denition to Riemannian manifolds:

Bias( Ŷ, Y ) = E Log Y Ŷ .

(5.6)

Chapter 5. Geometric Statistics on quotient spaces: bias in anatomical template shape estimation.

85 This is the asymptotic bias of the estimator Ŷ with respect to the (manifold-valued) parameter Y , which generalizes the corresponding denition for linear spaces:

Bias( Ŷ, Y ) = E Ŷ -Y (5.7)
In the Riemannian denition of the bias, Log Y Ŷ is the Riemannian logarithm of

Ŷ ∈ Q at Y ∈ Q, i.
Y Ŷ Y Ŷ Log Y Ŷ Log Y Ŷ Figure 5
.8: Illustration of the Riemannian denition of the asymptotic bias Log Y Ŷ for the plane example (a) and the sphere example (b). Log refers to the Riemannian logarithm [Postnikov 2001] and Log Y Ŷ is thus a tangent vector of the quotient space Q at Y . Log Y Ŷ represents how much one would have to shoot from Y to get the estimated parameter Ŷ . The norm of Log Y Ŷ , computed using the metric of Q at Y , represents the distance or the dissimilarity between Ŷ and Y , i.e. how far Ŷ is from estimating Y .

We could also consider the variance of the estimator Ŷ . The variance is dened as

Var n ( Ŷ ) = E[d M (Y, E[Y ]) 2 ].
In the limit of an innite sample, we have: Var ∞ ( Ŷ ) = 0. This is why we focus on the asymptotic bias.

Asymptotic bias of the template's estimator on examples

We rst compute the asymptotic bias for the examples of the plane and the sphere to give the intuition.

The probability distribution function of the X i 's comes from the generative model. This is a probability distribution on R 2 for the plane example, parameterized in polar coordinates (r, θ) like Figure 5.1. So we can compute the projected distribution function on the shapes, which are the radii r here. This is done simply by integrating out the distribution on θ, the position on the circles. This gives a probability distribution on R + for the plane example. We write it f : r → f (r). We remark that f does not depend on the probability distribution function on the θ i 's of Step 2 of the generative model. We can also compute f : θ → f (θ) in the sphere example: we integrate over φ the probability distribution function on (θ, φ).

Figure 7.7 (a) shows f for the plane example, for a template r = 1. We plot it for two dierent noise levels σ = 0.3 and σ = 3. Note that here f is the Rice distribution. Figure 7.7 (b) shows f for the sphere example, for a template θ = 1. We plot it for dierent noise levels and σ = 0.3 and σ = 3. In both cases, the x-axis represents the shape space which is R + for the plane example and [0, π] for the sphere example. The green vertical bar represents the template shape, which is 1 in both cases. The red vertical bar is the expectation of f in each case. It is Ŷ , the estimate of Y We see on these plots that f is not centered at the template shape: the green and red bars do not coincide. f is skewed away from 0 in the plane example and away from 0 and π in the sphere example. The skew increases with the noise level σ. The dierence between the green and red bars is precisely the bias of Ŷ with respect to Y . Figure 5.10 shows the bias of Ŷ with respect to Y , as a function of σ, for the plane (left) and the sphere (right). Increasing the noise level σ takes the estimate Ŷ away from Y . The estimate is repulsed from 0 in the plane example: it goes to ∞ when σ → ∞. It is repulsed from 0 and π in the sphere example: it goes to π/2 when σ → π, as the probability distribution becomes uniform on the sphere in this limit. One can show numerically that the bias varies as σ 2 around σ = 0 in both cases. This is also observed on the shiny applications [RStudio, Inc 2013] at https://nmiolane. shinyapps.io/shinyPlane and https://nmiolane.shinyapps.io/shinySphere.

These examples already show the origin of the asymptotic bias of Ŷ , for low noise levels σ → 0 or for high noise levels: σ → +∞ for the plane example and σ → π for the sphere example. As long as there is noise, i.e. σ = 0, there is a bias that comes from the curvature of the template's orbit. Figure 6.7 shows the template's orbit in blue, in (a) for the plane and (b) for the sphere. In both cases the black circle represents the level set σ of the Gaussian noise. In the plane example (a), the Chapter 5. Geometric Statistics on quotient spaces: bias in anatomical template shape estimation. probability of generating an observation X i outside of the template's shape orbit is bigger than the probability of generating it inside: the grey area in the black circle is bigger than the white area in the white circle. There will be more registered data that are greater than the template. Their expected value will therefore be greater than the template and thus biased. In the sphere example (b), if the template's shape orbit is dened by a constant θ < π/2, the probability of generating an observation X i "outside" of it, i.e. with θ i > θ, is bigger than the probability of generating it "inside". There will be more registered data that are greater than the template θ and again, their expected value will also be greater than the template. Conversely, if the template is θ > π/2, the phenomenon is inverted: there will be more registered data that are smaller than the template. The average of these registered data will also be smaller than the template. Finally, if the template's shape orbit is the great circle dened by θ = π/2, then the probability of generating an observation X i on the left is the same as the probability of generating an observation X i on the right. In this case, the registered data will be well-balanced around the template θ = π/2 and their expected value will be π/2: there is no asymptotic bias in this particular case. We prove this in the general case in the next section.

Asymptotic bias of the template's estimator for the general case

We show the asymptotic bias of Ŷ in the general case and prove that it comes from the external curvature of the template's orbit. We show it for Y a principal shape and for a Gaussian noise of variance σ 2 , truncated at 3σ. Our results will need the following denitions of curvature.

The second fundamental form h of a submanifold O of M is dened on 7.2) and Figure 7.5. There is a higher probability that the data are generated "outside" the orbit. The template shape estimate is biased towards greater radii (a) or towards angles closer to π/2 (b).

T X O × T X O by h(v, w) = (∇ v w) ⊥ ∈ N X O,
sures of extrinsic curvature of O in M . For example an hypersphere of radius R in R m has mean curvature vector ||H|| = m-1 R .

Theorem 5.2.1 The data X i 's are generated in the nite-dimensional Riemannian manifold M following the model: X i = Exp(g i • Y, ε i ), i = 1...n, described in Equation (7.2) and Figures 7.3-7.5. In this model: (i) the action of the nite dimensional Lie group G on M , denoted •, is isometric, (ii) the parameter Y is the template shape in the shape space Q, (iii) ε i is the noise and follows a (generalization to manifolds of a) Gaussian of variance σ 2 , see Section 6.2. Then, the probability distribution function f on the shapes of the X i 's, i = 1...n, in the asymptotic regime on an innite number of data n → +∞, has the following Taylor expansion around the noise level σ = 0:

f (Z) = 1 ( √ 2πσ) q exp - d 2 M (Y, Z) 2σ 2 F 0 (Z) + σ 2 F 2 (Z) + O(σ 4 ) + ε(σ)
where (i) Z denotes a point in the shape space Q, (ii) F 0 and F 2 are functions of Z involving the derivatives of the Riemannian tensor at Z and the derivatives of the graph G describing the orbit O Z at Z, and (iii) ε is a function of σ that decreases exponentially for σ → 0.

Proof The detailed proof is in Appendix A.

The exponential in the expression of f belongs to a Gaussian distribution centered at Z and of isotropic variance σ 2 I. However the whole distribution f diers from the Chapter 5. Geometric Statistics on quotient spaces: bias in anatomical template shape estimation.

89

Gaussian because of the Z-dependent term in the right parenthesis. This induces a skew of the distribution away from the singular shapes, as observed for the examples in Figure 7.7. This also means that the expectation of this distribution is not Z and that the variance is not the isotropic σ 2 I.

Theorem 5.2.2 The data X i 's are generated with the model described in Equation (7.2) and Figures 7.3-7.5, where the template shape Y is a parameter and under the assumptions of Theorem A.3.1. The template shape Y is estimated with Ŷ , which is computed by the usual procedure described in Equations (5.3).

In the regime of an innite number of data n → +∞, the asymptotic bias of the template's shape estimator Ŷ , with respect to the parameter Y , has the following Taylor expansion around the noise level σ = 0:

Bias( Ŷ, Y ) = - σ 2 2 H(Y ) + O(σ 4 ) + ε(σ) (5.8)
where (i) H is the mean curvature vector of the template shape's orbit which represents the external curvature of the orbit in M , and (ii) ε is a function of σ that decreases exponentially for σ → 0.

Proof The proof is in Appendix A.

This generalizes the quadratic behavior observed in the examples on Figure 5.10.

The asymptotic bias has a geometric origin: it comes from the external curvature of the template's orbits, see Figure 6.7. We can vary two parameters in equation A.15: Y and σ. The external curvature of orbits generally increases when Y is closer to a singularity of the shape space (see Section 1) [Lytchak 2010]. The singular shape of the two landmarks in R 2 arises when their distance is 0. In this case, the mean curvature vector has magnitude |H(Y )| = 1 d : it is inversely proportional to d, the radius of the orbit. d is also the distance of Y to the singularity 0.

Limitations and extensions

Beyond Y being a principal shape Our results are valid when the template Y is a principal shape. This is a reasonable assumption as the set of principal shapes is dense in the shape space. What happens when Y approaches a singularity, i.e. when Y changes stratum in the stratied space Q? Taking the limit d → 0 in the coecients of the Taylor expansion is not a legal operation. Therefore, we cannot conclude on the Taylor expansion of the Bias for d → 0. Indeed, the Taylor expansion may even change order for d → 0. We take M = R m with the action of SO(m) and the template Y = (0, ..., 0):

Bias( Ŷ, Y ) = √ 2 Γ( m+1 2 ) Γ( m 2 )
σ.

(5.9)

The bias is linear in σ in this case.

Chapter 5. Geometric Statistics on quotient spaces: bias in anatomical template shape estimation.

90 Beyond σ << 1 The assumption σ << 1 represents our hope that the noise on the shape data is not too large with respect to the overall size of the mean shape. Nevertheless it would be very interesting to study the asymptotic bias for any σ, including large noises (σ → +∞). The distribution over the X i 's in M will be spread on the whole manifold M . We cannot rely on local computations on M (at the scale of σ) anymore. We have to make global assumptions on the manifold M . The plane example is the canonical example of a at manifold. The sphere example is the canonical example of manifold with constant (positive) curvature. The bias as a function of σ is plotted in Figure 5.10. It leads us to the conjecture that the estimate converges towards a barycenter of shape space's singularities when the noise level increases. Singularities have a repulsive action on the estimation of each template's shape. Such repulsive force acts on each estimators. As a result, the estimators of the mean shape nds an equilibrium position: the barycenter.

Beyond one Dirac in Q: several templates We have considered so far that there is a unique template shape Y : the generative model has a Dirac distribution at Y in the shape space. What happens for other distributions? We assume that there are K template shapes Y 1 , ..., Y K . Observations are generated in M from each template shape Y k with the generative model of Section 2. Our goal is to unveil the structure of the shape distribution, i.e. the K template shapes here, given the observations in M . The distributions on shapes projected on the shape space is a mixture of probability density functions of the form of the density in Theorem A.3.1. Its modes are related to the template shapes. The K-means algorithm is a very popular method for data clustering. We study what happens if one uses K-means algorithms on shapes generated with the generative model above.

The goal is to cluster the shape data in K distinct and signicant groups. One performs a coordinate descent algorithm on the following function:

J(c, µ) = i d Q (X i , µ c i ) 2 .
(5.10)

In other words, the minimization of J is performed through successive minimizations on the assignment labels c's and the cluster's centers µ's. Given the c, minimizing J with respect to the µ's is exactly the simultaneous computation of K Fréchet means in the shape space. Meaningful well separated clusters (high inter-clusters dissimilarity) are chosen so that members are close to each other (high intra-cluster similarity). In other words, the quality of the clustering is evaluated by the following criterion: .11) which is the dissimilarity between clusters quotiented by the diameter of the clusters.

D = min clusters i,j d Q (c i , c j ) max i diam(c i ) , ( 5
In the absence of singularity in the shape space, the projected distribution looks like Figure 5.12 (a) and D ∝ 1 σ . The criterion is worse in the presence of singularities.

Chapter 5. Geometric Statistics on quotient spaces: bias in anatomical template shape estimation.

(a) (b) (c)

Figure 5.12: Two clusters of template shapes for the plane example: r 1 = 1 (blue) and r 2 = 2 (dark red). Noise levels: σ = 0.3 (left) and σ = 3 (right). The 2 clusters are hardly distinguishable when the noise increases.

Figure 5.12 illustrates this behavior for the plane example. We consider any two clusters i, j and call Ŷi , Ŷj the estimated centroids. The criterion D writes:

D ≡ ŷi -ŷj σ ∼ σ→+∞ Γ m+1 2 √ 2mΓ m 2 y 2 i -y 2 j σ 2 = O 1 σ 2 .
Even in the best case with correct assignments to the clusters i and j, the K-means algorithm looses an order of validation when computed on shapes.

Beyond the nite dimensional case Our results are valid when M is a nite dimensional manifold and G a nite dimensional Lie group. Some interesting examples belong to the framework of innite dimensional manifold with innite dimensional Lie groups. This is the case for the LDDMM framework on images [Joshi 2006]. It would be important to extend these results to the innite dimensional case.

We take M = R m with the action of SO(m). We have a analytic expression of f in this case [Miolane 2015a]. Figure 5.13 shows the inuence of the dimension m for the probability distribution functions on the shape space and for the Bias. The bias increases with m. This leads us to think that it appears in innite dimensions as well. Chapter 5. Geometric Statistics on quotient spaces: bias in anatomical template shape estimation. 92

Correction of the systematic bias

We propose two procedures to correct the asymptotic bias on the template's estimate. They rely on the bootstrap principle, more precisely a parametric bootstrap, which is a general Monte Carlo based resampling method that enables us to estimate the sampling distributions of estimators [Efron 1979]. We assume that we know the variance σ2 from the experimental setting.

Iterative Bootstrap

The rst procedure is called an Iterative Bootstrap. Algorithm 5.3.1 gives the details. Figure 5.14 illustrates it on the plane example. * , compute the bias Ŷ0 -Ŷ * 0 , (c) Correct Ŷ0 with the bias to get Ŷ1 , (d) Generate bootstrap sample from Ŷ1 and iterate as in (b), (e) Get Ŷ2 etc. Algorithm 5.3.1 starts with the usual template's estimate Ŷ0 = Ŷ , see Figure 5.14 (a). At each iteration, we correct Ŷ with a better approximation of the bias. First, we generate bootstrap data by using Ŷ as the template shape of the generative model. We perform the template's estimation procedure with the Fréchet mean in the shape space. This gives an estimate Ŷ * 0 of Ŷ0 . The bias of Ŷ0 * with respect to Ŷ0 is Bias( Ŷ * 0 , Ŷ0 ). It gives an approximation of the bias Bias( Ŷ, Ŷ ), see Figure 5.14 (b). We correct Ŷ by this approximation of the bias. This gives a new estimate Ŷ1 , see Figure 5.14 (c). We recall that the bias Bias( Ŷ, Ŷ ) depends on Y , see Theorem A.4.1. Ŷ1 is closer to the template Y than Ŷ0 . Thus, the next iteration gives a better approximation Bias( Ŷ * 1 , Ŷ1 ) of Bias( Ŷ, Ŷ ). We correct the initial Ŷ with this better approximation of the bias, etc. The procedure is written formally for a general manifold M in Algorithm 5.3.1.

Y σ σ Ŷ1 Ŷ0 = Ŷ Ŷ * 1 -Bias( Ŷ * 0 , Ŷ0 ) Ŷ * 0 -Bias( Ŷ * 1 , Ŷ1 ) Ŷ2 (a) (b) (c) (d) (e)
Chapter 5. Geometric Statistics on quotient spaces: bias in anatomical template shape estimation. Initialization:

Ŷ0 = Frechet({[X i ]} n i=1 ) k ← 0 Repeat: Generate bootstrap sample {X (k) * i } n i=1 from N M (Y k , σ 2 ) Y k = Fréchet({[X (k) * ] i } n i=1 ) Bias k = Log Y k Y k Ŷk = Exp Ŷ0 -Π Ŷ0 Ŷk (Bias k ) k ← k + 1 until convergence: ||Log Ŷk+1 Ŷk || < ε Output: Ŷk In Algorithm 5.3.1, Π B
A denotes the parallel transport from T A M to T B M . For linear spaces like R 2 in the plane example, Log P 1 P 2 = ---→ P 1 P 2 , Exp P 1 (u) = P 1 + u, For linear spaces and the parallel transport is the identity Π P 2 P 1 (u) = u. For other manifolds like S 2 in our sphere example, the parallel transport Π B A (u) can theoretically be computed by solving the parallel transport equation at any point on the chosen curve linking A to B: D t AB v = 0 in v, where D is the covariant derivative in the direction t AB , the tangent vector of the curve at the chosen point [Postnikov 2001]. In practice, the Schild's ladder [START_REF] Ehlers | [END_REF]] or the Pole ladder [Marco 2013] can be used to compute an approximation of the parallel transport.

Algorithm 5.3.1 is a xed-point iteration Y (k+1) = F (Y (k) ) where:

F (X) = Exp Ŷ (-Π Ŷ X (Bias))
where: Bias = Log X X.

(5.12)

In a linear setting we have simply F (X) = Ŷ ---→ X X. One can show that F is a contraction and that Y , the template shape, is the unique xed point of F (using the local bijectivity of the Riemannian exponential and the injectivity of the estimation procedure). Thus the procedure converges to Y in the case of an innite number of observations n → +∞. Figure 5.15 illustrates the convergence for the plane example, with a Gaussian noise of standard deviation σ = 1. The template shape Y = 1.2 was initially estimated at Ŷ = 4.91. Algorithm 5.3.1 corrects the bias.

Figures 5.16 and 5.17 show the iterations of Iterative Bootstrap for the plane and the sphere example.

Nested Bootstrap

The second procedure is called the Nested Bootstrap. Algorithm 4 details it. .16: Left: Implementation of the plane example: the green point is the template shape Y , the grey points are the data X i 's generated with the model (7.2), the black points are the registered data ĝi • X i 's, the orange point is the template shape estimate Ŷ . The quotient space R + is copied below, and the blue points show the iterations of the iterative bootstrap of Algorithm 5.3.1 that corrects the bias of Ŷ as an estimate of Y : the blue points go from the orange point Ŷ to the green point Y . Right: Convergence of the iterative bootstrap of Algorithm 5.3.1, for the plane example. The colors red, purple, blue represent dierent noises σ. The bias of Ŷ as an estimator of Y is shown on the ordinate axis: it converges to 0 in a few iterations.

replication Ŷ * 0 and the approximation Bias( Ŷ * 0 , Ŷ0 ) of Bias( Ŷ, Y ), see Figure 5.14 (b). Now Algorithm 4 diers from Algorithm 5.3.1. We want to know how biased is Bias( Ŷ * 0 , Ŷ0 ) as an estimate of Bias( Ŷ, Y )? This is a valid question as the bias depends on the template Y , see Theorem A.4.1. We want to estimate this dependence. We perform a bootstrap, nested in the rst one, with Ŷ (0) * as the template. We Chapter 5. Geometric Statistics on quotient spaces: bias in anatomical template shape estimation. .17: Left: Implementation of the sphere example: the green point is the template shape Y , the grey points are the data X i 's generated with the model (7.2), the black points are the registered data ĝi • X i 's, the orange point is the template shape estimate Ŷ . The quotient space [0, π] is copied below, and the blue points show the iterations of the iterative bootstrap of Algorithm 5.3.1 that corrects the bias of Ŷ as an estimate of Y : the blue points go from the orange point Ŷ to the green point Y . Right: Convergence of the iterative bootstrap of Algorithm 5.3.1, for the sphere example. The colors red, purple, blue represent dierent noises σ. The bias of Ŷ as an estimator of Y is shown on the ordinate axis: it converges to 0 in a few iterations. compute the estimate Ŷ * * 0 and the approximation Bias( Ŷ * * 0 , Ŷ * 0 ) of Bias( Ŷ * 0 , Ŷ0 ), see Figure 5.14 (c). We observe how far Bias( Ŷ * * 0 , Ŷ * 0 ) is from Bias( Ŷ * 0 , Ŷ0 ). This gives the blue arrow, which is the bias of Bias( Ŷ * * 0 , Ŷ * 0 ) as an estimate of Bias( Ŷ * 0 , Ŷ0 ), see Figure 5.14 (d). The blue arrow is an approximation of how far Bias( Ŷ * 0 , Ŷ0 ) is from Bias( Ŷ, Y ). We correct our estimation of the bias (in red) by the blue arrow. We correct Ŷ by the bias-corrected estimate of its bias, see Figure 5.14 (e).

Comparison

One may use the Iterative Bootstrap or the Nested Bootstrap depending on the experimental setting. We illustrate them both on the plane example in Figure 5.19. Figure 5.19 (a) shows the performance of both algorithms for a signal-over-noise ratio (SNR) of 1: the template shape in green is a r = 1 and the standard deviation of the noise is σ = 1, so that SNR = r σ = 1. Figure 5.19 (b) shows both algorithms for SNR = r σ = 1 3 = 0.33. In all four experiments: the template shape is the green dot at r = 1, the template shape estimate is in orange, and the successive steps of the bootstrap algorithms are the blue dots: we have several blue dots for the Iterative Bootstrap, and two blue dots for the Nested Bootstrap.

The advantages of the Iterative Bootstrap are the following. It corrects the bias of Ŷ perfectly in the case of a very large number of observations n, as we Chapter 5. Geometric Statistics on quotient spaces: bias in anatomical template shape estimation.

96 a) on top and (b) on top: the blue dots converge to the green dot for the two dierent SNRs. Thus, the Iterative Bootstrap can be used to experimentally compute the mean curvature vector H of each orbit of a group action. One probes the orbit's curvature by "feeling it" with a Riemannian Gaussian on M and projecting on the shape space. The drawbacks of the Iterative Bootstrap are the following. It works only with very large n. It is not robust as it uses the generative model several times. If the generative model is far from being true, then the iterative bootstrap fails.

Y Ŷ0 = Ŷ σ -Bias( Ŷ * 0 , Ŷ0 ) σ Ŷ * 0 Ŷ * 0 -Bias( Ŷ * * 0 , Ŷ * 0 ) (a) (b) (c) (d) (e)
The advantages of the Nested Bootstrap are the following. It is a standard statistical procedure that is more robust with respect to variations of the generative model. Even if generative model is dierent from the one that we assume, the Nested Bootstrap performs well. Moreover, it does not need as much data as the Iterative Bootstrap. Its drawback is that it does not correct perfectly the bias, especially when the noise is large. This can be seen in Figures 5.3.1 (a) on bottom and (b) on bottom. While the Nested Bootstrap gets close to the green dot on Figure 5.19 (a) on bottom for the SNR = 1, it stays signicantly far from the green dot on Figure 5.19 (b) bottom for the SNR = 0.33.

These simulations give a rule of thumb, i.e. some intuition, for when the bias needs to be corrected. They conrm what could already be observed in Figure 5.10. In Figure 5.10, the template is xed at r = 1 or θ = 1. A variation in the noise level σ corresponds to a variation in the SNR. In particular, we read the threshold SN R = 1 when σ = 1, i.e. when the noise σ is comparable to the distance of the template Y to the singularity. In both cases for SN R > 1, the template estimate is Chapter 5. Geometric Statistics on quotient spaces: bias in anatomical template shape estimation. 97 Algorithm 3 Corrected template shape estimation with Nested Bootstrap Input: Objects {X i } n i=1 , noise variance σ 2 Initialization:

Ŷ0 = Frechet({[X i ]} n i=1 ) Bootstrap: Generate bootstrap sample {X * i } n i=1 from N M ( Ŷ0 , σ 2 ) Ŷ * 0 = Fréchet({[X * ] i } n i=1 ) Bias = Log Ŷ0 Y * 0 Nested Bootstrap:
For each i: Chapter 5. Geometric Statistics on quotient spaces: bias in anatomical template shape estimation. 98 signicantly dierent from the template as the bias is of the order of magnitude of the template itself.

• Generate bootstrap sample {X * * i } n k=1 from N M ( Ŷ * 0 , σ 2 ) • Ŷ * * 0,i = Fréchet({[X * * ] i } n k=1 ) Bias(Bias) = Log Ŷ0 Y * 0 -Π Ŷ0 Ŷ * 0 Log Y * 0 Y * * 0 Ŷ1 = Exp Ŷ0 (-

Applications to simulated and real data

Simulated triangles

We perform a simulation using the iterative bootstrap on triangles. We randomly generate n = 10 5 triangles in R 2 through the generative model described in Equation (7.2) of Section 6.2. This is illustrated on Figure 5.20. We consider the isometric action of the Lie group SO(2) of 2D rotations on (R 2 ) 3 , the space of 3 landmarks in 2D. For Step 1 of the generative model of Section 6.2, the template triangle is chosen arbitrarily and then xed during the simulations. The template triangle is represented in green in Figure 5.20. For Step 2, we consider a Dirac distribution at the identity in the Lie group SO(2). In other words, we do not rotate the triangles. At the end of this step, each of the 10 5 triangles is exactly the green triangle of Figure 5.20. This simpler model does not decrease the impact of the simulation: the noise of Step 3 is independent of the position of the triangle on their orbit, and

Step (i) of the procedure is to quotient out the position of the orbit. For Step 3, we add bivariate Gaussian noise on each landmark, i.e. on each of the three points dening the green triangle. This gives a data set of 10 5 triangles. Some of them are represented in grey on Figure 5.20.

We then apply the procedure described in Equations (5.3) (i)-(ii) to estimate the (green) template triangle. In Step (i), we register the (grey) triangle data. This gives the registered the data, illustrated in black in Figure 5.20. We then compute the Fréchet mean of the black triangles by computing the Euclidean mean of each of their 3 landmarks. This gives the estimate of the template triangle, in orange on Figure 5.20.

The template estimate in orange is dierent from the template in green, even with a very high number of observations: n = 10 5 . We apply the iterative bootstrap to correct this bias. The number of iterations required for the convergence of Algorithm 1 with respect to the noise level are shown in Figure 5.20. We observe the convergence in the three experiments for less than 10 iterations.

Real triangles: shape of the Optic Nerve Head

Now we go to real triangle data. We have 24 images of Rhesus monkeys' eyes, acquired with a Heidelberg Retina Tomograph [Patrangenaru 2015]. For each monkey, an experimental glaucoma was introduced in one eye, while the second eye was kept as control. One seeks a signicant dierence between the glaucoma and the control eyes. On each image, three anatomical landmarks were recorded: S for the superior aspect of the retina, N for the nose side of the retina, and T for the side of the retina closest to the temporal bone of the skull. The data are matrices {X i } n i=1 where the landmark coordinates form the rows. For the ONH example, M is the space of 3 Chapter 5. Geometric Statistics on quotient spaces: bias in anatomical template shape estimation. landmarks in 3D, M = (R 3 ) 3 and the rotations act isometrically on each object X i .

Analysis This simple example illustrates the estimation of the template shape.

We use the following procedure to compute the mean shape for each group. We initialize Ŷ with X 1 and repeat the following two steps until convergence:

(1) ∀i ∈ {1, ..., n}, Ri = argmin

R∈SO(3)
|| Ŷ -X i .R|| 2 , (register to the current mean shape),

(2) Ŷ = 1 n n i=1 X i . Ri (update the mean shape estimate).

Figure 5.21 shows the mean shapes Ŷ control of the control group (left) and Ŷ glaucoma of the glaucoma group (right) in orange, while the initial data are in grey. The dierence between the two groups is quantied by the distance between their means: || Ŷ control -Ŷ glaucoma || = 21.84µm. We want to determine if this analysis presents a bias that signicantly changes the estimated shape dierence between the groups. We use the nested bootstrap to compute an approximation of the asymptotic bias on each mean shape, for a range of noise's standard deviation in {100µm, 200µm, 300µm, 400µm}. The asymptotic bias on the template shape of the glaucoma group is {0.1µm, 0.11µm, 0.12µm, 0.13µm} and of the control group is {0.27µm, 0.42µm, 0.55µm, 0.67µm}. The corrected template shape dierences are {22. 01µm, 22.08µm, 22.14µm, 22.18µm}. In particular, for σ = 400µm, we observe that the bias in the template shape are respectively 0.67µm for the healthy group Chapter 5. Geometric Statistics on quotient spaces: bias in anatomical template shape estimation. and 0.13µm for the glaucoma group. This follows the rule-of-thumb: the bias is more important for the healthy group, for which the overall size is smaller than the glaucoma group, for a same noise level. The bias of the template shape estimate accounts for less than 1µm in this case, which is less than 0.1% of the shapes' sizes. This computation guarantees that this study has not been signicantly aected by the bias.

Protein shapes in Molecular Biology

We estimate the impact of the bias on statistics on protein shapes. This subsection aims to suggest the potential importance of the results of this paper for Molecular Biology.

A standard hypothesis in Biology is that structure (i.e. shape) and function of proteins are related. Fundamental research questions about protein shapes include structure prediction -given the protein amino-acid sequence, one tries to predict its structure -and design -given the shape, one tries to predict the sequence needed.

One relies on experimentally determined 3D structures gathered in the Protein Data Base (PDB) [Berman 2000]. They contain errors on the protein's atoms coordinates. Average errors range from 0.01 Å to 1.76 Å, which is of the magnitude of the length of some covalent bonds. These values are averaged over the whole protein and in general, the main-chain atoms are better dened than the side-chain atoms or the atoms at the periphery. This is illustrated on Figure 5.22 where we have plot the B-factor (related to coordinates errors [START_REF] Tickle | [END_REF]]) as a colored map on the atoms for proteins of PDB-codes 1H7W and 4HBB.

Protein's radius of gyration A biased estimate of a protein shape has consequences for studies on proteins folding. Stability and folding speed of a protein depend on both the estimated shape of the denatured state (unfolded state) and of the native state (folded state). One may study if compact initial states yield to faster folding. The protein compactness is represented by the protein's Radius of Chapter 5. Geometric Statistics on quotient spaces: bias in anatomical template shape estimation. (5.13) where we also express this bias with respect to an adaptation of the signal-noise-ratio introduced in Section 7.5:

B(R 2 g ) = σ 2 3(N -1) N = Rg 2 (N -1) N 3 SNR 2 ,
SNR 2 = R 2 g σ 2 .
The radius of protein HJSJ (85 residues) is known to be around 10 Å. The error on R 2 g is of 0.3% with an average error of positions on the atoms of 0.3 Å. It is 8.6% for an error of 1.7 Å. The error will be greater if one considers binding sites at the periphery of the proteins rather than the whole protein. Indeed sites' size is smaller and they have less atoms.

One could think about doing clustering on radii of Gyration using the K-means algorithm on shapes. The index D of Section 7.5 is:

D = R σ 1 2 -R σ 2 2 σ = R 2 1 -R 2 2 σ + 3σ N 1 -1 N 1 - N 2 -1 N 2 .
(5.14)

Clustering on radii of gyration may lead to a misleading indicator. D indicates that the clustering performs better that it actually does.

False positive probability in protein's motif detection The relation between a protein's shape and function is linked to its motifs, which dene the supersecondary structure. Motifs have biological properties: for example the helixâturnâhelix motif [Brennan 1989]is responsible for the binding of DNA within several prokaryotic proteins. Automatic motif detection is another challenge in the study of protein shapes. We investigate the impact of bias on the false positive probability estimation in motif detection.

Let us consider a set {P i } n i=1 of proteins each with N i atoms. One is interested in the motifs of k atoms that can be detected in the protein's set, where k < N . We dene σ that represents an allowed error zone. The number of detected motifs increases if: (i) one decreases k, or (ii) one increases σ, or (iii) increases n. Thus how many detected motifs actually come from chance, with respect to the parameters k, σ, n? The false positives probability indicates when one detects truth and when one detects noise. The usual estimate of the false positive probability is P = V 0 V l . Here V 0 is the volume of the error zone allowed. V l is the total volume of the protein [Pennec 1998b], thus the a ball of radius the Radius of Gyration. Thus V l may be biased and overestimated. The probability of false positive is underestimated.

We consider the example of [Pennec 1998a]. One tries to nd motifs between the tryptophan repressor of Escherichia coli (PDB code 2WRP) and the CRO protein of phage 434 (PDB code 2CRO). These two proteins are known to share the helixturn-helix motif. The radius of Gyration of 2WRP is R g = 20 Å, the total volume is:

V l = 4 3 πR 3 g 33510 Å3 .
We assume an error zone that takes the form of a diagonal covariance matrix with standard deviations σ = 0.35 Å. We get the error zone volume: V 0 = χ 3 4 3 πσ 3 = 4.06 Å3 , where ξ 2 = 8 comes from a convention about how much error is allowed: the covariance of the error within the error volume shall be less than ξ 2 , see [Pennec 1998a] for details. The estimation of the false positive probability is: P = 1.2 × 10 -4 . We nd that P is underestimated by 0.27% using the expression of the Radius of Gyration's bias.

Brain template in Neuroimaging

We apply the rule of thumb of Section 7.5 to determine when the bias needs a correction in the computation of a brain template from medical images. Here M and G will be innite dimensional, so that the results of this paper do not apply directly. Nevertheless, this subsection allows us to gain intuition about how this paper may impact the eld of neuroimaging.

In neuroimaging, a template is an image representing a reference anatomy. Computing the template is often the rst step in medical image processing. Then, the subjects' anatomical shapes may be characterized by their spatial deformations from the template. These deformations may serve for (i) a statistical analysis of the subject shapes, or (ii) for automated segmentation by mapping the template's segmented regions into the subject spaces. In both cases, if the template is not centered among the population, i.e. if it is biased, then the analyzes and conclusions could be biased. We are interested in highlighting the variables that control the template's bias.

The framework of Large Deformation Dieomorphic Metric Mapping (LDDMM) [Younes 2012] embeds the template estimation in our geometric setting. The Lie group of dieomorphisms acts on the space of images as follows:

ρ : Di(Ω) × L 2 (Ω) → L 2 (Ω), (φ, I) → φ • I = I • φ -1 .
(5.15)

The isotropy group of I writes:

G I = {φ ∈ Di(Ω)|I • φ -1 = I}.
Its Lie algebra g I Chapter 5. Geometric Statistics on quotient spaces: bias in anatomical template shape estimation.

103 consists of the innitesimal transformations whose vector elds are parallel to the level sets of I:

g I = {v|∀x ∈ Ω, ∇I(x) T .v(x) = 0}. The orbit of I is : O I = {I ∈ L 2 (Ω)|∃φ ∈ Di(Ω) s.t. I • φ -1 = I}.
The "shape space" is by denition the space of orbits. Two images that are dieomorphic deformations of one another are in the same orbit. They correspond to the same point in the shape space. Topology of an image is dened as the image's properties that are invariant by dieomorphisms. Consequently, the shape space is the space of the images topology, represented by the topology of their level sets. We get a stratication of the shape space when we gather the orbits by orbit type. A stratum is more singular than another, if it has higher orbit type, i.e. larger isotropy group.

The manifold M has an innite stratication. One changes stratum every time there is a change in the topology of an image's level sets. Singular strata are connected to simpler topology. "Principal" strata are connected to more complicated topology. Indeed, the simpler the topology of the level sets is, the higher is the "symmetry" of the image. Thus the larger is its isotropy group. Note that strata with smaller isotropy group (more detailed topology) do not represent "singularities" from the point of view of a given image and do not inuence the bias. In fact, such strata are at distance 0: an innitesimal local change in intensity can create a maximum or minimum, thus complexifying the topology.

Using the rule-of-thumb of Section 7.5, the template's bias depends on its distance d to the next singularity, at the scale of σ the intersubjects variability. The template is biased in the regions where the dierence in intensity between maxima and minima is of the same amplitude as the variability. The template may converge to pure noise in these regions.

Conclusion

We introduced tools of statistics on manifolds to study the properties of template's shape estimation in Medical imaging and Computer vision. We have shown asymptotic bias by considering the shape space's geometry. The bias comes from the external curvature of the template's orbit at the scale of the noise on the data. This provides a geometric interpretation for the bias observed in [START_REF] Allassonnière | Towards a Coherent Statistical Framework for Dense Deformable Template Estimation[END_REF], Allassonnière 2015a]. We investigated the case of several templates and the performance K-mean algorithms on shapes: clusters are less well separated because of each centroid's bias. The variables controlling the bias are: (i) the distance in shape space from the template to a singular shape and (ii) the noise's scale. This gives a rule-of-thumb for determining when the bias is important and needs correction. We proposed two procedures for correcting the bias: an iterative bootstrap and a nested bootstrap. These procedures can be applied to any type of shape data: landmarks, curves, images, etc. They also provide a way to compute the external curvature of an orbit.

Our results are exemplied on simulated and real data. Many studies use the Chapter 5. Geometric Statistics on quotient spaces: bias in anatomical template shape estimation. 104 template's shape estimation algorithm in Molecular Biology, Medical Imaging or Computer vision. Their estimations are necessarily biased. But these studies often belong to a regime where the bias is not important (less than 0.1%). For example, the bias is important in landmark shapes analyses when the landmarks' noise is comparable to the template shape's size. Studies are rarely in this regime. We have considered shapes belonging to innite dimensional shape spaces. Our results do not apply to the innite dimensional case. We have used them to gain intuition about it. The bias might be more important in innite dimensions and needs a correction as we have suggested.

Chapter 6

Geometric Statistics on topologies: controlling the bias in brain template estimation

This Chapter applies the analyses of Chapter 5 to the estimation of brain templates.

As the Lie group involved is a group of dieomorphisms, the quotient structure leads us to consider spaces of topology. This Chapter is a practical illustration of tools of Geometric Statistics on real datasets of Computational Anatomy. This Chapter will be submitted to the SIAM Journal on Applied Algebra and Geometry under the title: "Topologically constrained template estimation via Morse-Smale complexes allows to control its statistical consistency".

Introduction

In neuroimaging, as well as in many other medical image analysis domains, a template is an image representing a reference anatomy. A template is computed from a database of brain images to serve as the brain image "prototype" for further analyses.

Computation of a brain template Various methods exist to compute a brain template [Evans 2012]. A rst practice selects one brain image from the database as the template. If the selected subject's anatomy is far from the population mean anatomy, the template is necessarily biased towards this specic individual. Thus, the template fails at being a prototype of the population. This is why researchers consider the computation of "unbiased template" that should better represent the mean anatomy.

Such an "unbiased" template is often constructed by performing an iterative averaging of intensities and deformations [Guimond 1998, Joshi 2004, Hadj-Hamou 2016]. One initializes with a template being one of the subject images. Then, during each iteration, one registers the subjects to the current template, and computes the mean deformation. The new template is computed as the mean intensity of the subjects' images, deformed with the mean deformation. This procedure does not favor any subject's image if it does not end in a local minimum. In this sense, the procedure is called "unbiased".

The computed brain template may look blurred or sharp depending on the design chosen for the registration in the above iterative procedure. If the algorithm is designed using linear registration, the template may appear blurred. In contrast, if one uses dieomorphic registration, the template is more likely to look sharp and the sharpness depends on the amount of regularization used [Evans 2012].

Purpose and desirable properties of the brain template Computing a template is often the rst step in medical image processing because of its many applications. The template is used as a standardized 3D coordinate frame where the subject brains can be compared. The subjects are then characterized by their spatial dieomorphic deformations from the template. These deformations may serve for a statistical analysis of the subject shapes [Ashburner 1998]. One studies the normal and pathological variations of the subjects with respect to the template. The deformations also facilitate automated segmentation, by mapping the template's already segmented regions into each subject space.

What are the desirable properties of the brain template, with respect to the applications mentioned above? First, the template should be representative of the population, to remove any bias toward a specic subject during the analysis [Bookstein 2001, Ashburner 2001, Baloch 2009]. Second, the template should be sharply dened, so that subtle anatomical structures can be easily observed or segmented.

The brain template, an inconsistent estimator of the unique brain anatomy However, the two desirable properties cannot be fullled simultaneously: there exists a trade-o.

First, this trade-o can be understood intuitively. Consider a database of brain images divided into two groups that have dierent topologies in the following sense. The rst group has subjects with three sulci -i.e. depressions or grooves in the cerebral cortex -in a specied brain region. The second group has subjects with only two sulci in the same region. A sharply dened template has to decide on a specic topology in this brain region, i.e. whether it shows two or three sulci. Therefore, it might not estimate correctly the brain anatomy of this population, which might be problematic for the following applications in neuroimaging. For example during a statistical analysis, registering the subjects with the three sulci topology to a template that has chosen a two sulci topology might not be reasonable [Baloch 2009]. A sharp template is only meaningful if the anatomical structures shown are representative of the population.

Second, the trade-o has been emphasized in recent studies investigating the bias of the template as an estimator of the population's shared anatomy. In the classical approach [START_REF] Allassonnière | Towards a Coherent Statistical Framework for Dense Deformable Template Estimation[END_REF]], an initial assumption states that there is a unique (brain) anatomy shared by the population. The subjects are then modeled through a generative model as random deformations of the unique brain anatomy, observed with additional noise. The unique brain anatomy is a parameter of this model. The template computation is interpreted as its estimation. One can ask about its asymptotic bias: does the template converge to the unique brain anatomy Chapter 6. Geometric Statistics on topologies: controlling the bias in brain template estimation. 107 for a database with an innite number of images?

This question has been investigated for signals, i.e. 1D images. Some authors prove the asymptotic unbiasedness of the template under the simplifying assumption of no measurement error on the observed signals [Kurtek 2011]. Other authors have already provided examples of asymptotic bias, and therefore inconsistency, when there is measurement error [START_REF] Allassonnière | Towards a Coherent Statistical Framework for Dense Deformable Template Estimation[END_REF]]. Their experiments show that the template may converge to pure noise when the measurement error on simulated signals increases. A bias is shown to occur in [Bigot 2011] for curves estimated from a nite number of points in the presence of noise.

Recently, the asymptotic bias has been shown in the setting of Lie group actions [Miolane 2015a, Miolane 2017]. Their argument, shown in an abstract geometric context but adapted here to brain images, is as follows. They look at the subspace dened by all brains with the same shape as the unique brain anatomy. They show that the curvature of this space, at the scale of the measurement noise, introduces the bias on the brain template.

Using topology to investigate the brain template's asymptotic bias We want to link: (i) the fact that a population with two groups of dierent brain topologies cannot be accurately represented by a sharp template, with (ii) the mathematical results on the template's bias as an estimate of the anatomy shared by the population. The framework of [Miolane 2015a] is based on the quotient of the space of the data space by the action of a Lie group. The data in our case are the brain images, and the Lie group action is the action of dieomorphisms on these images. Quotienting the images by the action of dieomorphisms amounts to ltering out any information that is invariant by dieomorphic deformations. Thus, the quotient gives the topology of the images' level sets. Using the intuition of [Miolane 2015a], we could quantify the brain template's asymptotic bias using a representation of its topology.

Quantifying the bias could enable us to decide when and where a sharply dened template makes sense. We could want a sharp template in brain regions where the intersubject anatomical variability is low and a fuzzier template when this variability is higher. Alternatively, we could consider computing several sharp brain templates using mixtures. This discussion boils down to the question: when is it reasonable to assume that a unique brain anatomy represents the whole subject population? Furthermore, we could think about controlling the brain template's asymptotic bias by constraining its topology. Topological representations of images have been used with various objectives in the literature. For example, [Chung 2009] uses a topological representation of brain images for classication of autism versus normals. Topological constraints have also been implemented for segmentation where the reconstruction of the cortical surface needs to match the brain anatomy [MacDonald 2000, Mangin 1995, Han 2003]. However, topological representation of images or topological constraints on images have not been used to study and enforce a statistical property, like the asymptotic unbiasedness.

Chapter 6. Geometric Statistics on topologies: controlling the bias in brain template estimation.

108 Contributions and Outline We use a topological representation of images -the Morse-Smale complex -to investigate and control the asymptotic bias of the brain template. We make three main contributions in this paper. First, we show how to combine Geometry and Topology to tackle a statistical problem in neuroimaging. We provide conjectures at the boundaries of the elds with sketches of their proofs. Second, we analyze the template as an estimator of the brain anatomy and quantify the asymptotic bias. This leads us to discuss the initial assumption of a unique anatomy. Third, we present an adaptation of the template computation algorithm that bounds the bias, through topological constraints, at the price of constructing a "smoother" template. Section 6.2 presents the geometry and the topology of the template computation. We emphasize the variables that describe the bias of the brain template. Section 6.3 presents the chosen computational representation of these variables through Morse-Smale complices. Section 6.4 leverages the previous computational model to spatially identify the biased regions of the template. We thus propose an adaptation of the template computation with topological constraints bounding the bias. In Section 6.5 our methodology is used on the Open Access Series of Imaging Studies (OASIS) database of T1-weighted MR brain images.

Geometry and Topology for template estimation

This paper does not aim at presenting mathematical results. We rather show how Geometry and Topology combine to formalize the template computation algorithm and highlight required directions for further mathematical developments.

Geometrization of the action of dieomorphisms on images

Brain images We consider two-and three-dimensional images, whose domain Ω ⊂ R d , with d = 2, 3, is supposed compact. We adopt the point of view of images as square-integrable functions I over the compact domain Ω, i.e. we write I ∈ L 2 (Ω), where L 2 (Ω) is a Hilbert space. The corresponding L 2 distance is invariant by volume preserving dieomorphisms (volumorphisms). Additionally, we assume that the images are in C ∞ (Ω), which is C ∞ dened on the compact support Ω. We denote Img(Ω) the set of images.

To illustrate the following concepts, we use the toy Hilbert space R 2 where one point schematically represents one image, see Figure 6.1.

Dieomorphisms A dieomorphism of Ω is a dierentiable map φ : Ω → Ω which is a bijection whose inverse φ (-1) is also dierentiable. We consider two sets of dieomorphisms. On the one side, we consider C ∞ (Ω), i.e. the smooth dieomorphisms that are the identity outside a compact support. C ∞ (Ω) can be seen as an innite dimensional manifold [Milnor 1976] and forms an innite dimensional Lie group [Kriegl 1997]. Its Lie algebra V is the set of smooth vector elds with com-Chapter 6. Geometric Statistics on topologies: controlling the bias in brain template estimation. 109 pact support [Kriegl 1997]. We use this set of dieomorphisms to present algebraic concepts.

On the other side, we consider the set

C Id (R d , R d ) dened as C Id (R 3 , R 3 ) = {φ = Id + u for u ∈ C 1 b (R d , R d )}
, where the subscript "b" refers to functions that are bounded with bounded derivatives. These dieomorphisms are "small", i.e. not too dierent from the identity. We use this set to formulate mathematical conjectures which need metric properties. If the specication between the two sets is not needed, we will refer to Di(Ω) to denote dieomorphisms.

Action of dieomorphisms on (brain) images

The Lie group of dieomorphisms Di(Ω) acts on the space of images L 2 (Ω) [Younes 2012]:

ρ : Di(Ω) × Img(Ω) → Img(Ω), (φ, I) → φ • I = I • φ -1 .
We note that the action of dieomorphisms on brain images is not isometric with respect to the L 2 distance on the images. Isotropy group G I of a (brain) image I Here, we consider Di(Ω) = C ∞ (Ω).

The notion of "isotropy group" dictates the dimension of a given orbit, i.e. whether we have a 1-dimensional submanifold like the blue dotted circle or a 0-dimensional submanifold like the red point on Figure 6.2. The isotropy group G I of a brain image I is dened as the subgroup of Di(Ω) formed by the dieomorphisms that leave I unchanged: G I = {φ ∈ Di(Ω)|I • φ -1 = I}. G I describes the intrinsic symmetry of the image I: the more symmetric is I, the larger its isotropy group. All images on the same orbit have conjugate isotropy groups. Moreover, the isotropy group (also called the stabilizer) and the orbit of an image are linked by the orbit-stabilizer theorem: O I ∼ Di(Ω)/G I in nite dimensions. The intuition is that the larger the isotropy group (and thus, the more symmetry the image has), the smaller the orbit.

The toy example of R 2 where the dieomorphisms are represented by rotations illustrates the notion, see Figure 6.3. The isotropy group of the image at the blue point on Figure 6.3(b) is the identity. Only the identity leaves this image at the same place. The isotropy group of the image represented by the red point is the whole Lie group of 2D rotations. Any rotation leaves this point invariant. Going back to real dieomorphisms, the isotropy group of a uniform image, i.e. I constant map over Ω, is the whole group Di(Ω). Figure 6.3(a) shows two brain images: the isotropy group of I 1 is larger than the isotropy group of I 2 , in the sense of the inclusion. As a consequence, the orbit of I 1 has smaller dimension than the orbit of I 2 .

From geometry to topology

Topology of (brain) images The topology of a brain image I is dened as the topology of its level sets, these surfaces of Ω with constant intensity. Topology refers to properties that are preserved under smooth deformations [Gerber 2012], Chapter 6. Geometric Statistics on topologies: controlling the bias in brain template estimation. The isotropy group is linked to the dimension of the image's orbit. I 2 has a smaller isotropy group, it is represented as a blue point which has a circle as orbit. I 1 has larger isotropy group: it is represented as the red point at (0, 0) whose orbit is itself.

i.e. conserved by the action of dieomorphisms on I: for example, the number of holes, or the number of connected parts, see Figure 6.4(a).

Geometry and Topology combine as follows. Two images I and I that are dieomorphic deformations of each other, i.e. that are on the same orbit, have the same topology. The orbit O I itself represents the topology of image I (and I ). The set of orbits Q = {O I |I ∈ Img(Ω)}, which is the quotient space of Img(Ω) by the action of Di(Ω) is the set of the topologies. Figure 6.4(b) shows how the space of images R 2 is partitioned into orbits: blue circles and one red "singular circle", the red point at (0, 0). Chapter 6. Geometric Statistics on topologies: controlling the bias in brain template estimation. 112 Gathering brain images with similar topologies: orbit types By denition, two brain images are of the same orbit type if their isotropy groups are conjugate subgroups in the Lie group of dieomorphisms. In particular, brain images that belong to the same orbit have same orbit type. The type corresponding to the smallest isotropy group -in the sense of the inclusion among the subgroups of the dieomorphism group -is sometimes called the principal type [Alekseevsky 2003]. We use this appellation in this paper. For example, if an orbit has type the identity of the group of dieomorphisms, then it is necessarily of principal type. Equivalently, the orbits of principal type are called principal orbits. Other orbits are called singular orbits.

The blue circles on Figure 6.4(b) have same orbit type: the images on these orbits have the identity {Id} of the Lie group as isotropy group. They are of principal type since the identity {Id} is the smallest subgroup -in the sense of the inclusion -of the group of rotations.

Stratication of the space of topologies In the space of brain images, we can gather orbits of same orbit type: we gather the blue circles of Figure 6.4(b) into R 2 \ {(0, 0)} on the one hand, and keep the red dot (0, 0) on the other hand. The orbit type itself is a submanifold of the space of brain images: R 2 \ {(0, 0)} or (0, 0) in the schematic brain images space R 2 . Furthermore, these orbit type submanifolds form a stratication, meaning they t together in a particularly nice way [Weinberger 1994].

The quotient space Q is also naturally partitioned into manifolds, and this partitioning is also a stratication. All in all, Q is not a manifold, but Q composed of manifold pieces, and those pieces are called strata. There is a partial ordering of the strata in the quotient space, using the inclusion [Hughes 1996]. Figure 6.4(b) shows the orbits grouped by orbit type: the color blue denotes one orbit type and the color red another orbit type. We see for example that Q = R + is stratied into one stratum being R * + -corresponding to the stratum R 2 \ {(0, 0)} in the space of brain images -and one stratum being {0} -corresponding to the stratum (0, 0) in the space of brain images.

Geometry of generative model and estimation procedure

Here: Di

(Ω) = C 1 b (R d , R d ).
Generative model The n brain images I 1 , ..., I n are interpreted with a generative deformable model:

I i = φ i • T + ε i , i = 1...n,
where each image I i ∈ Img(Ω) is a dieomorphic deformation φ i ∈ Di(Ω) of a unique brain anatomy T , to which variability ε i is added. The parameter T represents the brain anatomy shared by the population. The transformations φ i 's and the noises ε i 's are i.i.d. realizations of random variables. The transformations φ i 's follow a law that is a generalization of the Gaussian for Lie groups, i.e. a Riemannian Gaussian on the Lie group of dieomorphisms and the ε i 's represent Gaussian noise on the space of images. We Chapter 6. Geometric Statistics on topologies: controlling the bias in brain template estimation. 113 denote σ 2 its variance. Denitions of distributions on nite dimensional Riemannian manifolds are taken from [Pennec 2006] and the Gaussian distributions for innite dimensional spaces from [Lifshits 1995].

The model can be interpreted by a three step generative procedure illustrated schematically in Figure 6.5. First, there is only the shared anatomy T . Second, the template T is deformed with the dieomorphism φ i and gives a brain image φ i • T . Third, measurement noise is added through ε i , which gives the observed brain image Computing the template: an estimation procedure Computing the brain template amounts to invert the generative model: given the data, we want to estimate the parameter T . The transformations φ's are hidden variables of the model. The natural statistical procedure to estimate T in this context is the Expectation-Maximization (EM) algorithm [START_REF] Dempster | Maximum likelihood from incomplete data via the EM algorithm[END_REF]]. The EM is an iterative procedure that maximizes the log-likelihood of the generative model with hidden variables. As such, the EM gives an asymptotically unbiased and consistent estimation of the brain anatomy T . In practice, one does not use the EM algorithm because it is computationally expensive, especially when dealing with tridimensional images. Most neuroimaging pipelines rely on an approximation of the Expectation-Maximization algorithm to estimate the brain anatomy, called "Fast Approximation with Modes" in [START_REF] Allassonnière | Towards a Coherent Statistical Framework for Dense Deformable Template Estimation[END_REF]]. This approximation is the template estimation procedure described in the Introduction and the main purpose of this paper. It runs as follows.

I i . T T T σ φ i φ i • T ε i φ i • T + ε i (a) (b) (c)
Chapter 6. Geometric Statistics on topologies: controlling the bias in brain template estimation.

114 One initializes the estimate with T = I 1 , i.e. one of the brain images from the database. Then, one iterates the following two steps until convergence [Joshi 2004]:

(1) φi = argmin φ∈Di(Ω) d Img(Ω) ( T, φ • I i ) + λReg(φ), ∀i ∈ {1, ..., n}, (2) T = argmin T ∈Img(Ω) n i=1 d Img(Ω) (T, φi • I i ) 2 .
Step ( 1) is an estimation φi of the hidden observations, the dieomorphisms φ i , and an approximation of the E-step of the EM algorithm. In practice, each brain image I i is registered to the current template estimate and the φi is the result of this registration. The term Reg is a regularization that allows the optimization to have a solution. Showing that the estimates φi 's and T exist is technical and beyond the scope of this paper.

Step ( 2) is the M-step of the EM algorithm: the maximization of the surrogate in the M-step amounts to the maximization of the variance of the projected data. This computes the updated template estimate, as the mean intensity of the subjects images I i , deformed with the mean deformation of the φi 's.

The registration step (1) amounts to align the n subject images by transporting them on their orbit (see Figure 6.6(b)), i.e. projecting them in the quotient space (see Figure 6.6 (c)). Step (2) averages the n registered images (see Figure 6.6 (d)). Evaluation of the procedure: denition of asymptotic bias B ∞ We evaluate the template T as an estimator of the unique brain anatomy T (see Figure 6.6 (d)) given n observations I i , i = 1...n. We note that in other papers, T may be called the template directly and T the template's estimate. We consider two measures of the accuracy of this estimator: its variance V 2 n and Chapter 6. Geometric Statistics on topologies: controlling the bias in brain template estimation. 115 bias B n , which are dened as:

V 2 n = E(( T -E( T )) 2 ) and B n = E( T -T ).
The bias for n images B n is illustrated on Figure 6.6. We are interested in the asymptotic behavior n → +∞, i.e. when the number of brain images goes to innity which is an ideal situation. One would expect that the procedure converges the brain anatomy T it is designed to estimate. It is assumed that the estimator converges so that its variance is asymptotically zero: V 2 ∞ = 0.

Geometry of the template estimator's evaluation

Estimation procedure interpreted as the Fréchet mean in the quotient space First, we model the estimation procedure as follows.

(1) φi = argmin

φ∈G d M ( T, φ • I i ), ∀i ∈ {1, ..., n}, (2)
T = argmin T ∈M n i=1 d M (T, φi • I i ) 2 .
where M is a generic Riemannian manifold and G a Lie group acting on M isometrically. This represents well the actual estimation procedure whose regularization term makes the action of dieomorphisms close to isometric. This converges to a local minimum because it decreases at each step a cost bounded below by zero. The estimator computed with this procedure is:

T = argmin T ∈M n i=1 min φ∈G d 2 M (T, φ • I i ).
(6.1)

The term min

φ∈G d 2 M (T, φ • I i )
is the distance in the quotient space between T and I i . Thus Equation 6.1 is the the Fréchet mean on the quotient space [Pennec 2006].

Asymptotic bias B ∞ and curvature The authors of [Miolane 2015a] show that the asymptotic bias is non zero: B ∞ = 0. For an innite number of brain images n → +∞, the estimate converges, but not to the brain anatomy T it was designed to compute. This has a geometric origin explained in [Miolane 2015a, Allassonnière 2015a]. The authors of [Miolane 2015a] compute a Taylor expansion of the asymptotic bias B ∞ around the noise σ = 0, in the case of a nite dimensional manifold and isometric Lie group action [Miolane 2017]:

B ∞ = σ 2 2 H(T ) + O(σ 3 ) + ε(σ) (6.2)
where H(T ) denotes the mean curvature vector of the template's orbit. There is no bias when there is no measurement error σ = 0. It was observed experimentally that the bias was dependent on the measurement error [START_REF] Allassonnière | Towards a Coherent Statistical Framework for Dense Deformable Template Estimation[END_REF]].

Chapter 6. Geometric Statistics on topologies: controlling the bias in brain template estimation.

116

The coecient H(T ) depends on T that is being estimated. We investigate this dependency in the geometric framework of Subsection 6.2.2. Assume that there exists a xed point o of the Lie group action, i.e. a point that is left unmoved by the whole Lie group. Consider the orbit O T of T . As the action is isometric, the orbit belongs to a geodesic sphere S d with center o and radius d. A geodesic sphere of radius d in a manifold -like a hypersphere of radius d in R m -has mean curvature vector: ||H(T )|| = (m-1) d and a closed hypersurface in a manifold of dimension m obtained by perturbing the geodesic sphere has: [Pacard 2005, Lytchak 2010]. If we write the template's bias in the units of d, then the asymptotic bias depends on σ d 2 . In other words, the distance of the template to the singularity o, at the scale of the noise σ governs the asymptotic bias B ∞ .

||H(T )|| = (m-1) d + O(1)
Figure 6.7 shows the intuition behind this. On Figure 6.7(a), R 2 schematically represents the space of brain images -the black squares represent brain images from the database, the green square is T -and the green circle is the orbit of T . The dotted circles, that have their centers on the template's orbit, represent the probability distribution of the (2D isotropic) Gaussian noise in the generative model. More precisely, they represent the level set at σ of the noise distribution. The curvature H(T ) controls the area in grey on Figure 6.7, which is the area inside the Gaussian level set that is outside T 's orbit. This area is greater that the area inside T 's orbit. As a consequence, the probability that the brain images are generated "outside" T 's orbit is higher than the probability that they are generated inside T 's orbit.

Figure 6.7(b) shows the registration step of the template estimation: there is a higher probability that the registered images are away from T , as if repulsed from the singularity around which the orbits warp. When one averages the registered images, one sees that the template's estimate becomes biased as it will systematically give an image that is further away than T from the quotient space's singularity, i.e. from the red dot.

Quantifying the asymptotic bias B ∞ of the brain template In neuroimaging, the manifold is the space of brain images Img(Ω) and the Lie group is the group of dieomorphisms Di(Ω), both innite dimensional. This paper assumes that we can apply the geometry of [Miolane 2017], because there are indications that B ∞ appears in the same fashion in innite dimensions 6.8.

First, [Miolane 2017] studies the bias when the dimension of the manifold increases. They consider the nite dimensional manifold M = R m with the action of SO(m), i.e. a generalization of the toy example R 2 with the action of SO(2) from our illustrations. The authors of [Miolane 2017] show that B ∞ increases when m increases, see Figure 6.8 (a).

Then, the work of [Allassonnière 2015a] shows that there exists an asymptotic bias in an innite dimensional Hilbert space. [Devilliers 2017] provided an asymptotic behavior of the bias when the noise level σ tends to innity. This bias is Chapter 6. Geometric Statistics on topologies: controlling the bias in brain template estimation. Ultimately, [START_REF] Bigot | [END_REF]] gives a lower bound of the asymptotic bias for shapes of curves in 2D, where terms depend on derivatives of the functions representing the curve. These derivatives can be interpreted as the derivative of the action of translations, which leads to the curvature of the orbit of the given function under the translations' action.

As a consequence, we assume that the intuition provided by [Miolane 2017] applies to neuroimaging and that the bias depends on the ratio σ d 2 .

Computational representation of geometry and topology

This section shows how the topology of brain images -represented by Morse-Smale complexes -enables to estimate the geometric parameters d and σ introduced above.

Denition of Morse-Smale complexes for (brain) images

Morse-Smale (intensity) functions A real-valued smooth map I : Ω → R is a Morse function if all its critical points are non-degenerate (the Hessian matrix is non-singular) and no two critical points have the same function value. The intensity function I representing a bi-or tri-dimensional brain image is a Morse function, at least after a convolution with a smoothing Gaussian [Boscain 2012]. In the following, I represents a brain image. Morse theory traditionally analyzes the topology of a manifold by studying the Morse functions on that manifold. Here, the manifold is known: it is the image domain Ω. We are not interested in the topology of Ω but rather in the topology of the functions I themselves, that is: we would like to know the distribution of their critical points. Figure 6.9 shows a 2D slice of a 3D brain image I, where the intensity is represented as the height, to better emphasize its maxima and minima: the maxima are in red and the minima in blue.

We introduce the notions of integral lines, ascending and descending manifolds that are needed to dene Morse-Smale (intensity) functions. An integral line is a maximal path in the image domain Ω whose tangent vector correspond to the intensity gradient ∇I, the gradient of I, at every point. This notion comes from autonomous ordinary dierential equation, where it represents the trajectory of a system verifying:

dx dt (x) = ∇I(x) (6.3)
Each integral line starts and ends at critical points of I, where the gradient ∇I is zero. Ascending A(x i ) and descending D(x i ) manifolds of an extremum x i are Chapter 6. Geometric Statistics on topologies: controlling the bias in brain template estimation.

A(x i ) = {x ∈ Ω | The integral line going through x ends at x i } D(x i ) = {x ∈ Ω |
The integral line going through x starts at x i } Take the two manifolds A(x i ) and D(x i ) in Ω and assume they intersect at a point p ∈ Ω. Let T A (resp. T D ) denotes the set of all vectors tangent to A(x i ) (resp. D(x i ) at p. If every vector in Ω is the sum of a vector in T A and a vector in T D , then A(x i ) and D(x i ) are said to intersect transversely at the point p. The intensity function I dening the brain image is Morse-Smale if the ascending and descending manifolds only intersect transversely. We assume in the following that all brain images I are Morse-Smale.

Morse-Smale complex and persistence The Morse-Smale complex of a Morse-Smale function is the set of intersections A(x i ) ∩ D(x j ), over all combinations of extrema (x i , x j ) [Gyulassy 2008]. The Morse-Smale complex includes regions (i.e., sub-manifolds of Ω) of dimensions 0 through D, where D is the dimension of the domain Ω, i.e. D = 2 or D = 3 for our purposes. The Morse-Smale (MS) complex of I is a partition of the domain Ω into regions dened by the set of integral lines that share common starting and ending points. The interior of each region is monotonic with respect to the intensity I: a region contains no critical points and has a single local minimum and maximum on its boundary, see for example Figure 6.9(ii) where the maximum I max and minimum I min are shown on the boundary of the grey region.

The MS complex can also be seen as a graph on the brain image domain Ω whose nodes are the critical points of the brain image intensity.

The persistence of a critical point x i of I is the amount of change in intensity I Chapter 6. Geometric Statistics on topologies: controlling the bias in brain template estimation. 120 required to remove this critical point:

p(x i ) = |I(x i ) -I(n(x i ))| (6.4)
where n(x i ) is the critical point closest to x i in intensity, among the critical points connected to x i by an integral line [Gerber 2012]. The persistence of x i is a measure of its signicance as a critical point, i.e. importance of the topological feature. Figure 6.9 illustrates the denition of persistence on a 1D example. The function represented has 4 critic points: two minima and two maxima. The gure shows how they pair, as well as their persistence. On the x-axis, colors show the regions of the corresponding 1D Morse-Smale complex.

Beside this usual denition of persistence of a critical point, we dene here the persistence of a region of the Morse-Smale complex as the amount of change in intensity required to remove this region from the MS complex, and more precisely:

p(region) = |I max -I min | (6.5)
where I max and I min are respectively the maximum and the minimum in intensity of this region. In contrast to the denition of the persistence of a critical point, we do not rely on the saddle points, but only on the extrema i.e. the minima and maxima.

Hierarchy of Morse-Smale complexes The notion of persistence of a region enables the denition of a hierarchy of MS complexes of one brain image I [Gyulassy 2008, Gerber 2012]. One uses the ordering given by persistence to successively remove topological features from the image I. One starts with the MS complex of the brain image I dened above and one recursively removes the critical points with minimal persistence. This leads to a nested series of successively simplied Morse-Smale complexes. At each level, some of the MS regions are merged into a single region. Ultimately the Morse-Smale complex consists of only one region which is the entire domain Ω.

The persistence introduces a notion of scale at which the Morse-Smale complex of I is considered. One keeps only the nodes whose persistence is above the threshold. Figure 6.9 shows that the one dimensional domain is partitioned dierently if one takes a threshold p below p 1 or between p 1 and p 2 . We say that a Morse-Smale complex is represented at a given persistence level. At the scale of the persistence threshold p, the intensity is considered monotonic on each region of the MS.

We note that this MS hierarchy is dierent from a Gaussian scale space (GSS) hierarchy of images [Reininghaus 2011]. The latter takes critical points across smoothing scales and not across persistence levels.

Computing Morse-Smale complexes of (brain) images in practice

The previous denitions are relevant to (continuous) Morse-Smale theory and apply strictly for a continuous intensity function I. Nevertheless, the MS complex, introduced in terms of ascending and descending manifolds, can be computed for discrete brain images as follows [Gerber 2012].

Chapter 6. Geometric Statistics on topologies: controlling the bias in brain template estimation.

121

Computing the Morse-Smale of a brain image We compute the Morse-Smale complex of a brain image, which will later be the brain template image. Our input are {x i , I i }, i.e. the intensity values {I i } on a grid {x i } i of Ω. We compute the integral lines of the intensity gradient, which we then gather to get the regions of the Morse-Smale complex. For each element of the grid x i , following the gradient ∇I leads to computing the integral line going through x i and in particular its starting and ending points [Gerber 2012]. The domain Ω can be approximated via a k nearestneighbor graph and one computes the integral lines by considering the connectivity of the graph. Then, elements x i 's with same starting and ending points belong to the same Morse-Smale region. This gives the partition of the domain Ω and therefore the Morse-Smale complex. We remark that x i 's necessarily belong to a 3-dimensional (for a tridimensional image) component of the Morse-Smale complex because the 0-, 1-and 2-dimensional components have measure zero. Figure 6.9 shows the Morse-Smale complex of the 2D slice of a 3D brain image for level of persistence of p = 0.1. The image's 2D domain is divided in dierent regions, represented by the dierent colors. The quadrant shows part of the underlying Morse-Smale graph. The red dot represents a maximum in intensity, and the blue dot a minimum of intensity. They are nodes of the underlying graph on the domain Ω.

Morse-Smale (MS) graph and labeled Morse-Smale (MS) graph

There are two ways of representing the Morse-Smale graph corresponding to the computed Morse-Smale complex. Both will be useful for analyzing the template's asymptotic bias. One can consider the graph as the set of nodes and edges, without any intensity information at the nodes. We simply call this graph the Morse-Smale (MS) graph:

this is the graph illustrated on Figure 6.9(i). Alternatively, one can label the nodes with the intensity information. We call this graph the labeled Morse-Smale (MS) graph: this is the graph illustrated on Figure 6.9(ii).

Template's computation and Morse-Smale complexes

We show how the MS complex of an image can represent its geometry and in particular isotropy group.

Lie algebra of the isotropy group and intensity gradient of the brain template The template is an image I ∈ Img(Ω). We consider the Lie group of dieomorphisms Di(Ω) = C ∞ (Ω) and its Lie algebra V , see Subsection 6.2.1. Lemma 6.3.1 Take ε > 0 and consider the Lie algebra:

V I = v ∈ V s.t.: I • Exp(tv) = I, ∀t ∈] -ε, ε[ (6.6)
By construction, the exponential of the elements of V I are in the isotropy group of I. For v ∈ V I , we have:

∀x ∈ Ω, ∇I(x) T .v(x) = 0 (6.7)
Chapter 6. Geometric Statistics on topologies: controlling the bias in brain template estimation.

122 Proof The proof is given in Appendix 6.

We note that this lemma does not give a characterization of the vector elds in V I . It gives the inclusion: V I ⊂ {v|∀x ∈ Ω, ∇I(x).v(x) = 0}. Thus, it allows to control the complexity of V I , and thus of some of the isotropy group's Lie algebra. It represents a rst step towards the following conjecture.

Conjecture 1 The Lie algebra of the isotropy group G I of the brain image I is constituted of vector elds that are everywhere perpendicular to the image's gradient.

Sketch of Proof 1 The sketch of the proof is given in Appendix 6 and uses the above lemma. We note that there is a technical work needed to rigorously dene the dierential structure of G I and its Lie algebra.

To understand the intuition behind this conjecture, consider a uniform image, i.e. with a constant intensity. In this case, there is no restrictions a priori on the vector elds of the Lie algebra of the image's isotropy group. Thus, the isotropy group is as large as it can be. We get that the isotropy group of an image with constant intensity is the whole group of dieomorphisms.

Intensity gradient and MS graph

We now present a lemma showing that the MS graph can be used to computationally represent the isotropy group of an image. Take two images I 1 and I 2 . Assume that their MS graphs are the same, regardless of the nodes' positions. Lemma 6.3.2 Assume we can map the level sets of I 1 to the level sets of I 2 . This implies that we can map the partition of the domain I 1 induced by its Morse-Smale to the partition of the domain of I 2 .

Take a part w ⊂ Ω of this partition. There exists a dieomorphism ψ and a monotonic function κ such that:

∇I 2 (x) = κ(x).d * ψ(x).∇I 1 • ψ(x), ∀x ∈ w.
Proof The proof is given in Appendix 6. This lemma is the rst step towards the following conjecture.

Conjecture 2 Two images with same MS graphs have same isotropy group.

Sketch of Proof 2

The sketch of the proof is given in Appendix 6 and uses the above lemma. We note that there is technical work required to dene the assertion "having the same MS graph".

In others words: if two images I 1 , I 2 have the same MS graph, then I 1 can be dieomorphically deformed so that its intensity gradient is parallel at every point to the intensity gradient of I 2 .

From Lemma 6.3.1, the sets {v 1 |∀x ∈ Ω, ∇I 1 (x).v 1 (x) = 0} and {v 2 |∀x ∈ Ω, ∇I 2 (x).v 2 (x) = 0} control the isotropy groups of I 1 and I 2 . If I 1 and I 2 have Chapter 6. Geometric Statistics on topologies: controlling the bias in brain template estimation. 123 same MS graph, any vector eld in the rst set can be dieomorphically deformed to get a vector eld in the second set, and conversely. As a consequence, the MS graph represents the image's isotropy group. From Section 6.2, we know that the isotropy group controls, in turn, the orbit's type of the image i.e. to which stratum the image belongs.

Furthermore, we note that we could have considered the labeled MS graph of the image, i.e. the MS graph with intensities at the nodes, see Figure 6.9(ii). The labeled MS graph controls the orbit of the image: images in the same orbit have the same topology but also same intensities.

Topology quanties and controls the template's asymptotic bias

This Section gathers the elements of Sections 6.2 and 6.3 to quantify the asymptotic bias in the brain template computation. We use Morse-Smales complexes to quantify, and then control, the bias.

Quantify the template inconsistency

We express d and σ using the Morse-Smale complexes of Section 6.3.

Understand and estimate the geometric parameter d The distance d is the distance to a brain image with larger isotropy group, measured in sum of squared dierences of intensities, see Figure 6.7. How can we measure this distance d locally on the template's image? From Section 6.2, we know that the isotropy group becomes larger when the image is "more symmetric". From Section 6.3, we know that the isotropy group becomes larger when the image topology becomes simpler. Thus, the distance d is a distance in intensity from the template image to a similar image with simpler topology. We want to express this distance locally on the template image. Modifying the intensity locally on the template image modies the image itself and may simplify its topology. For example, modifying the intensity locally in a region of the image can suppress a min-max pair and the image becomes "more symmetric". Thus we describe the distance d locally on the image by the amount of intensity needed to be changed in this region, so that the topology is simplied.

We quantify the local intensity needed to simplify the template image's topology using the Morse-Smale complex representation of Section 6.3. Let be given the Morse-Smale complex of the template image. The intensity needed to simplify the image's topology is, by denition, the intensity needed to simplify the Morse-Smale graph. We consider the partition of the image's domain Ω induced by the Morse-Smale complex. For each region of the partition, the intensity needed to simplify the topology can be represented by the amount of intensity needed to remove the Chapter 6. Geometric Statistics on topologies: controlling the bias in brain template estimation. This quanties the importance of the region as a representative of the brain anatomy: if the intensity dierence between the region's min and max is low, then one can assume that this min-max pair has been created by chance because of the noise on the images. We see that the notion of persistence dened in Section 6.3 estimates the rst geometric parameter d.

Understand and estimate the geometric parameter σ Now we turn to the second geometric parameter that causes the asymptotic bias: the standard deviation of the noise σ, see again Equation A.15 and Figure 6.7. The noise σ is a parameter of the generative model that we assume has produced the observed images of the subjects brain anatomies. The parameter σ is unknown but it can be estimated from the observed images. Since we want to compute the asymptotic bias locally, we are interested in estimating the parameter σ locally, and for example on a region of the Morse-Smale complex of the template image. We estimate it as the average of the variability in intensity of the registered images in the region:

σ(region) 1 #x x∈region σ(x), (6.9)
where σ(x) is the variability in intensity of the registered images at the voxel x, and serves as an estimate of the noise at this voxel. This quanties the amount of noise in this region. The larger the noise, the more chances for the template to show min-max pairs that appeared by chance.

Compute the asymptotic bias using the persistence of the whitened brain template The local estimates of the geometric parameters d and σ enable us to estimate the asymptotic bias locally on a brain region:

B∞ (region) = d(region) σ(region) -2
, (6.10)

We emphasize here that B∞ is an estimate of the asymptotic bias B ∞ (of the brain template estimation), and not an exact computation.

We link the estimate B∞ to the denition of persistence in the Morse-Smale complex framework. First, we dene the whitened brain template estimate t of T as: .11) In other words, we divide the brain template intensity of each voxel x by the estimation of the standard deviation of the noise at this voxel σ(x). This whitens the noise all over the brain template.

∀x ∈ Ω, t(x) = T (x) σ(x) . ( 6
Chapter 6. Geometric Statistics on topologies: controlling the bias in brain template estimation.

125

We assume that the critical points of t are close to the critical points of T and consider the Morse-Smale complex of the whitened template. We further assume that: σregion σ(max) σ(min), where σ(max), σ(min) are the variabilities at the respective min and max of the Morse-Smale complex. We can write:

B∞ (region) Tmax σ(max) - Tmin σ(min) -2
= t(max) -t(min) -2 = p t(region) -2 (6.12) where we recognize the persistence p t(region) of the corresponding region of the whitened template t. This links the estimation of the asymptotic bias to the persistence of the whitened template's Morse-Smale complex. This shows how a topological property of the image in fact represents a statistical property of this image as the estimate of the brain template.

Hierarchy of the whitened template The persistence of the whitened template quanties locally the asymptotic bias, i.e. how far the brain template is from the unique brain anatomy of the generative model. Is there a statistical interpretation of the hierarchies of Morse-Smale complexes, introduced in Section 6.3? Let us consider another Morse-Smale of the whitened template's hierarchy, i.e. a Morse-Smale computed at a given persistence threshold p threshold . There is an asymptotic bias threshold that corresponds, which we can write: p -1/2 threshold . The regions kept in the new Morse-Smale are those having a persistence higher than the persistence threshold p threshold , i.e. those having an asymptotic bias lower than the asymptotic bias threshold p -1/2 threshold . Therefore, if we can impose the topology of the brain template to match the new Morse-Smale of threshold p -1/2 threshold , we control its asymptotic bias. This means that we preserve only the min-max pairs shown on the Morse-Smale graph chosen. It eliminates the min-max pairs that have been created by chance, because the noise on the images was at a similar level than the intensity signal on these regions. The next subsection explains how to impose the topology of a given Morse-Smale on the template's image.

Controlling the template's asymptotic bias by constraining its topology

We are given the template's image and we want to force its asymptotic bias to be below a threshold, so that it is closer to estimating the anatomy of the database, i.e. the anatomy shared by the subject brains. The development above suggests to compute the Morse-Smale complex with a persistence threshold corresponding to the desired bias threshold. Then, enforcing template's topology to match the Morse-Smale complex will control its asymptotic bias. This enforcement procedure is called "Topological denoising".

Chapter 6. Geometric Statistics on topologies: controlling the bias in brain template estimation. 126 6.4.2.1 Topological denoising Topological denoising is a procedure for smoothing an image, like our template image, while preserving topological features [Jacobson 2012[START_REF] Günther | [END_REF]]. The input of the procedure is the intensity function dening the template T : Ω → R and a MS complex with intensity values at its nodes. Enforcing the template's topology to match the MS complex means that we compute T : Ω → R which is a smoothed version of original template estimate T containing only the intensity min-max pairs specied by the MS complex chosen. T should be otherwise as close as possible to the original template estimate T in terms of intensity. The values and positions of the MS extrema are preserved, while all other extrema are removed from the brain template estimate. Such procedure provides control over the topology of the brain image T . Formally, the original Topological denoising problem is written as the minimization [Jacobson 2012]:

argmin T x i ∈Ω || T (x i ) -T (x i )|| 2 + Ω ||∆T || 2 s.t. T (x i ) = T (x i
) for x i a node of the MS complex T (x j ) > T (x i ) for x j neighbor of x i and x i minimum T (x j ) < T (x i ) for x j neighbor of x i and x i maximum T (x i ) > min neighbor x j T (x j ) for x i not an extremum T (x i ) < max neighbor x j T (x j ) for x i not an extremum

The non linear inequality constraints make this optimization problem hard to solve.

The solution suggested by [Jacobson 2012] is to compute a representative function u that veries the last four inequality constraints. Given this function u, the topological denoising problem becomes:

argmin T x i ∈Ω || T (x i ) -T (x i )|| 2 + Ω ||∆T || 2 s.t. T (x i ) = T (x i ) for x i a node of the MS complex (T (x i ) -T (x j ))(u(x i ) -u(x j )) > 0, for (x i , x j ) a pair of neighbors
where the last constraint means that the direction of T shall be aligned with the direction of u. This alternative optimization problem is easily solved [Jacobson 2012].

The representative function u can be computed by solving the Dirichlet problem:

argmin u Ω ||∇u|| 2 s.t. u(x i ) = 0 for x i a minimum u(x i ) = 1 for x i a maximum
Chapter 6. Geometric Statistics on topologies: controlling the bias in brain template estimation.

127 Minimizers of the Dirichlet energy are harmonic functions, and their properties guarantee that x i and x j are minima and maxima and that u contains no other extrema inside the MS regions. We refer to [Jacobson 2012] for further details. Figure 6.10 shows examples of topological denoising. The topology to be enforced is represented by the red and blue dots, which are nodes of the MS complex: red for intensity maxima and blue for intensity minima. On the left example, the circle motifs that were inducing undesirable minima and maxima are removed. On the right example, two of the initial four maxima in the center of the image are removed too. Only the topology dictated by the input Morse-Smale complex is preserved. The original template's computation is performed with the algorithm of [Joshi 2004] and use the LCC Log-daemons for the registrations [START_REF] Lorenzi | [END_REF]]. We adapt it by adding a Topological denoising step, in order to control the template's asymptotic bias.

Algorithm 4 shows the adapted procedure. One initiates with the template being one of the subject images: T1 = I 1 . At each iteration k of the template's computation, one registers the subject images to the current template Tk and performs the average of the registered images' intensities to get a rst version of the updated template Tk+1 . So far, this matches the usual template estimation procedure. Our adaptation is what follows. The MS complex of the updated template Tk+1 is com- puted, using the R package msr [Gerber 2012]. Then, the updated template Tk+1 is smoothed using Topological denoising, see Figure 6.12. These steps are iterated until convergence.

The main parameter controlling this adapted procedure is the asymptotic bias threshold, i.e. the persistence threshold p threshold for the MS complex computation. The next section discusses the choice of this parameter p threshold . Varying the threshold p threshold leads to the construction of a hierarchy of templates. The other parameter is σ, which is the noise on the subject images. Either one knows it from Chapter 6. Geometric Statistics on topologies: controlling the bias in brain template estimation.

Repeat:

Non-linearly register the images to Tk , i.e. compute φ i k :

J i k I i • φ i k
Compute the mean deformation: φk Register subject image:

L i k = I i • φ i k • φ-1 k
Compute the mean intensity image for template iteration:

Tk+1 = 1 n n i=1 L i k
Compute the MS complex of Tk+1 at persistence level p Topological denoising of T k+1 using the MS complex k ← k + 1

until convergence: || Tk -Tk+1 || < ε Output: Tk the experimental design, or one estimates it with the variability of the registered subject images, as we did in Section 6.3.

Experimental results

This section presents experimental results on the quantication of the template's asymptotic bias and the adapted algorithm that bounds this bias. We use the Open Access Series of Imaging Studies (OASIS) database consisting of 136 T1 weighted MR images of brains [START_REF] Marcus | [END_REF]].

Quantication of the template inconsistency

We quantify the asymptotic bias locally on the brain template computed from the OASIS database with the usual procedure. This shows how faithfully the computed template represents human brain anatomy for the neuroimaging studies. First, we produce maps showing the local asymptotic bias directly with a color code superimposed on the original tridimensional template image, see Figure 6.11.

We call these maps the asymptotic bias maps. A green color indicates a low asymptotic bias for the region and a red color indicates a high asymptotic bias on the region.

The scale for the color code corresponds to a logarithmic scale, and more precisely to SNR dB , where:

SNR dB = 10 log 10 d σ 2 (6.13)
The scale is thus in dB, as the decibel is the logarithmic unit that expresses the ratio of two values of a physical quantity, which is the squared intensity in our case. This unit emphasizes that the quantication of the asymptotic bias depends on a Chapter 6. Geometric Statistics on topologies: controlling the bias in brain template estimation.

129 signal-noise ratio (SNR). Indeed, one can consider that the signal is d, which is the template's intensities representing the brain anatomies and the "noise" is σ, the intersubject variability after registration. The larger is the SNR, the lower is the asymptotic bias on the brain template. We compute several maps, see (c)-( d)-(e) on Figure 6.11 for the same brain template. The dierence between the maps is the Morse-Smale complexes's persistence threshold used to compute the asymptotic bias. The threshold is increased from left to right on Figure 6.11 (c)-(d)-(e). Increasing the threshold makes more and more regions appear and these are more and more biased: they become colored in orange-red.

The asymptotic bias maps have the following interpretation with respect to neuroimaging. The maps show regions, in orange-red, where the template's brain structures are small with respect to the subjects' variability in the database. In these orange-red regions, it is not reasonable to have a sharply dened template, because the structures may have appeared by chance, by registration of noise between the dierent subjects. In other words, the maps reveal brain regions where the assumption of a unique anatomy in the subject population may break down. sents the asymptotic bias of the brain template we would obtain if we were constraining the image to the topology of the corresponding Morse-Smale. The persistence threshold gives a way to investigate the trade-o between asymptotic unbiasedness and sharpness of the template. On the one hand, a complex topology -i.e. a low persistence threshold -implies an important asymptotic bias on the template, which may not represent faithfully the brain anatomy shared by the subjects in the OASIS database. On the other hand, a topology that is too simple -i.e. a high persistence threshold -has no chance of representing a brain anatomy at all. If we want to look at small brain structures, we have to allow for some precision in the topology. Therefore, which topology shall we choose in this trade-o of asymptotic unbiasedness versus sharpness? If the local intensity of the computed template is below the noise, there is no hope to compute a consistent template. As in the 1D example of [START_REF] Allassonnière | Towards a Coherent Statistical Framework for Dense Deformable Template Estimation[END_REF]], if the noise is of the same order of magnitude as the signal, the template may estimate the noise instead of the signal. Thus it makes sense to choose an inconsistency threshold between -1 and 0 dB, that expresses the limit situation where signal (intensity on the brain image) and noise are of the same order of magnitude.

Applying topological denoising to control the brain template's bias We apply the methodology of Section 6.4.2 to enforce the asymptotic bias to be below a threshold, using Topological denoising. Enforcing the unbiasedness in the procedure enables us to build the template of Figure 6.12. As a proof of concept, we have run it on the subject coronal slices of the OASIS database. Following the development above, we bound the asymptotic bias by setting the SNR threshold to -0.8 dB. We observe that the brain regions that were the more biased -i.e. in orange-red in Figure 6.11 are now blurred. Thus, Topological denoising decides where the sharply dened brain template makes sense as a representative of the shared brain anatomy, and blurs it where it does not.

One could be interested in a template, that would be sharp and unbiased. In this case, one could consider dropping the assumption of a unique anatomy and consider multiple templates, i.e. use a mixture model. Further work is needed to investigate the construction of a stratied template, which would add a new stratication every time a region's asymptotic bias crosses the threshold B ∞ ∼ 1dB.

Conclusion and perspectives

Computations of templates have been used in the medical imaging literature for at least 15 years. This paper investigates such computations as the estimations of a unique anatomy shared by the population. We have presented a topological method to quantify the asymptotic bias of the template. This is, to our knowledge, the rst attempt to assess the bias of such procedures. Our methodology builds a bridge between the dieomorphic registration framework of Medical Imaging and Morse-Smale theory. This link is an interesting application of Topology in itself. There are some limitations from the technical point of view as we provide sketches of proves for our conjectures. As such, this paper opens the door to mathematical developments at the boundary of Dierential Geometry and Topology.

Our Morse-Smale framework identies biased regions in the brain template in Section 6.4. In these regions, a sharp template might not be desirable. We control the template's asymptotic bias by adding a Topological Denoising step in its iterative computation, creating a trade-o between sharpness and unbiasedness. Our methodology is illustrated on a real database of 136 brain images in Section 6.5. It shows how the Topological Denoising blurrs the regions that were the most biased. We control the template's bias at the price of dropping its sharpness.

It would be very interesting to be able to keep both the unbiasedness and the sharpness of the brain template. In fact, the template being biased can be seen as an indication that the assumption of a unique anatomy within the population should be relaxed. One could think about estimating a mixture of several templates or stratied templates. Each of the templates would represent only a subgroup of the brain population. This subgroup would have a lower variability. Therefore, the parameter σ will be decreasing and the bias too. This will allow for templates that are sharper and still unbiased.

Chapter 7

Geometric Statistics on manifolds with Lie group actions: geometric correction of the bias induced by registration of anatomical shapes Chapter 5 uses tools of Geometric Statistics on quotient spaces to analyze the algorithm of template shape computation. Chapter 6 shows the impact of the results on biased template in neuroimaging. In this Chapter, we generalize the use of Geometric Statistics to the analysis of other algorithms on shape data. More precisely, we focus on statistical analyses of shapes that use registration as a pre-processing step. We show that a bias also occurs in these analyses. We oer a theoretical geometric method to correct it, by enforcing a modication of the quotient space geometry.

In order for this Chapter to be self-contained, Section 7.2 is very similar to the rst section of Chapter 5. Thus Section 7.2 can be skipped in the reading of this thesis..

Introduction

Consider a dataset of n observations X i , i = 1, .., n in a nite dimensional manifold M . For example, each X i can represent a 3D conguration of an object, described by k landmarks on its surface: X i ∈ (R 3 ) k . Unsupervised learning seeks to discover the underlying structure of the X i 's, and to answer questions as: what is the probability distribution function of the 3D congurations of these objects? Both clustering or submanifold learning in M are techniques unveiling properties of such a probabilistic structure.

This paper considers a related problem. What if we are not interested in the 3D congurations of the objects, but rather in the shapes of these congurations?

In this case, the goal is to discover the probabilistic structure of some equivalence classes of the data: [X i ]'s. The shape of a set of landmarks, of a curve or of a surface can be dened as follows: the remainder after we have ltered out the position, the orientation [Kendall 1984] and the parameterization. Objects -landmarks, curves, surfaces, etc.-are the observations X i 's in a given manifold M . Their shapes are their equivalence classes [X i ]'s under a group of transformations, like rotations, translations, scalings, etc., or a group of reparameterizations. If one wants to do Chapter 7. Geometric Statistics on manifolds with Lie group actions: geometric correction of the bias induced by registration of anatomical shapes. 133 statistics on shapes, one does statistics on equivalence classes. We consider the specic case where the equivalence classes come from the isometric action of a Lie group on the manifold M .

Applications show various situations where one needs to perform statistics on equivalence classes [X i ]'s (the shapes) while observing only X i 's (the objects). Paleontologists consider human skulls described by the 3D coordinates of landmarks detected on the skull's surface. They seek to correlate the skull shapes with biogeographic data [Zalawadia 2010]. Biologists observe proteins, also described by the 3D coordinates of their atomes. They study how their shapes are related to their function. Statistics on misfolded proteins are used to understand diseases, like Parkinson's disease [Li 2008]. Orthopaedic surgeons analyze bones' shapes for surgical pre-planning [START_REF] Darmanté | Analyse biométrique de lánneau pelvien en 3 dimensions â à propos de 100 scanners[END_REF]].

The intuitive procedure to perform statistics on the equivalence classes (the shapes) goes as follows. First, one takes the equivalence class of each observation: X i → [X i ], i = 1, .., n. One goes from the object's parameterization to its shape's parameterization X i → [X i ] by "eliminating" the position, orientation or parameterization of the object. In other words, one projects the observations X i ∈ M in a quotient shape space Q:

[X i ] ∈ Q. We call π this projection from M to Q: X i π → [X i ].
In the literature, M is sometimes called the top space. Second, one conducts the statistical analysis on the [X i ]'s, for example subspace learning or clustering.

This two step procedure seems straightforward and is commonly used in the literature [Kendall 1989, Bookstein 1986]. But one may wonder if its rst step, i.e. the projection π from M to the quotient shape space Q: X i π → [X i ], i = 1..n, aects the consistency of the statistical analysis. This is a legitimate question as any inconsistency would make the conclusions of the study less accurate.

Our paper analyzes the procedure's consistency, under a generative model of the X i 's where the data are subject to a centered noise in the space of objects M . The generative model includes: (i) the probability distribution p Q of some shapes Y i 's in Q and (ii) the mechanism generating the objects X i with probability distribution p M in M . An important characteristic of (ii) is the noise model on the objects, which is taken to be in the space M and not in the quotient shape space Q. This means that the measurement error is made on the recording of the objects' congurations in 3D rather than on their shapes, in our previous example.

The goal is to learn the distribution of the shapes p Q . The two step procedure ultimately performs learning on the [X i ]'s, so it tries to unveil π(p M ). We demonstrate that the procedure is biased as a learning of p Q , in the sense that: π(p M ) = p Q .

We give a geometric interpretation. π(p M ) = p Q comes from the fact that the equivalence classes [X i ]'s are submanifolds in M , that have an external curvature. Thus the projection π : M → Q, which represents the registration pre-processing step, is a non-orthogonal projection. All in all: the geometry of the equivalence classes -called orbits in the case of the Lie group action -creates a bias in the estimation of the distribution. We leverage this geometric understanding to suggest a method that corrects for this bias. The correction performs a conformal transfor-Chapter 7. Geometric Statistics on manifolds with Lie group actions: geometric correction of the bias induced by registration of anatomical shapes. 134 mation of the Riemannian metric of M . As a consequence, the equivalence classes [X i ]'s loose their curvature in M and the projection π : M → Q becomes orthogonal.

Related work on the shape space's geometry as a quotient space Statistics on shape spaces have been studied since the 1980's. Kendall rst investigated the geometry of shape spaces of k labeled landmarks in R m [Kendall 1989]. A shape is represented as the equivalence class of the landmarks in (R m ) k under the action of translations, rotations and scalings. [Huckemann 2010] study statistics on general shapes, that are equivalence classes of objects in a manifold M under the action of a Lie group G. This unies the theory for shapes of landmarks, of curves and of surfaces.

Note that there is a distinction between "form" and "shape" is the literature. "Form" relates to the quotient of the object by rotations and translations only. "Shape" denotes the quotient of the object by rotations, translations, and scalings. Kendall shape spaces refer to "shape": the scalings are quotiented by constraining the size of the landmarks' set to be 1.

The models of [Kendall 1989] include the variability of the shapes, as opposed to the noise on the objects, which is a pervasive eect of imperfect measuring instruments. They do not consider that the data are observed in the space of landmarks (R m ) k i.e. in the ambient manifold M and projected in the shape space Σ k m i.e. the quotient space Q. The question of the bias of the procedure is not raised.

Procrustean analysis is another method for performing statistics on landmarks shape spaces [Dryden 1998]. Kendall analyses perform an explicit projection of the data in the shape space: they explicitly compute their coordinates in Σ k m . In contrast, Procrustean analyses have an implicit approach. They keep the coordinates in (R m ) k and stay in the ambient space: the projection of the data in the shape space is algorithmic and performed by "aligning" or "registering" them. For example, orthogonal Procrustes analysis "aligns" the sets of landmarks by rotating each set to minimize the Euclidean distance to the other sets. Procrustean analysis considers the fact that the data are observed in the space (R m ) k but does not consider the geometry of the shape space.

The issue of estimating the mean shape, which is the simplest form of learning on the shape space, has been studied in the literature. Computing the mean shape is one method for summarizing shape data. The mean shape is an average of equivalence classes of the data: one computes the mean after projection of the data in the shape space. Recent studies show that the statistical properties of the mean shape estimation are modied when one takes into account this noise [Miolane 2015a]. Consistency of the mean shape estimation has been discussed with and without the noise assumption. The mean shape was shown consistent for shapes of landmarks in 2D and 3D in [Lele 1993[START_REF] Le | On the Consistency of Procrustean Mean Shapes[END_REF]] and for shapes of signals in [Kurtek 2011], both under the simplifying assumption of no noise on the objects. [START_REF] Allassonnière | Towards a Coherent Statistical Framework for Dense Deformable Template Estimation[END_REF]] provided examples of inconsistency of the mean shape for noise on the objects, in examples with discretized signals and 2D images [Allassonnière 2015b]. [Bigot 2011] Chapter 7. Geometric Statistics on manifolds with Lie group actions: geometric correction of the bias induced by registration of anatomical shapes. 135 computed lower and upper bounds on the inconsistency of the mean shape for discretized curves in 2D. But this is restricted to the mean shape estimation and no geometric intuition nor correction strategy was given.

Recently, the inconsistency of the mean shape has been studied in the abstract setting of Lie group actions [Miolane 2017]. The authors provide a geometric interpretation of the bias of this procedure. Their result applies to the literature of shape statistics and to signals, curves and surfaces. For example, formulated in the Procrustean terminology it gives: the Generalized Procrustes Analysis (GPA) estimator of mean "form" is asymptotically biased. [Miolane 2017] identify the geometric variable that controls it and propose a correction. However, it corrects for the bias in the mean shape estimate and in the whole probability distribution function of shapes p Q .

Contributions and outline

We make three contributions. First, we show that the projection π : M → Q dened by X i π → [X i ], i = 1..n,, and representing the registration pre-processing step, biases the statistical analysis of shapes. This is done by computing π(p M ) and showing that π(p M ) = p Q . We oer an interpretation of the bias through the geometry of the shape space. Second, we leverage our understanding to suggest a geometric correction. These results are shown in the following mathematical framework: we assume a proper, eective and isometric action of a nite dimensional Lie group Gdening the registration process -on a nite dimensional Riemannian manifold M .

The paper has three Sections. Section 1 introduces the geometric framework of quotient spaces and details the geometry behind the projection π : X i π → [X i ], i = 1..n,. Section 2 shows that a statistical analysis of shapes starting with a registration step is biased and gives a geometric interpretation: the bias comes from the fact that the projection π is non-orthogonal. Section 3 describes the procedure to correct the bias: it amounts at changing the metric of the space M in order to force the projection π to be orthogonal.

Geometrization of unsupervised learning algorithms

on shape data

Two running examples

We introduce two simple examples of shape spaces as in [Miolane 2017]. We will refer to them constantly to provide intuition. First, we consider two landmarks in the plane R 2 as in Figure 7.1 (a). The landmarks are parameterized each with 2 coordinates. For simplicity we consider that one landmark is xed at the origin on R 2 . Thus the landmarks are now parameterized by the 2 coordinates of the second landmark only, e.g. in polar coordinates (r, θ). We are interested in the shape of the 2 landmarks, i.e. in their distance which is simply r.

Chapter 7. Geometric Statistics on manifolds with Lie group actions: geometric correction of the bias induced by registration of anatomical shapes. 136 Second, we consider two landmarks on the sphere S 2 as in Figure 7.1 (b). One of the landmark is xed at the origin of S 2 . The landmarks are now parameterized by the 2 coordinates of the second landmark only, i.e. (θ.φ). The shape of the two landmarks is the angle between them and is simply θ.

X = (r, θ) X = (θ, φ) (a

Elements of Dierential Geometry of shapes

The framework of Dierential Geometry of shapes that serves this study has been introduced in [Miolane 2017]. We only summarize the most important aspects here. The data are objects X i , i = 1...n, that are either sets of landmarks, curves, images, etc. We consider that each object X i is a point in a nite dimensional Riemannian manifold M , which may be called the top space in other studies [Allassonnière 2015a]. We have M = R 2 in the plane example: a at manifold of dimension 2. We have M = S 2 in the sphere example: a manifold of constant positive curvature and of dimension 2.

Shapes are orbits (equivalence classes) of a Lie group action By denition, the objects' shapes are their equivalence classes [X i ], i = 1...n, under the isometric action of some nite dimensional Lie group G: G is a group of continuous transformations that models what does not change the shape. A Lie group action of

G on M is a dierentiable map ρ : G × M (g, X) → g • X ∈ M, such that g 1 • (g 2 • X) = (g 1 • g 2 ) • X and Id • X = X,
where Id is the identity of the group. The action of G on M will be written with "•".

The equivalence classes [X i ]'s have a precise denition in the framework of Lie group actions. The equivalence class of X i is the orbit of X i under the action of G, i.e. the submanifold O X i of all objects in M that have the same shape as X. In other words: [X i ] = O X i and both notations and appellations may be found in the literature. The curvature of the orbit as a submanifold of M is the key point of the results in Section 7.4.

Chapter 7. Geometric Statistics on manifolds with Lie group actions: geometric correction of the bias induced by registration of anatomical shapes.

137

In our examples, the rotations are the transformations that leave the shape of the systems invariant. Let us take g a rotation. The action of g on the landmark X is illustrated by a blue arrow in Figures 7.2 (a) for the plane and (d) for the sphere. We observe that the action does not change the shape of the systems: the distance between the two landmarks is preserved in (a), the angle between the two landmarks is preserved in (d). The equivalence class of X i is also called its orbit and written O X i . The equivalence class/orbit of X is illustrated with the blue dotted circle in Figure 7.2 (a) for the plane example and in Figure 7.2 (b) for the sphere example.

Shape space is a quotient space The shape space is by denition the space of orbits, i.e. the space of equivalence classes. This is a quotient space denoted Q = M/G. One orbit in M , i.e. one circle in Figure 7.2 (b) or (e), corresponds to a point in Q. The shape space is Q = R + in the plane example. This is the space of all possible distances between the two landmarks, see Figure 7.2 (c). The shape space is Q = [0, π] in the sphere example. This is the space of all possible angles between the two landmarks, see Figure 7.2 (f).

The distance in M induces a quasi-distance d Q in the shape space Q:

d Q (O X 1 , O X 2 ) = inf g∈G d M (g • X 1 , X 2 ).
(7.1)

In the plane example, the distance between two shapes is the dierence in distances between the landmarks. One can compute it by rst aligning the landmarks, say on the rst axis of R 2 . Then, one uses the distance in R 2 .

Isotropy group of an object The isotropy group of X i is the subgroup of transformations of G that leave X i invariant. For the plane example, every X i = (0, 0) has isotropy group the identity and (0, 0) has isotropy group the whole group of 2D rotations. Objects on the same orbit, i.e. objects that have the same shape, have conjugate isotropy groups. The orbit type of an orbit is the corresponding conjugation class of the objects isotropy groups. The manifold M (resp. the quotient space) are stratied into orbit types (resp. projection of the orbit types) [Miolane 2017]: this means that both M and Q can be seen as a collection of submanifolds of various dimensions, called strata, glued together in a certain way.

Geometric statistics

Probability distribution on the shape space and on the object space Random points and random probability distributions on manifolds were dened for instance in [Pennec 2006]. In particular, for manifolds, the generalization of the uniform measure is the Riemannian measure dened on M by dM (X) = det(g AB (X))dX.

In our case, we will integrate over three manifolds: M , Q and O X for dierent orbits. Moreover, for a given orbit O X , we have the intrinsic measure dO int , which does not consider the embedding of O X in M , or the induced or extrinsic measure Chapter 7. Geometric Statistics on manifolds with Lie group actions: geometric correction of the bias induced by registration of anatomical shapes. dO ext which considers the embedding of O X in M as it comes from the pull-back on O X of the Riemannian metric on M . On Q, we have the induced measure. Because the action is isometric, the measures on M , Q and the extrinsic measure on O are related as follows:

dM (X) = det(g M (X))dX = dQ(x).dO ext (X).
where we emphasize that the extrinsic measure is used on O here.

The ambient space M is a manifold, the orbit O X is a submanifold of M. The quotient Q is a stratied manifold. To perform statistics on these spaces, we need reference measures. We use the Riemannian measure on M , the induced measure on O, and the induced measure on Q. These measures enable to dene probability distribution functions on manifolds, and for example the notion of a Riemannian Gaussian on Q, O and M [Pennec 2006].

We have focused on an intuitive introduction of the concepts. We refer to [Postnikov 2001, Alekseevsky 2003[START_REF] S Huckemann | Intrinsic shape analysis: Geodesic principal component analysis for Riemannian manifolds modulo Lie group actions[END_REF] for mathematical details. From now on, the mathematical setting is the following: we assume a proper, eective and isometric action of a nite dimensional Lie group G on a nite dimensional Riemannian manifold M .

Geometrization of generative models of shape data

and learning procedures

Generative models

We recall that the data are the X i , i = 1..n, which are sets of landmarks, curves, images, etc. We interpret the data X i 's as random realizations of the generative model:

X i = Exp(g i • Y i , ε i ) i = 1...n (7.2)
where Exp(p,u) denotes the Riemannian exponential of u at point p. The shape Y i ∈ M/G ⊂ M is given position/parameterization g i ∈ G and observed with noise

ε i ∈ T g i •Y i M . The Y i , g i , ε
i are realizations of random variables. Their individual laws are presented in the next paragraphs. They dene the nature of the generative model and the expression of the probability distribution of the X i 's in M , which we denote p M . Drawing the shape Y i , its position g i and the noise ε i lead to the following three step interpretation of the generative model of Equation 7.2. These steps are also illustrated with shiny applications available online at https://nmiolane. shinyapps.io/shinyPlane and https://nmiolane.shinyapps.io/shinySphere.

Step 1: Generate the shape Y i ∈ M/G We assume that there is an probability density of shapes p Q in Q = M/G, with respect to the measure on Q induced by the Riemannian measure of M . The Y i 's are assumed to be i.i.d. samples drawn from this distribution p Q . For example, p Q can be a Gaussian as illustrated in Figure 7.3 on the shape spaces for the plane and sphere examples. The variability of shapes described by p Q is the variability that is meaningful for the statistical study, whether we are analyzing shapes of skulls, proteins, bones, neural spike trains, handwritten digits or brains. Step 2: Generate its position/parameterization g i ∈ G We cannot observe shapes in Q = M/G. We rather observe objects in M , that are shapes posed or parameterized in a certain way. There are two classes of generative models, depending on the assumption we make on the poses: functional and structural models. This terminology comes from the literature of Error-in-Variables and will be used in Section 4.

In the functional model, the positions on the orbit are parameters which we want to estimate. In this model, the number of parameters increases with the number of data. In the structural model, the positions on the orbits are random variables generated by a probability distribution, which we denote p O . We assume that there is a probability distribution on the positions/parameterizations of G, or equivalently a probability distribution p O on principal orbits with respect to their Chapter 7. Geometric Statistics on manifolds with Lie group actions: geometric correction of the bias induced by registration of anatomical shapes. 140 intrinsic measure. In order to draw a position on the orbit, we draw an element g i from the probability distribution on the Lie group and make g i act on the Y i . We assume that the distribution does not depend on the shape Y i that has been drawn. The g i 's are assumed to be i.i.d. from this distribution. For example, p O can be a Gaussian as illustrated in Figure 7.4 on the shape spaces for the plane and sphere examples. In both cases, the probability distribution function on poses is illustrated with the black curve. (a) the distribution on orbits is a distribution in angles: each point is posed on its orbits through a distribution on φ 1 , ...φ m-1 . (b) the distribution on orbits is a distribution is angles: each point is posed on its orbits through a distribution on φ 1 ...φ m-1 .

Step 3: Generate the noise ε i ∈ T g i •Y i M The observed X i 's are the outcomes of noisy measurements. We assume that there is a probability distribution function on T g i •Y i M representing the noise. We further assume that this is a Gaussian centered at g i • Y i , the origin of the tangent space T g i •Y i M , and with standard deviation σ, see Figures 7.5. This assumes that the noise model is the same at g j • Y j and g i • Y i , i.e. that the noise is isotropic and stationary. The parameter σ representing its standard deviation will be extremely important in the developments of Section, as we will compute Taylor expansions around σ = 0, i.e. for low measurement error. All in all, these three steps produce observations X i with a given probability distribution p M that depends on the laws of Y , g and ε.

Learning the distribution of shapes

Our goal is to unveil the structure of shape data Y i 's, from the observations of noisy objects X i 's. Geometrically, the goal is to learn the underlying structure of the shapes Y i , represented by the probability distribution p Q in the quotient space Q, from the noisy observations X i drawn from p M on the manifold. We assume the structural model for the distribution of the g i 's. We do not introduce a new procedure to learn p Q , but we rather analyze the intuitive procedure usually performed Chapter 7. Geometric Statistics on manifolds with Lie group actions: geometric correction of the bias induced by registration of anatomical shapes. in the literature, which usually takes two steps [Kurtek 2011][Kendall 1989]: one registration step (like a pre-processing step) and the learning step.

First step: Registration step In a rst step, one registers (i.e. aligns, i.e.

puts in optimal position) the objects. By doing so, one lters out the position/parameterization component, i.e. one projects the objects in the shape space Q: X i π → [X i ], i = 1..n,. In practice, the projection is performed either explicitly, or implicitly by solving an optimization problem. The explicit approach computes the "shape coordinates" of each object. This is easily done for shapes like landmarks shapes, using Kendall or Bookstein coordinates. The implicit approach registers the objects. One takes representatives of each object that are aligned in a certain sense: two objects X 1 and X 2 are aligned if the X 1 is in optimal position with respect to X 2 , i.e. we have chosen an optimal g to act on X 1 so that it is as close as possible to X 2 . These representatives are objects and still belong to the manifold M . However, they belong to a subspace of M that is isomorphic to the shape space Q. As a consequence, on can perform statistics on them as if we were in the shape space. This step is illustrated on Figure 7.6.

Second step: Learning step In a second step, one proceeds with unsupervised learning techniques, for example subspace learning and clustering on the [X i ]'s. The standard situation would have been to perform the learning on the Y i 's. Here, one uses the [X i ]'s as proxies.

We could consider the mean shape estimation which is the simplest case of subspace learning, as it is a 0-dimensional subspace learning. We could also consider K-means on shapes which is a simple case of clustering algorithm.

Chapter 7. Geometric Statistics on manifolds with Lie group actions: geometric correction of the bias induced by registration of anatomical shapes.

(a) (b) X i ĝi • X i X i ĝi • X i Ŷ Ŷ (i) (ii) (i) (ii)
Figure 7.6: The 2 steps of Unsupervised learning on shapes. First, the observations are registered: the data follow their orbits until they are aligned. Then, unsupervised learning as for example the mean shape computation or K-means on shapes, is performed on the projected objects.

Bias on the usual procedure

This section shows that registration, as a pre-processing step for statistical shape analysis, biases the studies when the data are generated with the model of Equation 7.2. We recall that the registration is geometrically interpreted as the projection π of the data in the quotient shape space: π : X

i π → [X i ], i = 1..n,.
We thus show that π(p M ) = p Q : in other words, the statistical analysis is not even performed on the probability distribution function that we want to learn. In this sense, we say that unsupervised learning on shapes are biased. We show this result for a Riemannian Gaussian noise of variance σ 2 in M , truncated at 3σ.

Denitions of curvature

The geometric origin of this bias is the curvature of the orbits. Curvature can be dened in various ways, both intrinsic and extrinsic. We recall them here as they appear in the developments later. First, the extrinsic denitions describe the curvature of an orbit O, as embedded in a manifold M . The second fundamental form

h of a submanifold O of M is dened on T X O × T X O by h(v, w) = (∇ v w) ⊥ ∈ N X O,
where (∇ v w) ⊥ denotes the orthogonal projection of covariant derivative ∇ v w onto the normal bundle. The second fundamental form h is a (2, 1)-tensor which can be considered as a matrix in 3D. Thus, its trace will be a vector. The mean curvature vector H of O is dened as: H = T r(h).

Elementary bias induced by the registration step

We rst consider that the distribution of shapes that we want to learn, p Q , is a Dirac: p Q = δ Y . This means that the data X i 's are all generated from the same Chapter 7. Geometric Statistics on manifolds with Lie group actions: geometric correction of the bias induced by registration of anatomical shapes. Theorem 7.4.1 (Miolane et al. (2016) [Miolane 2017]) The data X i 's are generated in the nite-dimensional Riemannian manifold M following the model: in Equation (7.2) andFigures 7.3-7.5. In this model: (i) the action of the nite dimensional Lie group G on M , denoted •, is isometric, (ii) the parameter Y is the template shape in the shape space Q, (iii) ε i is the noise and follows a (generalization to manifolds of a) Gaussian of variance σ 2 , see Section 6.2.

X i = Exp(g i • Y, ε i ), i = 1...n, described
Then, the probability distribution function f on the shapes of the X i 's, i = 1...n, in the asymptotic regime on an innite number of data n → +∞, has the following Taylor expansion around the noise level σ = 0:

π(p M )(Z) = b(Z) = 1 ( √ 2πσ) q exp - d 2 M (Y, Z) 2σ 2 F 0 (Z) + σ 2 F 2 (Z) + O(σ 4 ) + ε(σ)
where (i) Z denotes a point in the shape space Q, (ii) F 0 and F 2 are functions of Z involving the derivatives of the Riemannian tensor at Z and the derivatives of the graph G describing the orbit O Z at Z, and (iii) ε is a function of σ that decreases exponentially for σ → 0.

We call π(p M ) = b the elementary bias, because it corresponds to the elementary eect of the registration step for the further statistical analysis. It shows that the isotropic Gaussian noise on the objects is integrated into a skewed distribution on the quotient shape space Q, because of the registration step. In other words, this elementary bias depends on the noise model. The exponential in the expression of π(p M ) in Theorem 7.4.1 belongs to a Gaussian distribution centered at Y . However the whole distribution π(p M ) diers from the Gaussian because of the Z-dependent term in the right parenthesis. This dependency in Z comes from the external curvature h a ij of the orbit of Y . Therefore, the curvature of the orbits induces a skew of the distribution π(p M ), as observed for the examples in Figure 7.7 taken from [Miolane 2017].

This skew biases the further statistical analysis. In particular, the usual procedure for estimating Y is to compute the mean shape, i.e. the expectation of π(p M ). As π(p M ) is skewed, the estimator is dierent from Y , the unique shape that it is designed to estimate [Miolane 2017]. This is shown on Figure 7.7: the shape Y is shown in green and the estimator is shown in red. We emphasize that the bias is asymptotic, as we are already in the limit of a sample size equal to innity.

Bias induced by the registration step

The previous subsection shows the bias that is induced by the registration step, when the distribution of shapes p Q is reduced to a unique shape Y . Here generalize Chapter 7. Geometric Statistics on manifolds with Lie group actions: geometric correction of the bias induced by registration of anatomical shapes. this result to any distribution of shapes p Q . Intuitively, a general distribution of shapes p Q can be seen as the continuous sum of Diracs. The elementary bias of the previous subsection thus impacts each of these Diracs. The induced distribution π(p M ), on which the statistical learning, is like a convolution of the elementary bias and the original distribution of shapes p Q .

Theorem 7.4.2 The projection π from the manifold M to the quotient shape space:

X i π → [X i ], i = 1.
.n, implies that the statistical analysis is performed on the distribution π(F ):

π(p M )(x) = E p Q [b(x)] (7.3)
where p Q is the distribution we want to learn and b is the bias function coming from Theorem 7.4.1. We could see this as a generalization of the convolution π

(p M )(x) = p Q * b(x).
The projection X i π → [X i ], i = 1..n, transforms the probability distribution p Q we seek to learn into p Q * b. This transformation is skewed, which induces a bias in the statistical moments of p Q * b, with respect to the statistical moments of p Q that one could be interested in. From the very start, the learning procedure tries to learn a biased structure. In this sense, the registration step through the projection π : M → Q biases the following statistical analysis or learning algorithm.

The systematic bias could impact the conclusions from learning algorithms on shapes. Correcting the bias may be preferable in order to ensure accurate conclusions from a statistical analysis or to unveil the expected structure in the shape data.

Correcting the usual procedure

We have highlighted the variables responsible for the bias: this is the external curvature of the orbits at the scale of the noise. This is induces an elementary bias function that is skewed: the registration step registers the Gaussian isotropic noise into a skewed noise on the quotient shape space. If we can correct this skew, we can Chapter 7. Geometric Statistics on manifolds with Lie group actions: geometric correction of the bias induced by registration of anatomical shapes. 145 correct the elementary bias and therefore unbias the further statistical analysis on shapes.

The intuition we have gained from the study enables us to suggest a geometric correction of the systematic bias. We perform a transformation of the orbits, through a conformal change of the Riemannian metric of M . The orbits loose their external curvature. The projection π : M → Q, corresponding to the registration step, becomes orthogonal and the skew disappears on the induced distribution π(p M ). We exhibit our geometric correction in two avors in Theorems 7.5.2 and Theorem 7.5.3, equivalently oering to transform the metric of the ambient space M in Theorem 7.5.2 or the distribution p M of the data X i 's in Theorem 7.5.3.

Correction of the bias by changing the metric M

We consider g M the Riemannian metric on M and φ a smooth real-valued function on M . Then:

g M = φ • g M = e 2ϕ g M = e 2ϕ g M (7.4)
is also a Riemannian metric on M . We say that g M is conformal to g M . We show that there exists a conformal change of metric, i.e. a function φ, that can make the orbits loose their external curvature. The notation • is purely schematic and does not represent the action of a Lie group. We conjecture the following lemma.

Lemma 7.5.1 There exists a smooth G-invariant φ : M → R that makes M a product space almost everywhere. Equivalently, by taking one point X ref on a principal orbit:

∀X ∈ M * , e 2φ(X) g O,ext (X) = g O,ext (X ref ) and φ(g • X) = φ(X) (G-invariance) (7.5)
where g O,ext is the extrinsic Riemannian metric on the orbit, i.e. induced by the metric of M .

This can be seen as follows. As the principal orbit type is dense, there are principal orbits almost everywhere. Thus we remove the singularities of the space. These principal orbits are dieomorphic, from the orbit-stabilizer theorem. The function φ takes a principal orbit, sets it as a reference, and then maps all principal orbits to it. Note that φ is not smooth at the singularities, thus of the set of measure 0: M -M * . Now we rely on this conformal transformation of the metric to modify the usual statistical analysis of shapes, by adding two geometric corrections.

Theorem 7.5.2 (Correction of the systematic bias -Metric version) Two successive transformations of the space correct the systematic bias in the procedure of statistical analysis on shapes:

(1) g M (X) → g M (X) = e 2φ(X) g M (X) (before the registration step) (2) g Q (X) → g Q (X) = e -2φ(X) g Q (X) (after the registration step) Chapter 7. Geometric Statistics on manifolds with Lie group actions: geometric correction of the bias induced by registration of anatomical shapes. 146 where φ of the conformal change of metric is given by Lemma 7.5.1. When taking the elementary shape distribution p Q , i.e. the Dirac at Y describing a unique shape in Q, the probability distribution of shapes π(p M ) in Q writes:

π(p M ) = 1 M 0 (Y ) exp - 1 2 d Q (Y, X) 2
Proof The proof is given in Appendix B.

The distribution π(p M ), which is the corrected version of the elementary bias of the previous section, is not skewed. As a consequence, its expectation gives a unbiased estimator of the unique anatomy Y in the case of a Dirac. For a general distribution of shapes p Q , it does not induce a global shift of the projected distribution π(p M ). In this sense, the geometric transformation has corrected the bias. We also note that this correction only focuses on adapting the registration step and does not depend on the statistical analysis that one wants to perform on the shapes. In particular, it corrects the bias for the mean shape estimate and K-means on shapes. This change of metric is illustrated in Figure 7.8 for the two running examples. More precisely, the change of metric amounts to replace the extrinsic metric on the orbit part by the metric of one of the orbit, which plays the role of an intrinsic metric. It puts the external curvatures of all orbits to 0 and send the singularities of Q to the innity, so Q is now a manifold. Moreover, π is now a Riemannian submersion from the whole M . The change of metric makes M * a principal ber bundle over Q * with ber G/G min , where G min is the minimal isotropy group.

Correction of the bias by transforming the distribution of the data

Now we show another method for correcting for the bias, by transforming the distribution of the data X i 's or, more precisely, by redistributing the X i 's. We consider p M the distribution of the data X i 's on M , as given by the generative model, and φ a smooth real-valued function on M . Then the following is also a distribution on Chapter 7. Geometric Statistics on manifolds with Lie group actions: geometric correction of the bias induced by registration of anatomical shapes.

M : p M (Z) = φ • p M (Z) = e mφ(Z) C M (σ) exp - d 2 M (Z, Y ) 2σ 2 (7.6)
where m is the dimension of M and the new geodesic distance d M is computed as:

d M (Z, Y ) 2 = g AB (γ(t)) dγ(t) A dt dγ(t) B dt dt = e 2φ(γ(t))) g AB (γ(t)) dγ(t) A dt dγ(t) B dt dt.
This amounts to changing the distances between data in the Riemannian Gaussian, and to weight by a factor which would correspond to the change in measure. Again, the notation • is purely schematic and does not represent the action of a Lie group. Taking the function φ of Lemma 7.5.1, we rely on this redistribution of the data to modify the usual statistical analysis of shapes as follows.

Theorem 7.5.3 (Correction of the systematic bias -Probability distribution version) Two successive transformations of the probability distribution function correct the systematic bias:

(1) p M (X) → p M (X) = φ * p M (before registration) (2) f (X) → f (X) = (-φ) * f (after registration)
where φ is given by Lemma 7.5.1. As a result, the probability distribution of shapes in Q, with the initial induced metric, writes:

f (X) = 1 M 0 (Y ) exp - 1 2 d Q (Y, X)
Proof Like for Theorem 7.5.2, the proof is given in Appendix B.

This transformation of the probability distribution is also illustrated in Figure 7.8 for the two running examples. It amounts to re-balance the distribution of objects with respect to the orbit they belong to, by applying a transformation depending on φ to each data, before and after registration. It corrects for the unbalanced distribution we had in Figure 6.7.

The two corrections are equivalent and boil down to computing φ

We consider again the probability distribution function p M of the data X i 's:

p M (Z) = 1 C M (σ) exp - d 2 M (Z, Y ) 2σ 2 (7.7)
The distribution p M depends on the Riemannian metric g M of M , through the exponential with the Riemannian geodesic distance d M . As a consequence, varying the Riemannian metric g M gives rise to a family of distributions on M .

Chapter 7. Geometric Statistics on manifolds with Lie group actions: geometric correction of the bias induced by registration of anatomical shapes.

148

We consider the family of distributions associated to conformal changes of metric g M . Formally, taking a conformal change g M (X) = e 2φ(X) g M (X) where φ is a smooth function φ : M → R, the corresponding probability density transformation writes:

p M (Z) ≡ φ * p M (Z) = e mφ(Z) C M (σ) exp - d 2 M (Z, Y ) 2σ 2 (7.8)
where m is the dimension of M and the new geodesic distance d M is as in the previous section. This shows the equivalence between our two corrections, i.e. between the geometry and the data distribution: the probability distribution p M on the manifold with metric g M is equivalent to the probability distribution p M on the new ambient manifold with metric g M . This is not surprising given the interpretation provided by Theoretical Physics. In General Relativity, the geometry of space-time is equivalent to the distribution of masses in space-time. Likewise in the correction we suggest, we may transform the metric of the ambient space or equivalently, transform the probability distribution function of the data.

All in all, the correction boils down to the functional equations 7.5 in φ. If we have expressions of the metric g M , we can perform the correction analytically. However, in real world examples, we may not have these expressions. This correction needs to be performed in a non-parametric way. In this case, our "correction" is closely related to metric learning methods, known to "enhance" the algorithms.

Here, we give it a geometric foundation to seek such a new metric on the quotient space.

Conclusion

We have investigated the consequence of the projection step in statistical analysis of shapes or unsupervised learning on shapes. We have shown that the curved geometry of the orbits -the equivalence classes -implies that the unsupervised learning tries to learn a biased probability distribution function. We have computed the induced probability distribution on the shape space and show how it diers from the probability distribution one tries to learn. We have leveraged this geometric understanding to propose a correction of the bias. By changing the metric on the shape space, we atten the orbits and thus eliminate the source of bias.

This geometric correction can be interestingly compared with the bootstrap corrections proposed in Chapter 5. The geometric correction relies on the knowledge of the space's metric. The bootstrap correction relies on the assumption that the generative model is not too far from reality. Therefore, one should use the geometric correction when one is condent about the geometric or metric modeling of the problem and, in contrast, use the bootstrap correction when one is condent about the statistical modeling of the problem.

Chapter 8 Geometric Statistics on ber bundles: processing of 3D medical images

In the previous Chapters, we have dened and used Geometric Statistics to analyze organ shapes. Now we show how the Geometric Statistics can be applied to medical imaging processing as well. This chapter has been published in the Proceedings of the workshop "Medical Computer Vision" of the MICCAI conference, under the title: "A Survey of Mathematical Structures for Extending 2D Neurogeometry to 3D Image Processing".

Introduction

Machine learning algorithms using big data are often "black-boxes". Thus, they can be hard to interpret. There is still a need of constructive models, so that the big data framework can be fed by new structures. The visual cortex oers inspiration for new methods in (medical) computer vision. From the biological model of human vision, one builds a geometric model of the visual cortex. The geometric model is in turn implemented for computer vision purposes. This is the eld of (2D)-Neurogeometry.

Biological Intuition behind Neurogeometry

The geometric model of the visual cortex's is built as follows. From the biological point of view, neurons of the primary visual cortex V 1 are local detectors called "point processors" [Koenderink 1987]. They are retinotopically connected to small domains of the retina, called their "receptive eld" [Jones 1987]. Mathematically, this structure is an isomorphic map from the 2D retina to the 2D cortical layer. It means that each neuron is associated to a position in our retina (x, y) ∈ R 2 , or equivalently in our visual eld.

Then, the neuron acts as a lter on the optical signal of the retina's photoreceptors. Its transfer function is called its "receptive prole". The so-called "simple neurons" of V 1 have a highly anisotropic prole [Jones 1987]. They are sensitive to the orientation θ ∈ S 1 of the optical signal, in terms of the intensity gradient. A simple neuron is thus represented by the corresponding position (x, y) ∈ R 2 of the retina and by the preferred orientation θ ∈ S 1 of its lter [Petitot 2013].

Chapter 8. Geometric Statistics on ber bundles: processing of 3D medical images. 150 Interestingly, Hubel and Wiesel have shown that neurons detecting all orientations at the same position (x, y) form an anatomical structure, called an "orientation hypercolumn" [Hubel 1977]. This discovery led to the Nobel Prize in 1981. It means that the ber bundle R 2 × S 1 is neurally implemented in the brain.

Ultimately, one models the neuronal activity propagation in R 2 × S 2 . The horizontal cortico-cortical connections of V 1 are represented by a horizontal distribution in sub-Riemannian geometry [Petitot 2013]. The propagation of the cortical activity is then a propagation along sub-Riemannian geodesics [START_REF] Citti | [END_REF]].

Implementations of Neurogeometry for Computer Vision

One nds implementations of 2D-Neurogeometry in computer vision. For example, a sub-Riemannian diusion process leads to algorithms for image completion or in-painting [Boscain 2014]. Fitting a sub-Riemannian geodesic enables contour completion [START_REF] Citti | [END_REF]]. Furthermore, a sub-Riemannian smoothing can smooth the image while preserving crossings [Duits 2010a]. But the framework lacks general applications in medical computer vision, although some exist [Duits 2011]. One reason is that Neurogeometry is essentially 2D, as the retina is 2D. And the generalization of 2D-Neurogeometry to 3D-Neurogeometry is conceptually subtle. There is a need of a theoretical survey summarizing the mathematical structures in the 3D framework. The purpose of this paper is to ll this gap.

Contribution and Outline of the Paper

This paper aims to be a guide for understanding and generalizing 2D-Neurogeometry to 3D-Neurogeometry. It is a theoretical toolbox of 3D-Neurogeometry for: 1) conceiving new algorithms in medical computer vision; and 2) interpreting existing algorithms. In Section 1, we recall briey some concepts of Dierential Geometry. In Section 2, we describe 2D-Neurogeometry and its applications, as an introduction to the 3D case. In Section 3, we describe 3D-Neurogeometry and its possible applications.

Requirements of Dierential Geometry

The following is summarized in Table 8.1 at the end of the section. We assume that the reader is familiar with the following concepts of Dierential Geometry: manifolds, (principal) ber bundles, (pseudo-) Riemannian manifolds [Postnikov 2001], Lie groups, Lie algebra, bi-invariant (pseudo-)metrics [START_REF] Bourbaki | Lie groups and lie algebras: Chapters 1-3. Elements of mathematics[END_REF]][Miolane 2015b], Lie group action on a manifold, homogeneous manifolds [Alekseevsky 2003], sub-Riemannian manifolds [Bellache 1996]. Some are illustrated in Fig. 8.1.

The aforementioned structures are present simultaneously in the computational framework of Neurogeometry. They arise with their set of related curves, as shown in Table 8.1. Depending on the application for image processing, one is interested Chapter 8. Geometric Statistics on ber bundles: processing of 3D medical images. 151 Figure 8.1: Left: S 2 and R 2 are manifolds. Center: tangent bundles of S 2 and R 2 . Right-top: principal bundle with base S 2 and structure group S 1 . Right-bottom: principal bundle with base R 2 and structure group R. In all cases, the bers are drawn in blue.

in computing one curve or another. Thus, on shall understand their dierences and relations. Some curves are illustrated in Fig. 8.2. Figure 8.2: From Left to Right. S 2 and R 2 with action of SO(2). The orbits are in blue and coincide with the curves created by the action of 1-parameter subgroups of SO(2) (as SO( 2) is 1-dimensional). Riemannian geodesics on S 2 and R 2 for standard induced metric from R 3 on S 2 and the Euclidean metric on R 2 .

The Example of 2D-Neurogeometry

This section serves as an introduction of the 3D case. An image processing pipeline using 2D-Neurogeometry usually follows three steps: 1. Lift (L), 2. Processing (P) and 3. Projection (P) (LPP-framework, see Fig. 8.3). These steps can be iterated [START_REF] Citti | [END_REF][START_REF] Sanguinetti | [END_REF] or not [Boscain 2012][Duits 2010a]. Biologically, the lift represents the activation of the neurons in V1. The processing of the lifted image represents the propagation of the neuronal activity in V1. The projection corresponds to our visual interpretation of the information given by the visual cortex after neuronal propagation.

First, we survey the mathematical structures (subsections 2.1, 2.2). We summarize them in Table 8.2. Then we present the LPP-frame of standard algorithms in 2D-Neurogeometry (subsection 2.3). Application-oriented readers can start with subsection 2.3, then go to subsections 2.1, 2.2.

Structures on the Lifted Space

SE(2) = R 2 × SO(2) = R 2 × S 1

Group Actions

The law of SE(2) is, for all (t 1 , R 1 ), (t 2 , R 2 ) ∈ SE(2):

(t 1 , R 1 ) * (t 2 , R 2 ) = (R 1 .t 2 + t 1 , R 1 .R 2 )
The second distinction is the existence of bi-invariant pseudo-metrics g BI in the 3D-case, but not in the 2D-case [Miolane 2015b]. As such, g BI could represent a new powerful tool of 3D-Neurogeometry. We note that in medical computer vision, the bi-invariant pseudo-metric g BI is rarely used as opposed to algorithms in robotics [Zefran 1995]. Considering its bi-invariance property, it would be interesting to consider it for the computations. For example, g BI characterizes the group geodesics of SE(3): this could simplify computations. g BI could replace the use of g R µ as an auxiliary metric, suppressing the need of a choice of µ.

Third Step: Projection (P)

The projection of the lifted image to an image dened on R 3 could be dened in two dierent ways, exactly as in the 2D-case.

Conclusion

This paper is a theoretical toolbox for creating new algorithms in 3D medical computer vision. We have described the mathematical structures arising in the generalization of (2D-)Neurogeometry to 3D images.

Chapter 9

Conclusion

This Ph.D thesis developed Geometric Statistics for Computational Anatomy. Computational anatomy seeks to uncover the normal and pathological variability of the human anatomy, through the statistical analysis of organ shapes in medical images. Organ shapes are complex data that typically live in manifolds with additional geometric structures, like Lie groups or quotient spaces. Geometric Statistics generalizes the theory of statistics to such data spaces. Geometric Statistics is a fusion of Dierential Geometry and Statistics that provides a successful framework for the computational analysis of the human anatomy.

Summary of the contributions

We summarize here the contributions of this Ph.D thesis with respect to the denition and use of Geometric Statistics for Computational Anatomy.

Theoretical developments on Geometric Statistics Geometric Statistics is a mathematical theory in construction. We have rst tackled its denition of the structure of nite dimensional Lie groups. Chapter 4 shows that a consistent theory of statistics, that works for all Lie groups, needs to be dened without (pseudo-)Riemannian metric. This is because a bi-invariant (pseudo-)Riemannian metric fails to exist for some Lie groups. Geometric Statistics on Lie groups may be instead dened with a bi-invariant ane connection, like the Cartan-Schouten connection, that exists on all Lie groups.

We also considered the case where one still wants to try Statistics on (pseudo-)Riemannian manifolds on a given Lie group. We have thus provided an algorithm that constructs a bi-invariant (pseudo-)metric on the group given as input, in case of existence. In other words, our algorithm tells if Riemannian statistics are applicable for the Lie group of the study. If they are, the algorithm outputs a Riemannian metric that respects the Lie group structure, i.e. that is bi-invariant.

The above contributions are examples where the geometry is given -the Lie group structure -but we need to dene the statistics. Our next contribution comes from the converse situation: the statistical principles are given, but we seek to construct the appropriate geometric structures. In this respect, Chapter 8 has set geometric foundations to use the sub-Riemannian Statistics, already applied in 2D image processing, for 3D image processing. The geometric structures involved in the 3D case are indeed signicantly dierent, and more complicated, than the structures involved in the 2D case. In particular, we have seen that a new quotient space Chapter 9. Conclusion 162 appears in the 3D case. We have thus provided a toolbox of Dierential Geometry needed to develop Geometric Statistics for the processing of 3D medical images.

Geometric Statistics for the analysis of statistics on shapes Then, we have shown that Geometric Statistics is an ecient formal setting to analyze statistical properties of algorithms on shape data. We can use Dierential Calculus to derive the statistical properties of procedures employed in Computational Anatomy. Chapter 5 has focused on the algorithm of template shape computation, which has been used for 15 years in the eld. We have evaluated its consistency as an estimator of the unique anatomy, which is a parameter of the generative model. We have shown that the algorithm is asymptotically biased.

Then, we have extended our analysis to algorithms that use registration as a preprocessing step for statistics on shapes. Chapter 7 has applied Geometric Statistics to emphasize that registration corresponds to a non-orthogonal projection in a quotient space. As a consequence, there is bias in such statistical analysis of shapes, just as there was bias in the computation of the template anatomy. Geometric Statistics thus enables to analyze a wide class of algorithms.

Geometric Statistics for the improvement of algorithms on shape data Geometric Statistics facilitates the analysis of estimators on manifolds with additional properties. As presented in Chapter 2, this analysis shall be used to improve the properties of the estimators. Geometric Statistics helps us to understand the geometric origin of pathological statistical properties of some algorithms in Computational Anatomy. As a consequence, we have leveraged this intuition to correct the algorithms.

First, we have proposed methods to correct for the bias in template shape estimation. These methods are inspired by estimation theory on vector spaces, reviewed in Chapter 2. Chapter 5 adapts bootstrap techniques to perform bias correction in the quotient space. The bootstrap methods proposed are very interesting because they are resampling methods and as such, they work on any type of (shape) data: landmarks, curves, surfaces, etc.

Second, we have suggested a theoretical approach to correct for the general bias occurring in every algorithm of statistical shape analysis that uses registration as a pre-processing step. This bias comes from the non-linearity of the projection in the quotient shape space. Our method, presented in Chapter 7 oers to modify the ambient Riemannian metric through a conformal transformation is order to make this projection linear. We deform the data space to overcome the bias induced by the geometry. This correction is therefore very general and applicable to a wide class of algorithms.

Applications to anatomical data Our last contributions focus on the applications of Geometric Statistics to simulated and real datasets in Computational Anatomy. We have validated our results on toy examples of landmarks in R m and the hypersphere S m to illustrate the apparition of the bias in the template shape estimation. Then, we have moved to real datasets and shown the bias on shapes of the optical nerve on monkeys in Chapter 5. We have further illustrated this bias on brain images in Chapter 6, where we have developed a topological method to model the quotient by the action of dieomorphisms. This has enabled us to show where the bias is the most important locally on the brain template.

We have quantied the bias on simulated and real datasets but we have also shown the usefulness of our methods to correct it. The two bootstrap methods have been validated on simulated datasets. Then we have used them to quantify and correct the bias on template shapes of landmarks describing optical nerves of monkeys in Chapter 5. For neuroimaging, we have introduced a new procedure relying on topological constraints in Chapter 6, in order to control the template's inconsistency by forcing it to stay below a threshold. These contributions show the impact of Geometric Statistics for Computational Anatomy.

Perspectives

This Ph.D thesis opens the door to other developments of Geometric Statistics, and to its applications in Computational Anatomy as well as in other elds. We present here some of its perspectives.

Extend Geometric Statistics to other manifolds and statistical procedures Geometric Statistics relies on Dierential Geometry and Statistics: the eld is therefore immensely rich as we can vary both the properties of the data space and the statistical procedures. For example, we can consider the sample mean on Lie groups dened with ane connection and study its asymptotic properties in this context. We can also dene a sub-Riemannian structure on Lie groups, dene a notion of mean and study its properties. Beyond Lie groups, we can explore notions of means on ber bundles, principal bundles etc.

While focusing on one geometric space, we can investigated statistical moments beyond the mean, like the variance for example. Eventually, statistical procedures like dimension reduction techniques or regression can be generalized to Lie groups, sub-Riemannian spaces, bundles, etc. We have shown that bootstrap techniques could be adapted to improve estimators on quotient spaces. We can also adapt them to improve future estimators on other manifolds with geometric properties, like ane connection spaces. The theoretical aspect of Geometric Statistics thus promises the development of various extensions.

Use Geometric Statistics to analyze and improve other algorithms We have investigated the asymptotic bias of the template estimation algorithm of Computational Anatomy. But we could also analyze other statistical properties of the same algorithm, and for example start with its bias in the low sample setting (small data). Again, the analysis of the bias in the low sample setting would lead to the development of methods designed to improve the estimator.

Then, we can investigate other algorithms of medical imaging or machine learning through the lens of Dierential Geometry. For example, general algorithms including trees or forests could get an interesting formulation in terms of stratied spaces. Such algorithms could be reformulated using Geometric Statistics so that their variance and bias could be studied.

Use intuition of Geometric Statistics to create new algorithms Just as

Statistics and Machine learning can be considered distinct, one could consider going from Geometric Statistics to "Geometric Machine Learning". Instead of dealing with input from Euclidean spaces, one would deal with input from spaces with geometric properties. This idea has already been exploited with the introduction of kernel in usual learning algorithms. Kernels enable to run the usual machine learning algorithm on spaces with another metric structure than the Euclidean one. Nevertheless, "Geometric Machine Learning" would go further by enabling to take into account other geometric structures of the space, non only the one related to its inner product, like algebraic structures.

Furthermore, we could build on the mathematical toolbox of Chapter 8 to develop some Geometric Statistics applied to Computer Vision. Sub-Riemannian geometry indeed allows to envision various applications regarding tridimensional image processing, like in-painting or dis-occlusion, anisotropic smoothing etc. Geometric Statistics and Heterogeneous data analysis The ultimate goal of Computational Anatomy within Computational Medicine is to describe the observed anatomical variability with a biological and biophysical model. Therefore, we could merge Geometric Statistics for anatomical data with other input data. We could integrate anatomical, functional and physiological or clinical parameters. Such an heterogeneous statistical theory holds promises with respect to the clinical applications.

Epilogue

We expressed a tendency of the medical imaging community to converge towards a theory of statistics adapted to the data spaces of Computational Anatomy. With this Ph.D thesis, we have proposed Geometric Statistics as a general framework to encompass the eorts made in this direction.

We have provided clear results on the current research boundaries of Geometric Statistics as a mathematical theory, as well as practical applications of the powerful tools it provides. We believe that this double exposition will inspire bright minds in order to keep on building this theory, with the aim to provide the medical community with novel and ecient statistical tools. Ultimately, we are certain that Geometric

We present the detailed proofs of the theorems. The precise statements of the theorems are given again in each section of this supplementary materials.

A.1 Notations

We denote Y the template shape and Ŷ its estimate. X is a point in the manifold M . We consider that X belongs to a principal orbit and we recall that the set of principal orbits is dense in M . We write Z = π(X) the projection of X in the shape space Q. We denote m the dimension of M , p the dimension of the principal orbits and q the dimension of the quotient space. Normal Coordinate Systems We often use Normal Coordinate Systems (NCS) to express the coordinates of tangent vectors. We refer to [Postnikov 2001] for theoretical developments about the NCS and to [Brewin 2009] for Taylor expansions of dierential geometric tensors in a NCS.

For example, we may consider a NCS centered at the point Y , with respect to the Riemannian metric of M . This NCS is valid on an open neighborhood of Y , that is start-shaped domain around Y . Moreover, we assume that M is geodesically complete : thus, this domain is equal to the whole manifold M , with the exception of the cut locus which is of null measure [Postnikov 2001]. The Riemannian logarithm of a point X in the NCS at Y is denoted:

--→ Y X.

Asymptotic behavior for σ → 0 We denote: (i) Θ(σ k ) a function that is proportional to σ k , (ii) O(σ k ) a function equivalent to σ k for σ → 0 and (iii) ε(σ) a function that is exponentially decreasing for σ → 0.

A.2 Preliminaries

A.2.1 A rst computation

First, we show a technical result that will be used throughout the proofs. We show that the following integral on T Y M , the tangent space of M at Y :

C Br d M (Y, X) k exp - d 2 M (Y, X) 2σ 2 d --→ Y X
is a function σ → ε(σ), i.e. decreases exponentially when σ → 0. In the integral above, the notation C Br denotes the complement in M of the geodesic ball B r of center Y and of radius r.

We split the coordinates in T Y M into (ρ, u) (polar coordinates):

|| C Br d M (Y, X) k exp - d 2 M (Y, X) 2σ 2 dX|| ≤ K u∈S m ρ(u) r ρ k exp - ρ 2 2σ 2 ρ m-1 dρdS m ≤ K u∈S m ρ(u) r ρ m+k-1 exp - ρ 2 2σ 2 dρdS m
where ρ(u) is the distance to the cutlocus in the direction u.

The positive integral is upper bounded by the same integral dened on the larger domain:

|| C Br d M (Y, X) k exp - d 2 M (Y, X) 2σ 2 dX|| ≤ K u∈S m +∞ r ρ m+k-1 exp - ρ 2 2σ 2 dρdS m
Integrating the volume of the unit hypersphere:

|| C Br d M (Y, X) k exp - d 2 M (Y, X) 2σ 2 dX|| ≤ K +∞ r ρ m+k-1 exp - ρ 2 2σ 2 dρ
where the right-hand-side is dominated by exp(-r 2 2σ 2 ) by the dominated convergence theorem. Therefore:

A.2.2.2 Second, we turn to the (unnormalized) truncated moments.

They write, with respect to the total moments:

M i 1 ...i k r (σ 2 A) = M i 1 ...i k (σ 2 A) - C Br X i 1 ...X i k exp - X T A -1 X 2σ 2 dX
The second term of the sum is negligible when σ → 0. To see this, we put an upper bound on its norm through triangular inequality:

|| C Br X i 1 ...X i k exp - X T A -1 X 2σ 2 dX|| ≤ C Br ||X i 1 ||...||X i k || exp - X T A -1 X 2σ 2 dX,
using triangular inequality on their coordinates:

|| C Br X i 1 ...X i k exp - X T A -1 X 2σ 2 dX|| ≤ C Br ||X|| k exp - X T A -1 X 2σ 2 dX,
and performing the change of variables X = A -1/2 X, i.e. taking the matrix square root of the positive denite matrix A -1 :

|| C Br X i 1 ...X i k exp - X T A -1 X 2σ 2 dX|| ≤ C B |||A 1/2 |||r |||A||| k/2 ||X || k exp - X T X 2σ 2 det A 1/2 dX, = |||A||| k/2 det A 1/2 C B |||A 1/2 |||r ||X || k exp - ||X || 2 2σ 2 dX.
By the computations in Subsection A.2.1, we have:

|| C Br X i 1 ...X i k exp - X T A -1 X 2σ 2 dX|| = ε(σ)
Therefore, the truncated moments are equivalent to the (full) moments for σ → 0:

M i 1 ...i k r (σ 2 A) = M i 1 ...i k (σ 2 A) + ε(σ) (A.3)

A.2.3 Isotropic Gaussian Moments on a Riemannian manifolds

We turn to computations of Gaussian moments in a m-dimensional Riemannian manifold M , using the notation M. We refer to the (unnormalized) moment of order k of the m-dimensional isotropic Gaussian of covariance σ 2 I as:

M i 1 ...i k σ 2 I = M --→ Y X i 1 ... --→ Y X i k exp - d 2 M (X, Y ) 2σ 2 dM (X)
and to the truncated (unnormalized) moment at radius r of the m-dimensional isotropic Gaussian of covariance σ 2 I as:

M i 1 ...i k r σ 2 I = Br --→ Y X i 1 ... --→ Y X i k exp - d 2 M (X, Y ) 2σ 2 d --→ Y X
where the integration domain is now the m-dimensional geodesic ball B r of radius r and centered at Y .

A.2.3.1 First, we consider the truncated moments:

M i 1 ...i k r σ 2 I = Br --→ Y X i 1 ... --→ Y X i k exp - d 2 M (X, Y ) 2σ 2 dM (X)
In a NCS at Y :

d 2 M (X, Y ) = --→ Y X T --→ Y X: M i 1 ...i k r σ 2 I = Br --→ Y X i 1 ... --→ Y X i k exp - --→ Y X T --→ Y X 2σ 2 dM (X)
On the small ball B r , dM

(X) = d --→ Y X + 1 6 R ab (Y ) --→ Y X a --→ Y X b d --→ Y X + O(|| --→ Y X|| 3 )d --→ Y X: M i 1 ...i k r σ 2 I = Br --→ Y X i 1 ... --→ Y X i k exp - --→ Y X T --→ Y X 2σ 2 d --→ Y X + 1 6 ab R ab (Y ) Br --→ Y X a --→ Y X b . --→ Y X i 1 ... --→ Y X i k exp - --→ Y X T --→ Y X 2σ 2 d --→ Y X + Br O(|| --→ Y X|| 3 ) --→ Y X i 1 ... --→ Y X i k exp - --→ Y X T --→ Y X 2σ 2 d --→ Y X
Now we recognize the (un-normalized) moments of a truncated isotropic Gaussian a B r in the vector space T Y M R m . We replace them by the expressions given in the previous subsection:

M i 1 ...i k r σ 2 I = M i 1 ...i k (σ 2 I) + ε(σ) + 1 6 ab R ab (Y ) M abi 1 ...i k (σ 2 I) + ε(σ) + O(σ m+k+4 )
The sum of two functions that decrease exponentially for σ → 0 is a function that decreases exponentially for σ → 0. So that:

M i 1 ...i k r σ 2 I = M i 1 ...i k (σ 2 I) + 1 6

ab R ab (Y )M abi 1 ...i k (σ 2 I) + O(σ m+k+4 ) + ε(σ)

We take the σ's out:

M i 1 ...i k r σ 2 I = σ k+m M i 1 ...i k (I) + σ k+m+2 6 ab R ab (Y )M abi 1 ...i k (I) + O(σ m+k+4 ) + ε(σ) (A.4) A.2.3.2 Second, we consider the (full) moments.

They write, with respect to the non-truncated moments:

M i 1 ...i k σ 2 I = M i 1 ...i k r σ 2 I + C Br X i 1 ...X i k exp - d 2 M (Y, X) 2σ 2 dM (X)
The second term of the sum is negligible when σ → 0. To see this, we put an upper bound on its norm through triangular inequality:

|| C Br X i 1 ...X i k exp - d 2 M (Y, X) 2σ 2 dM (X)|| ≤ C Br ||X i 1 ||...||X i k || exp - d 2 M (Y, X) 2σ 2 dM (X)
then using triangular inequality on the coordinates: We start by dividing the integral:

|| C Br X i 1 ...X i k exp - d 2 M (Y, X) 2σ 2 dM (X)|| ≤ C Br d M (Y, X) k exp - d 2 M (Y, X) 2σ 2 dM (X).
f (Z) = 1 C M (σ)
.

O Z ∩Br exp - d 2 M (Y, X) 2σ 2 dO Z (X) + O Z ∩C Br exp - d 2 M (Y, X) 2σ 2 dO Z (X)
In the parenthesis, we call the rst integral I Z (σ) and the second term is an η(σ):

f (Z) = C M (σ) -1 . (I Z (σ) + η(σ))

We compute the Taylor expansions of C M (σ) -1 and I Z (σ) for σ → 0 and show that η(σ) = ε(σ).

A.3.1 Taylor expansion of C M (σ) -1

The integration constant C M (σ) is the moment of order 0 of the m-dimensional isotropic Gaussian of variance σ 2 I in the m-dimensional manifold M . From Subsection A.2.3:

C M (σ) = ( √ 2πσ) m + σ m+2 6 ab R ab (Y )( √ 2π) m δ ab + O(σ m+3 ) + ε(σ) = ( √ 2πσ) m + σ m+2 6 R(Y )( √ 2π) m + O(σ m+3 ) + ε(σ)
where ε(σ) is exponentially decreasing wrt σ and R(Y ) is the scalar curvature of M at Y . Its inverse: We recognize the unnormalized truncated Gaussian moments of in R p , where p is the dimension of the orbit O Z , see Subsection A.2.2:

C M (σ) -1 = ( √ 2πσ) m + ( √ 2π) m 6 R (
I Z (σ) = exp - d 2 M (Y, Z) 2σ 2 . M r Z (σ 2 M -1 ) -N cb (Z)M bc r Z (σ 2 M -1 ) + O(σ p+4 ) - S abcd (Z) 2σ 2 M abcd r Z (σ 2 M -1 ) + O(σ p+6 ) 2σ 2
We express them in terms of the unnormalized Gaussian moments in R p , see Subsection A.2.2:

I Z (σ) = exp - d 2 M (Y, Z) 2σ 2 .
. σ p M(M -1 ) + N cb (Z)σ p+2 M(M -1 ) bc -S abcd 2σ 2 σ p+4 M(M -1 ) abcd + O(σ p+4 ) +

O(σ p+6 ) σ 2 + ε(σ)
We simplify the σ's:

I Z (σ) = exp - d 2 M (Y, Z) 2σ 2 .
. σ p M(M -1 ) 0 + σ p+2 N cb M(M -1 ) bc -S abcd (Z) 2 M(M -1 ) abcd + O(σ p+4 ) + ε(σ)

For convenience in the later computation, we dene the notations: where M ab and S abcd are given in the previous subsections.

I Z (σ) =

A.3.3 Upper bound on η(σ)

We proceed with an adaptation of the method in Subsection A.2.1:

η(σ) = O Z ∩C Br exp - d 2 M (Y, X) 2σ 2 dO(X)
Assuming that the Ricci curvature of the orbit is bounded by below, the measure of the orbit is bounded by above, by a constant that we write K O :

||η(σ)|| ≤ K O O Z ∩C Br exp - d 2 M (Y, X) 2σ 2 du
We gather the ε's and the O(σ 4 ):

Bias(Y, Ŷ ) a = F 01d (Y )σ 2 M Q r ad (I) + O(σ 4 ) + ε(σ)

We replace the normalized 2nd order moment by its expression which is simply δ ad , see Subsection A. where the trace of the second fundamental form is the external curvature vector H(Y ) by denition.

We convert this Taylor expansion in y, the coordinate of Y in a NCS at Z, into a Taylor expansion in z, the coordinate of Z in a NCS at Y . To express y with respect to z, we consider the geodesic γ ZY (t) from Z to Y and the geodesic γ Y Z (t) from Y to Z. When parameterized by the arclength s, they are related as follows:

γ ZY (s) = Exp Z s --→ ZY = γ Y Z (1 -s) = Exp Y (1 -s) --→ Y Z
Dierentiating this relation gives:

DExp Z | s -→ ZY . --→ ZY = -DExp Y | (1-s) -→ Y Z .
--→ Y Z

Taking the relation at s = 0:

DExp Z | 0 . --→ ZY = -DExp Y |-→ Y Z . --→ Y Z

Figure 1

 1 Figure 1.1: Left: Statue of Asclepius in the Epidaure's sanctuary. Middle: The ag of the World Health Organization with Asclepius rod. Right: Portrait of Andreas Vesalius (1514-1564) from De humani corporis fabrica.

Figure 1

 1 Figure 1.2: Number of MRIs (called "IRMs" in French) and scanners (equipment) per million of inhabitants worldwide for the year 2013 [Chasseing 2016].

Figure 1

 1 Figure 1.3: Evolution of the number of medical imaging equipment in France [Chasseing 2016].

Figure 1 . 4 :

 14 Figure 1.4: Number of MRIs (left, called "IRMs" in French) and scanners (right) exams in 2013 with respect to the population [Chasseing 2016].

  Figure 1.5: Market of Cloud Computing for medical imaging. Grey: Cloud based storage and Disaster Recovery Market. Red: Cloud based Sharing and Communication Software market. Orange: Picture Archiving and Communication System on cloud [GlobalData 2011].

Figure 2 . 1 :

 21 Figure 2.1: Data x i 's generated by underlying P θ in the sample space X .

Figure 2 . 2 :

 22 Figure 2.2: Data x i 's generated by underlying P θ in the sample space X .

  Figure 2.3: Probability distribution of θ for n data.

•

  13) Nested bootstrap The bootstrap procedure gives estimations of an estimator's properties: an estimate of the bias b ias B , an estimate of the standard error dB ( θ), etc. How accurate are these estimations? To answer this question, we can apply the same reasoning as before. We compute bootstrap replications of b ias B and dB ( θ)! This leads to a nested bootstrap [de La Rosa 2006]. • For each b = 1...B: • Take b-th bootstrap sample x b * and b-th bootstrap replication θb * computed above, For c = 1...C: Compute b, c-th bootstrap sample: With replacement from x b * (non-parametric) or from θb * (parametric) to get: x b,c * Compute b, c-th bootstrap replication: θb,c * • Compute the bootstrap distribution of θb * and bootstrapped desired property: b ias b * • This gives the bootstrapped distribution of the desired property: b ias b * b

Figure 4 . 1 :

 41 Figure 4.1: Left and right translation of a data set {g i } N i=1 on the Lie group G. The initial data set {g i } N i=1 has a mean represented in red. The left translated data set {h * g i } N i=1 has a mean represented in blue. The right translated data set {g i * h} N i=1

  Figure 4.2: Algebraic and Geometric structures. If we require compatible algebraic and geometric structures on the manifold, we get a quadratic Lie group: a Lie group with a bi-invariant pseudo-metric.

  making B 1 and B 2 subrepresentations of the adjoint representation of B, in other words: ideals of B.

  Figure 4.3.2 and Figure 4.3.2 illustrate the comparison between Lie algebras with bi-invariant metric and Lie algebras with bi-invariant pseudo-metric.

Figure 4 . 3 :Figure 4

 434 Figure 4.3: Structure of a Lie algebra with bi-invariant metric.

  61 algorithm, the following assertions are equivalents: (a) B is of Type (2), (b) B is quadratic, (c) ∀I minimal, I abelian, there is a double extension decomposition of B, (d) ∃I minimal, abelian such that there is a double extension decomposition of B.

Figure 4

 4 Figure 4.5: Tree structure of the algorithm.

  compute I minimal abelian ideal; if no I exists: "EXIT". compute its centralizer C B (I); if codim(C B (I)) = dim(I): "EXIT". compute S = B/C B (I), W = C B (I)/I; call algorithm recursively on B = W ; compute Hom S (S, I); solve Equation(4.7) for L ∈ Hom S (S, I) by plugging:

Figure 4

 4 Figure 4.7: Schematical result for H and U T (n). The top level indicates the direct sum decomposition step. Thus h is indecomposable and ut(n) decomposes into d andh. The bottom level for h indicates that h is neither 1-dimensional, nor simple, nor a double extension and therefore we exit the algorithm: h is not quadratic. The bottom level for ut(n) indicates that d is 1-dimensional and therefore quadratic but that h is not quadratic: ut(n) is not quadratic.

Figure 4

 4 Figure 4.8: Schematical result for SE(n). We recover the dierent cases depending on n.

Figure 5 . 1 :

 51 Figure5.1: Two landmarks, one in red and one in black, on the plane R 2 (a) and on the sphere S 2 (b). The landmark in red is xed at the origin of the coordinates. The system is entirely represented by the coordinates X of the landmark in black.

Figure 5 . 2 :

 52 Figure 5.2: First line: Action of rotations on R 2 , with (a): action of rotation g ∈ SO(2) on point X ∈ R 2 and orbit of X in blue dotted line; (b) Stratication of R 2 into principal orbit type (blue) and singular orbit type (red); (c) shape space R + = R 2 /SO(2) with a singularity (red dot). Second line: Action of SO(2) on S 2 with (d): action of rotation g ∈ SO(2) on point X ∈ S 2 and orbit of X in blue dotted line; (e) Stratication of S 2 into principal orbit type (blue) and singular orbit type (red) (f) shape space [0, π] = S 2 /SO(2) with two singularities (red dots).

  Figure 5.3:Step 1 of generative model of Equation (5.1) for the plane example (a) and the sphere example (b). The black curve illustrates the probability distribution function on shape space. This is a distribution on r ∈ R + for the plane example (a) and on θ ∈ [0, π] for the sphere example. The black square represents its expectation. For the simpler generative model of Equation (7.2), the probability distribution boils down to a single point at Y i.e. at the black square.

  Figure 5.4:Step 2 of generative model of Equation (7.2) for the plane example (a) and the sphere example (b). The blue dotted curve illustrates the orbit of the shape drawn in Step 1. The black curve illustrates the probability distribution function on this orbit. This is a distribution in angle θ ∈ [0, 2π] for the plane example (polar coordinates) and in angle φ ∈ [0, 2π] for the sphere example (spherical coordinates).

Figure 5 . 5 :

 55 Figure 5.5: Step 3 of generative model of Equation (7.2) for the plane example (a) and the sphere example (b). The dotted curve represents the isolevel at σ of the Gaussian distribution function on the ambient space.

  Figure 5.7: Screenshot of https://nmiolane.shinyapps.io/shinyPlane and https://nmiolane.shinyapps.io/shinySphere.Simulated data X i 's (grey points), template shape Y (green), registered data ĝi • X i (black points), template shape estimate Ŷ (orange). Induced distributions on the shapes, template shape Y (green), template shape estimate Ŷ (orange).

  (a)-(d) present what is meant by asymptotic bias: the estimate Ŷ , of the procedure, is in orange and the template shape Y , of the generative model, is in green. The estimator Ŷ (in orange) does converge when the number of data, i.e. the grey points in Figures 5.7

  (a)-(c), goes to innity, but Ŷ does not converge to the template shape Y it is designed to estimate. For Figures 5.7

  (a)-(d), this means

  e. a vector of the tangent space of Q at the real parameter Y , denoted T Y Q. The tangent vector Log Y Ŷ is illustrated on Figures 5.8(a) and (b) for the plane and sphere examples. Log Y Ŷ represents how much one would have to shoot from Y to get the estimated parameter Ŷ . The norm of Log Y Ŷ , computed using the metric of Q at Y , represents the dissimilarity between Ŷ and Y .

  1

  Figure5.9: (a) Induced distributions on the distance r between two landmarks in R 3 for real distance y = 1 (in green) and noise level σ = 0.3 and σ = 3. (b) Induced distributions on the angle x between the two landmarks on S 3 , for real angle y = 1 and noise levels σ = 0.3 and σ = 3. In both cases the mean shape estimate ŷ is shown in red.

  Figure5.10: Asymptotic bias on the mean shape estimate Ŷ with respect to the noise level σ for r = 1 in the plane example (a) and θ = 1 in the sphere example (b). The bias is quadratic near σ = 0. Increasing σ takes the estimate Ŷ away from 0 in shape space Q = R + (a) and away from 0 and π in shape space Q = [0, π] (b).

Figure 5

 5 Figure 5.11: The external curvature of the template's orbit creates the asymptotic bias, in the plane example (a) and the sphere example (b). The blue curve represents the template's orbit. The ball of radius σ represents a level set of the Gaussian distribution of the noise in R 2 (a) and S 2 (b). The grey-colored area represents the distribution of the noise that generates data outside the orbit of Y , in Step 3 of the generative model of Equation (7.2) and Figure7.5. There is a higher probability that the data are generated "outside" the orbit. The template shape estimate is biased towards greater radii (a) or towards angles closer to π/2 (b).

  Figure 5.13: Probability distributions functions (noise σ = 0.3) and bias for R m for m = 2, m = 10, m = 20 and m = 100. Template shape is r = 1.

Figure 5 .

 5 Figure 5.14: Algorithm 5.3.1 Iterative bootstrap procedure on the plane example for n → +∞. (a) Initialization, (b) Generate bootstrap sample from Ŷ0 and compute the corresponding estimate Ŷ0* , compute the bias Ŷ0 -Ŷ * 0 , (c) Correct Ŷ0 with the bias to get Ŷ1 , (d) Generate bootstrap sample from Ŷ1 and iterate as in (b), (e) Get Ŷ2 etc.

Figure 5

 5 Figure 5.15: F of the xed-point procedure and rst 2 iterations for σ = 1, m = 3. ∆ is the rst diagonal. The initial estimate is biased Ŷ0 = 4.91. The Iterative Bootstrap converges towards the template shape Y = 1.2.

  Figure5.17: Left: Implementation of the sphere example: the green point is the template shape Y , the grey points are the data X i 's generated with the model (7.2), the black points are the registered data ĝi • X i 's, the orange point is the template shape estimate Ŷ . The quotient space [0, π] is copied below, and the blue points show the iterations of the iterative bootstrap of Algorithm 5.3.1 that corrects the bias of Ŷ as an estimate of Y : the blue points go from the orange point Ŷ to the green point Y . Right: Convergence of the iterative bootstrap of Algorithm 5.3.1, for the sphere example. The colors red, purple, blue represent dierent noises σ. The bias of Ŷ as an estimator of Y is shown on the ordinate axis: it converges to 0 in a few iterations.

Figure 5 .

 5 Figure 5.18: Algorithm 4 Nested Bootstrap on the plane example for n → +∞. (a) Initialization, (b) Generate bootstrap sample from Ŷ0 ; compute the estimate Ŷ0 * , compute the bias Ŷ(0) -Ŷ * 0 , (c) Generate bootstrap sample from Ŷ0 * ; compute the estimate Ŷ0 * * , compute the bias Ŷ * 0 -Ŷ * * 0 , (d) compute the blue arrow, i.e. the bias of Bias( Ŷ * * 0 , Ŷ * 0 ) as an estimate of Bias( Ŷ * 0 , Ŷ0 ), (e) Correct Ŷ with the bias-corrected bias.

  (

  Figure 5.19: Comparison of the Iterative bootstrap and the Nested bootstrap on simulation with two dierent Signal-Noise ratio, which is SNR = Y σ = r σ , the ratio of the template Y , which is the radius r in the plane example, on the noise level σ. (a) shows SNR = 1 and (b) shows SNR = 0.33. In all four experiments: the template shape is the green dot at r = 1, the template shape estimate is in orange, and the successive steps of the bootstrap algorithms are the blue dots: we have several blue dots for the Iterative Bootstrap, and two blue dots for the Nested Bootstrap.

  Figure5.20: Left: Implementation for simulated triangles: the green triangle is the template shape Y , the grey triangles are (some of) the data X i 's generated with the model (7.2), the black triangles are (some of) the registered data ĝi •X i 's, the orange triangle is the template shape estimate Ŷ . The blue triangles show the iterations of the iterative bootstrap of Algorithm 5.3.1 that corrects the bias of Ŷ as an estimate of Y : the blue triangles go from the orange triangle Ŷ to the green triangle Y . Right: Convergence of the iterative bootstrap of Algorithm 5.3.1. The colors red, purple, blue represent dierent noises σ. The bias of Ŷ as an estimator of Y is shown on the ordinate axis: it converges to 0 in a few iterations.

  Figure 5.21: Triangles data in grey for the control group (left) and the glaucoma group (right). In orange, the estimated template shapes. Distances are measured in µm.

101Figure 5

 5 Figure 5.22: Errors on atoms coordinates represented by the B-factor, for proteins 1H7W (left) and 4HBB (right). Atoms at the periphery of the proteins tend to have more errors, which appear in yellow-red colors.

  Figure 6.1(b) illustrates the above notions on R 2 . The action of a dieomorphism φ on I is represented by the blue curved arrow, i.e. by the action of a 2D rotation. The action transforms the image I into another image φ • I, i.e. into a dierent point in the Hilbert space.

  Figure 6.1: Action of a dieomorphism φ on a brain image I. (a) the brain image before and after the action of φ, (b) schematic representation of the action of φ on the brain image I, represented as a dot in R 2 .

Figure 6

 6 Figure 6.3: (a) Two brain images with dierent isotropy groups: the left image shows more symmetry, thus a larger isotropy group, and the right image as more asymmetric details and hence smaller isotropy group. (b)The isotropy group is linked to the dimension of the image's orbit. I 2 has a smaller isotropy group, it is represented as a blue point which has a circle as orbit. I 1 has larger isotropy group: it is represented as the red point at (0, 0) whose orbit is itself.

  Figure 6.4(b) shows how the space of images R 2 is partitioned into orbits: blue circles and one red "singular circle", the red point at (0, 0). Figure6.4(b) also shows R + , the quotient space of R 2 by the group of 2D rotations, which schematically represents the quotient space of the space of brain images Img(Ω) by the Lie group of dieomorphisms Di(Ω). Each of the four blue circles in R 2 becomes a blue point in the quotient space R + .(b) (a)

Figure 6 . 5 :

 65 Figure 6.5: Schematic illustration of the generative model of the brain images data. As before, the space of brain images is represented by the plane R 2 . (a) First step of the generative model: generate a brain anatomy. One usually assumes that there is a unique brain anatomy: T , in green. (b) Second step of the generative model: generate a deformation φ i ∈ Di(Ω) which is used to deform the template. The brain image φ i • I belongs to the orbit of T , represented by the green circle. (c) Third step of the generative model: generate noise ε i in the space of images. The brain image φ i • T + ε i does not belong to the orbit of T anymore.

  Figure 6.6: Geometrization of an iteration of the template's computation: (a) n subjects images (black squares); (b) the n images are registered, they travel on their orbit (the blue circles) to get aligned; (c) registered images; (d) the empirical brain template T (in yellow) is computed as the Fréchet mean of the n registered images. How far is it from the unique anatomy T of the generative model (in green): can we quantify B n ?

  Figure 6.7: Schematic illustration of the asymptotic bias in the template computation algorithm of neuroimaging. (a) The images generated with the model described above have a higher probability to be "outside" (with respect to the curvature) of the template's orbit. (b) The registration during template's estimation aligns the images. The distribution of images is unbalanced with respect to the real template.

  Figure 6.9: (a) Intensity I on a 2D brain image visualized as height: maxima are in red, minima in blue. This is a Morse-Smale function I : Ω = R 2 → R. (b) Persistences p 1 < p 2 of two pairs min-max. A threshold p 1 < p < p 2 divides the domain into 3 regions (pink, violet, green), while p < p 1 divides it in 5 regions (pink, violet, brown, turquoise, green). (c) Computational representation of the geometry: Regions on a 2D domain, induced by a MSC with threshold p = 0.1: (i) The MS graph represents the isotropy group's class of the image, (ii) the labeled MS graph represents the image's orbit under the dieomorphisms

  124 min-max pair of the region: d(region) = Tmax -Tmin = p T (region) (6.8)

Figure 6 .

 6 Figure 6.10: Topological denoising on two toy examples. We impose topological constraints on the initial images, on the left in both cases: minima in blue and maxima in red. The arrows denote the action of the topological denoising and point to the output image.

128Algorithm 4

 4 Controlled brain template estimation Input: Images {I i } n i=1 , noise variance σ 2 , persistence threshold p threshold Initialization: T1 = I 1 (one of the subjects images) k = 1

Figure 6

 6 Figure 6.11: Investigation of the template's consistency as an estimator of a unique anatomy. (a) Template. (b) Template whitened by the intersubject variability. (c) Region-wise inconsistency for a threshold= 1.3, (d) for threshold = 2, (e) for threshold = 4 (dimensionless).

Chapter 6 .

 6 Figure 6.12: Results of Topological Denoising integration in the pipeline for the template's estimation. Left: Template from pipeline without topological denoising. Right: Template with the topologically constrained pipeline. Inconsistent regions from Figure 6.11 are now blurred.

Figure 7

 7 Figure7.1: Two landmarks, one in red and one in black, on the plane R 2 (a) and on the sphere S 2 (b). The landmark in red is xed at the origin of the coordinates. The system of landmarks is entirely represented by the coordinates X of the landmark in black.

  Figure 7.2: (a) Action of rotation g ∈ SO(2) on point X ∈ R 2 , and orbit of X in blue dotted line; (b) Action of rotation g ∈ SO(2) on point X ∈ S 2 , and orbit of X in blue dotted line.

  Figure 7.3: Step 1 of the Generative model illustrated for the Rrunning examples. In both cases, the probability distribution function p Q on shapes is illustrated with the black curve. Its expectation is illustrated by the green square. (a): Representative case R 2 , the distribution of shapes p Q is a distribution on r ∈ R + . (b): Representative case S 2 , the distribution of shapes p Q is a distribution on θ ∈ [0, π].

  Figure 7.4: Step 2 of the Generative model illustrated for the running examples.In both cases, the probability distribution function on poses is illustrated with the black curve. (a) the distribution on orbits is a distribution in angles: each point is posed on its orbits through a distribution on φ 1 , ...φ m-1 . (b) the distribution on orbits is a distribution is angles: each point is posed on its orbits through a distribution on φ 1 ...φ m-1 .

Figure 7 . 5 :

 75 Figure 7.5: Step 3 of the Generative model. In both cases, the probability distribution function on the ambient space is multivariate, one isolevel is illustrated with the dotted curve. (a) Noise in the ambient space R 2 . (b) Noise in the ambient space S 2 .

  142

  Both h and H are measures of extrinsic curvature of O in M . For example, an hypersphere of radius R in R m has mean curvature vector of magnitude: ||H|| = m-1 R . Second, taking the intrinsic point of view on the orbit, one can consider the Riemannian curvature tensor of the orbit R O ijkl and the associated Ricci curvature R O ij and scalar curvature R O [Postnikov 2001]. For example, an hypersphere of radius R in R m has intrinsic scalar curvature equal to 1.

  143shape Y , but possibly with dierent positions and with measurement error. The registration step induces a bias in the estimation of p Q , i.e. in the estimation of the unique shape Y , because of the extrinsic curvature of the orbit of Y[Miolane 2017]. More precisely, the distribution π(p M ) is not centered around the unique shape Y .

  Figure 7.7: (a) Induced distributions on the distance r between two landmarks in R 3 for real distance y = 1 (in green) and noise level σ = 0.3 and σ = 3. (b) Induced distributions on the angle x between the two landmarks on S 3 , for real angle y = 1 and noise levels σ = 0.3 and σ = 3. In both cases the mean shape estimate ŷ is shown in red.

Figure 7 . 8 :

 78 Figure 7.8: Illustration for the successive transformations of metric in R 2 /SO(2) and S 2 /SO(2). The reference orbit chosen is shown in light blue. In both cases, singularities are put at innity and the orbit have null external curvature in the new ambient space.

  147

Image

  Figure 8.4: The 3 steps of image processing for 3D-Neurogeometry: LPP framework.

  Figure A.1: Summary of the notations used in the proofs. Y is the template shape, X is a point in M , belonging to a principal orbit O X . Z = π(X) is the projection of X in the shape space.

Figure

  Figure A.2: Summary of the notations used in the proof of Theorem 1. The vectors of the tangent space T Z M are in red.

  Y )σ m+2 + O(σ m+3 ) + ε(σ) ) + O(σ 3 ) + ε(σ) m . 1 -σ 2 6 R(Y ) + O(σ 3 ) + ε(σ)By skew symmetry, the terms in T c u c (Z) and u a u b u c P abc (Z) integrate to 0. Moreover,O(||u|| 3 ), O(||u|| 5 ) become O(||u|| 4 ), O(||u|| 6 ): cb (Z)u b u c + O(||u|| 4 ) -u a u b u c u d S abcd (

  exp -d 2 M (Y, Z) 2σ 2 σ p m 0 (Z) + σ p+2 m 2 (Z) + O(σ 4 ) + ε(σ)where:m 0 (Z) = M(M ) 0 = (2π) p det ((M ab (Z) -1 ) m 2 (Z) = N cb (Z)M(M -1 (Z)) bc -1 2 S abcd (Z)M(M -1 (Z)) abcd (A.12)

  2.2:Bias(Y, Ŷ ) a = F 01d (Y )σ 2 δ ad + O(σ 4 ) + ε (σ) = F a 01 (Y )σ 2 + O(σ 4 ) + ε(σ) A.4.2 Computation of F a 01 (Y ): Taylor expansion of F 0 (Z) in the coordinate zThe term F a 01 (Y ) is the rst order coecient in the Taylor expansion of F 0 (Z) around Y in the coordinate z. Thus, we compute this Taylor expansion.We rst compute the Taylor expression of F 0 (Z) using the coordinate y = --→ ZY = Log Z Y in the NCS at Z. The expression of F 0 (Z) in the previous section gives:F 0 (Z) = Θ Z,0 (1) = (2π) p det ((M ab (Z) -1 )The previous subsections give:M ab (Z) = -y c h c ab (Z) + d ab (Z)and we replace d ab (Z) by the formula p.23 in[Brewin 2009] but keeping only the rst order:M ab (Z) = δ ab -y c h c ab (Y ) + O(||y|| 2 )So that:M ab (Z) -1 = δ ab + y c h c ab (Y ) + O(||y|| 2 )We plug this in F 0 (Z):F 0 (Z) = det δ ab + y c h c ab (Y ) + O(||y|| 2 ) = 1 + y c Trace(h c ab (Y )) + O(||y|| 2 ) = 1 + y c H c (Y ) + O(||y|| 2 ) = 1 + 1 2 y c H c (Y ) + O(||y|| 2 )

  This paper presents the spatial quantication of the bias in the estimation of the template anatomy. This paper is presented in Chapter 6.

	3. Miolane, N., Pennec, X.: Computing bi-invariant pseudo-metrics on Lie groups
	for consistent statistics. Entropy 17(4), 18501881 (2015). https://hal.
	inria.fr/hal-01133922. This paper presents an algorithmic procedure to
	compute a bi-invariant pseudo-metric on a Lie group, when it exists. Such
	a pseudo-metric enables to apply Riemannian statistics on Lie group while
	respecting the group structure. It completes conference paper "Statistics on
	Lie groups : a need to go beyond the pseudo-Riemannian framework" and is
	presented in Chapter 4.
	4. Darmanté, H., Bugnas, B., Dompsure, R.B.D., Barresi, L., Miolane, N.,
	Pennec, X., de Peretti, F., Bronsard, N.: Analyse biométrique de l'anneau
	pelvien en 3 dimensions à propos de 100 scanners. Revue de Chirurgie Or-
	thopédique et Traumatologique 100 (7, Supplement), S241 (2014). go beyond
	the pseudo-Riemannian framework. 34th International workshop on Bayesian 2. Miolane, N., Holmes, S., Pennec, X.: Template organ shape estimation in Inference and Maximum Entropy Methods in Science and Engineering. AIP Computational Anatomy: Correcting an asymptotic bias. SIAM Journal of Proceedings, 1641, pp.5966. (2014). https://hal.inria.fr/hal-01091515. Imaging Sciences (2016). Submitted. This paper shows a novel approach (Oral presentation). This paper advocates for the need to go beyond the using Geometric Statistics to study the properties of the template anatomy pseudo-Riemannian framework for dening consistent statistics on Lie groups. algorithm in Computational Anatomy. It completes conference paper "Biased estimators on quotient spaces" and is presented in Chapter 5. It has led to the journal paper presented in Chapter 4.

http:// www.sciencedirect.com/science/article/pii/S187705171400327X. This paper shows the applications of statistics on pelvis shapes in order to develop a new pelvis surgery. This paper is not presented in this thesis. Peer-reviewed conference papers 1. Miolane, N., Pennec, X.: A survey of mathematical structures for extending 2D neurogeometry to 3D image processing. 18th International Conference on Medical Image Computing and Computer Assisted Intervention. Medical Computer Vision: Algorithms for Big Data. Volume 9601 of the series Lecture Notes in Computer Science, pp 155-167. (2015). https://hal.inria.fr/ hal-01203518/document. This paper presents mathematical structures to perform new algorithms of 3D image processing. The paper is presented in Chapter 8. 2. Miolane, N., Pennec, X.: Biased estimators on quotient spaces. 2nd International Conference on Geometric Sciences of Information. Springer, 9389, pp.130-139, Lecture notes in computer science (LNCS). (2015). https: //hal.inria.fr/hal-01203805/document. (Oral presentation). This paper shows rst computations of Dierential Geometry to show the bias of template shape estimation. It has led to the submitted journal paper presented in Chapter 5. 3. Miolane, N., Pennec, X.: Statistics on Lie groups : a need to

•

  Miolane, N.: Noise Eects on Quotient Spaces M/G. Workshop on Innite-Dimensional Riemannian Geometry with Applications to Image Matching and Shape Analysis. Erwin Schröedinger International Institute for Mathematics and Physics, Vienna February 19, 2015. See: http://www.esi.ac.at/ activities/events/2015/files/bmb_ws2_schedule.

Miscellaneous

• Poster: Miolane, N., Pennec, X., Holmes, S.: Towards a unied geometric Bayesian framework for template estimation in Computational Anatomy. International Society for Bayesian Analysis World Meeting. (

2016

). (Young Researcher Travel Grant). • Poster: Miolane, N.: Statistics on Lie groups : can we obtain a consistent framework with pseudo-Riemannian metrics? Workshop on Geometrical Models in Vision, Institut Henri Poincaré, Paris (2014). http://gmvision.lsis. org/slides/miolane.pdf • Video: Miolane, N., Khanal, B.: Statistics on Lie groups for Computational

Table 2 .

 2 1: Properties of an estimator θ of a parameter θ, for θ ∈ V an Euclidean space.

	Property of θ	Euclidean denition
	Error	e = θ -θ
	Mean Squared Error	MSE

93

  Algorithm 2 Corrected template shape estimation with Iterative Bootstrap Input: Objects {X i } n i=1 , noise variance σ 2

estimation procedure -will be central in the context of Computational Medicine in the following of the thesis.

Statistics will also nd important developments in other elds and look forward to its expansion.

Acknowledgments

First of all, I am extremely grateful to my two supervisors who have guided me for the three years of this Ph.D. On the one hand, Xavier Pennec introduced me to the

Table 8.1: Curves related to the dierent structures of Dierential Geometry in Neurogeometry. For spaces: M is a manifold, (P, M ) a ber bundle of base M , (M, G) is M endowed with a G action, (P, M, G) is a principal bundle of base M and structure group G, G is a Lie group. For verticality/horizontality: V F , V O , V and V ∆ : vertical in the sense of bers, orbits, orbits=bers (same notion for principal bundle), ∆-distribution. Same notations using H for horizontal. For metric structures: g R is a (pseudo)-Riemannian metric, g SR is a sub-Riemannian metric. For curves: γ denotes a notion of geodesics. We have γ G , γ G e , γ R , γ SR for group geodesic, 1-parameter subgroup, Riemannian geodesics and sub-Riemannian geodesic.

On M On (P, M ) On (M, G)

No metric

In this law, we read the group actions on SE(2) and their general properties. SE(2) acts on itself through the left and right translations (freely and transitively). As a Lie subgroup, SO(2) acts on SE(2) on the left and right (freely but not transitively). Note that the right SO(2)-action is trivial on the R 2 part. Moreover, the right SO(2)-action makes SE(2) a principal bundle of base R 2 and structure group SO(2).

A Sub-Riemannian Metric and two Riemannian Metrics

To introduce the sub-Riemannian metric, one rst denes its horizontal distribution ∆. In 2D-Neurogeometry one takes the moving frame (X 1 , X 2 , X 3 ) on R 2 × S 1 :

The sub-Riemannian metric g SR is dened as the Euclidean metric on ∆. In the standard basis ∂ x , ∂ y , ∂ z , its inverse writes:

Chapter 8. Geometric Statistics on ber bundles: processing of 3D medical images. 153

In practice, the sub-Riemannian metric is usually approximated the Riemannian metric g R ε whose inverse is [START_REF] Citti | [END_REF]]:

In addition, one denes a left-invariant metric g R µ as:

on the Lie algebra se(2). Then, one propagates it on SE(2) through left translations. g R µ is (SE(2))-left-invariant by construction. But g R µ is not (SE(2))-right-invariant as there is no bi-invariant metric on SE(2) [Miolane 2015b]. g R µ is invariant by the SO(2)-left and SO(2)-right actions.

A Survey of Curves

From Table 8.1 and the aforementioned structures, we survey the curves on SE(2). We have group geodesics of SE(2), Riemannian geodesics of g R µ , sub-Riemannian geodesics of g SR and their Riemannian approximation through g R ε . Group and Riemannian geodesics dier as g R µ is not bi-invariant. w.r.t. the right SO(2)-action, some are vertical or horizontal (taken With respect to. g R µ ). Examples of vertical group geodesics, vertical Riemannian geodesics for g R µ and vertical sub-Riemannian geodesics for g SR are orbits of the SO(2)-action.

Examples of horizontal group geodesics, horizontal Riemannian geodesics for g R µ are straight lines between two translations. Examples of horizontal sub-Riemannian geodesics for g SR are integral curves of X 1 .

With respect to. ∆, some are ∆-vertical or ∆-horizontal. There is no ∆-vertical group geodesics, and no ∆-vertical Riemannian geodesic. Example of ∆-horizontal group geodesics and ∆-horizontal Riemannian geodesic for g R µ are orbits of the right SO(2)-action. Sub-Riemannian geodesics are always ∆-horizontal.

Structures on the Image Domain

We read the related left SO(2)-action. Regarding the metric structures, g R µ was right SO(2)-invariant. Thus the projection is a Riemannian submersion for g R µ . It induces a Riemannian metric on R 2 which is the Euclidean metric. Projecting the horizontal Riemannian geodesics gives linear curves in R 2 . The projection of the sub-Riemannian geodesics gives the elastica curves [START_REF] Citti | [END_REF]], which can be linear or curvilinear.

Chapter 8. Geometric Statistics on ber bundles: processing of 3D medical images. 154

Actions

Metrics

8.3.3 The Three Steps: LPP Framework for 2D [START_REF] Citti | [END_REF]] (positions and orientations taken with directions). The lift is implemented by detecting the direction of the intensity gradient:

At the end of this step, the intensity is a function of D.

Alternatively, one can lift to the projective tangent bundle P T R 2 = SE(2)/Z 2 (positions and orientations taken without directions) [Boscain 2012]. Whether one should use SE(2) of P T R 2 is discussed here [Boscain 2012] (rem. 4, 13).

Second

Step: Processing (P) First, the processing can be the evolution of partial dierential equations (PDEs) with sub-Riemannian operators. For example, the sub-Riemannian diusion is dened with the sub-Riemannian Laplacian ∆ SR = X 2 1 + X 2 2 . Depending on the goal of the processing, one adds drift (also called convection) to the PDE: there is drift for completion purposes [Boscain 2012] and for enhancement [Duits 2011]. Equivalently, one can formulate this step as an oriented random walk. One writes the corresponding Kolmogorov equations.

Some PDEs are computed with the lifted intensity I(x, y, θ). In-painting methods provide examples: one "paints" directly in the lifted space [Boscain 2012]. Others compute with the activity function: u(x, y, θ) = u(x, y, θ)δ Σ where u(x, y, θ) = |X 3 (θ).∇I(x, y)|. In-painting methods provide also examples of this approach [START_REF] Sanguinetti | [END_REF]]. The corrupted image has a hole in D. The activity propagation amounts to "ll the hole" by a minimal surface. Then, one "paints" the surface by linking the isolevel sets with sub-Riemannian geodesics.

Then, the processing can be curve tting. Which curve do we t? One can t a sub-Riemannian geodesic, as in the second example of in-painting above Chapter 8. Geometric Statistics on ber bundles: processing of 3D medical images. 155 [START_REF] Sanguinetti | [END_REF]]. Another example is contour completion [START_REF] Citti | [END_REF]]. One can also t a Riemannian geodesic or a group geodesic for enhancement of 1-dimensional structures. A comparison of the two suggests that one should prefer the group geodesic [Duits 2011][Duits 2010b](called the "exponential curve" here). 8.3.3.3 Third Step: Projection (P) The processed lifted image on R 2 × S 1 is projected to a "standard" image dened on R 2 . The projection can be done in two dierent ways. First, one can use the "verticality" along bers of the bundle R 2 × S 1 . In this case, one projects along the ber S 1 , choosing a θ that maximizes a likelihood criterion [Boscain 2014]. Second, one can use the "∆-verticality" of sub-Riemannian geometry. In this case, one projects along the normal of the horizontal distribution ∆ through a concentration scheme [START_REF] Sanguinetti | [END_REF]]. This allows for several maxima at each point, i.e. crossings on the image.

A Theoretical Toolbox for 3D-Neurogeometry

As implemented by now [Duits 2010a] [Duits 2011], image processing pipelines using 3D-neurogeometry also follows the three same steps: 1. Lift (L), 2. Processing (P) and 3. Projection (P) (see Fig. 8.4). This steps could be iterated or not. The dierence with the 2D-neurogeometry however is the Processing. In this step, there is an additional level of structure in 3D-Neurogeometry w.r.t. the 2D case.

As in the 2D-case, we rst survey the mathematical structures and summarize them in Table 8.5 (subsections 3.1,3.2, 3.3). Then we present the LPP-frame of a 3D-Neurogeometry (subsection 3.4). Application-oriented readers can read subsection 3.4 rst, and then go to subsections 3.1, 3.2, 3.3.

Chapter 8. Geometric Statistics on ber bundles: processing of 3D medical images. 156

We read the group actions on SE(3) and their properties (see Table 8.3). As a Lie group, SE(3) acts on itself through left and right translations. As subgroups of SE(3), SO(3) and SO(2) also act on SE(3), on the left and right. The right SO(3)-action makes SE(3) a trivial principal bundle over R 3 with structure group SO(3). The right SO(2)-action on SE(3) makes the SE(3) a principal bundle over R 3 × S 2 with structure group SO(2).

A Left-Invariant Metric and a Bi-Invariant Pseudo-Metric

As in the 2D case, one denes the left-invariant metric g R µ on SE(3). g R µ is leftinvariant by construction. But g R µ is not right-invariant. g R µ is invariant by the left and right SO(3)-actions. The left SO(3)-invariance comes from the left invariance of g R µ . The right SO(3)-invariance is shown considering the right action on the parts R 3 and SO(3) separately, as g R µ is diagonal. Consequently, g R µ is also invariant by left and right SO(2)-actions. As opposed as the 2D case, there exist bi-invariant pseudo-metrics on SE(3). We refer to [Miolane 2015b] for their explicit construction. A possible choice is:

known as the Klein form. Here I 3 is the 3D identity matrix.

Chapter 8. Geometric Statistics on ber bundles: processing of 3D medical images. 157 From Table 8.1 and the aforementioned structures, we survey the curves on SE(3).

We have the group geodesics of SE(3), the Riemannian geodesics of g R µ an the pseudo-Riemannian geodesics of g BI . The group geodesics coincide with the pseudo-Riemannian ones, but dier from the Riemannian ones [Postnikov 2001] (see Table 8.1).

With respect to. a SO(3)-or SO(2)-action, some of these curves are vertical, some are horizontal (taken w.r.t. g R µ ). Examples of vertical group geodesics are the orbits of the SO(2)-action or the action of the group geodesics of SO(3). Examples of horizontal group geodesics are those generated by an element of the Lie algebra of the translations. 8.4.2 Structures on the Lifted Space R 3 × S 2 and on R 3

We go from SE(3) to R 3 × S 2 , by quotienting the right SO(2)-action. The quotient is implemented by choosing an origin in R 3 × S 2 , usually (0, a). An element (x, n) ∈ R 3 × S 2 is represented as the result of the action of the corresponding (x, R) on (0, a), where R is precisely the rotation bringing a onto n.

Induced Group Actions

The induced action of SE(3

We read the group actions on R 3 × S 2 and their properties (see Table 8.4).

The SE(3)-action is transitive on R 3 × S 2 . It makes R 3 × S 2 a homogeneous space. As the isotropy group is SO(2) everywhere, the orbit-stabilizer theorem gives: R 3 × S 2 = SE(3)/SO(2). Moreover, it provides the justication of the choice of an origin (0, a) in computer vision algorithms. All points are equivalent in a homogeneous space. Computations do not depend on the choice of origin.

Chapter 8. Geometric Statistics on ber bundles: processing of 3D medical images. 158 8.4.2.2 Induced Riemannian and Pseudo-Riemannian Metrics g R µ was invariant by the right SO(2)-action. Thus, the projection onto R 3 × S 2 is a Riemannian submersion for g R µ . It induces a Riemannian metric on R 3 × S 2 , still denoted g R µ . g R µ is still SE(3)-and SO(3)-invariant. Similarly, g BI was invariant by the right SO(2)-action. It induces a Riemannian pseudo-metric on R 3 × S 2 , which is still SE(3)-and SO(3)-invariant.

A Sub-Riemannian Metric

As in 2D, one denes a sub-Riemannian metric g SR on R 3 × S 2 by rst dening ∆. We take (X 1 , X 2 , X 3 , X 4 , X 5 ) on R 3 × S 2 as:

g SR is dened as the Euclidean metric on ∆. As in the 2D-case, it would be approximated by a Riemannian metric in practice.

A Survey of Curves

From Table 8.1 and the aforementioned structures, we survey the curves on SE(3). However, we have a new class of curves in 3D-Neurogeometry w.r.t. 2D-Neurogeometry: the curves of the lifted space R 3 × S 2 that are projection of curves of SE(3), as the projection of the group geodesics.

In the following, "verticality" and "horizontality" are taken w.r.t. the right SO(2)-action. Projecting horizontal (g R µ ) Riemannian geodesics gives generalized Riemannian geodesics. Projecting horizontal (for g BI ) pseudo-Riemannian geodesics gives generalized pseudo-Riemannian geodesics. More precisely, a smooth horizontal curve in SE(3) is a (pseudo-) Riemannian geodesics if and only if it is a (pseudo-) Riemannian geodesics in R 3 × S 2 . The projection of vertical curves are points. The projection of a curve that is vertical at one point has a "cusp".

Ultimately, we have the curves that are ∆-horizontal in the sense of the sub-Riemannian geometry. Among them, we have sub-Riemannian geodesics.

Structures on the Image Domain R 3

The previous structures are projected to R 3 , using the projection of the trivial bundle R 3 × S 2 on the rst component. In particular, projecting the previous curves give curves in R 3 . We have: the projection of the sub-Riemannian geodesics (an equivalent of 2D elastica curves), the double-projection of the group geodesics (equivalently the double-projection of the pseudo-Riemannian curves for g BI ), the double-projection of the Riemannian geodesics for g R µ .

Chapter 8. Geometric Statistics on ber bundles: processing of 3D medical images. 159 Actions Metrics

8.4.4 The Three Steps: LPP Framework for 3D First, as in 2D, the processing could be performed on R 3 ×S 2 without taking into account the SE(3) structure. One would only consider the sub-Riemannian structure on the lifted space R 3 × S 2 . In doing so, one could dene sub-Riemannian partial dierential equations as in 2D-Neurogeometry, using the X i as dierential operators. For in-painting purposes, the 2D work of [Boscain 2014[START_REF] Sanguinetti | [END_REF]] provides intuition. Similarly, one could add drift (or convection) depending on the application.

Then, in contrast to 2D, the processing can be performed on SE(3). This is done by embedding R 3 × S 2 in SE(3) as the quotient of SE(3) by a SO(2)action. Then, performing SO(2)-invariant computations on SE(3) is equivalent to performing computations on R 3 ×S 2 . The advantage is that one has more structures, e.g. more curves for curve tting (compare subsections 2.2 and 3.4). This is the rst main distinction between the 2D and the 3D case. In 2D-Neurogeometry, we have one (trivial) quotient of R 2 × S 2 . In contrast in 3D-Neurogeometry, one has two successive quotients of SE(3) = R 3 × SO(3).

Appendix A

Proofs of the theorems of Chapter 5

i.e. decreases exponentially when σ → 0.

A.2.2 Truncated Gaussian moments in a Euclidean space

We give preliminary computations of Gaussian moments in a m-dimensional vector space R m , using the curved notation M. We refer to the (unnormalized) moment of order k of the m-dimensional Gaussian of covariance σ 2 A as:

and to the truncated (unnormalized) moment at radius r of the m-dimensional Gaussian of covariance σ 2 A as:

where the integration domain is now the m-dimensional ball B r .

A.2.2.1 First, we recall the expressions of some unnormalized Gaussian moments.

The order 0, 2 and 4 are:

where we recall that Θ(σ m+k ) denotes the proportionality to σ m+k .

189

We assume that the Ricci curvature of M is bounded from below. Therefore, the measure dM has an upper bound with respect to the Lebesgue measure on the tangent space, which we write K: dM (X) ≤ KdX:

By the computations in Subsection A.2.1, this inequality together with σ → 0 shows that:

i.e. is a function that decreases exponentially when σ → 0. Therefore, the (unnormalized) moment of order k of the m-dimensional isotropic Gaussian of covariance σ 2 I writes:

A.3 Proof of Theorem A.3.1: Induced probability density on shapes

In this section we prove the Theorem 1 of our paper. We recall Theorem 1 below.

Theorem A.3.1 The data X i 's are generated in the nite-dimensional Riemannian manifold M following the model:

, described in the paper. In this model: (i) the action of the nite dimensional Lie group G on M , denoted •, is isometric, (ii) the parameter Y is the template shape in the shape space Q, (iii) ε i is the noise and follows a (generalization to manifolds of a) Gaussian of variance σ 2 , see Section 1 of the paper. Then, the probability distribution function f on the shapes of the X i 's, i = 1...n, in the asymptotic regime on an innite number of data n → +∞, has the following Taylor expansion around the noise level σ = 0:

where (i) Z denotes a point in the shape space Q, (ii) F 0 and F 2 are functions of Z involving the derivatives of the Riemannian tensor at Z and the derivatives of the graph G describing the orbit O Z at Z, and (iii) ε is a function of σ that decreases exponentially for σ → 0.

We The generative model implies the following Riemannian Gaussian distribution on the objects:

where C M (σ) is the integration constant:

We compute the induced probability distribution f on shapes by integrating the distribution on the orbit of X out of f (Z):

A.3.2 Taylor expansion of I Z (σ)

Now we compute the integral I Z (σ) involved in the formula of f (Z):

We denote: B r Z = B r Z . We rst perform the computations for any σ. We recall that r is xed, and small enough in a sense made precise later.

The local graph u → G(u) is illustrated on Figure A.1 and has the following Taylor expansion around u = 0 in the NCS at Z:

The 0-th and 1-th order derivatives of u → G(u) are zero because the graph goes through Z and is tangent at

The second order derivative is by denition the second fundamental form h(Z) introduced in Section 2 of the paper: h(Z) represents the best quadratic approximation of the graph G. The third and fourth orders G 3 (Z) and G 4 (Z) are further renements on the shape of the graph G around Z.

Second, we compute the Taylor expansion of d 2 M (Y, X) with respect to u:

and our goal is to compute the dierent tensors.

Y and X are represented by their Riemannian Logarithms at Z: y = Log Z Y , and x u = Log Z X. We also recall: u T y = 0. Using these, we express the squared distance d 2 M (Y, X) in the NCS at Z. We use the formula p.23 in 192 [Brewin 2009] with the notations ∆x ← y -x u and x ← x u :

193

We express this in orders of x u :

194

We replace x a u by x a u = u a + G(u) a where the expression of G(u) is a O(||u|| 2 ), so that:

195

We replace G(u) by its Taylor expansion. Identifying the tensors gives: .9) and P abc (Z) and S abcd (Z) are tensors mixing the derivatives of the graph G and the derivatives of the Riemannian curvature R of M at Z.

A.3.2.2 Computing dO Z (X)

The second component of I Z (σ) is the measure of the orbit dO Z (X). We seek the Taylor expansion of the measure:

and our goal is, again, to express the tensors T c (Z) and N cb (Z).

The measure dO Z (X) is the restriction of the measure dM (X):

and we know that for X close enough to Z: dM (X) = dx u -1 6 Ric ab (Z)x a u x b u dx u . Thus:

We express dx u with respect to du:

Developing the determinant:

We develop by keeping up to the quadratic terms only:

We reorganize the terms, relabeling the mute labels:

Now we can factorize the quadratic terms:

So that we nd the expressions of the tensors:

We plug the expressions of d 2 M (Y, X) and dO Z (X), computed in the previous subsections, in the expression of I Z (σ):

Plugging the squared distance d 2 M (Y, X) rst:

We split the exponential and extract the part of the exponential that does not depend on u:

We perform the Taylor expansion of the term with O(||u|| 4 ), recalling that at this point, σ can still be anything.

Now we plug the dO Z (X):

We develop the product of the parenthesis on the right:

We integrate on the orbit by liating it with hyperspheres of radii ρ:

Now the volume of S ρ ∩ O Z is polynomial in ρ. We denote P this polynomial:

By dominated convergence theorem, the right-hand-side is dominated by exp(-r 2 2σ 2 ). Thus: .13) i.e. σ → η(σ) is exponentially decreasing when σ → 0.

A.3.4 Final result: Taylor expansion of f (Z)

Replacing the terms:

We put the ε inside the main parenthesis:

We recall that Z ∈ B r so that:

We dene:

and put it in the front, remembering that m = p + q:

We divide by σ p :

We develop everything except the Gaussian in front:

199

We write this:

where:

And:

So that:

and we refer to the previous subsections for the formula of M ab (Z) and m 2 (Z).

A.4 Proof of Theorem A.4.1: Bias on the template shape Now we prove the second theorem given in the paper "Template shape estimation".

Theorem A.4.1 The data X i 's are generated with the model described in the paper "Template shape estimation", where the template shape Y is a parameter and under the assumptions of Theorem 1. The template shape Y is estimated with Ŷ , which is computed by the usual procedure described the paper.

In the regime of an innite number of data n → +∞, the asymptotic bias of the template's shape estimator Ŷ , with respect to the parameter Y , has the following Taylor expansion around the noise level σ = 0: .15) where (i) H is the mean curvature vector of the template shape's orbit which represents the external curvature of the orbit in M , and (ii) ε is a function of σ that decreases exponentially for σ → 0.

We compute the bias Bias(Y, Ŷ ) of Ŷ as an estimator of Y . In the following, we take a NCS at Y . In particular, the vector --→ Y Z = Log Y Z has coordinates written z. The expectation of the distribution f of shapes in Q is Ŷ by denition. The point Ŷ , expressed in a NCS at the template Y , gives Bias(Y, Ŷ ), a tangent vector at T Y M that indicates how much one has to shoot to reach the estimator Ŷ :

First, we take a ball of small radius r in Q and x r. We split the integral:

By the result of the preliminaries, adapted to Q, the right part is a function σ → ε(σ) that is exponentially decreasing for σ → 0. We plug the expression of the density f using Theorem A.3.1:

Computing the a-coordinate of the bias:

We are on a ball of small radius r around Y . We write the Taylor expansions of the F 's terms around z = 0:

We replace these Taylor expansions in the expression of the bias:

Reorganizing:

By the dominated convergence theorem, we can put the inside ε(σ) outside the parenthesis.

We develop the measure on dQ: dQ

201

We develop:

We eliminate the odd terms that give 0 by skewsymmetry:

We delete the terms that will give more than O(σ 2 ) by integration (these are normalized moments) and put them in O(σ 4 ) which is the next order since there is noO(σ 3 ):

We gather the terms in O(σ 4 ):

We recognize the normalized truncated moment of order 2 in the q-dimensional Riemannian manifold Q, see Subsection A.2.3:

We express it with respect to the non-truncated normalized moment in R q , see Subsection A.2.3:

203

where DExp Z | 0 = Id, so that:

We use the denition of the NCS at Y . When Exp Y (u) = U , then U has coordinates u in the NCS at Y . So that: DExp Y | u .u = u. This gives, with u = z: y = -z and we get the Taylor expansion of F 0 (Z) expressed in the coordinate z:

And we identify the term F a 01 (Y ) needed:

A.4.3 Final result: Taylor expansion of the bias Replacing F a 01 (Y ) by its value computed above:

Proofs of the lemmas of Chapter 6

B.1 Lemma 6.3.1 and Conjecture 1 Lemma B.1.1 Take ε > 0 and consider the Lie algebra:

By construction, the exponential of the elements of V I are in the isotropy group of I. For v ∈ V I , we have:

Proof Take v inV I and t ∈] -ε, ε[. Its group exponential is a dieomorphisms in the isotropy group G I , which can be written:

Then, the equation above leads to:

The identication of the coecients in this Taylor expansion leads to: ∇I(x).v(x) = 0

A vector eld of V I , which is in the Lie algebra of the isotropy group of the image I, is perpendicular the image's gradient at any point x of the image's domain Ω.

Conjecture 3 The Lie algebra of the isotropy group G I of the brain image I is constituted of vector elds that are everywhere perpendicular to the image's gradient.

Sketch of Proof 3 First, one needs to rigorously dene the dierential structure of G I and its Lie algebra g I .

Then, the proof will study the relations between the following three sets of vector elds: the Lie algebra g I of G I , the set V I and the set

We have the inclusions V I ⊂ g I and V I ⊂ V ⊥ from Lemma 6.3.1. The proof needs to show: g I ⊂ V I and V ⊥ ⊂ V I . Lemma B.2.1 Assume there exists a dieomorphism ψ that maps the level sets of I 1 to the level sets of I 2 . This implies that ψ maps the partition of the domain I 1 induced by its Morse-Smale to the partition of the domain of I 2 .

Take a part w ⊂ Ω of this partition. There exists a function κ such that:

Proof We consider a part w ⊂ Ω of the partition of the domain of I 1 induced by its Morse-Smale complex. On this part, there exists a function f such that:

3)

The function f gives the mapping of intensity levels on each level set. Dierentiating the previous equation with the chain rule gives:

where df (I 2 • ψ) is a scalar which we note kappa. Taking the adjoint gives:

Conjecture 4 Two images with same MS graphs have same isotropy group.

Sketch of Proof 4 The images I 1 and I 2 have the same MS graph. The graph of I 1 , taken with the nodes and edges positions on Ω, can be dieomorphically deformed on the graph of I 2 . We take ψ 1 a dieomorphism that realizes the graphs' matching. Now, I 1 • ψ (-1) 1 and I 2 share the same MS graph, taken with the nodes and edges' positions on Ω.

We consider one cell of this graph. We consider the integral lines of the respective gradients ∇ψ 1 .∇I 1 • ψ 1 and ∇I 2 on the cell. Both dene a "parallel" partition of the cell. As a consequence, the set of integral lines of the gradient of I 1 • ψ (-1) 1 can be mapped dieomorphically to the set of integral lines of the gradient of I 2 . We take ψ 2 a dieormorphism that realizes the matching of the integral lines. We write ψ = ψ 2 • ψ 1 . At the end of this step, I 1 is transformed to I 1 • ψ (-1) .

The gradients of I 1 • ψ (-1) and I 2 share the same integral lines. Therefore ψ also maps the level sets of I 1 to the level sets of I 2 . Using Lemma 6.3.2, we conclude. As in [Miolane 2017], we can consider that the distribution on the orbits is a Dirac without changing the result of our computations. As we will project back, it does not matter where the data sits on the orbit.

The generative model implies the following Riemannian normal distribution on the objects:

(C.1) The distance d M (X, Y ) expressed in the normal coordinate system at Y is simply d M (X, Y ) 2 = X T X.

To use the result of [Miolane 2017], we truncate the Riemannian Gaussian:

where S is the normalization coecient coming of the univariate truncated Gaussian at σ/2. B M , B O and B Q refer to geodesic balls of radius σ/2 in their respective spaces. We denote C M = S m ( √ 2π) m σ m and remark that C M = C H C O = S q ( √ 2π) q σ q .S p ( √ 2π) p σ p where q is the dimension of the quotient space and p the dimension of the principal orbits.

Our goal is to compute the projected distribution on the shape space. This is done by integrating the distribution on the orbit of X out of F (X):

Plugging this into the expression of the projected distribution:

207

The integrals Q and O can be inverted. First, F (X)p Q (t) is a continuous function with a compact support induced by the truncated Gaussian. Thus, F (X)p Q (t) is dominated by a constant function which is its maximum on the compact. Fubini's theorem on integral inversion gives:

C.2 Proofs of Theorems 7.5.2 and 7.5.3.

We perform the change of metric:

3)

The metric g on M is also G-invariant. Thus we are under the same assumptions that in Theorem 1 and we can follow the computations of its proof. We will show that this new metric is chosen exactly in order to cancel the bias. Intuitively, this change of metric makes the manifold a product space almost everywhere. Also, it becomes a principal bundle. Therefore, there are no crossed terms and what is in the orbit does not aect what happens in the quotient space.

First, we show that under this new metric, there is no external curvature of the principal orbits. Recall that the external curvature is dened as:

where the second equality comes from the fact that orbit and quotient are perpendicular for a G-invariant metric. We show that the Christoel symbol is 0.

where the last line comes from the fact that g ij on the quotient coordinate anymore. Therefore the external curvature is null. As it represents the second derivative of the local graph of the orbit, and this is true everywhere on the orbit, we conclude that this graph is locally null.

208

Because G = 0, we have without Taylor expansion nor approximation:

Then, we add the weights w(X), and thus, following the computations of the proof of Theorem 1, we see that the integral dening the systematic bias can be split:

where the last equality comes from symmetry considerations on the rst integral.