Kuldeep Kumar
email: kkumar@livia.etsmtl.ca

Christian Desrosiers

Kaleem Siddiqi

Olivier Colliot

Matthew Toews

Fiberprint: a subject fingerprint based on sparse code pooling for white matter fiber analysis

Keywords: Subject fingerprint, dMRI, White matter geometry, fiber trajectories, Sparse Code Pooling, HCP, Twin data

White matter characterization studies use the information provided by diffusion magnetic resonance imaging (dMRI) to draw cross-population inferences.

However, the structure, function, and white matter geometry vary across individuals. Here, we propose a subject fingerprint, called Fiberprint, to quantify the individual uniqueness in white matter geometry using fiber trajectories. We learn a sparse coding representation for fiber trajectories by mapping them to a common space defined by a dictionary. A subject fingerprint is then generated by applying a pooling function for each bundle, thus providing a vector of bundle-wise features describing a particular subject's white matter geometry. These features encode unique properties of fiber trajectories, such as their density along prominent bundles. An analysis of data from 861 Human Connectome Project subjects reveals that a fingerprint based on approximately 3 000 fiber trajectories can uniquely identify exemplars from the same individual. We also use fingerprints for twin/sibling identification, our observations consistent with the twin data studies of white matter integrity. Our results demonstrate that the proposed Fiberprint can effectively capture the variability in white matter 1

Introduction

Diffusion magnetic resonance imaging (dMRI) is a powerful and non-invasive tool that provides key information on white matter organization and connectivity based on the diffusion of water molecules in white matter tissues [START_REF] Basser | MR diffusion tensor spectroscopy and imaging[END_REF]. Recent advances in dMRI acquisition protocols have lead to significant improvements in signal reconstruction [START_REF] Assemlal | Recent advances in diffusion MRI modeling: Angular and radial reconstruction[END_REF][START_REF] Tuch | Q-ball imaging[END_REF][START_REF] Wedeen | Mapping complex tissue architecture with diffusion spectrum magnetic resonance imaging[END_REF], driving the development of novel tools for processing and interpreting dMRI data. Among the many applications using dMRI data, the quantitative characterization of white matter geometry and its genetic basis [START_REF] Chiang | Genetics of white matter development: a DTI study of 705 twins and their siblings aged 12 to 29[END_REF][START_REF] Jahanshad | Discovery of genes that affect human brain connectivity: a genome-wide analysis of the connectome[END_REF][START_REF] Thompson | Genetics of the connectome[END_REF] is an important step in the study of the human brain, essential to understanding the mechanisms of neurological function and disease [START_REF] Chung | Principles and limitations of computational algorithms in clinical diffusion tensor MR tractography[END_REF][START_REF] Griffa | Structural connectomics in brain diseases[END_REF][START_REF] Hagmann | Understanding diffusion MR imaging techniques: from scalar diffusionweighted imaging to diffusion tensor imaging and beyond 1[END_REF][START_REF] Thomason | Diffusion imaging, white matter, and psychopathology[END_REF].

Over the years, several approaches have been proposed to provide a simplified quantitative description of white matter connections, to allow for crosspopulation inferences [START_REF] Guevara | Automatic fiber bundle segmentation in massive tractography datasets using a multi-subject bundle atlas[END_REF][START_REF] Jin | Automatic clustering of white matter fibers in brain diffusion MRI with an application to genetics[END_REF][START_REF] O'donnell | Automatic tractography segmentation using a high-dimensional white matter atlas[END_REF][START_REF] Prasad | Automatic clustering and population analysis of white matter tracts using maximum density paths[END_REF]. While numerous studies have focused on elucidating brain connectivity patterns that are shared across people, researchers have also acknowledged the high individual variability in brain structure [START_REF] Amunts | Brodmann's areas 17 and 18 brought into stereotaxic spacewhere and how variable?[END_REF][START_REF] Mangin | A framework to study the cortical folding patterns[END_REF][START_REF] Rademacher | Variability and asymmetry in the human precentral motor system[END_REF], function [START_REF] Barch | Function in the human connectome: task-fMRI and individual differences in behavior[END_REF][START_REF] Grabner | Individual differences in mathematical competence predict parietal brain activation during mental calculation[END_REF][START_REF] Mueller | Individual variability in functional connectivity architecture of the human brain[END_REF][START_REF] Ruiz-Blondet | CEREBRE: A novel method for very high accuracy event-related potential biometric identification[END_REF][START_REF] Rypma | The roles of prefrontal brain regions in components of working memory: effects of memory load and individual differences[END_REF][START_REF] Zilles | Individual variability is not noise[END_REF], and white matter geometry [START_REF] Bürgel | White matter fiber tracts of the human brain: three-dimensional mapping at microscopic resolution, topography and intersubject variability[END_REF][START_REF] De Schotten | Atlasing location, asymmetry and inter-subject variability of white matter tracts in the human brain with mr diffusion tractography[END_REF]. Motivated by this, the concept of connectome fingerprinting, which characterizes individuals using unique connectivity profiles, has recently drawn significant interest from the neuroscience community [START_REF] Armstrong | Brainprint: Assessing the uniqueness, collectability, and perma-nence of a novel method for ERP biometrics[END_REF][START_REF] Finn | Functional connectome fingerprinting: identifying individuals using patterns of brain connectivity[END_REF][START_REF] Miranda-Dominguez | Connectotyping: model based fingerprinting of the functional connectome[END_REF][START_REF] Mišić | From regions to connections and networks: new bridges between brain and behavior[END_REF][START_REF] Wachinger | Brainprint: a discriminative characterization of brain morphology[END_REF][START_REF] Yeh | Local connectome fingerprinting reveals the uniqueness of individual white matter architecture[END_REF][START_REF] Yeh | Quantifying differences and similarities in whole-brain white matter architecture using local connectome fingerprints[END_REF].

So far, most studies on subject fingerprinting have centered around functional [START_REF] Armstrong | Brainprint: Assessing the uniqueness, collectability, and perma-nence of a novel method for ERP biometrics[END_REF][START_REF] Finn | Functional connectome fingerprinting: identifying individuals using patterns of brain connectivity[END_REF][START_REF] Miranda-Dominguez | Connectotyping: model based fingerprinting of the functional connectome[END_REF] and structural data [START_REF] Wachinger | Brainprint: a discriminative characterization of brain morphology[END_REF][START_REF] Toews | How are siblings similar? how similar are siblings? large-scale imaging genetics using local image features[END_REF]. Recently, a novel approach was proposed for building individual connectome profiles based on dMRI data [START_REF] Yeh | Quantifying differences and similarities in whole-brain white matter architecture using local connectome fingerprints[END_REF][START_REF] Yeh | Connectometry: A statistical approach harnessing the analytical potential of the local connectome[END_REF]. This approach uses the Spin Distribution Function (SDF) at each voxel to obtain a fingerprint encoding the diffusion density along a set of prominent directions in cerebral white matter. While it captures key characteristics of white matter diffusivity, this voxel-level fingerprint lacks direct correspondence with white matter bundles, thus hindering an intuitive representation and analysis. As highlighted in [START_REF] Colby | Along-tract statistics allow for enhanced tractography analysis[END_REF], a direct voxelwise comparison of diffusion imaging data could also be challenging, since the high-contrast edges in diffusion MRI volumes (e.g., FA maps) make them more susceptible to small registration errors. Such comparison is also complicated by the anatomical variability of tract positions in subjects.

Building a fingerprint at the level of fiber trajectories, instead of voxels, could provide a more meaningful way of analyzing the unique connectivity properties of individuals from dMRI data. However, working with fiber trajectories also presents additional difficulties, due to the fact that the number and distribution of fiber trajectories may vary across subjects, and fiber trajectories may have very different lengths. Finding a common representation space of fiber trajectories, in different subjects, is essential to overcome these difficulties.

In recent work, we introduced a framework based on sparse coding for the compact representation and cross-population analysis of fiber trajectories [START_REF] Kumar | Brain fiber clustering using nonnegative kernelized matching pursuit[END_REF].

This framework utilizes dictionary learning to build an atlas of fiber bundles from multi-subject dMRI data. Via sparse coding, this atlas can then be used to encode new fiber trajectory data into a compact representation, common to all subjects, and segment these fiber trajectories into prominent bundles [START_REF] Kumar | A sparse coding approach for the efficient representation and segmentation of white matter fibers[END_REF].

In the current paper, we propose to use this framework to characterize the uniqueness in white matter connectivity exhibited by individual subjects, at the level of fiber trajectories. The key idea of our work is to represent each fiber trajectory as a sparse weighted combination of atlas bundles (i.e., the dictionary atoms), and use a pooling function [START_REF] Yang | Linear spatial pyramid matching using sparse coding for image classification[END_REF] to combine the sparse codes of a subject's fiber trajectories into a single feature vector representing bundle-wise properties of fiber trajectory geometry. The resulting fingerprint, called Fiberprint, is used to uniquely identify subjects, as well as to discover inheritable characteristics of fiber geometry by comparing the fingerprints of twins and non-twin siblings.

The use of fiber trajectories as a basis for the proposed subject fingerprint is supported by key studies, such as [START_REF] Bürgel | White matter fiber tracts of the human brain: three-dimensional mapping at microscopic resolution, topography and intersubject variability[END_REF][START_REF] De Schotten | Atlasing location, asymmetry and inter-subject variability of white matter tracts in the human brain with mr diffusion tractography[END_REF], which have shown that the geometry of fiber bundles varies across subjects. However, characterizing an individual subject's white matter fiber geometry via a signature has thus far been elusive.

The main contribution of our work is the use of sparse code pooling to build a subject fingerprint, called Fiberprint. To our knowledge, this is the first study to propose a fingerprint based on fiber geometry. Another notable contribution of this work is the large-scale analysis and validation of our fingerprint, involving a cohort of 861 subjects from Human Connectome Project.

The rest of this paper is organized as follows. We first give an overview of related work on brain fiber analysis, sparse coding, and subject fingerprinting. Section 3 then presents the proposed Fiberprint approach, based on nonnegative kernel sparse coding. In Section 4, we conduct an extensive experimental validation using the dMRI data of 861 subjects from the Human Connectome Project dataset, in which the impact of various parameters of our approach is measured. We also evaluate the usefulness of the proposed fingerprint on the task of subject, twin, and non-twin sibling identification, and use hypothesis testing to find bundles showing significant fingerprint dissimilarities across different subjects groups (i.e., males vs females). In Section 5, we discuss our main observations and experimental findings. We conclude with a summary of our contributions and a discussion of possible extensions.

Related work

Our presentation of relevant work is divided into three parts, focusing respectively on the representation and analysis of white matter fiber geometry, the application of sparse coding techniques in neuroimaging, and the topic of subject fingerprinting.

Representation and analysis of white matter fiber geometry

White matter fiber characterization often assumes an initial abstraction based on tractography, where local diffusion information is used to recover streamlines representing connectivity pathways in the brain [START_REF] Basser | In vivo fiber tractography using DT-MRI data[END_REF][START_REF] Conturo | Tracking neuronal fiber pathways in the living human brain[END_REF][START_REF] Mori | Three-dimensional tracking of axonal projections in the brain by magnetic resonance imaging[END_REF]. Since tractography may output thousands of fiber trajectories, early work has focused on finding simplified quantitative descriptions of white matter connections by grouping fiber trajectories into anatomically meaningful bundles [START_REF] O'donnell | Fiber clustering versus the parcellation-based connectome[END_REF].

Over the years, a wide range of approaches have been proposed to cluster fiber trajectories, including methods based on hierarchical clustering [START_REF] Corouge | Towards a shape model of white matter fiber bundles using diffusion tensor MRI[END_REF][START_REF] Gerig | Analysis of brain white matter via fiber tract modeling[END_REF] and spectral clustering [START_REF] Brun | Clustering fiber traces using normalized cuts[END_REF][START_REF] Jonasson | Fiber tracts of high angular resolution diffusion MRI are easily segmented with spectral clustering[END_REF][START_REF] Odonnell | White matter tract clustering and correspondence in populations[END_REF]. Most of these methods group fiber trajectories using problem-specific measures of similarity, such as the Hausdorff distance [START_REF] Corouge | Towards a shape model of white matter fiber bundles using diffusion tensor MRI[END_REF][START_REF] Gerig | Analysis of brain white matter via fiber tract modeling[END_REF][START_REF] Moberts | Evaluation of fiber clustering methods for diffusion tensor imaging[END_REF] or a mean of closest points (MCP) distance [START_REF] Corouge | Towards a shape model of white matter fiber bundles using diffusion tensor MRI[END_REF][START_REF] Ding | Classification and quantification of neuronal fiber pathways using diffusion tensor MRI[END_REF][START_REF] Gerig | Analysis of brain white matter via fiber tract modeling[END_REF][START_REF] Moberts | Evaluation of fiber clustering methods for diffusion tensor imaging[END_REF].

Various studies have also focused on the segmentation of white matter tracts, toward the goal of drawing cross-population inferences [START_REF] Guevara | Automatic fiber bundle segmentation in massive tractography datasets using a multi-subject bundle atlas[END_REF][START_REF] Jin | Automatic clustering of white matter fibers in brain diffusion MRI with an application to genetics[END_REF][START_REF] O'donnell | Automatic tractography segmentation using a high-dimensional white matter atlas[END_REF][START_REF] Prasad | Automatic clustering and population analysis of white matter tracts using maximum density paths[END_REF]. These studies either follow an atlas based approach [START_REF] Guevara | Automatic fiber bundle segmentation in massive tractography datasets using a multi-subject bundle atlas[END_REF][START_REF] Jin | Automatic clustering of white matter fibers in brain diffusion MRI with an application to genetics[END_REF][START_REF] O'donnell | Automatic tractography segmentation using a high-dimensional white matter atlas[END_REF] or align specific tracts directly across subjects [START_REF] Garyfallidis | Robust and efficient linear registration of white-matter fascicles in the space of streamlines[END_REF][START_REF] Odonnell | Unbiased groupwise registration of white matter tractography[END_REF]. Multi-step or multi-level approaches have also been proposed to segment fiber trajectories, for example, by combining both voxel and fiber trajectory groupings [START_REF] Guevara | Automatic fiber bundle segmentation in massive tractography datasets using a multi-subject bundle atlas[END_REF], fusing labels from multiple handlabeled atlases [START_REF] Jin | Automatic clustering of white matter fibers in brain diffusion MRI with an application to genetics[END_REF], using a white matter voxel-space atlas and a bundle representation based on maximum density paths [START_REF] Prasad | Automatic clustering and population analysis of white matter tracts using maximum density paths[END_REF], or using Gaussian processes [START_REF] Wassermann | Unsupervised white matter fiber clustering and tract probability map generation: Applications of a gaussian process framework for white matter fibers[END_REF]. A few studies have also investigated the representation of specific fiber trajectory bundles using different techniques such as gamma mixture models [START_REF] Maddah | A unified framework for clustering and quantitative analysis of white matter fiber tracts[END_REF], hierarchical Dirichlet processes [START_REF] Wang | Tractography segmentation using a hierarchical dirichlet processes mixture model[END_REF], and the computational model of rectifiable currents [START_REF] Durrleman | A statistical model of white matter fiber bundles based on currents[END_REF][START_REF] Gori | Parsimonious approximation of streamline trajectories in white matter fiber bundles[END_REF].

Sparse coding for neuroimaging

Sparse coding, which aims at encoding a signal as a sparse combination of prototypes in a data-driven dictionary, has been applied in various domains of computer vision and pattern recognition [START_REF] Elad | On the role of sparse and redundant representations in image processing[END_REF][START_REF] Wright | Robust face recognition via sparse representation[END_REF][START_REF] Wright | Sparse representation for computer vision and pattern recognition[END_REF][START_REF] Yang | Linear spatial pyramid matching using sparse coding for image classification[END_REF]. This technique has also shown promise for various neuroimaging applications, such as the reconstruction [START_REF] Lustig | Compressed sensing MRI[END_REF] or segmentation [START_REF] Tong | Segmentation of MR images via discriminative dictionary learning and sparse coding: Application to hippocampus labeling[END_REF] of MRI data, and for functional connectivity analysis [START_REF] Lee | SPARK: Sparsity-based analysis of reliable k-hubness and overlapping network structure in brain functional connectivity[END_REF][START_REF] Lee | Sparse SPM: Group sparse-dictionary learning in SPM framework for resting-state functional connectivity MRI analysis[END_REF]. For diffusion data, sparse coding has been used successfully for clustering white matter voxels from Orientation Density Function (ODF) data [START_REF] Wright | Segmentation of high angular resolution diffusion MRI using sparse riemannian manifold clustering[END_REF], and for finding a population-level dictionary of key white matter tracts [START_REF] Zhu | Population learning of structural connectivity by white matter encoding and decoding[END_REF].

To deal with the challenges of anatomic and tractographic variability, we have recently proposed a framework based on non-negative kernel dictionary learning for grouping fiber trajectories into prominent bundles [START_REF] Kumar | Brain fiber clustering using nonnegative kernelized matching pursuit[END_REF]. This framework encodes individual fiber trajectories as a sparse non-negative combination of dictionary prototypes corresponding to bundles. Unlike other fiber trajectory clustering approaches, which assign fiber trajectories to individual bundles, the proposed framework gives fiber trajectories a membership value to each bundle, thus providing a more intuitive way of dealing with overlapping bundles and inter-subject variability. In a later study, the same framework was used to learn a multi-subject atlas of fiber bundles and for the automatic segmentation of new fiber trajectory data [START_REF] Kumar | A sparse coding approach for the efficient representation and segmentation of white matter fibers[END_REF].

Subject fingerprinting

Most neuroimaging studies collapse multi-subject data to draw inferences about common patterns in a population. Although there are gross similarities, a substantial portion of a subject's connectome is unique to that individual [START_REF] Barch | Function in the human connectome: task-fMRI and individual differences in behavior[END_REF][START_REF] Bürgel | White matter fiber tracts of the human brain: three-dimensional mapping at microscopic resolution, topography and intersubject variability[END_REF][START_REF] Grabner | Individual differences in mathematical competence predict parietal brain activation during mental calculation[END_REF][START_REF] Mangin | A framework to study the cortical folding patterns[END_REF][START_REF] Mueller | Individual variability in functional connectivity architecture of the human brain[END_REF][START_REF] Rademacher | Variability and asymmetry in the human precentral motor system[END_REF][START_REF] Ruiz-Blondet | CEREBRE: A novel method for very high accuracy event-related potential biometric identification[END_REF]. A recent study has shown that functional connectivity profiles act as robust and reliable fingerprints that can identify individual subjects from a large group [START_REF] Finn | Functional connectome fingerprinting: identifying individuals using patterns of brain connectivity[END_REF]. In this study, a functional brain atlas was employed to define target brain regions. The Pearson correlation coefficients between the time courses of region pairs were then computed, and used as a functional connectivity profile. This fingerprint was able to identify individuals across scan sessions, both for task and rest conditions.

In [START_REF] Wachinger | Brainprint: a discriminative characterization of brain morphology[END_REF], Wachinger et al. proposed Brainprint, a subject fingerprint that characterizes brain morphology by calculating the spectrum of the Laplace-Beltrami operator on meshes from cortical and subcortical brain structures.

This fingerprint was used to study morphological similarity between brains, with applications in subject identification across multiple scans of the same subject (achieving a classification accuracy of up to 99.9%), and the analysis of potential genetic influences on brain morphology. While the majority of fingerprint studies have focused on functional and structural data, a local connectome fingerprint using Spin Distribution Function (SDF) voxel profiles obtained from dMRI data has recently been proposed in [START_REF] Yeh | Local connectome fingerprinting reveals the uniqueness of individual white matter architecture[END_REF][START_REF] Yeh | Connectometry: A statistical approach harnessing the analytical potential of the local connectome[END_REF]. This local fingerprint is built by sampling, at each voxel, the diffusion density of water along principal directions in the white matter, defined using a common fiber-direction atlas. The proposed fingerprint was used for quantifying the similarity between genetically-associated individuals, as well as measuring neuroplasticity over time, and was shown to vary substantially across individual subjects compared to traditional diffusivity measures like Fractional Anisotropy (FA). However, since this fingerprint is built using voxel-level information, it lacks a direct correspondence with white matter bundles, and a direct voxel-level comparison of diffusion imaging data could be challenging, as the high-contrast edges of diffusion MRI volumes (e.g., FA maps) make them more susceptible to small misregistration errors, as well as to anatomical variability of tract positions in health and disease [START_REF] Colby | Along-tract statistics allow for enhanced tractography analysis[END_REF]. Another point is that this fingerprint tries to capture both voxel-level diffusivity information and morphology. To our knowledge, the present study is the first to propose a white matter geometry fingerprint at the level of fiber trajectories and fiber bundles. Connectome Project [START_REF] Van Essen | The human connectome project: a data acquisition perspective[END_REF][START_REF] Van Essen | The WU-minn human connectome project: an overview[END_REF]. Second, a dictionary of prototype fiber trajectories is then learned from a subset of subjects, based on our non-negative kernel dictionary learning framework. This dictionary can be seen as an atlas for modeling and analyzing the geometry of fiber trajectories from multiple subjects, along prominent bundles. In the third step, the learned dictionary is used to encode the fiber trajectories of the remaining subjects in a common feature space, via a sparse coding method. A fingerprint is then obtained, for each subject, by applying a pooling function to the sparse codes corresponding to each subject's fiber trajectories. This pooling function allows the comparison of subjects having a different number of fiber trajectories by aggregating the information from all fiber trajectories in a single fixed-size vector. The resulting fingerprint corresponds to an estimate of fiber trajectory density along key bundles defined by the atlas. Finally, in the last step, fingerprints are used to identify unique characteristics of genetically-related subjects, or for finding bundles showing significant differences across various subject groups (e.g., male vs female). The following subsections describe each of these steps in greater detail.

Materials and methods

Data and pre-processing

We used the pre-processed dMRI data of 861 subjects (482 females, 378 male and 1 unknown, age [START_REF] Ruiz-Blondet | CEREBRE: A novel method for very high accuracy event-related potential biometric identification[END_REF][START_REF] Rypma | The roles of prefrontal brain regions in components of working memory: effects of memory load and individual differences[END_REF][START_REF] Zilles | Individual variability is not noise[END_REF][START_REF] Bürgel | White matter fiber tracts of the human brain: three-dimensional mapping at microscopic resolution, topography and intersubject variability[END_REF][START_REF] De Schotten | Atlasing location, asymmetry and inter-subject variability of white matter tracts in the human brain with mr diffusion tractography[END_REF][START_REF] Armstrong | Brainprint: Assessing the uniqueness, collectability, and perma-nence of a novel method for ERP biometrics[END_REF][START_REF] Finn | Functional connectome fingerprinting: identifying individuals using patterns of brain connectivity[END_REF][START_REF] Miranda-Dominguez | Connectotyping: model based fingerprinting of the functional connectome[END_REF][START_REF] Mišić | From regions to connections and networks: new bridges between brain and behavior[END_REF][START_REF] Wachinger | Brainprint: a discriminative characterization of brain morphology[END_REF][START_REF] Yeh | Local connectome fingerprinting reveals the uniqueness of individual white matter architecture[END_REF][START_REF] Yeh | Quantifying differences and similarities in whole-brain white matter architecture using local connectome fingerprints[END_REF][START_REF] Toews | How are siblings similar? how similar are siblings? large-scale imaging genetics using local image features[END_REF][START_REF] Yeh | Connectometry: A statistical approach harnessing the analytical potential of the local connectome[END_REF] from the Q3 release of the Human Connectome Project [START_REF] Glasser | The minimal preprocessing pipelines for the human connectome project[END_REF][START_REF] Van Essen | The human connectome project: a data acquisition perspective[END_REF][START_REF] Van Essen | The WU-minn human connectome project: an overview[END_REF], henceforth referred to as HCP data. All HCP data measure For signal reconstruction and tractography, we used the freely available DSI Studio toolbox. All subjects were reconstructed in MNI space using the Q-space diffeomorphic reconstruction (QSDR) [START_REF] Yeh | NTU-90: a high angular resolution brain atlas constructed by q-space diffeomorphic reconstruction[END_REF] option in DSI Studio. QSDR is an extension of generalized q-sampling imaging (GQI, [START_REF] Yeh | Generalized-sampling imaging[END_REF]), allowing the construction of spin distribution functions (SDF) in a given template space. DSI Studio first calculates the quantitative anisotropy (QA) mapping in the native space and then normalizes it to the MNI QA map using SPM normalization [START_REF] Ashburner | Computational neuroanatomy[END_REF]. We As a note, whether these fiber trajectories represent the actual white matter pathways remains a topic of debate [START_REF] Jones | White matter integrity, fiber count, and other fallacies: the do's and don'ts of diffusion mri[END_REF][START_REF] Thomas | Anatomical accuracy of brain connections derived from diffusion mri tractography is inherently limited[END_REF]. Fiber trajectories derived from DSI studio are hypothetical curves in space that represent, at best, the major axonal directions suggested by the orientation distribution functions of each voxel, which may contain tens of thousands of actual axonal fibers.

used

Learning the fiber trajectory dictionary

Out of the 861 available subjects, 10 unrelated ones [START_REF] Odonnell | Automated white matter fiber tract identification in patients with brain tumors[END_REF] were used to learn the dictionary of fiber trajectory prototypes, serving as a multi-subject atlas to map new fiber trajectory data to a common space. The learning process is based on the non-negative kernel dictionary learning method presented in [START_REF] Kumar | Brain fiber clustering using nonnegative kernelized matching pursuit[END_REF][START_REF] Kumar | A sparse coding approach for the efficient representation and segmentation of white matter fibers[END_REF], which we now summarize.

Let X be the set of n training fiber trajectories, represented as a set of 3D coordinates. For the purpose of explanation, we suppose that each trajectory i is encoded as a feature vector x i ∈ R d , and that X is a d × n feature matrix.

Since our dictionary learning method is based on kernels, a fixed set of features is however not required, and fiber trajectories having a different number of 3D coordinates could be compared with a suitable similarity measure (i.e., the kernel function).

In the proposed model, each fiber trajectory can be described as a sparse linear combination of m prototype fiber trajectories in a dictionary D. Formally, we write this as x i ∼ Dw i , where w i is a sparse vector of non-negative weights representing the fiber trajectory's relationship to each prototype. Since fiber trajectories may have very different lengths and endpoints, encoding them using a fixed set of features can be challenging. To avoid this problem, we embed them into a q-dimensional Hilbert space via a mapping function φ :

R d → R q , such that φ(x) ⊤ φ(x ′ ) = k(x, x ′ ) is a kernel function.
The main advantage of this approach is that fiber trajectories can now be represented based on a similarity measure tailored to this type of data, such as the Hausdorff distance [START_REF] Corouge | Towards a shape model of white matter fiber bundles using diffusion tensor MRI[END_REF][START_REF] Gerig | Analysis of brain white matter via fiber tract modeling[END_REF][START_REF] Moberts | Evaluation of fiber clustering methods for diffusion tensor imaging[END_REF], the mean of closest points (MCP) distance [START_REF] Corouge | Towards a shape model of white matter fiber bundles using diffusion tensor MRI[END_REF][START_REF] Ding | Classification and quantification of neuronal fiber pathways using diffusion tensor MRI[END_REF][START_REF] Gerig | Analysis of brain white matter via fiber tract modeling[END_REF][START_REF] Moberts | Evaluation of fiber clustering methods for diffusion tensor imaging[END_REF] or the Minimum average Direct Flip (MDF) distance [START_REF] Garyfallidis | Quickbundles, a method for tractography simplification[END_REF]. In this work, we considered the MDF distance, which computes the average distance between points on a fiber trajectory and corresponding points in a second fiber trajectory, or in the reverse point sequence of the second fiber trajectory if it leads to a smaller distance. A Gaussian kernel was used to convert distances to similarities, i.e.

k(x, x ′ ) = exp -γ • dist MDF (x, x ′ ) .
The fiber trajectories were sampled to 15 equidistant points for distance computation [START_REF] Garyfallidis | Quickbundles, a method for tractography simplification[END_REF] and the kernel bandwidth parameter was set empirically to γ = 0.0001.

Using Φ ∈ R q×n to denote the matrix of mapped training fiber trajectories, the kernel matrix of pairwise similarities then corresponds to K = Φ ⊤ Φ. Using the idea proposed in [START_REF] Nguyen | Kernel dictionary learning[END_REF], we express the dictionary as a non-negative linear combination of training examples, i.e., D ∼ ΦA, and formulate the dictionary learning task as the following optimization problem:

arg min A, W ≥ 0 1 2 ∥Φ -ΦAW ∥ 2 F s.t. ∥w i ∥ 0 ≤ S max , i = 1, . . . , n, (1)
where ∥w i ∥ 0 is the L 0 norm (i.e., number of non-zero elements) of w i , constraining each fiber trajectory to be encoded using at most S max prototypes,

A ∈ R n×m is the dictionary coefficient matrix, and W ∈ R m×n is the sparse code matrix for all fiber trajectories. When S max = 1, this formulation corresponds to the kernel K-means problem [START_REF] Dhillon | Kernel k-means: spectral clustering and normalized cuts[END_REF]. As shown in Section 4.1.4, expressing fiber trajectories using more than one prototype (i.e., S max > 1) provides a better representation of complex bundles, leading to a more discriminative fingerprint.

This problem is solved using the method described in [START_REF] Kumar | Brain fiber clustering using nonnegative kernelized matching pursuit[END_REF], which updates the sparse codes W and dictionary matrix A iteratively, until convergence. In the sparse coding step, each column of W is updated independently by optimizing the following sub-problem:

arg min wi ≥ 0 1 2 w ⊤ i A ⊤ KAw i -k ⊤ i Aw i s.t. ∥w i ∥ 0 ≤ S max , (2)
where k i ∈ R n is the vector containing the similarities between fiber trajectory i and all training fiber trajectories. This problem is solved heuristically using a non-negative kernel Orthogonal Matching Pursuit (NKOMP) algorithm [START_REF] Kumar | Brain fiber clustering using nonnegative kernelized matching pursuit[END_REF].

The dictionary matrix A is then obtained using a kernel version of the nonnegative matrix tri-factorization approach proposed in [START_REF] Ding | Orthogonal nonnegative matrix tfactorizations for clustering[END_REF], which applies the following update scheme until convergence:

A ij ← A ij • KW ⊤ ij (KAW W ⊤ ) ij , ∀ i, j. (3)
Due to machine precision, the above update scheme produces small positive values instead of zero entries. To resolve this problem, a small threshold is applied on A.

Since the kernel contains the similarities between each pair of fiber trajectories (50 000 × 10 fiber trajectories, squared), computing it directly is impracticable. Instead, we start with 5 000 fiber trajectories sampled uniformly from each subject, and approximate the resulting kernel matrix (50 000 × 50 000) using Nystrom's method [START_REF] Fowlkes | Spectral grouping using the nystrom method[END_REF][START_REF] O'donnell | Automatic tractography segmentation using a high-dimensional white matter atlas[END_REF]. This method starts with defining a subset of fiber trajectories and computing the pairwise similarities between each training fiber trajectory and this sampled subset. The missing entries in kernel matrix K are then estimated using a low-rank approximation process based on SVD. Using this technique, the entire dictionary learning process takes about 1 000 seconds on a quad-core 3.6 GHz computer with 32 GB of RAM. 2), each one corresponding to a different color. To generate this figure, we convert the soft assignment defined in A to a hard clustering, by assigning each fiber trajectory i to the prototype j for which a ij is maximum 3 . We see that the fiber trajectory clusters defined by the dictionary are reasonably consistent with prominent neuroanatomical bundles, such as the corpus callosum, cingulum, corticospinal tract and superior cerebellar penduncle. Note, however, that a one-to-one relationship does not always hold between these prototypes and neuroanatomical bundles: complex bundles may be represented using multiple prototypes. Nonetheless, to simplify the presentation, we use the term bundle dictionary when referring to the output of the dictionary learning step.

Generating the subject fingerprints

The generation of a fingerprint from the fiber trajectory data of a new subject is composed of two steps: sparse coding of fiber trajectories and sparse code pooling.

Sparse coding of fiber trajectories

In the first step, the learned dictionary is used to map the fiber trajectories of a given subject to a common feature space defined by the dictionary's bundles. This encoding process consists of solving the sparse coding problem of Eq.

(2), which has been used for dictionary learning. Since each fiber trajectory is represented using at most S max coefficients, this re-encoding of a subject's fiber trajectory data is very compact.

The fiber trajectory sparse codes of four different subjects, obtained using the dictionary of Figure 2, are illustrated in Figure 3. We represent bundles using the same colors as in Figure 2, and assign each fiber trajectory i to the bundle for which w ji is maximum, where W is the sparse code matrix of a given subject. This hard assignment of fiber trajectories to dictionary bundles corresponds to the fiber trajectory segmentation approach presented in [START_REF] Kumar | A sparse coding approach for the efficient representation and segmentation of white matter fibers[END_REF]. The strength of the relationship between fiber trajectories and individual bundles can also be visualized by considering the values in each row of W . In Figure 4, the sparse code values (i.e., rows of W ) corresponding to the left and right corticospinal bundles are color coded such that fiber trajectories having a high membership to a bundle are red and those having a low membership are green (fiber trajectories with zero membership are not shown). These figures highlight the implicit correspondence of bundles across subjects, as well as the variability in the fiber trajectory geometry of bundles, observed for different subjects.

Sparse code pooling

Because subjects may have a different number of fiber trajectories, to allow comparison across subjects, the sparse codes for fiber trajectories obtained in the previous step must be aggregated in a fixed-size feature vector. This is achieved using a sparse code pooling function [START_REF] Yang | Linear spatial pyramid matching using sparse coding for image classification[END_REF] that combines, for each dictionary bundle, the relationship between this bundle and all fiber trajectories of a subject into a single value. Let W ∈ R m×n be the sparse code matrix obtained 310 in the previous step, each column corresponding to a different fiber trajectory of the subject to encode. We consider three pooling functions frequently used in the literature, based on the root mean square (RMS), mean and maximum: and right (bottom row) corticospinal bundles from four subjects. Green and red represent, a low and a high membership of a fiber to a bundle, respectively. Fibers with a zero membership to the bundle are removed for a simplified visualization.

f RMS (W ) j = 1 n n i=1 w 2 ji (4) f Mean (W ) j = 1 n n i=1 |w ji | (5) f Max (W ) j = max |w j1 |, |w j2 |, . . . , |w jn | . (6)
1 where [f (W )] j is the pooled feature corresponding to the j-th dictionary bundle.

Each of these pooling functions encodes a different property of a subject's fiber trajectory distribution along the dictionary bundles. Function f mean computes the average sparse code value of fiber trajectories belonging to a bundle, thus giving an estimate of the bundle's density. f RMS is another measure of density, which gives a greater importance to large magnitude values in W . Finally, f max selects the maximum sparse code value over all fiber trajectories in relationship to a given bundle. In practice, this value will be low for dictionary prototypes which are not useful for encoding a subject's fiber trajectories.

Figure 5 shows a bar plot representation of fingerprints obtained using the three pooling functions, for four different subjects. We observe small but meaningful differences when comparing these fingerprints, supporting the hypothesis that they encode unique characteristics of fiber trajectory geometry. Moreover, we see that the pooling functions capture different properties (in particular the max pooling function) and have varying responses across bundles. The uniqueness of subject fingerprints can be further visualized in Figure 6, which color codes the fiber trajectory bundles of the four subjects based on the magnitude of their corresponding RMS pooling function values. We observe that the bundles showing the highest response are consistent across subjects, although the magnitude of these responses differs from one subject to another.

RMS Mean Max

Experiments and results

335

In this section, we test the hypothesis that the proposed subject fingerprint can effectively capture a particular subject's white matter fiber geometry. Because there are many parameters and factors involved in the generation of fingerprints (e.g., pooling function, dictionary size, and fiber tracking approach), we first perform an analysis to assess the robustness of our fingerprint to these 340 various parameters and factors. We then validate our main hypothesis using the task of subject identification and twin identification. Specifically, we try to determine if an individual can be identified using the proposed fingerprint, and whether this fingerprint can discriminate between twin and non-twin siblings.

In the process, we also analyze important properties of our fingerprint, such spheres, required to characterize a subject's fiber trajectory geometry. Finally, we conduct a significance testing analysis to identify fiber trajectory bundles which show important differences related to the genetic proximity of siblings (i.e., twins vs non-twins), and subject gender (i.e., males vs females).

Impact of method parameters

We first analyze the impact of various parameters on the proposed subject fingerprint's ability to discriminate between subjects. The following parameters are considered in our analysis: the pooling function (i.e., RMS, Mean or Max), the dictionary size (i.e., m), the sets of dictionary learning subjects, the fiber trajectory representation sparsity (i.e., S max ), the inclusion/exclusion of cerebellar white matter, the fiber tracking parameters, and the number of fiber trajectories used to generate the fingerprint.

The fingerprint's discriminability is measured quantitatively as follows. First, the 50 000 fiber trajectories of each subject (i.e., the 851 subjects not used for training the dictionary) are randomly divided into 5 instances, each one containing 10 000 fiber trajectories. These instances are then converted to subject fingerprints using the sparse coding and pooling process of Section 3.3, giving a total of 851 × 5 = 4 255 fingerprints. Each of these fingerprints is a vector of m features, one for each dictionary bundle. We use the Euclidean distance between two fingerprints to measure their similarity, and evaluate the separability of the proposed approach by comparing the distribution of distances between samesubject instances and instances obtained from different subjects. The d-prime sensitivity index [START_REF] Gale | A basal ganglia pathway drives selective auditory responses in songbird dopaminergic neurons via disinhibition[END_REF] is used to obtain a quantitative measure of separability: Overall, this analysis shows that fingerprints obtained using the RMS and Mean pooling functions are significantly more similar for same-subject instances than instances from different subjects, and that the RMS function slightly outperforms Mean. As mentioned above, both functions estimate the fiber trajectory density along prominent bundles defined by the dictionary. In contrast, the Max function leads to a poorly discriminative fingerprint. This could be due to the fact that features corresponding to each bundle are estimated using a single fiber trajectory with maximum sparse code magnitude, which does not capture the full variability in bundle geometry across subjects. The RMS pooling function was used for the remaining experiments of this study.

d-prime = µ 1 -µ 2 1 2 σ 2 1 + σ 2 2 , (7) where

Dictionary size

The size of the dictionary (i.e., parameter m), which reflects the number of different bundles that can be captured by the encoding, can also impact the quality of the fingerprint: a small number of bundles may be insufficient to capture subtle differences between subjects, while having a large number of bundles can affect the performance of the dictionary learning and sparse coding steps.

We tested seven different dictionary sizes, i.e. m = 10, 25, 50, 75, 100, 125, 150, while keeping the number of fiber trajectories per subject to 50 000. Note that varying m affects the number of fiber trajectories per bundle, as well as the number of features in subject fingerprints. Figure 8 (left) shows the box plot of Euclidean distances between same-subject (red) and different-subject (blue) instances, for the tested dictionary sizes. We observe that the separation between same-subject and different-subject distance distributions increases slightly with the number of bundles, mostly due to a decrease in variance for distances between different-subject instances. In summary, the separability of our subject fingerprint remains significant for dictionaries sizes of m ≥ 50, and using a higher number of bundles may improve the consistency of the fingerprint. A dictionary size of m = 50 was used for the remaining experiments.

Independent dictionary sets

Since white matter geometry varies across individuals, changing the subjects used for learning the dictionary can also impact our fingerprint. To measure this impact, we created 5 different dictionaries learned from independent sets of 10 subjects, while keeping the sampling strategy and other parameters to their default values (m = 50). Figure 9 (top left) shows the box plot of Euclidean distances between same-subject and different-subject instances using each of these dictionaries. We observe no significant difference across dictionaries, demonstrating the robustness of our fingerprint to the choice of dictionary subjects.

Encoding sparsity

In the fiber trajectory encoding process, parameter S max controls the level of sparsity, i.e., the maximum number of dictionary prototypes used to encode a given fiber trajectory. This parameter can also be interpreted as the maximum number of bundles to which a fiber trajectory can be assigned, thereby providing a soft fiber-to-bundle assignment for S max > 1.

To evaluate the impact of sparsity, we varied parameter S max from 1 to 6, both for learning the dictionary and encoding new fiber trajectory data. Figure 8 (right) shows the box plots of distances between same-subject and differentsubject instances, obtained from 10 subjects. We observe that the separability increases with S max and saturates around S max = 4 (Box plots for m = 100 can be found in the supplementary materials). These results indicate that having a soft fiber-to-bundle assignment is necessary to capture the complex topology of bundles, which may cross or overlap one another. Since a maximum d-prime value was obtained for S max = 4, this sparsity level was kept for the following experiments.

Fiber tracking parameters

We analyzed the robustness of the proposed method to various fiber tracking parameters, for a given QSDR based signal reconstruction (in MNI space) and a fixed dictionary. For this purpose, we generated fingerprints based on the fiber trajectories of 10 subjects, obtained by varying the following parameters: the number of output fiber trajectories (from 30 000 to 150 000), the deterministic fiber tracking approach (Runge-Kutta -RK4 or Euler [START_REF] Basser | In vivo fiber tractography using DT-MRI data[END_REF][START_REF] Yeh | Deterministic diffusion fiber tracking improved by quantitative anisotropy[END_REF]), the turning angle threshold (from 15 to 75 degrees), and the minimum length of fibers (from 20 to 250 mm). A single parameter was varied at a time, all other ones set to the value used in the previous experiments.

Figure 9 summarizes the results of this analysis, leading us to the following observations. First, we notice that the separation between same-subject (red) and different-subject (blue) instances remains similar for numbers of output fiber trajectories of 30 000 or more. Moreover, the separability of our fingerprint is nearly the same for both the RK4 and Euler fiber tracking approaches. For the turning angle threshold, the separation between the medians of the two distributions decreases as we increase the threshold's value. Increasing this threshold may lead to the generation of fibers with large curvature or very small length, which are significantly different from other fibers in the same bundle.

Encoding these fibers can therefore add noise to the sparse code representation of subjects, resulting in a reduced separability.

450

Results also show a higher separation for larger values of minimum fiber trajectory length. As highlighted in several fiber-related studies [START_REF] Garyfallidis | Quickbundles, a method for tractography simplification[END_REF][START_REF] Odonnell | Automated white matter fiber tract identification in patients with brain tumors[END_REF], fiber trajectories below 40 mm in length represent short-range connections, having lower clinical relevance (e.g., surgical planning). In applications like automated fiber grouping, such fiber trajectories may pose a considerable challenge [START_REF] Garyfallidis | Quickbundles, a method for tractography simplification[END_REF]. For long fiber trajectories (i.e., 80 mm to 250 mm), we observe a similar trend where the distance between distribution medians increases with minimum fiber length.

However, the separation in terms of d-prime does not increase monotonically due to a higher variance in different-subject distances. Note that this phenomenon could also be explained by the fact that the dictionary used in this experiment was generated with a minimum fiber length of 40 mm. Overall, we observe that the fingerprints are quite separable across a large range of variations in these parameters.

Inclusion of cerebellum

The inclusion of fiber trajectories from cerebellar white matter could also impact the proposed fingerprint, due to the variability in cerebellum slice coverage across subjects. Figure 10 gives the normalized histograms and box plots of distances between same-subject and different-subject instances of all 851 subjects, obtained with and without considering the cerebellum. Fingerprints without cerebellum were obtained from the full fingerprints by removing the features corresponding to fiber trajectory bundles in the cerebellum. These bundles were determined by visual inspection of bundles in the dictionary. These results

show a small decrease in separability when excluding cerebellum fiber trajectories (d-prime from 4.347 to 3.995), which could be due to the reduction in the number of bundles from 50 to 44, and also the reduction in total number of fiber trajectories contributing to the fingerprint. Nevertheless, the fingerprints generated without information from the cerebellum still exhibits significant differences across subjects.

Number of fingerprint fiber trajectories

Since the fingerprint (with RMS or Mean pooling) estimates the fiber trajectory density along specific bundles, another relevant question is the impact of the number of fiber trajectories n used to generate the fingerprint. If this number is low, relative to the number of bundles, it may not be possible to get an accurate measure of fiber trajectory density. To determine how this parameter affects the fingerprint's separability, we generated fingerprints for all 851 subjects using sub-samples of the subject's fiber trajectories. For every subject, five instances were created for fiber trajectory sub-sample sizes ranging from n = 100 to 10 000.

Figure 11 (left) gives the box plot of distances between same-subject and different-subject instances. We observe that the separability (i.e., d-prime) increases steadily with the number of fiber trajectories n. Moreover, we notice that separability measures increase only slightly after n = 3 000, suggesting this to be the minimum number of fiber trajectories necessary to obtain a discriminative fingerprint (for a dictionary size of m = 50). To understand how

d-prime

Figure 11: Impact of the number of fiber trajectories used to generate a subject fingerprint.

Box plot for Euclidean distances between same-subject (red) and different-subject (blue) instances for all 851 subjects (left). Bar plot of RMS pooled features corresponding to four different bundles of a subject, obtained with varying numbers of fiber trajectories (right).

the number of fiber trajectories affects the fingerprint, Figure 11 (right) shows the RMS pooled features corresponding to four different bundles of a subject, obtained with varying numbers of fiber trajectories. We observe that pooled features stabilize for n ≥ 3 000, confirming our previous hypothesis.

Subject identification

The experiments presented in previous sections showed the robustness of the proposed subject fingerprint to various parameters. In this section, we apply our fingerprint to the task of identifying subjects and pairs of genetically-related subjects (i.e., twins and non-twin siblings). The objective of this analysis is two-fold: to demonstrate that the fingerprint captures characteristics of white matter geometry which can uniquely identify a subject, and to show that some of these characteristics are inheritable.

Toward this goal, we use the fingerprints obtained from each of the 4255 instances of fiber trajectory data (i.e., 851 subjects with 5 instances each), and perform a ranked retrieval analysis based on the k-nearest neighbors of a fingerprint. Given a subject and a target group (i.e., same subject, twins or non-twin siblings), we consider each of the subject's instances individually, and rank the remaining 4254 instances by their similarity to this subject instance (using the Euclidean distance between their fingerprints). Denote as T the set of instances in the target group, and let S k be the set containing the k most similar instances. We evaluate the retrieval performance of the fingerprint, for a specific value of k, using the measures of precision and recall:

precision@k = |T ∩ S k | k , recall@k = |T ∩ S k | |T | . ( 8
)
We report the mean precision@k and recall@k, computed over all subjects and instances.

Same subject identification

Table 1 gives the mean precision of the fingerprint for identifying same subject instances, using a single nearest neighbor (i.e., precision@1). In other words, we measure the frequency at which the nearest neighbor of an instance belongs to the same subject. Precision values are reported for a varying number of fiber trajectories used to generate the fingerprints (i.e., parameter n), as well as for fingerprints generated with and without cerebellum fiber trajectories. Furthermore, to evaluate the contribution of fiber trajectories across brain hemispheres, we also report the precision of fingerprints obtained using only fiber trajectories from the left hemisphere (17 bundles) or right hemisphere (15 bundles), as well as those obtained using only inter hemispheric fiber trajectories (12 bundles located mostly in the corpus callosum). Note that we obtained hemispherespecific fingerprints from the full brain fingerprint by keeping only the features corresponding to bundles within these hemispheres. As mentioned earlier, these bundles were identified by visualization of all dictionary bundles. Finally, to evaluate the chance factor, we also computed the precision obtained from 1 000 random lists of nearest neighbors (i.e., the first k entries in a random permutation), using all n = 10 000 fiber trajectories.

We observe that a mean precision@1 of 100% is achieved, both with and without cerebellum fiber trajectories, when n = 3 000 or more fiber trajectories are used to generate the fingerprints. Below this number, the precision decreases monotonically to 1.0% for n = 100. Since a maximum precision@1 of 0.4% was obtained for the randomly generated lists of k-nearest neighbors, we conclude that these results are significant. Furthermore, we see that the precision reduces significantly when considering only fiber trajectories from the left or right hemispheres, or just inter-hemispheric fiber trajectories. Once again, this could be due to the smaller number of features in these hemisphere-specific fingerprints, 535 which reduces their ability to differentiate subjects. Nevertheless, for n = 10 000 full-brain fiber trajectories, fingerprints generated using only single-hemisphere or inter-hemispheric fiber trajectories achieve a mean precision@1 above 95%, suggesting that characteristics unique to a subject are located in both hemispheres, as well as in crossing bundles like the corpus callosum. Comparing 540 values across hemispheres, we notice a higher precision in the left hemisphere (e.g., precision@1 of 77.6 for n = 3 000, versus 67.4 for the right hemisphere).

To determine whether handedness could be a factor in this difference (i.e., 781 of the 851 subjects are right-handed), we repeated this experiment using 80 left-handed and 80 right-handed subjects. Results obtained with this setup are similar to those observed for the entire set of subjects (see Table 1 of Supplementary materials), indicating that this bilateral asymmetry is independent of subject handedness.

To analyze the robustness of our fiberprint to alignment and signal reconstruction, we generated new fingerprints for two subjects using different methods for these pre-processing steps, and tried to re-identify these two subjects with their original fingerprints. The new fingerprints were obtained by aligning the diffusion data of the subjects to the HCP 842 template4 (MNI space, 1mm resolution, similar to the QSDR reconstruction output) using FSL [83] flirt with 12 DOF affine transform (first aligning T1w images, and then applying the affine transform to diffusion data using the applyxfm4D option). We then performed DTI signal reconstruction followed by RK4 streamline tracking (FA threshold 0.2, other parameters are kept the same). Five fingerprint instances were generated for each subject, each one obtained by randomly subsampling 5 000 fiber trajectories (see Section 3.3 for details). Note that the same dictionary as in previous experiments was employed for obtaining these fingerprints. For every subject having a MZ, DZ or NT sibling, we used a single instance, and obtained a measure of recall@k, for k = 1, . . . , 30, by counting the ratio of MZ, DZ or NT sibling subjects within the list of k-nearest neighbors. As in the previous experiment, the chance factor was considered by computing the maximum recall@k value obtained from 1 000 random lists of nearest neighbors. Moreover, a higher recall is obtained for DZ twins, in comparison to non-twin on fingerprint similarity are significantly higher than those computed from random lists of nearest neighbors, validating the significance of these results.

Unlike non-twin siblings, DZ twins have the same age, a confound which might bias our analysis. To measure the true impact of this factor, we divided pairs of NT siblings in two groups based on their age difference: 0 ≤ ∆age 1 ≤ 3 and 3 < ∆age 2 ≤ 11. Figure 14 (right) gives the recall@k values obtained for these two groups. It can be seen that NT siblings having greater age differences lead to a slightly higher recall (not statistically significant), and that recall values in both groups are significantly smaller than those observed for DZ twins, thereby eliminating age as a possible bias.

To substantiate these observations, Figure 15 and [-0.0154, -0.0113], for BMZ vs BDZ, BMZ vs BNT, and BDZ vs BNT, respectively. Overall, this analysis shows that the proposed fingerprint captures genetically-related information on the geometry of white matter.

Comparison with a global fingerprint based on T1-weighted images

To compare our Fiberprint with a standard morphological approach, we used the T1-weighted images (rigidly aligned to MNI space) of subjects as fingerprint and computed nearest neighbors based on the sum of squared differences (SSD) between aligned images. Figure 14 (middle) shows the mean recall@k, for k = 1, . . . , 30, obtained by this fingerprint for identifying MZ, DZ and NT siblings.

We observe higher recall values for the fingerprint using T1-weighted images, compared to our Fiberprint, the most substantial differences obtained for monozygotic twins. This confirms that global brain geometry, as captured by versus m = 50 features for our Fiberprint), and contains a lot of information unrelated to connectivity (e.g., skull, non-white matter brain regions, etc.). In contrast, the proposed Fiberprint is highly compact and thus suitable for largescale datasets. Moreover, it can be employed to compare subjects specifically on the level of structural connectivity, rather than global geometry.

To further assess the informativeness of our fiberprint compared to a fingerprint based on whole T1-weighted images, we computed the number of distinct and common sibling pairs (MZ/DZ/NT) identified by these two fingerprints.

Toward this goal, we used the same lists of nearest neighbors as in Figure 14 and considered the identification of a sibling as successful if this sibling's fingerprint is found within the k = 30 nearest neighbors.

Table 2 reports the proportion of subjects for each category (mean over 5 fiberprint instances). It can be seen that the proposed fiberprint provides information complementary to the fingerprint based on raw T1 intensities, finding around 15% of siblings not identified by this fingerprint. Conversely, about 20% of siblings are identified only by the whole-image fingerprint. In summary, both fingerprints capture unique information of the similarity of genetically-related subjects.

Bundle-wise significance analysis

As mentioned before, the proposed fingerprint encodes fiber trajectory geometry along bundles defined by the dictionary. In this section, we evaluate the significance of individual bundles by comparing the distribution of fingerprint features in instances corresponding to different subject groups (e.g., DZ twins vs non-twin siblings, male vs female, etc.).

This bundle-wise analysis of significance uses the distributions of fingerprint features corresponding to specific bundles, in instances belonging to two different subject groups. For each of the 50 dictionary bundles, we obtain a p-value using a Wilcoxon rank-sum test5 , representing the confidence at which we can reject the hypothesis that the two distributions are equal. To account for multiple comparisons, we correct these p-values using the Holm-Bonferroni method [START_REF] Holm | A simple sequentially rejective multiple test procedure[END_REF] and consider as significant the bundles with corrected p < 0.05.

Differences across genetically-related subjects

We first identify the bundles which show a statistically significant difference across two groups of genetically-related subjects. As in the subject identification experiment, we compute the pairwise distances between instances corresponding to MZ twins, DZ twins and non-twin siblings, considering each fingerprint feature (i.e., bundle) individually. The significance of a bundle is measured based on the null hypothesis that the distances in two groups are equally distributed.

Figure 16 shows the Holm-Bonferroni corrected p-values (in -log 10 scale) of each bundle, for MZ twins compared to non-twin siblings. The results identify three separate bundles with significant differences (-log 10 (p-value) > 1.

3) corresponding to the corticospinal bundles, with fiber trajectories in the parietal lobe and dorsal regions of the brain. Furthermore, bundle-wise differences between DZ and NT siblings, occurring mainly in frontal cortex areas, can also be seen in Figure 17.

Differences related to gender

A similar analysis was conducted to find bundles showing statistically significant differences between male and female subjects. For this analysis, we used the data from 332 males (age: 28.05 ± 3.65) and 436 females (age: 29.33 ± 3.55), all of them right-handed. While the analysis on genetically-related subjects compared distance distributions, in this case, we compared features directly.

That is, for each bundle, we computed the distribution of feature values corresponding to this bundle, and compared the distributions obtained in instances of male and female subjects.

Figure 18 reports the corrected p-values (in -log 10 scale) obtained for each bundle. We can see several significant bundles (14 in total), with corrected p < 0.05, with the most prominent differences occurring in the frontal cortex.

Specifically, significant bundles include fiber trajectories in the pre-frontal area, and around the precuneus.

0 0.5 1 1.5 2 2.5

Discussion

We now summarize and discuss the findings related to our parameter study, subject identification experiments, and bundle-wise significance tests. We then highlight limitations and additional considerations of this study.

Findings related to the parameter study

An extensive set of experiments was conducted to determine the impact of various parameters on the fingerprint's ability to uniquely characterize a subject.

These experiments showed that pooling functions estimating the fiber trajectory density along dictionary bundles, such as the RMS and Mean functions, provided fingerprints that were significantly more similar for same-subject instances than those from different subjects. Moreover, fingerprints obtained using RMS pooling were found to give significant separability for dictionaries containing 50 bundles or more, a number consistent with previous studies on the topic of fiber trajectory clustering and segmentation [START_REF] Guevara | Automatic fiber bundle segmentation in massive tractography datasets using a multi-subject bundle atlas[END_REF][START_REF] O'donnell | Automatic tractography segmentation using a high-dimensional white matter atlas[END_REF]. Our experiments have also shown the advantage of using a soft assignment of fiber trajectories to bundles, via our non-negative sparse coding framework, which offers a more precise description of complex bundles that may overlap and cross each other. Specifically, we observed that fiber trajectories can be encoded as a sparse combination of S max = 4 bundle prototypes. This sparsity level was also found to be optimal in our previous work on fiber trajectory segmentation [START_REF] Kumar | A sparse coding approach for the efficient representation and segmentation of white matter fibers[END_REF].

0 0.5 1 1.5 2 2.5
We evaluated the robustness of the proposed method by varying the fiber tracking parameters. Our method provides separability for 30 000 or more output fiber trajectories, both using the RK4 and Euler fiber tracking approaches.

The tracking parameters having the highest impact are the turning angle threshold and minimum fiber trajectory length, although significant separability was achieved for all tested values of these parameters. In another experiment, we found that excluding cerebellum fiber trajectories resulted in a small decrease in separability. However, the fingerprint without information from the cerebellum still exhibit significant differences across subjects.

Varying the number of fiber trajectories used for fingerprint generation, we observed that our fingerprint could uniquely identify a subject with only 3 000 fiber trajectories uniformly sampled over the whole brain. Moreover, we found that fiber trajectories from both hemispheres and inter-hemispheric fiber trajectories contributed in a synergic manner to characterize a subject, the highest separability obtained using left-hemisphere fiber trajectories. This suggests that unique characteristics of a subject, in terms of fiber trajectory distribution, are present in the entire brain. Overall, the small variations found in individual bundles, across subjects, suggest a common blueprint of connectivity, but also an overall pattern that is unique to each individual. This is consistent with previous work in the literature, showing that each individual is unique in terms of brain structure [START_REF] Mangin | A framework to study the cortical folding patterns[END_REF], function [START_REF] Barch | Function in the human connectome: task-fMRI and individual differences in behavior[END_REF][START_REF] Mueller | Individual variability in functional connectivity architecture of the human brain[END_REF], and white matter micro-structure [START_REF] Bürgel | White matter fiber tracts of the human brain: three-dimensional mapping at microscopic resolution, topography and intersubject variability[END_REF][START_REF] De Schotten | Atlasing location, asymmetry and inter-subject variability of white matter tracts in the human brain with mr diffusion tractography[END_REF].

Findings related to subject identification tests

Our experiments on subject identification have also lead to useful observations. Using fingerprint similarity to define the k-nearest neighbors of a subject instance, we obtained results consistent with previous work from the literature,

showing that MZ twins are significantly more similar at the fingerprint level than DZ twins, and DZ twins more similar than non-twin siblings [START_REF] Kochunov | Heritability 53 of fractional anisotropy in human white matter: a comparison of human connectome project and ENIGMA-DTI data[END_REF]. Results also showed a greater similarity between DZ twins than between non-twin siblings, although both types of siblings have the same amount of shared genetic information. A deeper analysis revealed that the higher similarity of DZ twins was not fully explained by age difference. While studies have shown the impact of various environmental factors on white matter development [START_REF] Chiang | Genetics of white matter development: a DTI study of 705 twins and their siblings aged 12 to 29[END_REF], in particular during adolescence, the link between the fetal environment and brain development remains largely unknown. Further investigation is required to determine whether prenatal development factors, like the mother's nutrition and stress levels during pregnancy, could play a role in our observations.

There are many factors to be considered while interpreting these results, for example, environmental factors, gender differences, aging effects, limitations of fiber tracking processes, non-rigid alignment process, etc. Note the twin zygosity labels used in this analysis were self reported (HCP Q3 release). The impact of aging was addressed indirectly by the HCP dataset recruitment policies, which limited the allowed age of subjects to the 22-35 years range, corresponding to a plateau in the FA-aging curve [START_REF] Kochunov | Fractional anisotropy of cerebral white matter and thickness of cortical gray matter across the lifespan[END_REF][START_REF] Kochunov | Heritability 53 of fractional anisotropy in human white matter: a comparison of human connectome project and ENIGMA-DTI data[END_REF][START_REF] Van Essen | The human connectome project: a data acquisition perspective[END_REF]. We also considered the effect of aging for identifying twins and non-twin siblings by dividing pairs of non-twin siblings into two groups, using the median age difference as the separation threshold.

No significant difference was observed across age groups, for age differences up to 11 years.

Findings related to bundle significance tests

Our bundle-wise fingerprint analysis revealed several bundles showing significant differences, when comparing groups of genetically-related subjects, or different sex subjects. For the comparison between MZ twins and Non-Twin siblings, we find three significant bundles (p < 0.05 after Holm-Bonferroni correction), corresponding roughly to the corticospinal bundles. The differences between DZ twins and Non-Twin siblings were most prominent in the frontal cortex, suggesting that variations between individuals sharing the same amount of genetic material are linked to higher processing areas. Although a direct comparison is not feasible, these results are consistent with white matter regions in a recent heritability study, based on the voxel-wise analysis of fractional anisotropy (FA) [START_REF] Kochunov | Heritability 53 of fractional anisotropy in human white matter: a comparison of human connectome project and ENIGMA-DTI data[END_REF].

Moreover, gender-related differences were found to be significant in 14 different bundles, connected mostly to the pre-frontal cortex and precuneus. Again, several of these bundles correspond to regions shown to have significant genderrelated effects on FA, in studies using tract based spatial statistics (TBSS) [START_REF] Chou | Sex-linked white matter microstructure of the social and analytic brain[END_REF][START_REF] Gong | Brain connectivity gender makes a difference[END_REF] or structural network analysis [START_REF] Yan | Sex-and brain size-related small-world structural cortical networks in young adults: a DTI tractography study[END_REF].

Informativeness of fiberprint compared to fingerprints based on whole T1weighted images

Comparing the proposed fiberprint with a brain fingerprint generated from intensities in aligned T1w volumes, the two fingerprints yield a similar performance (measured in terms of recall@k) for the task of identifying geneticallyrelated subjects. However, analyzing the list of sibling pairs (MZ/DZ/NT) identified by these two fingerprints indicates that each one provides complimentary information, with 15% to 20% of sibling pairs identified by only one of these fingerprints.

Although using raw intensities as fingerprint also allows to capture both local and global differences in structural or diffusion geometry, the proposed fiberprint provides a more compact and high-level representation of white-matter connectivity. Thus, our fiberprint can effectively encode this information in a vector of about m = 50 features, compared to 157 × 189 × 136 features for T1-weighted volumes. This makes our framework particularly attractive for handling large datasets. Moreover, direct voxelwise comparison of diffusion data (e.g., FA maps) could also be challenging, since high-contrast edges in such volumes make them more susceptible to small registration errors and to the variability of local tract geometry [START_REF] Colby | Along-tract statistics allow for enhanced tractography analysis[END_REF]. In contrast, our experiments have shown the proposed fiberprint to be robust to differences in the alignment and signal reconstruction process. Lastly, unlike voxelwise fingerprints, our framework allows comparing subjects on the level of structural connectivity (i.e., fiber bundles), rather than unspecified global structure.

Additional considerations

In this study, we analyzed the impact of various factors on our fingerprint's ability to describe unique characteristics of subjects. However, additional factors could be considered in our analysis. For instance, other distances metrics can be used to measure the similarity between fiber trajectories, such as the Mean of Closest Points (MCP) or the Hausdorff distance. The flexibility of the proposed framework allows its potential extension to various computational models or representations for fiber trajectories, for which a similarity measure can be computed. These measures could help capture additional information on fiber trajectories (e.g., along-tract diffusion signal), which may be not possible to encode with a geometric representation, leading to a more discriminative fingerprint.

Partial volume effects and other tractography-related effects, such as fiber tracking or registration errors, could also impact our fingerprint. Moreover, as highlighted in [START_REF] Jones | White matter integrity, fiber count, and other fallacies: the do's and don'ts of diffusion mri[END_REF], caution must be used to when interpreting results obtained from diffusion MRI. For instance, since there is no gold standard for calibrating DWI measures, the reliability of tractography outputs cannot be evaluated.

However, these factors are in part minimized by the large number of subjects used in our study (i.e., 851 subjects), the pre-processing done by the HCP pipeline and the QSDR signal reconstruction approach.

In our experiments, we have created multiple instances of the same subject using fiber trajectories derived from a single scan. Another aspect could be to test same subject identification using repeat scans of the same subject, as done in [START_REF] Yeh | Local connectome fingerprinting reveals the uniqueness of individual white matter architecture[END_REF] for the study of white matter structure. Since we use the same reconstruction approach and toolbox (DSI studio), the results after fiber tracking should extend to repeat scan data. Moreover, because our experiments have demonstrated that fingerprints generated from the scans of identical siblings are more similar than those from other sibling types, we expect repeat scans of the same subject to have highly similar fingerprints.

Although aging effects were considered in our analysis of bundle-wise significance, a deeper study is needed to fully understand the impact of neuroplasticity on fingerprints. This could also be achieved using longitudinal data, by measuring how a subject's fingerprint changes over time. Our bundle-wise significance analysis could also be extended to find differences related to additional phenotypic variables, such as cognitive score.

Conclusion

We presented a new subject fingerprint, called Fiberprint, which uses sparse code pooling to characterize the unique properties of subjects at the level of fiber trajectories. The proposed fingerprint measures the fiber trajectory density along specific bundles, which are defined using dictionary learning. Experiments using the dMRI data of 861 subjects from the HCP dataset were conducted to evaluate the impact of our method's parameters, to demonstrate that the proposed fingerprint can be used to identify subjects, pairs of twins, or nontwin siblings, and to find bundles showing significant differences across various subject groups.

Our results show that a fingerprint capable of uniquely identifying subjects can be obtained using only 3 000 fiber trajectories sampled across the brain.

Moreover, such a fingerprint is robust to parameters related to fiber tracking, dictionary learning and sparse code pooling. Experiments on the identification of genetically-related subjects demonstrate that the proposed fingerprint can correctly retrieve instances belonging to a given subject. Our experiments also suggest that subjects sharing the same genetic information (i.e., monozygotic twins) have more similar fingerprints than siblings sharing half of their genetic material (i.e., dizygotic twins and non-twin siblings). Furthermore, our bundlewise analysis of significance showed that corticospinal bundles had significantly different fingerprint features when comparing monozygotic twins with non-twin siblings, and that differences between dizygotic twins and non-twin siblings were most prominent in the pre-frontal cortex. A similar comparison across male and females subjects identified 14 significant bundles, most of them connected to the pre-frontal cortex and precuneus. Several of these results are consistent with recent heritability studies based on the voxel-wise analysis of FA.

This work could be extended by evaluating the impact of additional factors related to the tracking and encoding of fiber trajectories. Likewise, a deeper analysis of aging effects could help better understanding the effect of neuroplasticity on individual characteristics of white matter fiber geometry.

1. Supplemental Results: Figures

Dictionary bundle visualization

We observed bundles from the dictionary, and assigned the hemisphere label to a bundle based on where majority of fiber trajectories lie. We had 6 bundles in the cerebellum (Figure 4), 12 inter-hemispheric bundles (Figure 1), 17 bundles in the left hemisphere (Figure 2), and 15 bundles in the right hemisphere (Figure 3).

Impact of Handedness on subject identification

Table 1 shows mean precision@1 (in %) for a varying number of fiber trajectories using the RMS pooling function and 80 Left handed and 80 Right handed subjects, in a nearest neighbor analysis.

Impact of age on twin identification

Figure 6 (left) measures the impact of age on MZ-twin identification, where twin pairs were divided based on the median age. We observe that age does not impact MZ-twin identification, pairs in different age groups having similar mean recall@k values. This is consistent with HCP study design, which aims to minimize the impact of age by selecting subjects in the plateau of white matter development. Figure 6 (right) gives the results of a similar analysis of DZ twins. While group mean recall@k plots are significantly higher than random mean recall@k, differences between the mean recall@k values obtained in the two age groups are not statistically significant. p-values (-log 10 scale), corresponding to each dictionary bundle. These results validate those obtained using the Wilcoxon rank sum tests.

Twin fingerprint analysis

Figure 1 :

 1 Figure 1: Pipeline of the proposed Fiberprint approach based on sparse code pooling.

Figure 1

 1 Figure 1 summarizes the pipeline of the proposed Fiberprint method, comprised of three steps. In the first step, signal reconstruction and fiber tracking

  diffusivity along 270 directions distributed equally over 3 shells with b-values of 1000, 2000 and 3000 s /mm 2 , and were acquired on a Siemens Skyra 3T scanner with the following parameters: sequence = Spin-echo EPI; repetition time (TR) = 5520 ms; echo time (TE) = 89.5 ms; resolution = 1.25 × 1.25 × 1.25 mm 3 voxels. Further details can be obtained from HCP Q3 data release manual 2 .

Figure 1 : 1 Figure 2 :Figure 2

 1122 Figure 1: Dictionary visualization. Visualization of m = 50 fiber prototypes learned from 10 subjects, with an unique color assigned to each dictionary prototype. For this simplified visualization each fiber is assigned to a single prototype by taking the maximum for each row of the matrix A. (superior axial, left sagittal, and anterior coronal views respectively)

Figure 1 : 1 Figure 3 :

 113 Figure 1: Visualization of sparse code representation of fibers from four subjects. Each fiber is assigned to a single bundle by taking the maximum of the sparse code vector. Bundles are represented using the same colors as in Figure ??. (superior axial (top), left sagittal (middle), and anterior coronal (bottom) views respectively)

Figure 1 :

 1 Figure 1: Color coded visualization of sparse code memberships of fibers for the left (top row)

Figure 4 :

 4 Figure 4: Color coded visualization of sparse code memberships of fiber trajectories for the left (top row) and right (bottom row) corticospinal bundles from four subjects. Green and red represent, a low and a high membership of a fiber trajectory to a bundle, respectively. Fiber trajectories with a zero membership to the bundle are removed for a simplified visualization.

Figure 5 :

 5 Figure 5: Subject fingerprint visualization. Color coded bar plot representation for four subjects (rows) and three pooling functions (RMS, Mean, and Max; columns), plotted as a value per bundle ID.

Figure 1 : 1 Figure 6 :

 116 Figure 1: Subject fingerprint visualization. Color coded bundles from four subjects representing the magnitude of their corresponding RMS pooling function values. We use the same color code scheme as in Figure ??. (superior axial (top), left sagittal (middle), and anterior coronal (bottom) views respectively)

, µ 1 Figure 7 :

 17 Figure 7: Impact of pooling functions. Euclidean distance between fingerprints of 10 subjects with 5 instances each (top). Probability normalized histogram (middle) and box plot (bottom) for distances between same subject (SS) and different subject (DS) instances for all 851 subjects. Pooling functions: RMS, Mean, and Max (left to right columns respectively)

Figure 8 :

 8 Figure8: Impact of the size of the dictionary and the level of sparsity Smax on subject fingerprint. Box plot of Euclidean distances between same-subject (red) and different-subject (blue) instances for seven different dictionary sizes using all 851 subjects (left); and for varying level of the sparsity parameter Smax using 10 subjects (right).

Figure 9 :

 9 Figure9: Impact of independent dictionary sets and fiber tracking parameters on subject fingerprints. Box plots of Euclidean distances between same-subject (red) and different-subject (blue) instances using 10 subjects for: independent sets of dictionaries; the number of output fiber trajectories; the fiber tracking approach; the turning angle threshold; and the minimum length of fiber trajectories. (d-prime values are reported along the right axis of each plot)

Figure 10 :

 10 Figure 10: Impact of cerebellum exclusion on subject fingerprint. Probability normalized histogram (top) and box plot (bottom) for Euclidean distances between same subject (SS) and different subject (DS) instances for all 851 subjects. Note that the fingerprint without cerebellum is obtained by removing the bundles corresponding to cerebellum from the full subject fingerprint.

Figure 12 (

 12 Figure 12 (left) compares the two subjects' tractography output obtained using the different alignment and reconstruction approaches. We can observe clear differences in the produced tractographies, highlighted by the non-overlapping red-and blue-colored fiber trajectories. Examples of fingerprint instances generated using the two processes are shown in Figure 13, the first column corresponding to an instance obtained with QSDR and rigid alignment (QSDR+rigid), and columns two and three showing two fingerprint instances based on DTI and affine alignment (DTI+affine). Although small differences are present, we

Figure 13 :

 13 Figure 13: Color-coded bar plot representation of a subject's fiberprint, compared across the different alignment and signal reconstruction methods. Column 1 is a fiberprint based on QSDR and rigid alignment (Figure 5); columns 2 and 3 show fiberprint instances obtained with DTI and affine alignment.

Figure 14 :

 14 Figure14: Genetically-related subject identification. The mean recall@k for MZ-twin (82pairs), DZ-twin (82-pairs), Non-Twin siblings (166 pairs) using Fiberprint (left) and full T1w images rigidly aligned to MNI space as fingerprint (middle). The age difference impact on Non-Twin sibling identification, with 0 ≤ ∆age1 ≤ 3, and 3 < ∆age2 ≤ 11, 3 being the median age difference (right). In all plots, the chance factor is measured via a random list of nearest neighbors (rnd).

Figure 14 (

 14 Figure 14 (left) summarizes the results of this analysis. As expected, higher recall values are observed for MZ twins compared to DZ twins and non-twin siblings, reflecting the fact that such subjects have identical genetic material.

  gives the normalized histogram and box plots of Euclidean distances between instances belonging to MZ, DZ and NT siblings. We observe that the mean of distances corresponding to MZ twins is smaller than the mean of DZ twin distances, which is itself less than the mean distance between NT instances (d-prime values of 0.47, 0.64, and 0.26 for BMZ vs BDZ, BMZ vs BNT, and BDZ vs BNT). Note that these differences are significant in an unpaired t-test, with p < 0.01. Confidence intervals on the difference of distribution means are [-0.0190, -0.0158], [-0.0327, -0.0287],

Figure 15 :

 15 Figure 15: Differences between fingerprints of genetically-related subjects. Probability normalized histogram and box plot of Euclidean distances between instances belonging to MZ, DZ, and Non-Twin siblings

Figure 1 : 1 Figure 16 :

 1116 Figure 1: MZ vs NT. Differences between MZ-twin and Non-Twin siblings. Color coded bundle visualization for Holm-Bonferroni corrected p-values (in -log 10 scale) obtained using Wilcoxon rank-sum test. (superior axial, anterior coronal, and left sagittal views (top row); inferior axial, posterior coronal, and right sagittal views (bottom row);)

Figure 1 : 1 Figure 17 :

 1117 Figure 1: DZ vs NT. Differences between DZ-twin and Non-Twin siblings. Color coded bundle visualization for Holm-Bonferroni corrected p-values (in -log 10 scale) obtained using Wilcoxon rank-sum test. (superior axial, anterior coronal, and left sagittal views (top row); inferior axial, posterior coronal, and right sagittal views (bottom row);)

Figure 1 : 1 Figure 18 :

 1118 Figure 1: Male vs Female. Differences related to gender. Color coded bundle visualization for Holm-Bonferroni corrected p-values (in -log 10 scale) obtained using Wilcoxon rank-sum test. (superior axial, anterior coronal, and left sagittal views (top row); inferior axial, posterior coronal, and right sagittal views (bottom row);)

Figure 3 :

 3 Figure 3: Left hemisphere dictionary bundles with respective bundle IDs. (Right sagittal view)

Figure 4 :

 4 Figure 4: Cerebellum dictionary bundles with respective bundle IDs. (Left sagittal view in top row; right sagittal and posterior coronal view in bottom row)

Figure 7 1 . 6 .

 716 Figure 7 gives the distributions of Euclidean distances between fingerprint instances of subject pairs corresponding to MZ, DZ and non-twin siblings. We observe that distances between instances of MZ twins (BMZ) are smaller than those between instances of DZ twins (BDZ) and distances between instance of Non-Twin siblings (BNT). This figure also highlights that fingerprints of MZ twins are more similar than those of DZ and non-twin siblings, but same-subject instances are still more similar to one another. Thus, subject signatures preserve individual differences, and show an expected trend for MZ, DZ and non-twins siblings.

Figure 5 :

 5 Figure 5: Fingerprint comparison across two instances of a subject. Color coded bar representation of subject fingerprint (subject 1, Instance 1 and 2); with Columns representing: RMS, Mean, and Max pooling functions respectively

Figure 6 :

 6 Figure 6: Impact of age on MZ/DZ-twin identification; and the level of sparsity Smax on subject fingerprint for m = 100. (Left) The mean recall@k for MZ-Twin identification (82 pairs); where, 22 ≤ age1 ≤ 29, and 30 ≤ age2 ≤ 35, 30 is the median age for MZ-twin pairs. (Middle) The mean recall@k for DZ-Twin identification (82 pairs); where, 22 ≤ age1 ≤ 29, and 30 ≤ age2 ≤ 35, 29 is the median age for DZ-Twin pairs. (Right) Impact of the level of sparsity Smax on subject fingerprint for m = 100. Note: mean recall@k for random lists of nearest-neighbors is identified by rnd.

Figure 7 :Figure 1 :Figure 8 :Figure 1 :Figure 9 :Figure 1 :Figure 10 :

 71819110 Figure 7: Euclidean distance based differences between fingerprints of genetically-related subjects w.r.t. same-subject (SS) and different-subject (DS) instances. Probability normalized histogram and box plots of Euclidean distances for MZ twins (164 subjects), DZ twins (164 subjects), and Non-Twin siblings (215 subjects) using RMS pooling function

Table 1

 1

	: Same-subject instance identification. Mean precision@1 (in %) for a varying number
	of fiber trajectories using the RMS pooling function and all 851 subjects, in a nearest neighbor
	analysis. The second column shows results for fingerprints generated from the full brain. The
	third column shows result for without-cerebellum fingerprints. The right columns evaluate the
	contribution of fiber trajectories from a specific hemisphere. Note that the without-cerebellum
	fingerprints are obtained by removing cerebellum bundles from the full brain fingerprint, and
	the hemisphere specific fingerprints are obtained from the full brain fingerprints by keeping
	hemisphere-specific bundles only. Also, the first column indicates the number of fiber trajec-
	tories used for generation of the full brain fingerprint. Maximum precision@1 of 0.4% was
	obtained for the randomly generated lists of k-nearest neighbors using the full brain finger-
	print.
	# Fibers	Cerebellum	Hemisphere
		Yes	No	Left Right Inter
	100	1.4	1.0	0.4	0.4	0.4
	500	36.9	21.7	5.1	3.9	3.2
	1 000	85.7	68.3	17.4	14.0	10.5
	2 000	99.7	97.8	54.0	41.5	27.5
	3 000	100.0	99.9	77.6	67.4	46.9
	4 000	100.0	100.0	88.6	81.5	61.4
	5 000	100.0	100.0	94.7	89.5	73.1
	6 000	100.0	100.0	97.7	93.6	81.8
	8 000	100.0	100.0	99.3	98.3	91.2
	10 000	100.0	100.0	99.8	99.3	95.3

Table 2 :

 2 Informativeness of our fiberprint compared to a fingerprint based on whole T1weighted images for identifying genetically-related subjects. Column 1 gives the proportion of twins/siblings identified by both fingerprints, Column 2 and 3 the proportion of twins/siblings identified by only one fingerprint, and column 4 the proportion of twins/siblings not identified by any of the fingerprints. A sibling is considered as identified if his/her fingerprint is within the list of k = 30 nearest neighbors. Number of identification tasks: 164-MZ, 164-DZ, and 215-NT. We report mean over 5 fiberprint instances.

	Sibling	Both	T1w	Fiberprint	None
	MZ	50.12%	22.44%	15.37%	12.07%
	DZ	18.17%	19.02%	15.24%	47.56%
	NT	11.35%	19.81%	14.51%	54.33%

Table 1 :

 1 Same-subject instance identification. Mean precision@1 (in %) for a varying number of fiber trajectories using the RMS pooling function and 80 Left handed and 80 Right handed subjects, in a nearest neighbor analysis. The second column shows results for fingerprints generated from the full brain. The third column shows result for without-cerebellum fingerprints. The right columns evaluate the contribution of fiber trajectories from a specific hemisphere. Note that the without-cerebellum fingerprints are obtained by removing cerebellum bundles from the full brain fingerprint, and the hemisphere specific fingerprints are obtained from the full brain fingerprints by keeping hemisphere-specific bundles only. Also, the first column indicates the number of fiber trajectories used for generation of the full brain fingerprint. Maximum precision@1 of 0.6% was obtained for the randomly generated lists of k-nearest neighbors using the full brain fingerprint.

	# Fibers	Cerebellum	Hemisphere
		Yes	No	Left Right Inter
	100	6.0	3.4	2.3	1.8	1.9
	500	58.3	40.3	12.3	13.8	9.3
	1000	94.3	85.0	33.8	30.0	23.8
	2000	99.6	98.9	74.4	65.4	47.0
	3000	100.0	99.9	92.0	86.9	69.0
	4000	100.0	100.0	96.9	93.8	78.8
	5000	100.0	100.0	98.6	96.0	87.6
	6000	100.0	100.0	99.4	97.9	92.4
	8000	100.0	100.0	99.5	99.5	96.9
	10000	100.0	100.0	99.9	99.9	98.6
			6

http://www.humanconnectome.org/documentation/Q3/

A separate visualization of each fiber trajectory cluster can be found in the supplementary material.

as the number of fiber trajectories, from the whole brain or individual hemi-

http://dsi-studio.labsolver.org/download-images/hcp-842-template)

siblings. Note that, for MZ, DZ and NT pairs, the recall values obtained based

Results obtained using an unpaired t-test can be found in the supplementary materials.

Acknowledgements

Data were provided in part by the Human Connectome Project, WU-Minn Consortium (Principal Investigators: David Van Essen and Kamil Ugurbil; 1U54MH091657) funded by the 16 NIH Institutes and Centers that support the NIH Blueprint for Neuroscience Research; and by the McDonnell Center for Systems Neuroscience at Washington University.

can see that our fiberprint preserves the location and relative importance of the principal fingerprint values (i.e., "peaks") across the two different alignment and reconstruction approaches. This can be explained by the fact that the fiberprint models fiber trajectory density along prominent bundles, which is weakly affected by differences in the local geometry of individual fibers.

These results are substantiated in Figure 12 (right), where we report mean recall@k for the task of identifying the DTI+affine fingerprints using the 851 originally generated QSDR+rigid fingerprints. The mean recall@k is computed over 10 identification tasks (two subjects with 5 instance each). We observe that a mean recall@k of 100% is achieved within k = 10 nearest neighbors, further demonstrating the robustness of our fiberprint to alignment and signal reconstruction methods.

Genetically-related subject identification

A similar analysis was performed to identify genetically-related subjects. For this analysis, we used the Mother ID, Age, Twin stat, and Zygosity fields of the Twin HCP dataset to identify 82 pairs of monozygotic twin (MZ) subjects, 82 pairs of dizygotic twin (DZ) subjects, and 166 pairs of non-twin siblings (NT).