Sorbonne Universit

Docteur De

Federico Bertoni

Alessandro Sarti

Giovanna Citti

Govanna Citti

[ Sarti

LIE SYMMETRIES IN CORTICAL INSPIRED CNNS

 studied the dynamics of neural population, modeling the cortex in the same SE(2)

 which describes the evolution of the cortical activity depending on a connectivity kernel and the eigenvectors of the affinity matrix. In particular, they proved that the eigenvectors, ordered w.r.t. the corresponding eigenvalues in decreasing order, represent the most salient objects in the image. ces modèles. Dans les deux derniers chapitres, nous testons la capacité des filtres et des noyaux appris dans les CNNs à reproduire les phénomènes perceptifs typiques des architectures cérébrales. Dans le sixième chapitre, nous étudions les effets Retinex sur le filtre CGL inspiré appris, alors que dans le dernier nous montrons les résultats de regroupement sur les noyaux appris dans différents CNNs.

Introduction

The visual system is one of the most studied part of the brain and, in particular, the LGN and V1, which are the first layers that analyze the visual stimulus, are some of the best understood visual areas. The first description of the early visual pathway and its corresponding geometry was proposed by D. H. Hubel and T. N. Wiesel in the '60s and further developed in the subsequent years [START_REF] Hubel | Eye, brain, and vision[END_REF]Wiesel, 1962, 1977;[START_REF] Hubel | Eye, brain, and vision[END_REF]. The retinal input is first processed by the radially symmetric families of cells present in the LGN who are responsible of human contrast perception. Then it reaches V1, whose neurons are not only sensitive to the light intensity of the visual stimulus, but they also show a selectivity to other features, such as e.g. orientation, scale, velocity. Indeed, every cell presents in the visual cortices reacts to a local area of the visual stimulus called receptive field (RF), whereas the function that describes its activation in the presence of a visual stimulus is called receptive profile (RP) and it will be denoted as Ψ. Every retinal location is associated to an entire set of neurons of V1 called hyper-columns, sensitive to all possible instances of the corresponding variable, organized in the so called orientation maps.

From the mathematical point of view, several neuromathematical models based on classical differential geometry and Lie groups have been proposed. In the '80s Jan Koenderink in [START_REF] Koenderink | Representation of local geometry in the visual system[END_REF] studied perceptual spaces using differential geometry and William Hoffman in [START_REF] Hoffman | The visual cortex is a contact bundle[END_REF] developed a model of the visual cortex as a fiber bundle equipped with a contact structure. In the same period Steven Zucker observed experimentally the relation between the measurement of Euclidean curvature and the role of end-stopping cells [START_REF] Dobbins | Endstopped neurons in the visual cortex as a substrate for calculating curvature[END_REF]. i

INTRODUCTION

These first models raised the idea that the visual system was a "geometric machine" leading in the '90s to the development of several phenomenological models of vision based on calculus of variations and parabolic partial differential equations: to cite just some pioneering papers, the segmentation model of [START_REF] Mumford | Optimal approximations by piecewise smooth functions and associated variational problems[END_REF]] and the multiscale analysis of Alvarez, Lions, Morel [START_REF] Alvarez | Image selective smoothing and edge detection by nonlinear diffusion. ii[END_REF]. Moreover, in the same years, some neuromathematical models presenting a double neuro-psycho nature have been proposed. David Mumford in [START_REF] Mumford | Elastica and Computer Vision[END_REF] modeled perceptual illusory contours through its elastica curves which are still at the center of contemporary research, but rethought in new mathematical setting. Thus, in '95 in [START_REF] Williams | Stochastic Completion Fields: A Neural Model of Illusory Contour Shape and Salience[END_REF]] a first stochastic model of illusory contours in the space of position and orientation was proposed by Williams and Jacobs. At the end of the '90s a fundamental contribution of Jean Petitot and Yannick Tondut in [START_REF] Petitot | Vers une neuro-géométrie. Fibrations corticales, structures de contact et contours subjectifs modaux[END_REF] reconsidered the Hoffman model of the cortex as a contact bundle, computing the geodesic curves of the non-integrable structure and showing that the curves modeled the perceptual association fields, measured by Fields, Heyes and Hess in [START_REF] Field | Contour integration by the human visual system: evidence for a local association field[END_REF]. In this way, they were able to predict the shape of an illusory contour, given its inducturs, and Petitot explicitly introduced the word Neurogeometry to denote the inner geometry of cortical connectivity. In the subsequent years Jean Petitot has further developed his model contributing massively to the progress of this field of research [START_REF] Petitot | The neurogeometry of pinwheels as a sub-riemannian contact structure[END_REF][START_REF] Petitot | Neurogéométrie de la vision -Modèles mathématiques et physiques des architectures fonctionnelles[END_REF][START_REF] Petitot | Elements of Neurogeometry: Functional Architectures of Vision[END_REF].

INTRODUCTION iii

as reported in classical literature. In [Duits and Franken, 2010a,b] Remco Duits shifted the focus by working in the SE [START_REF] Williams | Stochastic Completion Fields: A Neural Model of Illusory Contour Shape and Salience[END_REF] group from an image processing point of view. He lifted and propagated the visual signal in the R 2 × S 1 space and, by applying invertible kernels, was able to reconstruct the stimulus without loss of information in the 2D image plane. A phenomenological approach can be found in the entire work of Jean-Michel Morel and his computational Gestalt group [see e.g. [START_REF] Delsolneux | From Gestalt Theory to Image Analysis: A Probabilistic Approach[END_REF][START_REF] Morel | A pde formalization of retinex theory[END_REF][START_REF] Limare | Retinex poisson equation: a model for color perception[END_REF]. Indeed, he found in the Helmoltz principle the basis of the classical theory of Gestalt obtaining deep theoretical and computational results. Daniel Bennequin approached several neuromathematical problems in terms of invariance, symmetry and ambiguity: in particular, he formalized these problems with principles of "information topology" theory, a new co-homology theory of information based on probability and entropy theory [see e.g. [START_REF] Pham | Affine invariance of human hand movements: a direct test[END_REF][START_REF] Bennequin | Remarks on invariance in the primary visual systems of mammals[END_REF]. James Bednar reproduced the geometric morphologies of the visual cortex through brain plasticity principles, recovering the main characteristics of the functional architecture of the visual cortex by a process of learning on a suitable set of stimuli [START_REF] Stevens | Mechanisms for stable, robust, and adaptive development of orientation maps in the primary visual cortex[END_REF][START_REF] Antolík | Model constrained by visual hierarchy improves prediction of neural responses to natural scenes[END_REF]. In more recent years, further research has been carried out leading to several developments in this field: in [START_REF] Citti | A gauge field model of modal completion[END_REF][START_REF] Sarti | The constitution of visual perceptual units in the functional architecture of V1[END_REF][START_REF] Favali | Local and Global Gestalt Laws: A Neurally Based Spectral Approach[END_REF]Boscain et al., 2018;[START_REF] Boscain | Cortical-inspired image reconstruction via sub-riemannian geometry and hypoelliptic diffusion[END_REF][START_REF] Baspinar | Multi-frequency image completion via a biologically-inspired sub-riemannian model with frequency and phase[END_REF][START_REF] Galyaev | Liouville integrability in a four-dimensional model of the visual cortex[END_REF][START_REF] Jumakulyyev | Fourth-Order Anisotropic Diffusion for Inpainting and Image Compression[END_REF] the authors faced the perceptual completion of figures task on SE [START_REF] Williams | Stochastic Completion Fields: A Neural Model of Illusory Contour Shape and Salience[END_REF] group; perception of contours in motion and trajectories has been studied in [START_REF] Barbieri | A cortical-inspired geometry for contour perception and motion integration[END_REF]; applications in vessel segmentation have been analyzed in [START_REF] Zhang | Robust retinal vessel segmentation via locally adaptive derivative frames in orientation scores[END_REF][START_REF] Bekkers | Vessel tracking via sub-riemannian geodesics on the projective line bundle[END_REF][START_REF] Abbasi-Sureshjani | Curvature integration in a 5d kernel for extracting vessel connections in retinal images[END_REF][START_REF] Yu | 3d vessel reconstruction in oct-angiography via depth map estimation[END_REF]; an orientation-dependent contrast perception model inspired by Wilson-Cowan-type equations has been proposed in [START_REF] Bertalmío | A cortical-inspired model for orientation-dependent contrast perception: A link with wilson-cowan equations[END_REF][START_REF] Bertalmío | Cortical-inspired wilson-cowan-type equations for orientation-dependent contrast perception modelling[END_REF] a model of the functional architecture of V1 from the RPs of simple cells have been proposed in [Montobbio et al., 2019b[START_REF] Montobbio | A metric model for the functional architecture of the visual cortex[END_REF].

In particular, in these models the feature space G usually has the product form R 2 × F, where the parameters (x 0 , y 0 ) ∈ R 2 indicate the retinal location where each RP is centered, while f ∈ F encodes the selectivity of the neurons to other

INTRODUCTION local features of the image.

A visual stimulus can be represented as a function I = I(x, y), where (x, y) represents a retinal location and I(x, y) the corresponding light. Its action z, associated to a kernel defined as Ψ = Ψ(x, y), can be modeled as a linear integral operator as follows z(I) := I * Ψ.

In the case of simple cells of V1 F = S 1 , i.e. each neuron responds maximally to a certain orientation θ at a specific retinal location (x, y).

Furthermore, in V1 there is experimental evidence of the existence of connections between simple cells of different hyper-columns with similar orientation, called long-range horizontal connectivity [see e.g. [START_REF] Ts'o | Relationships between horizontal interactions and functional architecture in cat striate cortex as revealed by cross-correlation analysis[END_REF][START_REF] Bosking | Orientation selectivity and the arrangement of horizontal connections in tree shrew striate cortex[END_REF].

The connectivity has been described in [START_REF] Citti | A cortical based model of perceptual completion in the roto-translation space[END_REF]] as families of integral curves of two vector fields in the sub-Riemannian structure on R 2 × S 1 . This model has been further developed in [START_REF] Sanguinetti | A model of natural image edge co-occurrence in the rototranslation group[END_REF] where the authors proposed to model connectivity as suitable geometric kernels. In particular, in [Montobbio et al., 2019b] it is proposed to directly relate the shape of connectivity kernels in a family of cells to their RPs {Ψ p } p∈G as follows:

K(p, p 0 ) := Re R 2
Ψ p (x, y)Ψ p 0 (x, y)dxdy ,

Many efforts have been made to relate cortical architectures with Convolutional

Neural Networks (CNNs). Indeed, the first neural networks have been inspired by a simplification of the structure of the visual system, presenting a hierarchical structure, where each layer receives input from the previous one and provides output to the next one. Despite this simplification, they reached optimal performances in processes typical of the natural visual system, as for example object-detection [START_REF] Redmon | You only look once: Unified, real-time object detection[END_REF][START_REF] Ren | Faster r-cnn: Towards real-time object detection with region proposal networks[END_REF] or image classification [START_REF] He | Deep residual learning for image recognition[END_REF]INTRODUCTION v and DiCarlo, 2016] the authors were able to study higher areas by focusing on the encoding and decoding ability of the visual system. Recurrent Neural networks have been introduced to implement the horizontal connectivity [e.g. [START_REF] Sherstinsky | Fundamentals of recurrent neural network (rnn) and long short-term memory (lstm) network[END_REF], or feedback terms [e.g. [START_REF] Liang | Recurrent convolutional neural network for object recognition[END_REF]. A modification of these nets, more geometric and more similar to the structure of the brain, has been recently

proposed in [Montobbio et al., 2019a]. Furthermore, it is well known that both V1 RPs and the first convolutional layer of a CNN are mainly composed by Gabor filters. Biological based models of V1 in terms of Gabor filters have been made in [START_REF] Serre | Robust object recognition with cortex-like mechanisms[END_REF][START_REF] Zhang | Convolutional neural network models of v1 responses to complex patterns[END_REF] and the statistic of the RFPs of a macaque's V1 was studied in [START_REF] Ringach | Spatial structure and symmetry of simple-cell receptive fields in macaque primary visual cortex[END_REF], but a comparison between these results and the statistics of learned filters is still missing.

Our scope in this thesis is to propose architectures of CNNs in such a way to model the early visual pathway, including the Lateral Geniculate Nucleus and the Horizontal Connectivity of the primary visual cortex. Moreover, we will show how cortically inspired architectures allow to perform contrast perceptual invariance as well as grouping and the emergence of visual percepts. Particularly, the LGN is modeled with a first layer 0 containing a single filter Ψ 0 that pre-filters the image I. Since the RPs of the LGN cells can be modeled as a LoG, we expect to obtain a radially symmetric filter with a similar shape; to this end, we prove the rotational invariance of Ψ 0 and we study the influence of this filter to the subsequent layer. Indeed, we compare the statistic distribution of the filters in the second layer 1 of our architecture with the statistic distribution of the RPs of V1 cells of a macaque studied in [START_REF] Ringach | Spatial structure and symmetry of simple-cell receptive fields in macaque primary visual cortex[END_REF].

Then, we model the horizontal connectivity of V1 implementing a transition kernel K 1 to the layer 1 . In this setting, we study the vector fields and the association fields induced by the connectivity kernel K 1 . To this end, we first approximate the filters bank in 1 with a Gabor function and use the parameters just found to re-parameterize the kernel. Thanks to this step, the kernel is now re-parameterized into a sub-Riemmanian space R 2 × S 1 . Now we are able to compare the vector and association fields induced by K 1 with the models of the INTRODUCTION horizontal connectivity.

The Retinex theory, proposed by Land [START_REF] Land | The retinex[END_REF], describes the contrast perception phenomenon. Later, he developed this theory with J. McCann in [START_REF] Land | Lightness and retinex theory[END_REF], proposing an algorithm and a computer program designed to simulate the Retinex processes taking place in the visual system : it associated to an image I the perceived I. In the following years several works have contributed to the development of the Retinex theory and its application [among others [START_REF] Enroth-Cugell | The contrast sensitivity of retinal ganglion cells of the cat[END_REF][START_REF] Brainard | Analysis of the retinex theory of color vision[END_REF][START_REF] Provenzi | Mathematical definition and analysis of the retinex algorithm[END_REF][START_REF] Solomon | Suppressive surrounds and contrast gain in magnocellular-pathway retinal ganglion cells of macaque[END_REF][START_REF] Lei | An investigation of retinex algorithms for image enhancement[END_REF][START_REF] Valberg | Neurophysiological correlates of color vision: A model[END_REF][START_REF] Yeonan-Kim | Analysis of retinal and cortical components of Retinex algorithms[END_REF]. Some variational approaches have been proposed for example by [START_REF] Kimmel | A variational framework for retinex[END_REF][START_REF] Morel | A pde formalization of retinex theory[END_REF][START_REF] Limare | Retinex poisson equation: a model for color perception[END_REF]. In the last two works the authors have formalized the Retinex theory as the solution of the following discrete Poisson PDE

-∆ d I = M (I)
where ∆ d is the classical discrete Laplacian and M is a modified version of the discrete Laplacian. More recently, a geometrical model relating the architecture of the visual system and the invariances of RPs has been presented in [START_REF] Citti | A gauge field model of modal completion[END_REF]. Our scope will be to adapt this approach by using kernels and structures learned by the CNN, instead of modelled kernels.

We will apply cortical inspired CNN to grouping, i.e. the ability of the visual system to perceive the elements in the image as distinct. The main principles that rule how we perceive the elements were described by [START_REF] Kanizsa | Organization in vision: Essays on gestalt perception[END_REF] through the Gestalt laws. Then, several models have been proposed to connect the cortical activity with these phenomena [see e.g. [START_REF] Hoffman | The visual cortex is a contact bundle[END_REF][START_REF] Weiss | Segmentation using eigenvectors: a unifying view[END_REF][START_REF] Shi | Normalized cuts and image segmentation[END_REF]. In [START_REF] Sarti | The constitution of visual perceptual units in the functional architecture of V1[END_REF][START_REF] Favali | Local and Global Gestalt Laws: A Neurally Based Spectral Approach[END_REF] the authors put in

INTRODUCTION vii

The last part of our work is aimed to show that filters and kernels learned by the LGN-CNN with the connectivity layer are able to reproduce visual perceptual phenomena, namely contrast perception, whose main model is the Retinex theory and grouping, whose properties are governed by Gestalt laws. Our purpose is to substitute the action of the RP with the filter learned by the LGN-CNN. If the filter becomes a good good approximation of the associated ∆G σ , then we shall expect that its inverse will allow to recover the perceived image I in problems of contrast perception. Indeed, we propose a Retinex algorithm for a learned kernel and apply it to the LGN inspired learned filter Ψ 0 comparing the results with the classical ones. We show that our model reaches similar Retinex abilities with respect to the classical theory and the LoG, the model of the LGN RPs.

On the other hand, we perform the grouping with learned kernels. Indeed, we propose an algorithm that allows to generate an affinity matrix from a bank of filters and, then, finding the first eigenvectors, to locate the perceptual units.

We first test our algorithm with a modeled bank of Gabor filters, showing good performances in the case of position-oriented elements but bad ones in the case of clouds of points. Then, we test our algorithm on several learned CNN architecture, generating the affinity matrix from one or more convolutional layers. We show that, thanks to the presence of both Gaussian and Gabor like filters, the kernel generated from these banks of filters are able to detect both position-oriented elements and clouds of points. Furthermore, the use of more convolutional layers enhances the grouping ability of the kernels.

The thesis is organized as follows. The first three chapters contain an overview of the existing literature. Namely, the first chapter contains an overview of the structure of the visual pathway and the group symmetries presents there, as well as a review of the mathematical models of the functional architecture of the visual cortex. In the second chapter, we present the contrast perception phenomena and the grouping, whereas in the third one the main structures of the CNN architectures and the analogies with the visual system. Then, in the fourth and fifth chapters we describe the biologically inspired CNN architectures, one with the LGN layer

INTRODUCTION

[proposed in [START_REF] Bertoni | Lgn-cnn: A biologically inspired cnn architecture[END_REF] and one with the transition kernel too [proposed in [START_REF] Bertoni | Emergence of lie symmetries in functional architectures learned by cnns[END_REF]. Here we show the invariances arising in these models.

In the last two chapters we test the ability of filters and kernels learned in CNNs to reproduce perceptual phenomena typical of brain architectures. In the sixth chapter we study the Retinex effects on the LGN inspired learned filter, whereas in the last one we show the grouping results on kernels learned in different CNNs.

Resumé

Le système visuel est l'une des parties du cerveau les plus étudiées et, en particulier, le corps géniculé latéral (CGL) et le cortex visuel primaire (V1), qui sont les premières couches qui analysent le stimulus visuel, sont parmi les zones visuelles les mieux comprises. La première description du premier parcours visuel et de sa géométrie correspondante a été proposée par D. H. Hubel et T. N. Wiesel dans les années 60 et les années suivantes [START_REF] Hubel | Eye, brain, and vision[END_REF]Wiesel, 1962, 1977;[START_REF] Hubel | Eye, brain, and vision[END_REF]. Dans les années 80, Jan Koenderink dans [START_REF] Koenderink | Representation of local geometry in the visual system[END_REF] a étudié les espaces perceptifs à l'aide de la géométrie différentielle et William Hoffman dans [START_REF] Hoffman | The visual cortex is a contact bundle[END_REF] [START_REF] Dobbins | Endstopped neurons in the visual cortex as a substrate for calculating curvature[END_REF].

Ces premiers modèles ont soulevé l'idée que le système visuel était une "machine géométrique", ce qui a conduit dans les années 90 au développement de plusieurs modèles phénoménologiques de la vision basés sur le calcul des variations et les équations aux dérivées partielles paraboliques : pour ne citer que quelques articles pionniers, le modèle de segmentation de Mumford-Shah [START_REF] Mumford | Optimal approximations by piecewise smooth functions and associated variational problems[END_REF] et l'analyse multi-échelle d [START_REF] Alvarez | Image selective smoothing and edge detection by nonlinear diffusion. ii[END_REF]. De plus, dans les mêmes années, des modèles neuromathématiques aiant une double nature neuro-psychologique ont été proposés. David Mumford dans [START_REF] Mumford | Elastica and Computer Vision[END_REF] a modélisé des contours illusoires perceptifs à travers ses courbes élastiques qui sont toujours au centre de la recherche contemporaine, mais repensées dans un nouveau cadre mathématique. Ainsi, en 1995 dans [START_REF] Williams | Stochastic Completion Fields: A Neural Model of Illusory Contour Shape and Salience[END_REF] un premier modèle stochastique de contours illusoires dans l'espace de position et d'orientation a été proposé par Williams et Jacobs. A la fin des années 90, une contribution fondamentale de Jean Petitot et Yannick Tondut dans [START_REF] Petitot | Vers une neuro-géométrie. Fibrations corticales, structures de contact et contours subjectifs modaux[END_REF] a reconsidéré le modèle de Hoffman du cortex comme structure de contact, calculant les courbes géodésiques de la structure non intégrable et montrant que les courbes modélisées les champs d'association perceptifs, mesurés par Fields, Heyes et Hess dans [START_REF] Field | Contour integration by the human visual system: evidence for a local association field[END_REF]. De cette façon, ils ont pu prédire la forme d'un contour illusoire, compte tenu de ses inductances, et Petitot a explicitement introduit le mot Neurogéométrie pour désigner la géométrie interne de la connectivité corticale. Au cours des années suivantes, Jean Petitot a fait évoluer son modèle en contribuant massivement aux progrès de ce domaine de recherche [START_REF] Petitot | The neurogeometry of pinwheels as a sub-riemannian contact structure[END_REF][START_REF] Petitot | Neurogéométrie de la vision -Modèles mathématiques et physiques des architectures fonctionnelles[END_REF][START_REF] Petitot | Elements of Neurogeometry: Functional Architectures of Vision[END_REF].

Au nouveau siècle, Govanna Citti et Alessandro Sarti dans [START_REF] Sarti | From neural oscillations to variational problems in the visual cortex[END_REF][START_REF] Citti | A cortical based model of perceptual completion in the roto-translation space[END_REF] [START_REF] Delsolneux | From Gestalt Theory to Image Analysis: A Probabilistic Approach[END_REF][START_REF] Morel | A pde formalization of retinex theory[END_REF][START_REF] Limare | Retinex poisson equation: a model for color perception[END_REF]. En effet, il a trouvé dans le principe de Helmoltz la base de la théorie classique de la Gestalt obtenant des résultats théoriques et informatiques approfondis. Daniel Bennequin a abordé plusieurs problèmes neuromathématiques en termes d'invariance, de symétrie et d'ambiguïté : il a notamment formalisé ces problèmes avec les principes de la théorie de la "topologie de l'information", une nouvelle théorie de la cohomologie de l'information basée sur la théorie des probabilités et de l'entropie

[voir par exemple [START_REF] Pham | Affine invariance of human hand movements: a direct test[END_REF][START_REF] Bennequin | Remarks on invariance in the primary visual systems of mammals[END_REF]. James Bednar a reproduit les morphologies géométriques du cortex visuel à travers les principes de plasticité cérébrale, récupérant les principales caractéristiques de l'architecture fonctionnelle du cortex visuel par un processus d'apprentissage sur un approprié ensemble de stimuli [START_REF] Stevens | Mechanisms for stable, robust, and adaptive development of orientation maps in the primary visual cortex[END_REF][START_REF] Antolík | Model constrained by visual hierarchy improves prediction of neural responses to natural scenes[END_REF]. Ces dernières années, d'autres recherches ont été menées aboutissant à plusieurs développements dans ce domaine : dans [START_REF] Citti | A gauge field model of modal completion[END_REF][START_REF] Sarti | The constitution of visual perceptual units in the functional architecture of V1[END_REF][START_REF] Favali | Local and Global Gestalt Laws: A Neurally Based Spectral Approach[END_REF]Boscain et al., 2018;[START_REF] Boscain | Cortical-inspired image reconstruction via sub-riemannian geometry and hypoelliptic diffusion[END_REF][START_REF] Baspinar | Multi-frequency image completion via a biologically-inspired sub-riemannian model with frequency and phase[END_REF][START_REF] Galyaev | Liouville integrability in a four-dimensional model of the visual cortex[END_REF][START_REF] Jumakulyyev | Fourth-Order Anisotropic Diffusion for Inpainting and Image Compression[END_REF], les auteurs ont été confrontés à la tâche d'achèvement perceptif des figures sur le groupe SE(2) ; la perception des contours en mouvement et des trajectoires a été étudiée dans [START_REF] Barbieri | A cortical-inspired geometry for contour perception and motion integration[END_REF] ; applications dans la segmentation des vaisseaux ont été analysées dans [START_REF] Zhang | Robust retinal vessel segmentation via locally adaptive derivative frames in orientation scores[END_REF][START_REF] Bekkers | Vessel tracking via sub-riemannian geodesics on the projective line bundle[END_REF][START_REF] Abbasi-Sureshjani | Curvature integration in a 5d kernel for extracting vessel connections in retinal images[END_REF]Yu et al., xii RESUM É 2021] ; un modèle de perception de contraste dépendant de l'orientation inspiré des équations de type Wilson -Cowan a été proposé dans [START_REF] Bertalmío | A cortical-inspired model for orientation-dependent contrast perception: A link with wilson-cowan equations[END_REF][START_REF] Bertalmío | Cortical-inspired wilson-cowan-type equations for orientation-dependent contrast perception modelling[END_REF] un modèle de l'architecture fonctionnelle de V1 à partir des PR de cellules simples a été proposé dans [Montobbio et al., 2019b[START_REF] Montobbio | A metric model for the functional architecture of the visual cortex[END_REF].

En conséquence, l'espace de fonction G a généralement la forme du produit En outre, dans V1 il y a des preuves expérimentales de l'existence des connexions entre des cellules simples qui appartiennent à différentes colonnes corticales avec une orientation similaire, appelée connectivité horizontale à longue portée [par example [START_REF] Ts'o | Relationships between horizontal interactions and functional architecture in cat striate cortex as revealed by cross-correlation analysis[END_REF][START_REF] Bosking | Orientation selectivity and the arrangement of horizontal connections in tree shrew striate cortex[END_REF]. La connectivité a été décrite dans [START_REF] Citti | A cortical based model of perceptual completion in the roto-translation space[END_REF]] comme des familles de courbes intégrales des deux champs vectoriels dans la structure sous-riemannienne sur R 2 × S 1 . Ce modèle a été développé dans [START_REF] Sanguinetti | A model of natural image edge co-occurrence in the rototranslation group[END_REF] où les auteurs ont proposé de modéliser la connectivité comme noyaux géométriques appropriés. En particulier, dans [Montobbio et al., 2019b] il est proposé de relier directement la forme des noyaux de connectivité dans une famille de cellules à leur PRs {Ψ p } p∈G comme suit : [START_REF] Redmon | You only look once: Unified, real-time object detection[END_REF][START_REF] Ren | Faster r-cnn: Towards real-time object detection with region proposal networks[END_REF] ou la classification d'images [START_REF] He | Deep residual learning for image recognition[END_REF][START_REF] Simonyan | Very deep convolutional networks for large-scale image recognition[END_REF].

K(p, p 0 ) := Re R 2 Ψ p (x, y)Ψ p 0 (x,
Plus récemment, les relations entre les CNNs et le système visuel humain ont été largement étudiées, dans le but ultime de rendre le CNN encore plus efficace dans des tâches spécifiques. Un modèle de premières couches corticales décrites comme couches d'un CNN a été étudié dans [START_REF] Serre | Robust object recognition with cortex-like mechanisms[END_REF], alors que dans [START_REF] Yamins | Hierarchical modular optimization of convolutional networks achieves representations similar to macaque it and human ventral stream[END_REF][START_REF] Yamins | Using goal-driven deep learning models to understand sensory cortex[END_REF] les auteurs ont pu étudier des zones plus élevées en se concentrant sur l'encodage et la capacité de décodage du système visuel. Des réseaux neuronaux récurrents ont été introduits pour mettre en oeuvre la connectivité horizontale [par exemple [START_REF] Sherstinsky | Fundamentals of recurrent neural network (rnn) and long short-term memory (lstm) network[END_REF], ou termes de rétroaction [par exemple [START_REF] Liang | Recurrent convolutional neural network for object recognition[END_REF]. Une modification de ces filtres, plus géométrique et plus similaire à la structure du cerveau, a été récemment proposée dans [Montobbio et al., 2019a]. De plus, il est bien connu que les V1 PRs et la première couche convolutionnelle d'un CNN sont principalement composées de filtres de Gabor. Des modèles biologiques de V1 en termes des filtres de Gabor ont été réalisés dans [START_REF] Serre | Robust object recognition with cortex-like mechanisms[END_REF][START_REF] Zhang | Convolutional neural network models of v1 responses to complex patterns[END_REF] et la statistique des PRs de la V1 d'un macaque a été étudiée dans [START_REF] Ringach | Spatial structure and symmetry of simple-cell receptive fields in macaque primary visual cortex[END_REF], néanmoins une En effet, nous comparons la distribution statistique des filtres de la deuxième couche 1 de notre architecture avec la distribution statistique des PR des cellules V1 d'un macaque étudiée dans [START_REF] Ringach | Spatial structure and symmetry of simple-cell receptive fields in macaque primary visual cortex[END_REF].

Ensuite, nous modélisons la connectivité horizontale de V1 implémentant un noyau de transition K 1 vers la couche 1 . Dans ce cadre, nous étudions les champs vectoriels et les champs d'association induits par le noyau de connectivité K 1 . À cette fin, nous rapprochons d'abord la banque de filtres en 1 avec une fonction de Gabor et utilisons les paramètres trouvés pour reconfigurer le noyau. Grâce à cette étape, le noyau est maintenant reconfiguré dans un espace sous-Riemmanien R 2 × S 1 . Nous sommes maintenant en mesure de comparer les champs vectoriels et d'association induits par K 1 avec les modèles de la connectivité horizontale.

La théorie Retinex, proposée par Land [START_REF] Land | The retinex[END_REF], décrit le phénomène de perception de contraste. Plus tard, il a développé cette théorie avec J. McCann dans [START_REF] Land | Lightness and retinex theory[END_REF], proposant un algorithme et un programme informatique conçu pour simuler les processus Retinex se déroulant dans le système visuel: il associait à une image I la perception I. Dans les années suivantes, plusieurs travaux ont contribué au développement de la théorie Retinex et son application [entre autres [START_REF] Enroth-Cugell | The contrast sensitivity of retinal ganglion cells of the cat[END_REF][START_REF] Brainard | Analysis of the retinex theory of color vision[END_REF][START_REF] Provenzi | Mathematical definition and analysis of the retinex algorithm[END_REF][START_REF] Solomon | Suppressive surrounds and contrast gain in magnocellular-pathway retinal ganglion cells of macaque[END_REF][START_REF] Lei | An investigation of retinex algorithms for image enhancement[END_REF][START_REF] Valberg | Neurophysiological correlates of color vision: A model[END_REF][START_REF] Yeonan-Kim | Analysis of retinal and cortical components of Retinex algorithms[END_REF]. Certaines approches variationnelles ont été proposées par exemple par [START_REF] Kimmel | A variational framework for retinex[END_REF][START_REF] Morel | A pde formalization of retinex theory[END_REF][START_REF] Limare | Retinex poisson equation: a model for color perception[END_REF] [START_REF] Kanizsa | Organization in vision: Essays on gestalt perception[END_REF] à travers les lois de la Gestalt. Ensuite, plusieurs modèles ont été proposés pour relier l'activité corticale à ces phénomènes [voir par ex. [START_REF] Hoffman | The visual cortex is a contact bundle[END_REF][START_REF] Weiss | Segmentation using eigenvectors: a unifying view[END_REF][START_REF] Shi | Normalized cuts and image segmentation[END_REF].

Dans [START_REF] Sarti | The constitution of visual perceptual units in the functional architecture of V1[END_REF][START_REF] Favali | Local and Global Gestalt Laws: A Neurally Based Spectral Approach[END_REF] les auteurs mettent en relation l'équation "meanfield" d'Ermentrout et de Cowan [START_REF] Ermentrout | Large scale spatially organized activity in neural nets[END_REF] et Bressloff et Cowan [Bressloff et al., 2002;[START_REF] Bressloff | The functional geometry of local and long-range connections in a model of V1[END_REF] between the filter Ψ f with orientation θ f = 2π 5 and the other filters Ψ g centered at the same point (i.e. with relative displacement = (0, 0)) as a function of their orientation θ g . The curve has been smoothed using the MatLab built-in function smoothdata. . . . . . . . . . . . 62 5.8 (A) The 7 × 7 filter Ψ f , with orientation θ f = 3π 10 . (B) The vector field V and its integral curves obtained from the kernel K 1 computed around Ψ f , with θ f = 3π 10 . (C) The association field of the transition kernel K 1 (blue), and its approximation using the integral curves defined in (1.7) (red). For better visualization the kernel has been resized by a factor of 10 using the built-in Matlab function imresize. 65 5.9 (A) In blue: the vector field of the transition kernel K 1 around Ψ f . In red: the vector field of the best-fitting fundamental solution of The functional architecture of the

Visual System

In this chapter we recall some of the main properties of the visual system that are useful for our research and refer the reader for a more general description to [E. R. [START_REF] Kandel | Principles of neural science, third edition[END_REF][START_REF] Nolte | The Human Brain: An Introduction to Its Functional Anatomy[END_REF]. We briefly summarize the structure of the visual system focusing on the Lateral Geniculate Nucleus (LGN)

and the Primary Visual Cortex (V1) and we describe the geometric properties of the Receptive Profiles (RPs) of cells in both structures.

Architecture of the visual system

The visual system is one of the most studied and most understood part of the brain, composed by many cortices that elaborate the visual signal. The first element of this structure is the retina, a light-sensitive layer of tissue which receives the visual stimulus and translates it into electrical impulses. The latter first reach the

LGN whose cells pre-process the visual stimulus and send the elaborated impulse to V1. Here the cells of V1 process the information and send the output to all the other layers of the visual system. Furthermore, each cortex receives information from other cortices, processes it through horizontal connectivity that allows cells of the same cortex to communicate with each others, forwards it to higher areas and sends feedback to previous ones. The structure is very complex and not totally ordered as physiologically described for example in [START_REF] Hubel | Eye, brain, and vision[END_REF].

Receptive Profiles of the LGN and V1 cells

Our main interest lies in the cells of the LGN and in the simple cells of V1.

Each cell in the visual system receives the electrical impulse from a portion of the retina Ω called receptive field (RF). The RF of each cell is divided in excitatory and inhibitory areas, activated by the light. As a result, the behavior of each cell can be modeled as a function Ψ :

Ω ⊂ R 2 → R called receptive profile (RP),
positive in excitatory areas and negative in the inhibitory ones. If the excitatory areas are activated the firing rate of the cell increases whereas it decreases in case of inhibitory areas activation.

As concern the LGN, figure 1.1 on the left shows a recording of the RP of one cell and on the right its approximation [see e.g. [START_REF] Deangelis | Receptive-field dynamics in the central visual pathways[END_REF]. Indeed, the RP of LGN cells can be modeled as a Laplacian of Gaussian (LoG):

Ψ LoG (x, y) = - 1 πσ 4 1 - x 2 + y 2 2σ 2 e -x 2 +y 2 2σ 2 , (1.1)
where σ denotes the standard deviation of the Gaussian function [see e.g. [START_REF] Petitot | Vers une neuro-géométrie. Fibrations corticales, structures de contact et contours subjectifs modaux[END_REF][START_REF] Petitot | Neurogéométrie de la vision -Modèles mathématiques et physiques des architectures fonctionnelles[END_REF][START_REF] Petitot | Elements of Neurogeometry: Functional Architectures of Vision[END_REF]. Furthermore, in the LGN there are two types of cells RPs, respectively with excitatory or inhibitory internal region. In the second case the RP can be modeled by minus the LoG.

As concern V1, there exist several types of cells with different RPs, called simple cells and complex cells. We are mainly interested in the first kind that has sharply-tuned RPs with a preferred orientation. Figure 1.2 shows in the first row the RPs of two simple cells of V1 and in the second one their approximation.

Indeed, an established model for V1 simple cells is represented by a bank of Gabor filters {Ψ x 0 ,y 0 ,θ,σ } (x 0 ,y 0 ,θ,σ)∈R 2 ×S 1 ×R 2 + built by translations T (x 0 ,y 0 ) , rotations R θ and dilations D σx,σy of the filter

Ψ 0,0,0,1 (x, y) = Ae -x 2 +y 2 2 cos(2πf x + φ),
(1.2) where A is the amplitude and f is the frequency of the filter and φ is the phase which indicates if the Gabor filter is even or odd. See e.g. [START_REF] Daugman | Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters[END_REF] and [START_REF] Lee | Image representation using 2d Gabor wavelets[END_REF].

Furthermore, several classes of visual cells, as e.g. the ones just described, are shown to act, to a first approximation, as linear filters on an optic signal: the response of this kind of cells to a visual stimulus, defined as a function on the retina From: [START_REF] Hubel | Eye, brain, and vision[END_REF].

I : Ω ⊂ R 2 → R, is

The hyper-columnar structure and non maxima suppression

In the 70s Hubel and Wiesel discovered that V1 is organized in the so called hyper-columnar structure [START_REF] Hubel | Receptive fields, binocular interaction and functional architecture in the cat visual cortex[END_REF]. This means that for each retinal point (x, y) there is an entire set of cells each one sensitive to a specific instance of the considered feature, organized in "pinwheel" arrangements. The set of RPs of cells reacting to a specific retinal region is usually represented by a bank of linear filters {Ψ p } p∈G ⊆ L 2 (R 2 ) [for some references see e.g. [START_REF] Petitot | Vers une neuro-géométrie. Fibrations corticales, structures de contact et contours subjectifs modaux[END_REF][START_REF] Citti | A cortical based model of perceptual completion in the roto-translation space[END_REF][START_REF] Sarti | The symplectic structure of the primary visual cortex[END_REF][START_REF] Petitot | Neurogéométrie de la vision -Modèles mathématiques et physiques des architectures fonctionnelles[END_REF][START_REF] Petitot | The neurogeometry of pinwheels as a sub-riemannian contact structure[END_REF]. The feature space G is typically described as a group of transformations of the plane {T p , p ∈ G} under which the entire filter bank is invariant. Indeed, each profile Ψ p can be obtained from any other profile Ψ q through the transformation T p-q .

Furthermore, the feature space G usually has the product form

R 2 × F (x, y, f ),
where the parameters (x, y) ∈ R 2 indicate the retinal location where each RP is centered, while f ∈ F encodes the selectivity of the neurons to other local features of the image, such as orientation and scale.

The figure 1.3 is a scheme of the hyper-columnar structure when the considered feature is orientation. In this case the space of feature can be modelled by F = S 1 and so the higher dimensional cortical space is given by G = R 2 × S 1 . At a certain scale and resolution there exists a set of neurons centered at the same retinal position (x, y) in which each neuron responds maximally to a certain orientation θ.

All the possible orientations are expressed and a triple (x, y, θ) can be associated to each neuron.

When a visual stimulus is projected on a retinal point (x, y) all neurons belonging to the hyper-column over the point (x, y) will be activated and provide the response given by the formula (1.3). The one with the same orientation of the stimulus is maximally activated, giving rise to orientation selectivity. Indeed the cortex is equipped with a neural circuitry, called intracortical circuitry, that is able to keep the orientation of maximal response of cells' outputs. Furthermore, among the cells in the same hyper-column is present a short-range connectivity that induces excitation between cells with close orientation and inhibition between cells with different orientation. This kind of connectivity for V1 simple cells has been modelled as a "Mexican hat"-like function in [Bressloff and Cowan, 2002]. For a general feature described by a space F the process is the same and the non maxima suppression principle is modelled by:

z(I)(x 0 , y 0 , f 0 ) = max f 0 ∈F z(I)(x 0 , y 0 , f 0 ) (1.4)
where (x 0 , y 0 ) is the considered retinal position and F represents the set of all possible instances. Within each hyper-column the feature associated to the maximum output is selected and all the others are suppressed, leading to the selection of the maximal responding feature. In this way our brain is able to locally detect a specific feature and to discard all the other possibility.

Statistics of V1 simple cells RPs

As just described, the RPs of V1 cells can be modeled as Gabor functions (1.2).

Recently, Ringach in [START_REF] Ringach | Spatial structure and symmetry of simple-cell receptive fields in macaque primary visual cortex[END_REF] has proved that the RPs are not uniformly distributed with respect to all the Gabor parameters, but they have a very particular statistic. Indeed, Ringach defines two coefficients n x and n y which estimate the elongation in x and y directions respectively

(n x , n y ) = (σ x • f, σ y • f ). (1.5)
1. The functional architecture of the Visual System

In particular, if f = 0 the Gabor function in (1.2) simplifies to a Gaussian since the cosine becomes a constant. Otherwise the Gabor function is elongated in its preferred orientation θ. In order to study the statistics of recorded RPs of V1 simple cells, Ringach has followed these steps:

• Fitting the RPs with Gabor function Ψ x 0 ,y 0 ,θ,σ built by translation T (x 0 ,y 0 ) , rotation R θ and dilation D σx,σy of the filter defined in eq. (1.2);

• Comparing the results on (n x , n y ) = (σ x • f, σ y • f ) plane.
Figure 4.10c shows the statistical distribution of RPs of V1 cells in monkeys in (n x , n y ) plane obtained by Ringach in [START_REF] Ringach | Spatial structure and symmetry of simple-cell receptive fields in macaque primary visual cortex[END_REF]. In [Barbieri et al., 2014a] the authors have studied the same statistical distribution with respect to the Uncertainty Principle associated to the task of detection of position and orientation.

The horizontal connectivity

From the neurophysiological point of view, there is experimental evidence of the existence of connections between simple cells of different hyper-columns. It is the so called long-range horizontal connectivity that is responsible for the cortico-cortical propagation of the neural activity between hyper-columns.

In [START_REF] Ts'o | Relationships between horizontal interactions and functional architecture in cat striate cortex as revealed by cross-correlation analysis[END_REF] correlation techniques have been used in order to estimate the correlation between connectivity and orientation of cells. Recently techniques of optical imaging allowed to measure the horizontal connectivity (see figure 1.4) by injecting a tracer (biocytin) in a simple cell and recording the trajectory of the tracer [see [START_REF] Bosking | Orientation selectivity and the arrangement of horizontal connections in tree shrew striate cortex[END_REF]. In this way it has been possible to clarify properties of horizontal connections on V1 of the tree shrew and it has been revealed that the linked cells of different hyper-columns not only share the angle of tuning but also the axis corresponding to the orientation is roughly the same.

It is believed that the long-range horizontal connections of V1 constitutes the neural implementation of contour completion, i.e. the ability to group (C) Enlarged portions of the map show linear two zones (left) and two pinwheel arrangements (right). From: [START_REF] Bosking | Orientation selectivity and the arrangement of horizontal connections in tree shrew striate cortex[END_REF].

local edge items into extended curves. This perceptual phenomenon has been described through association fields [START_REF] Field | Contour integration by the human visual system: evidence for a local association field[END_REF], characterizing the geometry of the mutual influences between oriented local elements. See for example figure 1.5A from the experiment of Field, Heyes and Hess. Then, Petitot and Tondut in [START_REF] Petitot | Vers une neuro-géométrie. Fibrations corticales, structures de contact et contours subjectifs modaux[END_REF] proposed a geometric model of the functional architecture of V1 that also models the association fields. More recently, association fields have been described in [START_REF] Citti | A cortical based model of perceptual completion in the roto-translation space[END_REF] as families of integral curves of the two vector fields

X 1 = (cos θ, sin θ, 0), X 2 = (0, 0, 1) (1.6)
that generates the sub-Riemannian structure on R 2 × S 1 . Figure 1.5B shows the 3-D constant coefficients integral curves of the vector fields (1.6) and figure 1.5C their 2-D projection. These integral curves can be also modeled as the solution of the following ordinary differential equation:

γ (t) = X 1 (γ(t)) + kX 2 (γ(t)).
1. The functional architecture of the Visual System The curves starting from (0, 0, 0) can be rewritten explicitly in the following way:

x = 1 k sin(kt), y = 1 k (1 -cos(kt)), θ = kt. (1.7)
while integral curves starting from a general point (x 0 , y 0 , θ 0 ) can be obtained from equations (1.7) by translations T (x 0 ,y 0 ) and rotations R θ .

The probability of reaching a point (x, y, θ) starting from the origin and moving along the stochastic counterpart of these curves can be characterized as the fundamental solution of a second order differential operator expressed in terms of the vector fields X 1 , X 2 . This is why the fundamental solution of the sub-Riemannian heat kernel or Fokker Planck (FP) kernel have been proposed as alternative models of the cortical connectivity.

This viewpoint based on connectivity kernels was further exploited in [START_REF] Montobbio | A metric model for the functional architecture of the visual cortex[END_REF]: the model of the cortex was reformulated in terms of metric spaces,

A metric model of cortical connectivity

Let us recall the metric model of the cortex proposed in [Montobbio et al., 2019b]. First, we would like to choose a bank of linear, real or complex valued, filters that model the RPs of a set of simple cells of V1. In order to do so we can fix a feature space G that in general has the form G = R 2 × F. Then we can introduce the following bank of filters:

{Ψ p } p∈G ⊆ L 2 (R 2 )
where each point (x, y) ∈ R 2 indicates the retinal position at which the filter is centered whereas f ∈ F encodes the specific feature that our bank of filters will detect. In the case of Gabor filters the space F = S 1 but we can consider any other spaces. For example, we can consider filters modelling other types of cells or even filters learned by automatic algorithms.

The idea is now to understand how each pair of filters is correlated with each other. In the case of Gabor filters we know for example that filters that share a similar orientation can communicate. Then we can define for every p, p 0 ∈ G the 1. The functional architecture of the Visual System generating kernel as:

K(p, p 0 ) := Re R 2 Ψ p (x, y)Ψ p 0 (x, y)dxdy , (1.8)
which expresses the strength of correlation between them. Since we would like to obtain a connectivity kernel that mimics the behaviour of the horizontal connectivity we have introduced an iterative procedure. This idea follows from the reproducing property of the kernel of a heat equation [see [START_REF] Montobbio | A metric model for the functional architecture of the visual cortex[END_REF].

Indeed, fixing a point p 0 and considering the generating kernel computed around it

p -→ K(p, p 0 ) =: K p 0 1 (p),
we can define the connectivity kernel at the n-th step as

K p 0 n+1 (p) := G K(p, q)ν(K p 0 n (q))dµ(q) (1.9)
where ν is a non-linear activation function and µ is a suitable measure defined on G [see [START_REF] Montobbio | A metric model for the functional architecture of the visual cortex[END_REF]. Furthermore this approach allows to gain more information about the correlation between pairs of filters and in presence of a visual stimulus I we can use the connectivity kernel to express the correlation between points of I. Note that we will apply this method also to learned filters which are represented as small matrices (e.g. 7 × 7, 5 × 5) and it will allow to enlarge the area in which points can communicate with each other.

Group symmetries in the early visual pathway

Over the years, numerous models of the functional architecture of the early visual pathway has been proposed in terms of geometric invariances arising in its organization, e.g. in the spatial arrangement of cell tuning across retinal locations, or in the local configuration of single neuron selectivity. Many classes of visual cells act as linear filters on a visual stimulus I defined as the function (1.3). This is the case for cells in the LGN and for simple cells in V1 [see e.g. [START_REF] Petitot | Vers une neuro-géométrie. Fibrations corticales, structures de contact et contours subjectifs modaux[END_REF][START_REF] Citti | A cortical based model of perceptual completion in the roto-translation space[END_REF][START_REF] Petitot | Neurogéométrie de la vision -Modèles mathématiques et physiques des architectures fonctionnelles[END_REF][START_REF] Petitot | Elements of Neurogeometry: Functional Architectures of Vision[END_REF]] that reacts to specific retinal regions, i.e. the RFs. Each localized area of the retina is known to be associated with a bank of similarly tuned cells [see e.g. [START_REF] Hubel | Ferrier lecture: Functional architecture of macaque monkey visual cortex[END_REF][START_REF] Sarti | The constitution of visual perceptual units in the functional architecture of V1[END_REF], yielding an approximate invariance of their RPs under translations.

Furthermore, the RPs of the cells can usually be embedded in a feature space G = R 2 × F, where F encodes the selectivity of the neurons to other local features of the image w.r.t. the bank of cells is usually invariant.

Rotational symmetry in the LGN

An essential elaboration step for human contrast perception is represented by the processing of retinal inputs via the radially symmetric families of cells present in the LGN [START_REF] Hubel | Ferrier lecture: Functional architecture of macaque monkey visual cortex[END_REF][START_REF] Hubel | Eye, brain, and vision[END_REF]. The RPs of such cells can be approximated by a LoG (1.1). A model of human contrast perception, called

Retinex theory, has been proposed by Land in [START_REF] Land | The retinex[END_REF]. We describe it more in detail in chapter 2.

Roto-translation symmetries in V1 and the lateral connectivity

Several mathematical models have been proposed to describe the invariances in the functional architecture of V1. The sharp orientation tuning of simple cells is the starting point of most descriptions. This orientation selectivity characterizes the response of each neuron to feedforward inputs; however, it is also present in the horizontal connections taking place between neurons of V1. These connections grant facilitatory influences for cells that are similarly oriented; furthermore, the connections departing from each neuron spread anisotropically, concentrating along the axis of its preferred orientation [see e.g. [START_REF] Bosking | Orientation selectivity and the arrangement of horizontal connections in tree shrew striate cortex[END_REF].

The evolution in time t → h(p, t) of the activity of the neural population at p ∈ R 2 × F is assumed in [START_REF] Bressloff | The functional geometry of local and long-range connections in a model of V1[END_REF] to satisfy a Wilson-Cowan

1. The functional architecture of the Visual System equation [START_REF] Wilson | Excitatory and inhibitory interactions in localized populations of model neurons[END_REF]:

∂ t h(p, t) = -α h(p, t) + s K(p, p )h(p , t)dp + z(p, t) .
(1.10)

Let us observe that, if K(p, p ) is of the special form K(p -p ), then the integral in Eq. (1.10) becomes a convolution as follows

∂ t h(p, t) = -α h(p, t) + s(K * h + z).
(1.11)

In the Wilson-Cowan equation, s is an activation function; α is a decay rate; z is the feedforward input corresponding to the response of the simple cells in presence of a visual stimulus, as in Eq. (1.3); and the kernel K represents the strength of horizontal connections between p and p . The form of this connectivity kernel has been investigated in several studies employing differential geometry tools. A breakthrough idea in this direction has been the description of the feature space as a fiber bundle with basis R 2 and fiber F. This approach appeared firstly in the works of [START_REF] Koenderink | Representation of local geometry in the visual system[END_REF] and [START_REF] Ferraro | Lie transformation groups, integral transforms, and invariant pattern recognition[END_REF]]. Then, it was further developed by [START_REF] Petitot | Vers une neuro-géométrie. Fibrations corticales, structures de contact et contours subjectifs modaux[END_REF][START_REF] Petitot | Neurogéométrie de la vision -Modèles mathématiques et physiques des architectures fonctionnelles[END_REF][START_REF] Petitot | Elements of Neurogeometry: Functional Architectures of Vision[END_REF], where they showed that the simple cells of V1 induce a fibration of orientations. Then, in [START_REF] Citti | A cortical based model of perceptual completion in the roto-translation space[END_REF] the model is described as a sub-Riemannian structure on the Lie group R 2 × S 1 by requiring the invariance under roto-translations. Other works have extended this approach by inserting other variables such as scale, curvature, velocity [see e.g. [START_REF] Sarti | The symplectic structure of the primary visual cortex[END_REF][START_REF] Barbieri | A cortical-inspired geometry for contour perception and motion integration[END_REF][START_REF] Abbasi-Sureshjani | Curvature integration in a 5d kernel for extracting vessel connections in retinal images[END_REF].

Chapter 2

Contrast perception phenomena and grouping

In this chapter we introduce some visual perception phenomena that we will study in the following chapters. We first describe the Retinex theory of Land and some recent developments in this field. Moreover, we recall the Gestalt laws and a model and the corresponding numerical approach that describes how the visual system analyzes globally a visual stimulus, finding the perceptual units or percepts, i.e. the elements in the image perceived as distinct.

Contrast perception 2.1.1 The Retinex theory of Land

The Retinex theory has been formulated by E. H. Land in [START_REF] Land | The retinex[END_REF] to describe the phenomenon of color constancy, in particular how the perceived color of objects, varying illumination conditions, remains relatively constant. Indeed, Land realized that, even if no green or blue wavelengths are present in a visual stimulus, the visual system would still perceive them as green or blue by overlooking the red illumination.

He has demonstrated this effect performing a famous experiment. First, he 13 showed to a subject a display called a 'Mondrian', i.e. an image composed by several colored patches named after Piet Mondrian, whose paintings are really similar. This display is illuminated by three white lights, whose intensity can be modified by the subject, each one projected through a different filter, a red, a green and a blue one respectively. Thus, Land asked the subject to adjust the intensity of the lights so that a specific patch in the image appears white and measured the intensities of red, green, and blue lights reflected from this white-appearing patch.

After this step, the subject had to identify the color of a neighboring patch, which, e.g., appeared red. Then Land adjusted the light intensities in such a way that the red patch reflected the same red, blue, and green light intensities measured from the white patch. Eventually, The subject showed color constancy: indeed, the white patch continued to appear white, the red patch continued to appear red and all the other patches continued to show their original colors.

Some years later, Land further developed his theory with J. McCann in [START_REF] Land | Lightness and retinex theory[END_REF], proposing an algorithm and a computer program designed to simulate the Retinex processes taking place in the visual system. Indeed, the algorithm associated to an image I : Ω ⊂ R 2 → R the perceived image I.

The development of the Retinex model

After the work of Land, many developments and algorithms have been proposed

to describe the Retinex theory [e.g. [START_REF] Brainard | Analysis of the retinex theory of color vision[END_REF][START_REF] Provenzi | Mathematical definition and analysis of the retinex algorithm[END_REF]Lei et al., 2007, among others]. A further analysis of the retinal and cortical components causing Retinex effects has been performed by several research teams in the past years [as e.g. [START_REF] Valberg | Neurophysiological correlates of color vision: A model[END_REF][START_REF] Yeonan-Kim | Analysis of retinal and cortical components of Retinex algorithms[END_REF]]. Furthermore, it has been demonstrated in [START_REF] Enroth-Cugell | The contrast sensitivity of retinal ganglion cells of the cat[END_REF][START_REF] Solomon | Suppressive surrounds and contrast gain in magnocellular-pathway retinal ganglion cells of macaque[END_REF] that the Magnocellular cells are the ones involved in contrast perception.

Some variational approaches have been proposed for example by [START_REF] Kimmel | A variational framework for retinex[END_REF][START_REF] Morel | A pde formalization of retinex theory[END_REF][START_REF] Limare | Retinex poisson equation: a model for color perception[END_REF]. In particular, in the last two works the authors have formalized the Retinex theory as the solution of the following discrete Poisson PDE

-∆ d I = M (I) (2.1)
where ∆ d is the classical discrete Laplacian and M is a modified version of the discrete Laplacian where at each difference between the central pixel and one in proximity is applied a threshold function.

More recently, a geometrical model making a first step in relating the architecture of the visual system and the invariances of RPs has been presented in [START_REF] Citti | A gauge field model of modal completion[END_REF]. The RP of the LGN cell which takes in input the visual signal acting by convolution on it, modeled as a modified discrete LoG, M (G σ ) ≈ ∆G σ , where G σ is a Gaussian bell. Thus, the action on an input image I is the following:

Out LGN (I) = ∆(G σ * I) ≈ ∆I
Also the horizontal connectivity in this layer is radially symmetric and modeled as the fundamental solution T = log( x 2 + y 2 ) whose associated operator is the inverse of the Laplacian ∆ -1 and allows to recover the function I:

I = T * Out LGN (I).
As a result

∆ I = Out LGN (I) ≈ ∆I,
becomes the Retinex equation. In general, I does not coincide with I, but differs by a harmonic function.

Perceptual grouping 2.2.1 The Gestalt laws

So far we have described only local aspects of perception but now we introduce some global perception phenomena, as e.g. the individuation of perceptual units.

In particular, V1 is able to analyze more global feature of the visual stimulus thanks to the long-range connectivity and distinguishes the different percepts. Indeed, a model for the functional architecture of V1 should reproduce this mechanism.

The main principles that describe how our visual system perceives the elements in a scene are called Gestalt laws of grouping. Here we summarize some of them, following [START_REF] Kanizsa | Organization in vision: Essays on gestalt perception[END_REF]]:

• Law of Similarity states that perception tends to group together elements that physically resemble each other as part of the same object and to recognize as distinct elements that are different. The visual system is then able to distinguish between adjacent and overlapping objects thanks to similar features;

• Law of Proximity states that objects or shapes that are close to one another appear to form a group. In particular, we can group together close objects even if they are different in shape and size;

• Law of Good continuation states that objects intersecting with each other are perceived as single uninterrupted objects. Indeed, we have a tendency to group lines and curves with similar direction over those with sharp changes of direction;

• Law of Closure states that the visual system tends to complete figures or forms even if they are incomplete, partially hidden or some information misses.

Therefore, there exists a natural tendency to recognize patterns familiar to us and to fill missing information;

• Law of Symmetry states that elements symmetrical with each other are perceived as a unified group. This principle has a central role in determining figure-ground perception;

• Law of periodicity states that the visual system tends to detect periodicity patterns grouping them together. Therefore, if an element in the image breaks this periodicity the visual system easily distinguishes it among all the other elements.

In figure 2.1 there are some examples of Gestalt laws just listed. At this point we do not want to give a deep explanation of all Gestalt laws but only the idea behind them in order to describe which are the main tasks the visual system faces. On the other hand, we display a famous example by Kanizsa [see [START_REF] Kanizsa | Organization in vision: Essays on gestalt perception[END_REF]. Figure 2.2 shows the square of Kanizsa in which there are 4 black circles with a white quarter; the radii that delimit the white quarters are collinear and forms a square shape. If we look at this image we should not detect the four circles as distinct perceptual units but we should see the white square delimited by the radii of the squares. This is due to the good continuation law.

A neural model of perceptual units

In this section we present a neural mechanism responsible to perceptual unit constitution, and provide a model in terms of spectral analysis The most classical equation describing the cortical activity is the mean field equation (1.10) of Ermentrout and Cowan [START_REF] Ermentrout | Large scale spatially organized activity in neural nets[END_REF] and Bressloff and Cowan [Bressloff et al., 2002;[START_REF] Bressloff | The functional geometry of local and long-range connections in a model of V1[END_REF]. This equation describes the evolution of the cortical activity depending on a connectivity kernel.

In the work of [START_REF] Sarti | The constitution of visual perceptual units in the functional architecture of V1[END_REF] the authors find a relation between the stable states of this equation and perceptual units of the input We refer also to [ [START_REF] Hoffman | The visual cortex is a contact bundle[END_REF][START_REF] Weiss | Segmentation using eigenvectors: a unifying view[END_REF][START_REF] Shi | Normalized cuts and image segmentation[END_REF][START_REF] Favali | Local and Global Gestalt Laws: A Neurally Based Spectral Approach[END_REF].Let us fix the cortical space G = R 2 × F where p = (x, y, f ) is a point on this space and consider the mean field eq. (1.10).

The input configurations are constituted by a finite number N of elements

p i = (x i , y i , f i )
and their responses to a visual stimulus I is given by the function z defined in equation (1.3). We can define the set of cells activated by the image I as the set where the output of simple cells is over a fixed threshold:

Ω I = {p 0 |z I (p 0 ) > } ⊂ G.
In this way the responses of all the cells that have a low response or a negative one are set to zero; from a neurophysiological point of view a neuron fires only if its excitation is strong enough and our model mimics this particular behaviour.

Furthermore, the cells communicate with each others through the horizontal connectivity (as described in section 1.2). Indeed, between each couple of cells RPs Ψ p i , Ψ p j it can be defined a kernel K(p i , p j ), as in eq. (1.9), expressing this relation.

Therefore, the cortical connectivity on an image I restricted on this set defines a neural affinity matrix A I :

A I (i, j) = K(p i , p j )I(x i , y i )I(x j , y j ), i, j = 1, . . . N (2.2)
In general the kernel K will have the anysotropic pattern of the considered family of cells. Kernels can be sensible to position alone, or position and orientation, curvature or movement if the metric of simple cell is considered [START_REF] Citti | A cortical based model of perceptual completion in the roto-translation space[END_REF][START_REF] Barbieri | A cortical-inspired geometry for contour perception and motion integration[END_REF][START_REF] Abbasi-Sureshjani | Curvature integration in a 5d kernel for extracting vessel connections in retinal images[END_REF] In this discrete setting, the mean field equation (1.10) reduces to:

dh(i) dt = -λ I h(i) + N j=1 A I (i, j)h(j) i = 1, . . . , N, (2.3)
where λ I is a physiological parameter, whose value can be modulated by the image. We can also perform the stability analysis of equation ( 2.3), namely we set

dh dt = 0 obtaining N j=1 A I (i, j)h(j) = λ I h(i), i = 1, . . . , N, (2.4)
and then we can rewrite it as the eigenvalue problem

A I h k = λ k I h k , (2.5)
where h k is the k-th eigenvectors of the affinity matrix A I associated to the eigenvalue λ k I . The eigenvectors describe the stationary states of the mean field equation hence the emergent perceptual units; in this way it is possible to select them in the scene and segment it. Indeed, we can order the eigenvectors in decreasing order with respect to the corresponding eigenvalues and, in particular, the first one will correspond to the most salient object in the image.

Grouping with the geometric kernel

The first family of cells that can be considered is the retinal one, whose RPs are radially symmetric. Thus, the connectivity kernel associated to these cells is, in turn, radially symmetric and we can assume it is a Gaussian one defined on

G = R 2 : K x 0 ,y 0 (x, y) = 1 4πt exp - (x -x 0 ) 2 + (y -y 0 ) 2 4t (2.6)
centered at the point (x 0 , y 0 ). This particular connectivity kernel has only the position and t as parameters; indeed it will correlate points that are close with each others. The parameter t will enlarge or shrink the area of influence of the kernel.

Indeed this connectivity kernel can detect objects with different sizes depending on the choice of t. We can calculate the affinity matrix between each pair of points p i = (x i , y i ) and p j = (x j , y j ) using the formula (2.6) for the kernel:

A I (i, j) = K x i ,y i (x j , y j ) • I(x i , y i ) • I(x j , y j ),
where I(x, y) is the value at the position (x, y) of the image. This matrix is equivalent to the affinity matrix introduced by Perona and Freeman in [START_REF] Perona | A factorization approach to grouping[END_REF]] to perform perceptual grouping. They modeled the affinity matrix in term of an heuristic distance d(p) that gives high correlation to collinear and cocircular couple of elements. In particular, the affinity matrix was defined as [ [START_REF] Barbieri | A cortical-inspired geometry for contour perception and motion integration[END_REF] the authors proposed an algorithm for the individuation of perceptual units in images with position-orientation elements through a stochastic kernel (see figure 2.4). On the other hand, in [START_REF] Abbasi-Sureshjani | Curvature integration in a 5d kernel for extracting vessel connections in retinal images[END_REF] the authors modeled a kernel that was able to detect the curvature at each spatial location and then performed grouping (see figure 2.5). This necessity of introducing specific geometry and kernels for each kind of images places a problem on the grouping task. [START_REF] Rosenblatt | The perceptron: a probabilistic model for information storage and organization in the brain[END_REF] who also developed some of the main concepts of deeplearning in [START_REF] Rosenblatt | Principles of Neurodynamics: Perceptrons and the Theory of Brain Mechanisms[END_REF]. However, the first working learning algorithm for supervised, deep, feedforward, multilayer perceptrons was published in 1967 by Alexey Ivakhnenko and Lapa [START_REF] Ivakhnenko | Cybernetics and Forecasting Techniques[END_REF]. Only after the introduction of CPU these models have been systematically implemented with great results on different tasks. We refer to [START_REF] Gidas | Symmetry of positive solutions of nonlinear elliptic equations in r[END_REF]Cucker and Smale, 2001;[START_REF] Szegedy | Going deeper with convolutions[END_REF][START_REF] Higham | Deep learning: An introduction for applied mathematicians[END_REF] for a recent review. A Convolutional Neural Network (CNN) is a particular class of Artificial Neural Network (ANN) that can be used to face different tasks, usually related to visual imagery as for example object-detection [START_REF] Redmon | You only look once: Unified, real-time object detection[END_REF][START_REF] Ren | Faster r-cnn: Towards real-time object detection with region proposal networks[END_REF] or image classification [START_REF] He | Deep residual learning for image recognition[END_REF][START_REF] Simonyan | Very deep convolutional networks for large-scale image recognition[END_REF]. For more detailed information about

A(i, j) = e - d 2
CNNs [see e.g. [START_REF] Lawrence | Face recognition: a convolutional neural-network approach[END_REF][START_REF] Lecun | Gradient-based learning applied to document recognition[END_REF]].

The architecture of a CNN

The CNN architecture is a machine learning procedure inspired by the visual cortices that approximates a functional y : V 0 → H, known only on a subset of the domain called training set, through a minimization process of a loss functional.

More precisely, a CNN is obtained by the sequentially application of a linear functional, a non linear one and a pooling operator. Then, to describe the CNN we recall:

• The structure of an artificial neuron that, as just said, has inspired the CNNs;

• The description of the layers that compose the CNN: the convolutional layer, the non linear layer and the pooling one. The last layers of a CNN are instead fully-connected;

• The description of a loss functional

• The training and test sets;

• The overfitting problem and the choice of the hyperparameters.

The multilayered structure

Since the whole neural network is composed by multiple layers we can express it as a composition of different functionals. If T Ψ i f : V i-1 → V i represents the i-th convolutional layer, Θ : V i → V i a non linear layer, Π : V i → V i a pooling layer and Υ 1 : V L → H and Υ j : H → H, (j = 2, . . . , k) the fully connected layers, then the CNN can be expressed as a functional

F Ψ 0 f ,...,Ψ L f ,ϕ 1 ,...,ϕ k : V 0 -→ H F Ψ 0 f ,...,Ψ L f ,ϕ 1 ,...,ϕ k (I) = Υ k • • • • Υ 1 • Π • Θ • T Ψ L f • • • • Π • Θ • T Ψ 0 f (3.1)

The artificial neuron

Considering the linear filter model (1.3) and the subsequent suppression step

(1.4), the output of each neuron is given by the function

z(x, y, f ) = max((Ψ f * I)(x, y) + b f (x, y), 0), (3.2)
where Ψ f is the f -th filter and b f the corresponding bias term. Indeed, an artificial neuron, which formally generalizes this structure, is composed by a linear transformation, followed by a non-linear one. Considering the i-th artificial neuron, the linear operator becomes

T Ψ f : V i-1 → V i T Ψ f (I) = Ψ f * I + b f . (3.3)
On the other hand, the non linear step becomes

Θ : V i → V i Θ(z)(x) = σ(z(x)),
where σ is the associated activation function. As result the model of the i-th artificial neuron will be

Θ • T Ψ i f : V i-1 → V i
The convolutional layer

The discrete formalization of the linear step is the convolutional layer which gives the name to this particular kind of ANNs. Indeed, it is a tensor of size s × s × n i × n o where s is the spatial size of the layer, n i is the number of channels of the previous convolutional layer and n o is the number of channels of the current convolutional layer. Fixing one channel of the last dimension of the tensor, we obtain a tensor of size s × s × n i called filter. The filter is indeed convolved with the input of the layer, allowing the structure to be equivariant to translation [see e.g. [START_REF] Cohen | A general theory of equivariant cnns on homogeneous spaces[END_REF][START_REF] Cohen | A general theory of equivariant cnns on homogeneous spaces[END_REF]. The output of the i-th convolutional layer, following eq. (3.3) becomes

z i I (x, y, f ) = T Ψ i f (I)(x, y)
where

z i I is a [n × n × n o ]
tensor, where n is the spatial dimension of the input.

The non linear layer

The non linear step can be modelled via the application of a function σ called activation function. A typical choice of this function can be the ReLU (rectified linear unit) function:

σ : R → R, σ(x) = max(0, x).

(3.4) Indeed, a neuron fires if the received input is large enough, otherwise remains inactive; indeed a step function is a natural model for a neuron, giving x when firing, zero when not. We can generalize the definition of the ReLU function in a multi dimensional space. Thus, considering x ∈ R m , σ : R m → R m is defined by applying the one dimensional ReLU to each component, i.e.

(σ(x)) i = σ(x i ).

The pooling operation

The pooling operation allows the CNN to decrease the spatial dimensions of the input and can be modeled as a functional Π :

V i → V i .
Since the image I has a compact support, we can assume that the input to Π has still a compact support Ω ⊆ R 2 in the spatial dimensions. Then, we can split Ω in r mutually disjoint open sets {Ω j } j=1,...,r and we can choose a function τ (e.g. the max function). We should also consider a function : V i → V i that rescales of a parameter α the support of a given function, i.e. (z(x, y)) = z(αx, αy). Then, the output of the pooling operator becomes

Π(z)(x, y, f ) = (τ (z(ξ, η, f ))) , for (ξ, η) ∈ Ω j (x, y) , j = 1, . . . , r
The pooling layer

The discrete formulation of the pooling operation is the pooling layer. A common approach is to split the spatial dimensions (width, height) in s × s boxes and shrink them to a single value applying the function τ (e.g. mean or max) to the function z I (x, y, f ) on each box B x,y,f where x, y = 1, . . . , n and are even and

f = 1, . . . , m. In particular, Π(z I )(x, y, f ) = τ (z I (ξ, η, f ) : (ξ, η) ∈ B x,y,f ),
As a result this operator changes the size of the tensor to [n/s × n/s × n o ]. The pooling layer is not always applied after an artificial neuron and a specific choice can be done for each CNN architecture; however, if it is applied, the output becomes

h i I (x, y, f ) = Π(Θ(T Ψ i f (z i-1 I (x, y, f ))(x, y)).

The fully-connected step

At the end of the CNN architecture, several fully-connected layers are usually applied to modify the output sizes. The fully-connected layer can be modeled as a function

Υ j (z)(η) = G ϕ η (p)z(p)dµ(p)
where G = V L for j = 1 and G = H for j = 2, . . . , k. Similarly, ϕ η : V L → V L for j = 1 and ϕ η : H → H for j = 2, . . . , k.

The fully-connected layer

The fully connected layer is a classical layer that connects each neuron in one layer to any neuron in another layer. The discrete formulation of the j-th one is

h j (z)(η) = p∈G ϕ η (p)z(p)
for the different choice of G. The output of each fully-connected layer becomes a tensor of size [1 × 1 × M j ]. In the case of j = k, i.e. the last fully-connected layer, M k = M , i.e. the number of the categories of the classification task.

The loss functional

The CNN, modeled as a functional F = F Ψ 0 f ,...,Ψ L f ,ϕ 1 ,...,ϕ k , contains unknown a-priori weights, in particular in the convolutional and fully-connected layers; furthermore, it should approximate the operator y : V 0 -→ H. The latter is a-priori known over a subset Γ of the domain space V 0 , called training set. This is a finite set of N samples in V 1 , (I i ), i = 1, . . . , N and the corresponding outputs function is commonly used. This is defined as follow:

L(F (I), y(I)) = log( M u=1 e (Fu(I)-F u (I)) ) + F u (I) -F y(I) (I) (3.5)
where u is the label selected by the neural network among M labels and y(I) is the true one. The choice of this training set and the regularity of the operator F should ensure that the operator itself remains a good approximating on the whole set of images. This extension process is called generalization.

The learning phase

During the learning phase all the weights in the CNN are updated in order to diminishing the loss function (3.5). This is done through some optimization Furthermore, the architecture of a CNN has to be determined before the training phase. Modifying the number of layers, the sizes of the tensors of the convolutional layers and any other parameters of the architecture could lead to an improvement or a worsening of the performance of the architecture. All these hyperparameters are hard to determine and many trainings are usually done to find the ones that lead to the highest performance. There exists some other methods to increase performance, e.g. the dropout (that also avoids overfitting). At each training stage, each node is either dropped out (i.e. ignored) with probability 1 -p or kept with probability p. Indeed, a different reduced CNN is left at each stage; this allows to reduce node interactions, leading them to learn more robust features and then, generalizing better to new input data.

Analogy between the visual system and CNNs

We are now interested to analyze the connections between the visual system and CNNs. The first neural nets have been inspired by a simplification of the structure of the biological visual system, presenting a hierarchical structure, where each layer receives input from the previous one and gives output to the next one.

Despite such simplification, CNNs have reached optimal performances in processes typical of the human visual system, as e.g. object-detection [START_REF] Redmon | You only look once: Unified, real-time object detection[END_REF][START_REF] Ren | Faster r-cnn: Towards real-time object detection with region proposal networks[END_REF] or image classification [START_REF] He | Deep residual learning for image recognition[END_REF][START_REF] Simonyan | Very deep convolutional networks for large-scale image recognition[END_REF]. Along with the hierarchical organization, translation invariance is present in CNNs thanks to local convolutional windows shifting over the spatial domain.

This structure was first inspired by the localized RF of cells in the early visual pathways, and by the approximate translation invariance in their tuning.

In more recent years relationships between CNNs and the visual system have been further studied, with the eventual goal of making the CNNs even more efficient in specific tasks. To this end, a model of the first cortical layers described as layers of a CNN has been proposed in [START_REF] Serre | Robust object recognition with cortex-like mechanisms[END_REF]. In [START_REF] Yamins | Hierarchical modular optimization of convolutional networks achieves representations similar to macaque it and human ventral stream[END_REF][START_REF] Yamins | Using goal-driven deep learning models to understand sensory cortex[END_REF] the authors were also able to study higher areas by focusing on the encoding and decoding ability of the visual system. In the last years, Recurrent Neural Networks (RNNs) have been proposed to implement horizontal connections [e.g. [START_REF] Sherstinsky | Fundamentals of recurrent neural network (rnn) and long short-term memory (lstm) network[END_REF], or feedback terms [e.g. [START_REF] Liang | Recurrent convolutional neural network for object recognition[END_REF]. A modification of these neural networks, more geometric and closer to the structure of the visual system, have been recently proposed in [Montobbio et al., 2019a]. In particular, they have modeled the horizontal connectivity of V1 cells, implementing a connectivity kernel between each pair of filters in a single layer that estimates their similarities.

Convolutional Neural Networks

Moreover, it is well known that both V1 RPs and the first convolutional layer of a CNN are mainly composed by Gabor filters. To this end, biological based models of V1 cells RPs in terms of Gabor filters have been proposed in [START_REF] Serre | Robust object recognition with cortex-like mechanisms[END_REF][START_REF] Zhang | Convolutional neural network models of v1 responses to complex patterns[END_REF] and the statistic of the RPs of a macaque's V1 was studied in [START_REF] Ringach | Spatial structure and symmetry of simple-cell receptive fields in macaque primary visual cortex[END_REF]; however a comparison between these results and the statistics of learned filters in CNNs is still missing.

Invariance properties of CNNs

In this last section we are going to describe some invariance properties of CNNs.

We first remind that Gabor functions, i.e. a model of V1 cells RPs, are mainly invariance with respect to translation and rotations. On the other hand CNNs are translation equivariant since they are defined in terms of convolutional kernels [see [START_REF] Cohen | A general theory of equivariant cnns on homogeneous spaces[END_REF][START_REF] Cohen | A general theory of equivariant cnns on homogeneous spaces[END_REF].

Other invariance properties, strictly related to the invariances of the Gabor filters, can be imposed to optimize the training phase. In [START_REF] Wu | Flip-rotate-pooling convolution and split dropout on convolution neural networks for image classification[END_REF][START_REF] Marcos | Learning rotation invariant convolutional filters for texture classification[END_REF] the authors learned just a small number of filters, and then obtained an entire bank of filters by rotation. In [START_REF] Barnard | Invariance and neural nets[END_REF] the authors have studied invariances of neural nets with respect to specific feature spaces.

Another method for introducing invariances is data augmentation, which consists in incrementing the number of training images by flipping, rotating, rescaling, cropping, adding gaussian noise to them. In this way the neural network is trained on a larger set of images and can learn different kinds of invariances. A way to achieve rotation invariance in classification task is to rotate every test image by S different angles as performed in [START_REF] Fasel | Rotation-invariant neoperceptron[END_REF][START_REF] Dieleman | Rotation-invariant convolutional neural networks for galaxy morphology prediction[END_REF]. In [START_REF] Gens | Deep symmetry networks[END_REF][START_REF] Dieleman | Exploiting cyclic symmetry in convolutional neural networks[END_REF][START_REF] Laptev | Ti-pooling: Transformation-invariant pooling for feature learning in convolutional neural networks[END_REF] the authors introduce one or more novel layers to face the rotation invariance problem. In [START_REF] Dieleman | Exploiting cyclic symmetry in convolutional neural networks[END_REF] they propose four different layers that work on the input rotated by 0, 90, 180 and 270 degrees. A different kind of pooling is used in [START_REF] Laptev | Ti-pooling: Transformation-invariant pooling for feature learning in convolutional neural networks[END_REF] before the fully connected layers in order to obtain invariance with respect to different features. A kernel that finds out symmetries in the input connecting consecutive layers is introduced in [START_REF] Gens | Deep symmetry networks[END_REF].

Chapter 4

A biologically inspired CNN architecture:

LGN-CNN

In this chapter we introduce one of the novelty of our work, a CNN architecture inspired by the structure of the visual system, proposed in [START_REF] Bertoni | Lgn-cnn: A biologically inspired cnn architecture[END_REF]. To begin with, we will introduce a first convolutional layer that mimics the role of the

LGN and that eventually will approximate the LGN cells' RP, i.e. a LoG. Then, in next chapter, we will add a kernel to the second convolutional layer mimicking the horizontal connections of the V1 cells as proposed by Montobbio in [Montobbio et al., 2019a]. Finally, we will study how the filters approximate LGN and V1 cells RPs, how the kernel approximates the connectivity kernel of V1, if some Retinex effects occur and we perform grouping.

LGN in a CNN

As far as we know, the action of the LGN has not been implemented in a CNN.

As we have already discussed in section 1 the LGN receives the visual stimulus from the retina and processes it through its cells. Each cell, connected to its RF, acts as a linear filter that can be modeled as a LoG. Our aim is to build a CNN architecture that takes into account this behavior; indeed, it should contain a first layer that pre-processes the visual stimulus before it reaches the second layer that, in turn, should mimic V1. We have called this neural network architecture as Lateral Geniculate Nucleus Convolutional Neural Network (LGN-CNN).

The RPs of LGN cells can be modeled as a LoG which is a rotationally symmetric function. Thus, the first convolutional layer 0 , that models the LGN, should contain just a single filter that eventually should obtain a LoG shape. On the contrary, the second convolutional layer 1 , that models V1, should contain several filters that eventually should obtain Gabor shapes with different orientations. As described in chapter 3, CNNs are architectures composed by several convolutional layers, each containing many filters. Indeed, given a classical CNN architecture, we can add before the other convolutional layers, a layer 0 composed by only one filter Ψ 0 of size s 0 × s 0 and a ReLU function. In this way the first two convolutional layers 0 and 1 of this new CNN architecture model the first stages of the visual pathway. Furthermore, the introduction of 0 does not change the rest of the neural network since the input of 1 keeps the same dimensions as before.

Thus, our model of LGN can be easily implemented to any CNN architecture by adding 0 and a ReLU function. Moreover, the number of parameters of the neural network is just increased by s 0 × s 0 .

We expect the first filter Ψ 0 to be a good model of the RP of LGN cells thank to a simple result on rotational symmetric convex functionals. Indeed, let us consider a rotational symmetric convex functional F with a unique minimum ω;

then ω is also rotational symmetric. In particular, since F is rotational symmetric,

F (ω • g) = F (ω)
for a rotation g of an angle θ. Furthermore, since the minimum is unique, ω = ω • g and this implies the rotation symmetry of the solution. Many results on symmetries of minima of functionals can be found for example in [START_REF] Gidas | Symmetry of positive solutions of nonlinear elliptic equations in r[END_REF][START_REF] Lopes | Radial symmetry of minimizers for some translation and rotation invariant functionals[END_REF]. Our purpose is to extend these theoretical results on the

LGN-CNN architecture. Thus, the filter Ψ 0 should attain a rotational symmetric pattern and should eventually approximate a LoG if the LGN-CNN architecture we have proposed is a good model of the first stages of the visual pathways. More in details, the first convolutional layer 0 is composed by a single filter Ψ 0 of size 7 × 7, followed by a batch normalization layer b 1 and by a ReLU R. Thus, the second convolutional layer 1 receives as input a tensor of size 64 × 64 × 1. We do not insert a pooling layer after the first layer 0 , i.e. this layer does not change the dimension of the input image.

The second convolutional layer 1 is composed by 64 filters of size 11 × 11 and its stride is 2, halving the spatial dimensions. After 1 it is applied a batch normalization layer b 64 , a ReLU R and a max POOLING p 2 m with squares of size 2 × 2. Then, the third convolutional layer 2 , composed by 32 filters of size 5 × 5 × 64, is followed by a batch normalization layer b 32 , a ReLU function R and a max POOLING p 2 m . Thus, the fourth convolutional layer 3 is composed by 32 filters of size 3 × 3 × 32 followed by a batch normalization layer b 32 , a ReLU function R and a max POOLING p 2 m . Finally, the last convolutional layer 4 has 32 filters of size 3 × 3 × 32 followed by a batch normalization layer b 32 , a ReLU function R and a max POOLING p 2 m . Eventually three fully-connected (F C 1 , F C 2 , F C 3 ) layers of size 1000, 500 and 10 respectively are applied giving as output a vector of length 10. As a last step, a softmax σ is applied to the output vector of size 10 giving a probability distribution over the 10 classes. The functional that models this neural network is the following

F (I) :=(σ • F C 3 • F C 2 • F C 1 • p 2 m • R • b 32 • 4 • p 2 m • R• b 32 • 3 • p 2 m • R • b 32 • 2 • p 2 m • R • b 64 • 1 • R • b 1 • 0 )(I) (4.1)
A cross-entropy loss for softmax function defined as in equation (4.2) is applied to the functional (4.1) where u is the label selected by the neural network and y(I) is the true label.

L(F (I), y(I)) = log( u e (Fu(I)-F u (I)) ) + F u (I) -F y(I) (4.2)

As just described, after each convolutional layer there is a batch normalization layer b with the same size of the number of filters of the corresponding convolutional layer. In particular, the batch normalization layer b 1 after 0 performs a normalization similar to the one that the retina performs as described in [START_REF] Carandini | Normalization as a canonical neural computation[END_REF]. One of the main difference is the subtraction of the mean value µ performed by the batch normalization layer defined as follows

x i = x i -µ √ σ 2 + ,
where x i is the element to normalize, µ is the mean value of the batch, σ is the standard deviation of the batch and is a small value that prevents bad normalization in case of small standard deviation. However, the input images have zero mean which imply that the convolution with Ψ 0 have still zero mean.

Thus, the batch normalization layer between 0 and 1 has similar characteristics as the biological one. In spite of this, the two approaches differ since the statistical parameters (µ, σ) of b 1 are calculated channel-wise over all input of the batch, while in the biological normalization described in [START_REF] Carandini | Normalization as a canonical neural computation[END_REF] the σ value is calculated from a single input instance over a restricted spatial neighborhood. However, we expect that the final result will not differ heavily applying the batch normalization layer.

We trained our LGN-CNN architecture on a dataset of natural images called STL-10 [see [START_REF] Coates | An analysis of single-layer networks in unsupervised feature learning[END_REF] that contains 5000 training images divided in 10 different classes. The steps we have followed for modifying the training set are:

• Changing the images from RGB color to grayscale color using the built-in function rgb2gray of MatLab;

• Applying a circular mask to each image, leaving unchanged the image in the circle and putting the external value to zero;

LGN • Rotating each image by 5 random angles augmenting the dataset to 25000 images; thanks to the previous step no boundary effects are present;

• Cropping the 64 × 64 centered square that does not contain the black boundaries;

• Subtraction of the mean value in order to have zero mean input images.

Thus, after these steps we have obtained a rotation invariant training set composed by 25000 64 × 64 images. We have applied the same steps to the test set but we have rotated each image to just one random angle. Since the images are 64 × 64 we have decided to use quite large filters in the first and second layer (7 × 7 and 11 × 11 respectively) in order to obtain more information about their shapes.

We have trained the neural network for 30 epochs with an initial learning rate of 0.01, a learning rate drop factor of 0.97 and a piecewise learning rate schedule with a learning rate drop period of 1. The mini batch size is 128 with an every-epoch shuffle, the L 2 regularization term is 0.02 and the momentum is 0.9.

In Table 4.1 there are summarized the mean performances over 10 different trainings of three CNN architectures with the LGN layer 0 plus several convolutional layers (4, 8 and 12 respectively). As expected, the performance increases when the architecture is deeper. We stress out that the results obtained on the

LGN layer and the first convolutional layer are not affected from the rest of the neural network. Indeed, it is possible to add more layers and build more deeper In this section we focus on the filter Ψ 0 of the layer 0 . Indeed, we are interested in analyzing its symmetry properties. So far we have modeled the RPs of LGN cells with a single filter Ψ 0 . However it is well known that in the LGN there exists both cells with on-center/off-surround as well as off-center/on-surround RPs. In order to model these two kinds of cells we modify the current LGN-CNN architecture by adding a second filter in the layer 0 . This is not the case and figure 4.5 shows the on-center/off-surround and off-center/on-surround obtained after the training phase. However, for simplicity, we consider the model with a single filter.

Since we are interested in the rotational properties of Ψ 0 , we aim to quantify them. To this end, we can generate a new filter Ψ 0 S from Ψ 0 via a rotation invariance symmetrization. In particular, the normalized rotational invariant filter Ψ 0 S is obtained by the following procedure:

• Using the function imresize with scale 3 and bilinear method to enlarge the filter;

• Rotating the filter with imrotate by 360 discrete angles between 1 and 360 degrees and summing them up;

• Applying again the function imresize with scale 1/3 and nearest method to recover a filter with the same size of Ψ 0 ;

• Normalizing the filter by subtracting the mean and dividing it by L 2 norm.

Then, we compute the correlation between Ψ 0 S and the normalized Ψ 0 using the Matlab function corr2 in order to quantify the rotational invariance of Ψ 0 . Indeed, the computed correlation estimates how close the normalized filter Ψ 0 is w.r.t its corresponding normalized rotational invariant filter.

Thank to this tool we are now able to test the behavior of the first filter Ψ 0 of LGN-CNN for different values of the L 2 regularization term and adding more convolutional layers. In particular, we add to the previous architecture described in chapter 4.2 two convolutional layers composed by 32 filters of size 3 × 3 × 32 (each one followed by a batch normalization layer b and ReLU R ) after the third layer 3 . We also add other two convolutional layers with the same characteristics after the layer 4 . LGN-CNN influences directly the properties of the filters. Furthermore, the Table 4.2 shows that the rotational symmetry of Ψ 0 is stable with respect to variations of L 2 regularization term and adding convolutional layers.

L 2 term
Furthermore, we study the properties of Ψ 0 varying the data augmentation (DA) applied to the dataset. Table 4.3 shows the correlation of Ψ 0 with the LoG and the correlation of Ψ 0 with Ψ 0 S with three different DA applied to the dataset. In the first one we do not apply any DA (No DA); in the second one we randomly rotate the images of an angle of 0, π 2 , π, 3π 2 radiant (Mild DA); in the third one we apply the DA described in section 4.2 (Hard DA). From Table 4.3 it emerges that the introduction of rotation invariances in the dataset by rotating the images lightly affects the correlation with the LoG but gives stability to Ψ 0 allowing the filter to be more rotational symmetric.

In conclusion, thanks to the analysis performed on the properties of Ψ 0 , we can argue that the structure of the architecture itself influences the shape of the filters and that the introduction of 0 with a single filter Ψ 0 is a good model of the

LGN. In the next section we are going to prove that Ψ 0 is rotational invariant for a specific architecture whereas in chapter 6 we will study the Retinex effects of Ψ 0 .

Rotation invariance proof of Ψ 0

We first define the setting in which we study the rotation symmetry of Ψ 0 .

Let us consider the architecture of an LGN-CNN in which we can split the first convolutional layer composed by only one filter from the rest of the neural network which will be fixed. Thus, this first layer can be approximated by a function

Ψ 0 : R 2 → R, assuming Ψ 0 ∈ L 1 (R 2 ).
A general image is defined as a function I : R 2 → R where we assume I ∈ L 1 (R 2 ). We consider a subset Γ of all the images where, for each image I, is defined a labelling y : Γ → R, where y(I) is the corresponding label to the image I.

We require some rotational invariant properties on this set Γ. In particular, let us consider a rotation R θ of an angle θ on R 2 plane around its center; then, the composition

I θ = R θ (I) = I(R -θ (x)
) is still an image and we can also assume that I θ ∈ Γ (i.e., that the subset Γ is close under rotation). Furthermore, the rotated image should maintain the same label, i.e., y(I θ ) = y(I). Since the images we consider in this problem are gray-scale ones (thus with values between 0 and 1) we can also assume that the images are normalized with ||I|| L 1 ≤ 1. Summarizing all these properties we assume that Γ = {I ∈ L 1 (R 2 ) : ||I|| L 1 ≤ 1} Thus, the rest of the neural network is defined by a convex regularization C of the ReLU function σ defined in eq. (3.4)

C : R 2 → R 2 , s.t. ||C(x) -σ(x)|| L 1 < (4.3)
for a small enough > 0. And then we can define

F : L 1 (R 2 ) × L 1 (R 2 ) → R F (I, Ψ 0 ) := R 2 C (I * Ψ 0 )(z) dz.
(4.4)

Then F (I, Ψ 0 ) is the label that the LGN-CNN architecture associates to the image I and should eventually approximates y(I).

Thus, our aim is to minimize the following functional min

Ψ 0 ∈L 1 ( Ω) Γ |F (I, Ψ 0 ) -y(I)|dµ(I) (4.5)
where the integral done over the set Γ is a Bochner's integral (see e.g., section 5 of chapter V of [START_REF] Mikusiński | The Bochner Integral. Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften: Mathematische Reihe[END_REF][START_REF] Yosida | Functional Analysis[END_REF].

Thus, the function Ψ 0 attains the minimum of (4.5) where F is defined in (4.4)

and C is defined in (4.3). We aim to find out if there exist any rotational invariant properties on the function Ψ 0 . Indeed, the functional defined in (4.5) is convex thanks to the convexity of the function C defined in (4.3) and continuous. Then, the existence and uniqueness of a solution is guaranteed [see e.g., section 1.4 of [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF].

Remark 1. Let us consider two function f, g ∈ L 1 (R 2 ) and a rotation R θ of an angle θ. Then

f * R θ (g)(x) = (R -θ (f ) * g)(R -θ (x)) Proof. f * R θ (g)(x) = R 2 f (x -y)R θ (g(y))dy = R 2 f (x -y)g(R -θ (y))dy =
then we substitute y = R -θ (y) whose Jacobian has determinant equal to 1

= R 2 f (x -R θ (y ))g(y )dy = R 2 f (R θ (R -θ (x)) -R θ (y ))g(y )dy = R 2 f (R θ (R -θ (x) -y ))g(y )dy = R 2 R -θ (f (R -θ (x) -y ))g(y )dy = then we substitute y = R -θ (x) -y whose Jacobian has determinant equal to 1 = R 2 R -θ (f (y ))g(R -θ (x) -y )dy =(R -θ (f ) * g)(R -θ (x))
and this concludes the proof.

Thanks to this Remark we are now able to demonstrate the rotational invariance of Ψ 0 .

Theorem 4.3.1. Let Ψ 0 be a solution to the problem (4.5) where F is defined in (4.4) and C is defined in (4.3). Then Ψ 0 is rotational invariant.

Proof. Let us consider the rotated solution R θ (Ψ 0 ) of an angle θ ∈ [0, 2π].

Γ | R 2 C (I * R θ (Ψ 0 ))(z) dz -y(I)|dµ(I) =
and because of remark 1 Testing the theorem on the LGN-CNN architecture

= Γ | R 2 C (R -θ (I) * Ψ 0 )(R -θ (z)) dz -y(I)|dµ(I) Since for a general f ∈ L 1 (R 2 ) it holds R 2 f (R θ (x))dx = R 2 f (x)dx, then = Γ | R 2 C (R -θ (I) * Ψ 0 )(z)
We aim now to test the theorem just proved on the LGN-CNN architecture used in the theorem itself. In particular, we aim to see if 0 becomes rotational invariant;

however, since we train a simple architecture composed by a single convolutional layer 0 with a single filter Ψ 0 and a ReLU we do not expect to obtain a LoG shape filter as for the LGN-CNN defined in chapter 4.2. Indeed, in the previous case the

LGN-CNN architecture has other convolutional layers whose purpose is to further analyze the image and in particular the contours of the objects. For this reason we expect in that case that the LGN-CNN behave similarly to the LGN and the V1, i.e., Ψ 0 has a LoG shape. On the other hand, in the test we perform now we can only expect a rotational invariant filter as stated in theorem 4.3.1.

To with a single filter Ψ 0 of size 13 × 13 followed by a ReLU and by a fully connected layer. We train this neural network for a total of 25 epochs obtaining an accuracy of 98.55 % on the classification task. Figure 4.6 shows Ψ 0 of this LGN-CNN architecture. We can observe that it has a rotational invariant shape as we expect from theorem 4.3.1. Then, we approximate the filter Ψ 0 with a Gaussian function defined by the following formula G(x, y) = αe -x 2 -y 2 2σ 2

. Figure 4.7 shows Ψ 0 and its approximation. The rotation invariance of Ψ 0 is now enforced thanks to the approximation obtained with a rotational invariant function as the Gaussian.

The layer 1 of the LGN-CNN

Now we aim to study the second layer 1 of the architecture. As described in chapter 3, the first layer of a classical CNN contains filters approximated by Gabor functions (1.2). Thus, to enforce the relation between our architecture and the structure of the visual system, we compare the filters in 1 with the statistics of RPs shape from [START_REF] Ringach | Spatial structure and symmetry of simple-cell receptive fields in macaque primary visual cortex[END_REF]. To this end, we train two different CNNs, an

LGN-CNN defined by the functional (4.6)

F (I) := (σ • F C 1 • R • 3 • p 4 a • R • 2 • p 4 m • R • 1 • R • 0 )(I) (4.6)
and a classical CNN defined by the functional (4.7) in which we eliminate the first convolutional layer 0 and its following ReLU R, characteristic of our architecture.

F (I) := (σ • F C 1 • R • 3 • p 4 a • R • 2 • p 4 m • R • 1 )(I) (4.7)
As expected, in both architectures 1 contains filters with Gabor-like shapes after the training phase. Thus, we study the statistical distribution of these banks of filters comparing the results with the real data of Ringach. We aim to see if the introduction of 0 influences in any way the filters of 1 . Since the filters in a standard CNN have input on the retina, we will construct analogous filters in the LGN-CNN architecture. Consequently we do not consider the filters in 1 but the filters obtained by the convolution with Ψ 0 .

As a first step, we need to approximate the filters in 1 using the function (1.2);

in this way we are able to obtain for each filter the parameters defined in eq. (1.5) that Ringach has used for the statistical comparison as described in chapter 1.

Figure 4.8 shows some of the 1 filters of the LGN-CNN and their approximation; the same occurs in figure 4.9 in the case of classical CNN. As a first result, the mean correlation estimated with the built-in MatLab function corr2 increases from classical CNN to LGN-CNN from just 71.62% to 93.50%. This suggests that introducing the layer 0 with a single filter better regularize the filters in the following convolutional layer 1 .

To compare the statistical distributions of the 1 filters of the classical CNN and the LGN-CNN with the real data from Ringach we follow the same step in [START_REF] Ringach | Spatial structure and symmetry of simple-cell receptive fields in macaque primary visual cortex[END_REF], described in chapter 1. Indeed, we plot the distribution in the (n x , n y ) plane and approximate the data with two lines in the following way. We first approximate the points closer to the origin with a line y = αx and then the rest of the points are approximated with a line starting from the end of the previous one. Chapter 5

The LGN-CNN architecture with the transition kernel

Horizontal connectivity of V1 in a CNN

Even if the analogy with biological vision is strong, CNN architectures contains a simplified feedforward mechanism that ignores many of the main processes allowing the visual system to interpret a visual scene. Thus, we insert a mechanism of horizontal propagation first proposed in [Montobbio et al., 2019a]. In particular, this mechanism is defined by entirely learned connectivity kernels which will be analyzed in terms of invariance patterns we expect to arise as a result of learning from natural images. The connectivity kernel will be implemented only in the layer 1 and it should express long range connections between the learned filters of 1 . Some implementations of horizontal connections of convolutional type have been already proposed by [START_REF] Liang | Recurrent convolutional neural network for object recognition[END_REF][START_REF] Spoerer | Recurrent convolutional neural networks: a better model of biological object recognition[END_REF]. In particular, they have applied a recurrent formula analogous to (1.10), describing an evolution in time. However, the horizontal kernels employed in these works are very localized, so that the lateral kernels applied at each step only capture the connections between neurons very close-by in space.

More recently, Montobbio in [Montobbio et al., 2019a] has proposed a two-step version of this model that allows the kernel to express long range connections 51 between neurons. In the first step, the output of the first layer 1 is mapped to the corresponding feature space through the feedforward stream, leading to an activation pattern h 1 = ReLU (z 1 ); then it is convolved with a connectivity kernel K 1 with a wider support. In the second step, the new output h1 is obtained by averaging between this propagated activation K 1 * h 1 and the original activation h 1 . Indeed, the update rule reads:

h1 = 1 2 K 1 * h 1 + h 1 . (5.1)
We stress out that the entire CNN architecture is first trained over a classification task, with the exception of the lateral kernel K 1 . In a second step, the rest of the architecture is kept fixed and only the connectivity kernel K 1 is updated.

From a physiological point of view it means that in an embryo first there is the formation of filters, and then the connectivity between them. This is proved by experiments which show that there is no lateral connectivity in animal who do not seen alignments in the beginning of their life. Thanks to this procedure it is possible to easily add the lateral kernel to a pretrained CNN without any strong modifications. Furthermore, this two-step training allows the filters of 1 to be trained freely without any constraints from K 1 . In fact, since at the beginning of the first training all the filters are initialized randomly, training K 1 simultaneously with the filters of 1 could force them to attain particular shapes. However, our purpose is to extract long range information from the bank of filters without imposing any constraints on them.

Let us describe in more details the connectivity kernel K 1 . K 1 is a 4-dimensional tensor parameterized by 2-D spatial coordinates (i, j) and by the indices (f, g) corresponding to all pairs of 1 filters. Indeed, for fixed f and g filters, the function

(i, j) → K 1 (i, j, f, g) (5.2)
expresses the strength of connectivity between the filters Ψ f and Ψ g , where the spatial coordinates (i, j) indicate the displacement in space between the two filters. In order to express all possible displacements between filters, the spatial dimensions of K 1 are almost doubled with respect to the filters ones. The idea behind this model is that K 1 behaves like a 'transition kernel' on the feature space of 1 and modifies the feedforward output according to the learned connectivity between filters. Thus, similarly to the horizontal connectivity of cells in V1, a filter activation increases the activation of the filters strongly related with it.

Let us also observe that K 1 takes the form of a linear propagation applied before the nonlinear activation function of 1 . Indeed, an even wider connectivity may be modelled by iterating the process, obtaining a larger kernel in spatial dimensions.

This can be obtained via 'self-replication' by convolving K 1 against itself in the following way:

1 2 K 1 * h1 + h1 = 1 4 K 1 * K 1 * h 1 + 2K 1 * h 1 + h 1 .

The proposed architecture

In this section we give a detailed overview of the LGN-CNN architecture with a transition kernel in the second convolutional layer,proposed in [START_REF] Bertoni | Emergence of lie symmetries in functional architectures learned by cnns[END_REF], as well as the data and training scheme employed.

training stage: we re-update the whole architecture including the lateral kernel K 1 for a maximum of 800 epochs with early stopping as in the first phase. The initial purely feedforward training is intended both to obtain more stable learning of the receptive profiles, and to simulate the pre-existing orientation tuning of receptive profiles in V1 prior to the development of horizontal connections [START_REF] Espinosa | Development and plasticity of the primary visual cortex[END_REF]. However, after the "initialization" stage, all the network weights are trained jointly: this allows the feedforward weights to possibly readjust in the presence of lateral connections.

In order to enhance the generalization ability of the neural network, along with the weight regularization and the batch normalization layers, we apply dropout with dropping probability equal to 0.5 after the convolutional layer 10 . This dropout is intended to reduce overfitting in the final classification layers by weakening the reciprocal dependencies between weights of the same layer: this allows the network to have a more stable selection of the features relevant for image classification [START_REF] Srivastava | Dropout: A simple way to prevent neural networks from overfitting[END_REF]. We also apply dropout with dropping probability 0.2 when applying the kernel K 1 [see also [START_REF] Semeniuta | Recurrent dropout without memory loss[END_REF] in the second training.

Randomly dropping 20% of the lateral connections at each weight update has the purpose of avoiding co-adaptation of the connectivity weights, thus reinforcing their dependency on the intrinsic geometric properties of the receptive profiles. Applying dropout increase the performance on the test set from 72.24% to 80.28%. The network architecture and optimization procedure are coded using Pytorch [START_REF] Paszke | Automatic differentiation in pytorch[END_REF].

The performance of the neural network are quantified as accuracy (fraction of correctly predicted examples) on the CIFAR-10 testing dataset. We train several CNN architectures varying the number of layers and features. For the analyses described in section 5, we select the architecture that have reached the best mean performances (80.28%± 0.17 over 20 trained instances of the shows the accuracy values for architectures that have 16, 32 and 64 filters in 1 respectively). Let us note that, since we are mainly interested in the emergence of symmetries, we do not focus on reaching state-of-the-art performances on classifying the CIFAR-10 dataset. However, our architecture reaches fair performances that can be increased by adding further convolutional layers and/or increasing the number of units.Furthermore, we observe that all the architectures examined show comparable behaviors as regards the invariance properties of the convolutional layers 0 and 1 and the connectivity kernel K 1 . Therefore, it is reasonable to expect a similar behavior also when further increasing the complexity of the model.

Results on the LGN-CNN with transition

kernel K 1

Emergence of rotational symmetry in the LGN layer

As described in [START_REF] Bertoni | Lgn-cnn: A biologically inspired cnn architecture[END_REF] and shown in chapter 4 the introduction of a convolutional layer 0 containing a single filter Ψ 0 models the role of the LGN.

This first layer acts as a pre-filtering step of the input visual stimulus before it reaches the V1 cells. We have also shown that a rotational invariant pattern is attained by Ψ 0 , arguing that such behavior should be present also for deeper and more complex architectures. In this section we are going to see if this behavior persists.

Figure 5.3A shows the filter Ψ 0 obtained after the training phase. As expected it has a radially symmetric pattern and its maximum absolute value is attained in the center. Thus, we approximate the filter with the classical LoG model for the RP of an LGN cell. To do so, we find the optimal value for the parameter σ in Eq.

(1.1) by using the built-in function optimize.curve fit from the Python library SciPy. This result enforces the one obtained in the LGN-CNN architecture described in chapter 4.2. Indeed, Ψ 0 spontaneously evolves into a radially symmetric pattern during the training phase, and more specifically its shape approximates the typical geometry of the RPs of LGN cells.

Emergence of Gabor-like filters in the first layer

As introduced in section 1, the RPs of V1 simple cells can be modeled as Gabor functions by Eq. (1.2). Furthermore, the first convolutional layer of a classical CNN architecture usually shows Gabor-like filters [see e.g. [START_REF] Serre | Robust object recognition with cortex-like mechanisms[END_REF]. This behavior suggests that the first convolutional layer of classical CNNs assumes a role analogous to V1 orientation-selective cells. In this section, we first approximate the filters of 1 as a bank of Gabor filters. This allows us to obtain a parameterization in terms of position (x 0 , y 0 ), orientation θ and parity φ of each filter. Thus, the parameterization will provide a suitable set of coordinates for studying the corresponding lateral kernel in the R 2 × S 1 domain in the following sections.

After the training phase, Gabor-like filters emerge spontaneously in 1 as expected (see figure 5.4A). We approximate the filters in 1 with the Gabor function in Eq. (1.2), where all the parameters are found using the built-in function optimize.curve fit from the Python library SciPy. The obtained approximation is significant; indeed, the mean Pearson's correlation coefficient (obtained using the built-in function pearsonr from the Python library SciPy) between the filters and their Gabor approximations is 89.14%. Furthermore, all correlation values are statistically significant (p < .001). This step is crucial for the analysis of the kernel K 1 since a good approximation of 1 filters will allow us to re-parameterize the kernel itself in the space R 2 × S 1 .

We also expect that the introduction of the pre-filtering layer 0 should influence the filters of the following layer 1 . Since Ψ 0 acts as a LoG on the input image we expect that 1 filters should not show any Gaussian-like filters. Indeed, as shown in figure 5.4A, the layer 1 only contains filters sharply tuned for orientation, with no Gaussian-like filters. On the other hand, by removing 0 , the two types of filters are mixed up in the same layer. Figure 5.4B shows the 1 filters of a trained classical CNN with the same architecture, but without 0 , where there are several Gaussian-like filters. However, we can observe from figure 5.4A that some filters of the LGN-CNN architecture with transition kernel have more complex shapes that are neither Gaussian-, nor Gabor-like; indeed, since we have not introduced other geometric structures on following layers, it is reasonable to observe the emergence of more complex patterns.

As a last step, we decide to re-organized the filters of 1 in a different way. We first split the bank in odd and even filters using the parameter φ and then we reordered them w.r.t. the orientation θ. The interest of the organization in odd and even filters comes from comparison with neurophysiological data. Indeed, it is well known that the RPs of simple cells are organized in odd and even profiles, with different functionality. The odd ones are responsible for boundary detection, the even ones for the interior. The reorganization w.r.t. θ is useful to see if all the orientations are analyzed by the filter bank and to rearrange the kernel K 1 w.r.t.

that parameter.

Indeed, we split the filter bank of 1 w.r.t. the parity of their approximation, indicated by the parameter φ, that was forced to be between -π and π. Specifically, we labelled a filter as odd if π 4 < |φ| < 3π 4 , as even if 0 < |φ| < π 4 or 3π 4 < |φ| < π. Figures 5.5 and 5.6 show the odd and even filters rearranged w.r.t. the orientation θ.

For the sake of visualization, the even filters whose central lobe have negative values are multiplied by -1. The orientation values are quite evenly sampled, allowing the neural network to detect even small orientation differences. Most of the filters have high frequencies, allowing them to detect thin boundaries, but some low-frequency filters are still present.

Re-parameterization of the connectivity kernel using the first layer approximation

In this section, we examine the learned transition kernel K 1 , to investigate whether it shows any invariances compatible with the known properties of the lateral connectivity of V1. In order to study the selectivity of the connectivity kernel to the properties of 1 filters, we first rearrange it based on the set of coordinates in R 2 × S 1 induced by the Gabor approximation of the filters.

We first split the kernel w.r.t. the parity of 1 filters, resulting in two separate connectivity kernels for even (figure 5.6A) and odd (figure 5.5A) filters. We then adjust the spatial coordinates of each kernel using the estimated Gabor filter centers (x 0 , y 0 ) in order to re-center the kernel of each filter w.r.t. any other filter of 1 . Specifically, for each f, g in {1, . . . , n}, we shift the kernel K 1 (•, •, f, g) of Eq. (5.2) so that a displacement of (i, j) = (0, 0) corresponds to the situation where the centers of the filters Ψ f and Ψ g coincide.

So far we have re-organized each kernel w.r.t. the spatial domain R 2 without modifying the third dimension. Indeed, the original ordering of f, g in {1, . . . , n} has no geometric meaning. However, each filter Ψ f is now associated with an orientation θ f obtained from the Gabor approximation. Therefore, we are able to rearrange the slices of K 1 so that the f and g coordinates are ordered by the corresponding orientations θ f and θ g . Indeed we expect a high correlation between filters with similar orientation and a low correlation between filters with different orientation as it happens in the horizontal connectivity of V1 cells. By fixing one filter Ψ f , we then obtain a 3-D kernel (i, j, g) → K 1 (i, j, f, g)

(5.3) defined on R 2 × S 1 , describing the connectivity between Ψ f and all the other 1 filters, each shifted by a set of local displacements (i, j) ∈ {-6, . . . , 6}×{-6, . . . , 6}.

Non-maximal suppression within orientation hypercolumns

The re-parameterized connectivity kernel K 1 describes the strength of interaction as a function of relative displacement and relative orientation of the profiles.

In this section, we restrict ourselves to the case of a displacement (i, j) = (0, 0), i.e. the case of profiles belonging to the same hypercolumn of orientation. We thus study, for fixed f , the function

θ g → K 1 (0, 0, f, g) = K 1 (0, 0, θ f , θ g ).
(5.4)

This function, plotted in figure 5.7, describes the learned pattern of excitation and inhibition between the profiles Ψ f and Ψ g as a function of the orientation θ g . Notably, the resulting interaction profile showed a "Mexican hat"-like shape, indicating an enhancement of the response to the optimal orientation and a suppression of the response to non-optimal orientations. This type of profile has been shown to yield a sharpening of the orientation tuning [Bressloff and Cowan, 2002].

Association fields induced by the connectivity kernel

We are now interested in the analysis of the re-parameterized kernel in terms of the association fields induced by the kernel itself. Indeed, starting from the re-parameterized kernel centered around a filter Ψ f as in Eq. ( 5.3), we use the θ-coordinates to construct a 2-D association field as in [START_REF] Sanguinetti | A model of natural image edge co-occurrence in the rototranslation group[END_REF].

We first define a 2-D vector field by projecting down the orientation coordinates weighted by the kernel values. Specifically, for each spatial location (i, j), we define

V (i, j) := max g K 1 (i, j, f, g) • n g=1 K 1 (i, j, f, g)v g n g=1 K 1 (i, j, f, g)v g , (5.5)
where v g ∈ R 2 is a unitary vector with orientation θ g . This yields for each spatial point (i, j) a vector whose orientation is essentially determined by the leading θ values in the fiber, i.e. the ones where the kernel has the highest values. The norm of the vector is determined by the maximum kernel value over (i, j). Finally, we defined the association field as the integral curves of the so-obtained vector field V starting from points along the trans-axial direction in a neighborhood of (0, 0).

Figure 5.8B shows the association field obtained from the kernel computed around the filter Ψ f in figure 5.8A, with orientation θ f = 3π 10 . The vector field V is plotted using the Matlab function quiver, and the integral curves are computed using the Matlab function streamline. The vectors and curves are plotted over a 2-D projection of the kernel. The latter is obtained by first resizing the kernel by a factor of 10 using the built-in Matlab function imresize for better visualization, and then projecting down on the spatial coordinates by taking the maximum over g. Note that around (0, 0) the association field is aligned with the orientation of the starting filter Ψ f , while it starts to rotate when it moves away from the center. This behavior is consistent with the typical shape of psychophysical association fields of the horizontal connectivity of V1 cells, see section 1.

We can also note that a similar pattern arises with the integral curves of [START_REF] Sanguinetti | A model of natural image edge co-occurrence in the rototranslation group[END_REF], see figure 1.5D. However, in our case the rotation is less evident since the spatial displacement encoded in the kernel K 1 is more localized than the edge co-occurrence kernel constructed in [START_REF] Sanguinetti | A model of natural image edge co-occurrence in the rototranslation group[END_REF]. In spite of that, we approximate each integral curve as a circular arc, obtained by fitting the parameter k of Eq. (1.7) to minimize the distance between the two curves. Figure 5.8C shows in red the approximation of each integral curve. The empirical curves induced by the learned connectivity kernel are very close to the theoretical curves, with a mean Euclidean distance of 0.0036. Indeed, the learned kernel generates a good approximation of the theoretical integral curves; in the next section we are going to see if also the vector field is well modelled by the theoretical one.

Comparison of transition kernel with the solution of the heat equation

As recalled in chapter 1, there exists a relationship between connectivity kernel and the fundamental solution of Sub-Riemannian operators. In analogy with this relation, we compare the learned transition kernel K 1 with the fundamental solution The association field of the transition kernel K 1 (blue), and its approximation using the integral curves defined in (1.7) (red). For better visualization the kernel has been resized by a factor of 10 using the built-in Matlab function imresize. (5.6) where the sub-Laplacian operator is generated by the vector fields introduced in [START_REF] Citti | A cortical based model of perceptual completion in the roto-translation space[END_REF] and defined by Eq. (1.6). To this end, we fit the parameter α by comparing the vector field V generated from the kernel K 1 described in the previous section with the 2-D vector field obtained with the same procedure from the fundamental solution of (5.6). More precisely, the optimal parameter α is chosen as the one minimizing the mean angular difference between the two vector fields. We obtain an optimal value α = 4.40, with a mean angular difference of 0.0340. Figure 5.9A shows the vector field V in blue, and the sub-Riemannian vector field with optimal α in red. As already observed, the learned kernel K 1 captures the connections over a localized spatial area. Figure 5.9B shows in green the sub-Riemannian vector field over a wider spatial region, with the local area that best fits the vector field V highlighted in red. Chapter 6

of the sub-Riemannian heat equation in R 2 × S 1 ∂u ∂t = -α∆ X 1 ,X 2 u = -α X 2 1 u + X 2 2 u ,
The Retinex effects of LGN-type learned filters

In this chapter we focus on testing the rotational symmetric filter Ψ 0 of the LGN-CNN architecture described in chapter 4 on Retinex effects and contrast-based illusions. In chapter 2 we have described how the model of [START_REF] Morel | A pde formalization of retinex theory[END_REF] have been neurally interpreted in [START_REF] Citti | A gauge field model of modal completion[END_REF] and applied to LoG. Our purpose is to extend this algorithm to a general operator and to test it on Ψ 0 .

Retinex algorithm via learned kernels

We now introduce the Retinex algorithm we apply to the operator Ψ 0 . Indeed, our approach aims to find the perceived image I starting from a visual stimulus I.

We represent the action of the cortical RPs as a convolutional operator from the inverse operator itself. In order to compare our algorithm with the Morel one we face the problem for a Laplacian operator as performed in [START_REF] Morel | A pde formalization of retinex theory[END_REF]. As we will see the results for a Laplacian operator are comparable.

O M : L 2 (R 2 ) → L 2 (R 2 ), ( 6

Application of the algorithm

We can now test our algorithm to different operators and see if Retinex effects occur. We start by finding the inverse operators of some modeled and learned operators; then, we test the algorithm on two images. We point out that the Retinex model is an algorithm of contrast perception and does not try to improve the image quality.

Detect the inverse operators

Applying the steepest descent method, we obtain an iterative process, which can be formally expressed as

M t+1 = M t + dt • (M * M t -δ). (6.8)
The algorithm stops at time T when

|| M T +1 -M T || L 1 dt
< , for a fixed error > 0.

Thus, ||M * M T -δ|| L 1 < and indeed M T becomes a good approximation of M . -0.9 and 0.9 (indeed really close to completely black and white dots) whereas the entire background is gray with 0 value.

Thus, we apply the algorithm to the exact inverse of the Laplacian operator log( x 2 + y 2 ). From figure 6.2b it is clear that the Retinex effects occur to the gray dots. Furthermore, we can observe that the contours of the two dots are more defined w.r.t. the discrete Laplacian operator. Also in this case the values in the position of the dots are close to -0.9 and 0.9.

the LoG operator. In particular, the value of square 'A' changes to -0,6553 whereas the value of square 'B' changes to 0,02351. Thus, we study the behavior of Ψ 0 operator whose results are shown in figure 6.3e. Even in this case Retinex effects occur where the value of square 'A' changes to -0,6035 and the value of 'B' changes to 0,2348. Figures 6.3d, 6.3d and 6.3f highlight the two squares 'A' and 'B' in the starting image and in the recovered images using the LoG and Ψ 0 operators. In this way it is clearer that the Retinex effects occur in both cases.

To summarize, we have shown that the filter Ψ 0 considered as a convolution operator shows Retinex effects really closed to the Retinex effects of the LoG. This enforces again the link between the structure of the LGN-CNN architecture and the structure of LGN.

Information transmission efficiency

We are now interested to analyze the information transmission efficiency properties of the filter Ψ 0 and compare them to the LGN ones, expecting a similar behavior. Indeed, it is well established (see e.g. [see e.g. [START_REF] Reinagel | Temporal coding of visual information in the thalamus[END_REF][START_REF] Zaghloul | Different circuits for on and off retinal ganglion cells cause different contrast sensitivities[END_REF][START_REF] Uglesich | Stimulus size dependence of information transfer from retina to thalamus[END_REF][START_REF] Im | Indirect activation elicits strong correlations between light and electrical responses in on but not off retinal ganglion cells[END_REF][START_REF] Pregowska | Information processing in the lgn: a comparison of neural codes and cell types[END_REF] that the average firing rate of the retinal neurons that drive information to the LGN is much bigger than that of the LGN. In particular, LGN is able to delete spikes preserving the more informative ones leading to a loss of information.

Thus, we study the information loss on the 8000 images of STL10 test set by convolving each image with Ψ 0 and computing the entropy from the histogram of the gray-scale values via the built-in MatLab function entropy. It turns out that on average the entropy decreases from 7.04 to 5.97 with a loss of 15.27 % of the information. Thus, we reconstruct the images using the Retinex algorithm described in section 6.1. The average entropy increases to 6.92 leading to a loss of just 1.83% of the information w.r.t. the original dataset. This suggests that almost the entire information contained in the visual stimulus can be reconstructed via some feedback or horizontal connections, where the reconstructed stimulus Chapter 7

Perceptual grouping with learned kernels

In this chapter we propose a generalization of the algorithm proposed in chapter 2, for detection of perceptual units. Up to now it has been proved that different kernels have different grouping capabilities. Hence, for every different problem it was necessary to introduce an ad hoc kernel. We propose to learn directly the kernels from a set of images with a neural network and to combine the kernels found at different layers.

A general method of group detection

Thus, we can now describe the general algorithm we can use to find the perceptual units on the images.

The method:

1. Choose a bank of filters;

2. Define a kernel with respect to the chosen bank;

3. Apply at each spatial location of the image all the filters and select the one with the best response (if necessary); 5. Find the eigenvectors with the highest eigenvalues that represent the main perceptual units of the image.

Modeled kernel 7.2.1 Kernel computed from Gabor filters

In this section we will generate a kernel from a bank of Gabor filters using the procedure introduce in section 1.2.1. First, we consider a bank of Gabor filters We can now compute the generation kernel defined by equation (1.8) which can be expressed in the case of Gabor filters by the following analytic formula:

(Ψ p ) = (Ψ x,y,θ ),
K(p, 0) = σ 2 π exp - x 2 + y 2 4σ 2 - 2σ 2 π 2 (1 -cos θ) λ 2 cos π x(1 + cos θ) + y sin θ λ . (7.1)
Note that this formula expresses the correlation between a generic point p = (x, y, θ) and the point (0, 0, 0). If we compute the kernel K(p, p 0 ) for a generic point p 0 = (x 0 , y 0 , θ 0 ) we can apply a rotation and a translation similarly to the generation of a Gabor filter described in section 1.3.2. Denoting (x 0 , y 0 , θ 0 ) -1 G the inverse in the group G, we impose: K((x, y, θ), (x 0 , y 0 , θ 0 )) = K((x 0 , y 0 , θ 0 ) -1 G (x, y, θ), (0, 0, 0)).

Then, starting from this kernel we can define the connectivity kernel through the iterating procedure introduced in equation (1.9). In figure 7.2 it is displayed the second iteration starting from the filter with θ = 0; in particular, we select at each position the filter with the highest correlation and we visualize only the ones that reach a certain minimum value of correlation. Note that there is a central connectivity area that has a similar behaviour with respect to the integral curves displayed in figure 1.5. Furthermore, there are two lateral lobes that represent the connectivity with filters with a similar direction at a certain distance; indeed neurally there exists a link between cells with collinear receptive profiles at a certain frequency distance. From a perceptive point of view this correlation permits to analyze visual stimuli containing parallel lines at the same distance.

In figure 7.3 we can see that the isosurface computed on this connectivity kernel is a good approximation of the solution of the heat equation. In particular the isosurface in the center is the approximation we consider whereas the lateral surfaces are due to the connections with collinear filters we have just explained.

Then, in presence of an image I, we apply the spectral grouping algorithm explained in section 2.2.2 with the resulting kernel K. The main difference between applying this connectivity kernel and the previous example is the presence of the direction of Gabor filters, so that the associated kernel acts on R 2 × S 1 . Indeed

Gabor filters lift the input image defined on R 2 to an output z I = z I (x, y, θ) defined on the cortical plane via equation (1.3). Equation (1.4) allows to associate an Figure 7.4 displays the first two eigenvectors in red and green obtained through the algorithm; in particular, the image contains both collinear and cocircular orientation elements that represent a line and a arc of a circumference and this kernel clearly detects them. However, if we try to use this kernel in presence of clouds of points we can see that the results are completely different as the ones obtained using the Gaussian kernel. Figure 7.5 displays the first three eigenvectors.

We can note that the three clouds are not perceived as distinct but the eigenvectors cross each other and the kernel seems to select orientation patterns. This result is reasonable since Gabor filters are able to detect the orientation of boundaries but not clumps of points.

Finally, we test this kernel with an image containing two clouds of points and a set of collinear segments. In order: red, green and blue eigenvectors.

algorithm has considered the remaining points and the line as the same perceptual unit. The third eigenvector has an orientation shape similar to the first one. With such an image the algorithm is not able to distinguish the three percepts but tries to find orientation patterns in the image.

Bank of learned filters

Now we test our algorithm with some learned banks of filters. We first describe the architectures of the neural networks and then we analyze their behavior on different images. We use different neural networks: a pre-trained one called GoogLeNet model imported from the Princeton version [DagNN format] [see [START_REF] Szegedy | Going deeper with convolutions[END_REF] and two neural networks we train on a set of gray-scale images containing numbers from 0 to 9 called MNIST [START_REF] Deng | The mnist database of handwritten digit images for machine learning research[END_REF].

GoogLeNet is a CNN trained on a set of RGB images for a classification task and it is composed of 2 convolutional layers: the first one is a bank of 64 filters of [7 × 7 × 3] sizes and the second one a bank of 192 filters of [3 × 3 × 64] sizes.

After each convolutional layer it is applied a POOLING; then there is a sequence of 9 inception layers with a POOLING after the first two and before the last two. On the other hand, the two CNN architectures are trained for a classification task on MNIST In particular, MNIST is a set of 70 000 gray-scale images of sizes 28 × 28 representing a number from 0 to 9, divided in a training set of 60 000 images and a test set of 10 000 images. The objective of the neural networks is to detect the number that is displayed on the image and give it as the answer.

The first architecture is composed of three convolutional layers without 0padding followed by a RELU or a POOLING. In particular, the first convolutional layer, which receives a 28 × 28 input, is composed of 128 filters of 5 × 5 sizes. Finally, it is applied a fully-connected layer, giving an output of length 10, and the error is calculated using a softmax loss.

Perceptual grouping with GoogLeNet

In this section we analyze the kernels generated from the first two banks of learned filters of GoogLeNet. We consider the first bank whose red canal is displayed in figure 7.7. The space in which the bank is defined is G 1 = R 2 × F 0 × F 1 where F 0 = 1 . . . 3 refers to 3 color canals (RGB) and F 1 = 1 . . . 64 labels the entire set of filters. Indeed, we can translate in the x-y direction each filter obtaining the following set

(Ψ 1 p ) = (Ψ 1 x,y,f 0 ,f 1 ),
where p = (x, y, f 0 , f 1 ) ∈ G 1 . Then we compute the connectivity kernel applying the iterating procedure introduced in equation (1.9). Similarly, considering the second convolutional layer, we can define the space

G 2 = R 2 × F 1 × F 2
where F 1 is the space of features of the first layer defined above and F 2 = 1 . . . 192 labels the second bank of filters. Therefore, the second layer is composed of 192 [3 × 3 × 64] filters that can be translated in the x-y direction, obtaining

(Ψ 2 p ) = (Ψ 2 x,y,f 1 ,f 2 ),
where p = (x, y, f 1 , f 2 ) ∈ G 2 . Also in this case we compute the connectivity kernel applying the iterating procedure introduced in equation (1.9). Now let us consider a visual stimulus I to which we apply the spectral grouping algorithm explained in section 2.2.2. In the present case we should consider the presence of the parameter f 1 ∈ F 1 that labels the set of filters and also the presence of the 3 canals of colours. Since we are not interested in the differences between each colour in the image we ignore that parameter and, thus, we can associate at each position (x, y) ∈ R 2 the filter that gives the highest response, i.e. the maxima suppression principle becomes:

z 1 I (x, y, f 1 ) = max f 1 i ∈F 1 z 1 I (x, y, f 1 i ), (7.2)
where, analogously to equation (1.3), we can define

z 1 I (x 0 , y 0 , f 1 0 ) = D Ψ 1 x 0 ,y 0 ,f 0 ,f 1 0 (x, y, f 0 )I(x, y, f 0 )dxdydµ(f 0 ), ( 7
z 2 I (x 0 , y 0 , f 2 0 ) = D Ψ 1 x 0 ,y 0 ,f 1 ,f 2 0 (x, y, f 1 )z 1 I (x, y, f 1 )dxdydµ(f 1 ), (7.4) z 2 I (x, y, f 2 ) = max f 2 i ∈F 2
z 2 I (x, y, f 2 i ).

(7.5)

Now we can build the two affinity matrices, A 1 I for the first layer applying equations (7.3) and (7.2) and A 2

I for the second one applying equations (7.4) and (7.5) as described in section 2.2.2. In this way we can use one matrix at a time to find the perceptual units of the image or weighting them together. In particular we define the two layers affinity matrix as (7.6) where 0 ≤ α ≤ 1. Note that A 0 I = A 2 I and A 1 I = A 1 I . In figure 7.8 we can see the first perceptual unit detected using A 1 I = A 1 I , i.e. considering only the first layer, and A 1 2 I = 1 2 (A 1 I + A 2 I ), i.e. a combination of the two layers. Note that there is a good improvement considering two layers instead of only one; however, none of them gives good results as the ones obtained using a kernel generated from a bank of Gabor filters. This is reasonable since some of the learned filters have a Gabor shape but not many orientations are present as we can see in figure 7.7. In section 7.2.1, the kernel computed from a bank of Gabor filters was unable to detect clouds of points (see figure 7.5). Then, the idea is to use the two layers In the center first eigenvector obtained with only the first layer. On the right first eigenvector obtained with a weighted combination of two layers.

A α I = αA 1 I + (1 -α)A 2 I ,
of GoogLeNet to approach the same task since in this case we do not have only Gabor filters. Figure 7.9 represents the three eigenvectors obtained in this case. If we compare this result with respect to the Gaussian kernel one (see figure 2.3A) we can see that they are really similar and the eigenvectors are also in the same order.

Eventually, we use the two layered structure with an image containing both clouds of points and collinear elements. Figure 7.10 displays the grouping. We can see that the first two eigenvectors are the two clouds of points (however the first one does not contain the upper part of the cloud) and the third one is the line. Therefore the two layers structure of GoogLeNet is extremely flexible and can face completely different problems with better results with respect to the ones obtained with the models of section 7.2 thanks to the heterogeneity of the filters. Furthermore we have seen that using two layers gives better results than a single one; in section 7.3.2 we are going to study deeply this correlation for two multi-layered architectures. I (x, y, f 1 ) = max

f 1 i ∈F 1 z 1 I (x, y, f 1 i ).
For the second layer we can give similar definitions z 2 I (x 0 , y 0 , f 2 0 ) = D Ψ x 0 ,y 0 ,f 1 ,f 2 0 (x, y, f 1 )z 1 I (x, y, f 1 )dxdydµ(f 1 ),

z 2 I (x, y, f 2 ) = max f 2 i ∈F 2 z 2 I (x, y, f 2 i ).
Then, as in section 7.3.1 we can apply the perceptual grouping algorithm defined in section 2.2.2 for each layer obtaining two affinity matrices A 1 I and A 2 I that can be combined in a unique matrix The second eigenvector contains the smaller cloud of points with a missing part and the last one the set of collinear segments. The grouping results are comparable with the one obtained using GoogLeNet architecture but they can be improved using a structure with more layers. Now let us consider the second architecture and, in particular, the first three banks of filters Ψ 1 , Ψ 2 and Ψ 3 . In this case we have three spaces

G 1 = R 2 × F 1 , G 2 = R 2 × F 1 × F 2 and G 3 = R 2 × F 2
× F 3 and we can give similar definitions for h 1 I , h 2 I and h 3 I and the corresponding maxima suppression principles as for the first architecture. We can apply the perceptual grouping algorithm defined in section where 0 ≤ α, β, α + β ≤ 1. In this setting we approach the same problem as before using the image with two clouds of points and a set of collinear segments to test the flexibility of the three layers structure. Figure 7.12 shows the grouping results.

We can see that the three eigenvectors represents the two clouds of points and the line. Therefore also in this case the kernels are extremely flexible, adapting to different type of percepts, and they give better results than the ones obtained using the two layered architecture of GoogLeNet and the previous one.

Comparison between the two trained architectures

Here we test the two architectures on some simple images in order to understand how the first eigenvector changes and how far points can communicate.

Let us start with the 2 layers architecture. First, we consider a set of images containing two collinear segments at different distances, then a set containing the four corners of a square at different distances. The perceptual units obtained on the first set of images are displayed in figure 7.13 in which the first column represents the set of images whereas the other 3 columns the first eigenvector obtained using different choices of α in equation (7.7). In particular the second column uses only the first layer (i.e. α = 1), the third column only the second layer (i.e. α = 0) and the last column a weighted combination of the two layers (i.e. α = 1 2 ). Looking carefully to the perceptual units, we can see that in the first three rows the two segments are seen as the same percepts for each choice of α. Then, increasing a bit the distance in the fourth row, the first layer becomes unable to link the two segments but can perceive only one. This means that at this distance for the first layer is hard to connect collinear segments. Besides, the second layer and the combination of them are still able to detect the two segments. When the segments become too far the second layer and the combination of layers can only detect a single segments as we can see in the last two rows. Note that this is an important result since there are experimental clues that suggest that the second cortical layer can connect points that are more distant than the first cortical layer alone.

A similar result is displayed in figure 7.14. Indeed in the second and third row the first eigenvector of the first layer is only a corner whereas the second layer and the combination can detect the entire square. In the last row the second layer and the combination can perceive three corners which is still far better than the eigenvector obtained using only the first layer. Note also that in this row the combination of the first two layers seems to work slightly better than only the second layer. This suggests that using many layers at the same time should improve the performances of the algorithm. However this is not enough to give such a conclusion and we should try to use an architecture with more layers.

Let us consider the 3 layers architecture and approach the same problem as before using a set of images representing the corners of a square at different distances.

The perceptual grouping results are displayed in figure 7.15 considering the affinity matrix defined in equation (7.8). The second, third and fourth columns represent the first eigenvector obtained using respectively the first layer (i.e. α = 1, β = 0), the second layer (i.e. α = 0, β = 1) and the third layer (i.e. α = 0, β = 0). Moreover, ) and the last one a combination of all three layers (i.e. α = 1 3 , β = 1 3 ). The hypothesis suggested by the first architecture in this case are reinforced.

Indeed we can see that the first layer gives worse results than the others two even applied alone. Moreover, if we use the first two layers or all three layers at the same time we obtain better results in general. Finally, the last row gives us a strong suggestion. Indeed, we can see that the first two layers detect only three corners whereas the combination of all three layers is able to perceive the entire square. This result is even clearer than the previous one and suggests that using more layers leads to better result on perceptual grouping.

Chapter 8 Conclusions

We introduced a biologically inspired CNN architecture able to correctly model the early visual pathway and tested the efficiency of the learned kernels on two perceptual problems: Retinex effects and individuation of perceptual units.

We have first described some state-of-the-art topics useful in the subsequent chapters. We started from the main structures of the visual system, focusing on the

LGN and V1 and analyzing the group symmetries present in these layers. Then, we have recalled the Gestalt laws with the related grouping and the Retinex theory of Land with its more recent developments. Thus, we have introduced the main structures composing the CNNs, analyzing the similarities with the visual system and the invariance properties.

Then, we have moved towards our contributions. To begin with, we have proposed a biologically inspired CNN with a first layer 0 , mimicking the role of the

LGN. This layer contained a single filter Ψ 0 that has turned out to be rotational invariant and to approximate a LoG. We have proved the rotation invariance of Ψ 0 and we have studied the statistical distribution of the subsequent layer 1 with respect to the recordings of the neurons of a macaque's V1. Thus, we have introduced an LGN-CNN architecture with a transition kernel K 1 , mimicking the role of the long-range horizontal connectivity of V1. We have studied the emergence of a rotational symmetric filter in 0 and of Gabor filters in 1 . Then, thanks to the approximation of 1 filters with a Gabor function, we have re-parameterized the transition kernel and we have studied its properties. We have first analyzed the association fields induced by the connectivity kernel and, then, the vector fields with the solution of the heat equation.

In the last chapters we have presented two applications on learned filters. We have described an algorithm for the Retinex effects via learned kernels, through the application of an inverse operator. Thus, we have applied our algorithm to two different images using different kernels and, in particular, the filter Ψ 0 , comparing the corresponding Retinex effects. We have also studied the information transmission efficiency of the filter Ψ 0 . Then, we have moved towards the problem of grouping. We have first analyzed the grouping obtained with a modeled kernel generated from a bank of Gabor filters. Then, we have put forward the results obtained with different banks of learned filters, showing a good flexibility due to the presence of different kinds of filters.

In the future, there are several possible research paths. Indeed, we could face the theoretical problem regarding the rotation symmetry of the first convolutional layer for a general LGN-CNN architecture. We could also introduce an autoencoder associated to this architecture, which can reconstruct perceived images with the Retinex effect. Furthermore, since the learned connectivity kernel K 1 describes the strength of interactions between filters shifted by up to 6 pixels w.r.t. one another in each direction, we could introduce a wider connectivity, modelled by taking further propagation steps. Thanks to the linearity, this composition would be equivalent to propagate the long-range kernel 2K 1 + K 1 * K 1 , obtained via "self-replication" from K 1 through convolution against itself. Indeed, the update rule becomes:

1 2 h1 + K 1 * h1 = 1 4 h 1 + 2K 1 * h 1 + K 1 * K 1 * h 1 .
Therefore, we could consider larger natural images and examine wider horizontal connectivity obtained via multiple diffusion steps -as well as to compare the propagated long-range connectivity kernels with theoretical kernels.

As other future perspectives, we would like to extend our study by considering a wider range of features; therefore, we could find a finer characterization of the geometry of first layer filters, including more complex types of receptive profiles.

Finally, we could extend our model to other convolutional layers, comparing their properties with higher cortices of the visual system.

L

  'entrée rétinienne est d'abord traitée par les familles radialement symétriques des cellules présentes dans le CGL qui sont responsables de la perception du contraste humain. Puis il atteint V1, dont les neurones ne sont pas seulement sensibles à l'intensité lumineuse du stimulus visuel, mais ils montrent également une sélectivité à d'autres caractéristiques, telles que l'orientation, l'échelle, la vitesse. En effet, chaque cellule présente dans les cortices visuels réagit à une zone locale du stimulus visuel appelé champ récepteur (CR), alors que la fonction qui décrit son activation en présence d'un stimulus visuel est appelée profil réceptif (PR) et sera désignée comme Ψ. Chaque emplacement rétinien est associé à un ensemble entier de neurones de V1 appelés colonne corticale, sensibles à toutes les instances possibles de la variable correspondante, organisés dans les soi-disant emphcartes d'orientation. Du point de vue mathématique, plusieurs modèles neuromathématiques basés sur la géométrie différentielle classique et les groupes de Lie ont été proposés.

R 2 ×

 2 F, où les paramètres (x 0 , y 0 ) ∈ R 2 indiquent l'emplacement rétinien où chaque PR est centré, tandis que f ∈ F encode la sélectivité des neurones vers d'autres caractéristiques locales de l'image. Un stimulus visuel peut être représenté comme une fonction I = I(x, y), où (x, y) représente un emplacement rétinien et I(x, y) la lumière correspondante. Son action z, associée à un noyau défini comme Ψ = Ψ(x, y), peut être modélisée en tant qu'un opérateur intégral linéaire comme suit z(I) := I * Ψ. Dans le cas des cellules simples de V1 F = S 1 , i.e. chaque neurone répond au maximum à une certaine orientation θ à un emplacement rétinien spécifique (x, y).

  the sub-Riemannian heat equation, with α = 4.40. (B) In green: the vector field of the best-fitting fundamental solution of the sub-Riemannian heat equation over a wider spatial region. In red: the local vector field fitted to the learned kernel. . . . . . . . . . . . . . 66 6.1 Comparison between inverse operators. First row: inverse of the discrete Laplacian and its exact inverse operator. Second row: inverse operators of a discrete LoG and the first filter Ψ 0 of an LGN-CNN. 69 6.2 Retinex effects of some inverse operators on starting image 6.2a. . . 71 INDEX xxv 6.3 Comparison between the Retinex effects of some operators on grayscale Adelson's checker shadow illusion. . . . . . . . . . . . . . . . . . . . 6.4 On the left: a gray-scale image I, the convolved image I and the reconstructed image I via eq. (6.7). On the right: the corresponding histograms of the gray-scale values. . . . . . . . . . . . . . . . . . . 7.1 Bank of 31 11 × 11 Gabor filters with σ = 1 and λ = 1. . . . . . . . 7.2 2 iterations Kernel with respect to the filter with θ = 0. . . . . . . . 7.3 (a) Solution of the heat equation. From: [Favali, 2017]. (b) Isosurface computed on the connectivity kernel iteratively generated from equation (7.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 On the left image containing collinear and cocircular oriented elements. On the right the grouping operated via a connectivity kernel generated by a bank of Gabor filters. In order: red and green eigenvectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 On the left image containing three clouds of points. On the right its grouping operated via a connectivity kernel generated by a bank of Gabor filters. In order: red, green and blue eigenvectors. . . . . . . 7.6 On the left image containing two clouds of points and a set of collinear segments. On the right its grouping operated via a kernel generated from a bank of Gabor filters. In order: red, green and blue eigenvectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Bank of 64 [7 × 7] filters of the red canal of the first convolutional layer of GoogLeNet. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 On the left image containing collinear and cocircular oriented elements. In the center first eigenvector obtained with only the first layer. On the right first eigenvector obtained with a weighted combination of two layers. . . . . . . . . . . . . . . . . . . . . . . . . . 7.9 On the left image containing three clouds of points. On the right its grouping operated via a weghted combination of two layers. In order: red, green and blue eigenvectors. . . . . . . . . . . . . . . . . xxvi LIST OF FIGURES 7.10 On the left image containing two clouds of points and a set of collinear segments. On the right its grouping operated via a weighted combination of two layers of GoogLeNet. In order: red, green and blue eigenvectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 7.11 On the left image containing two clouds of points and a set of collinear segments. On the right its grouping operated via a weighted combination of two layers. In order: red, green, blue and yellow eigenvectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 7.12 On the left image containing two clouds of points and a set of collinear segments. On the right its grouping operated via a weighted combination of three layers. In order: red, green and blue eigenvectors. 91 7.13 First column: starting images with 2 segments at different distances. Second column: first eigenvector of first layer. Third column: first eigenvector of second layer. Fourth column: first eigenvector of first and second layers applied together. . . . . . . . . . . . . . . . . . . 92 7.14 First column: starting images with corners of a square at different distances. Second column: first eigenvector of first layer. Third column: first eigenvector of second layer. Fourth column: first eigenvector of first and second layers applied together. . . . . . . . . 93 7.15 First column: starting images with corners of a square at different distances. Second column: first eigenvector of first layer. Third column: first eigenvector of second layer. Fourth column: first eigenvector of third layers. Fifth column: first eigenvector of first and second layers applied together. Sixth column: first eigenvector of first, second and third layers applied together. . . . . . . . . . . . 94 List of Tables 4.1 Mean performances of several architectures over 10 different trainings with LGN layer plus 4, 8 and 12 other convolutional layers (CL). . 4.2 Correlation between Ψ 0 and Ψ 0 S varying the L 2 regularization term and the number of layers of the LGN-CNN architecture. Best rotational symmetric filters are selected in cyan for both architectures.4.3 Correlation between Ψ 0 and its LoG approximation and between Ψ 0 and Ψ 0 S varying the data augmentation (DA) applied to the dataset. xxvii Chapter 1

Figure 1 . 1 -

 11 Figure 1.1 -On the left: RFP of an LGN cell where the excitatory area is in white and the inhibitory one is in gray. On the right: Its approximation by a LoG. From: [DeAngelis et al., 1995].

Figure 1 . 2 -

 12 Figure 1.2 -First row: RFPs of two simple cells of V1 where the excitatory area is in white and the inhibitory one is in black. Second row: their approximations by Gabor functions. From: [Sarti and Citti, 2011].

  Figure 1.3 -Cube scheme of V1 by Hubel and Wiesel. Cells belonging to the same column have a similar receptive profiles, the orientation hyper-columns are arranged tangentially to the cortical sheet.From:[START_REF] Hubel | Eye, brain, and vision[END_REF].

Figure 1

 1 Figure 1.4 -(A) Images obtained for four stimulus angles θ = 0, 45, 90, 135 degrees. Dark areas were preferentially activated by a stimulus with orientation θ; light areas were active during presentation of the orthogonal angle. (B) The orientation map obtained by vector summation of data obtained for each angle. The orientation preference of each location (x, y) is color-coded according to the key shown below.

Figure 1

 1 Figure 1.5 -(A) Association fields from the experiment of Field, Hayes and Hess, from [Field et al., 1993]. (B) 3D representation of the association field with contact planes of integral curves of the fields (1.6) with varying values of the parameter k, from [Citti and Sarti, 2006]. (C) Integral curves of the fields (1.6) with varying k,from[START_REF] Citti | A cortical based model of perceptual completion in the roto-translation space[END_REF]]. (D) The vector field of unitary vectors oriented with the maximal edge co-occurrence probability (red) with superimposed its integral curves (blue), from[START_REF] Sanguinetti | A model of natural image edge co-occurrence in the rototranslation group[END_REF].

  figure 1.5D , whose integral curves (depicted in blue) offer an alternative model of association fields, learned from images.

Figure 2 .

 2 Figure 2.1 -Examples of Gestalt laws.

Figure 2 . 2 -

 22 Figure 2.2 -Square of Kanizsa.

  Figure 2.3 -First line: starting images. Second line: corresponding grouping with Gaussian kernel. If the first three eigenvectors are displayed they are red, green and blue respectively. Columns: (A) image containing three clouds of points; (B) image containing collinear and cocircular oriented elements; (C) image containing two clouds of points and a set of collinear segments.

Figure 2 .

 2 Figure 2.4 -Grouping with stochastic kernel performed on image with positionorientation elements. From: [Barbieri et al., 2014b].

Figure 2 . 5 -

 25 Figure 2.5 -Grouping with modeled kernel on different image. This kernel allows to recover the curvature at each spatial location. From: [Abbasi-Sureshjani et al., 2018].

  algorithm, for example the steepest descent. However, the reduction of the loss function on the training set Γ does not always imply good performances on the entire set V 0 . To control if the neural network is able to generalize to never seen images, it is tested during the training phase on a test set Γ ⊆ V 0 disjoint with the training set, i.e. Γ ∩ Γ = ∅. Indeed, it is possible that the neural network performance increases on the training set but remains stable or decreases on the test set, leading to overfitting. If the performances continue to decrease on the test set is possible to adjust some parameters of the training or stop the training and selecting the CNN weights that give highest performances.

Figure

  of the first layers of the visual pathway in comparison with the first layers of the LGN-CNN architecture.

Figure 4 . 1 -

 41 Figure 4.1 -Scheme of the LGN and V1 in parallel with the first two layers 0 and 1 of the LGN-CNN architecture.

Figure 4 . 2 -

 42 Figure 4.2 -Architecture of LGN-CNN.

  architectures to obtain better results on the classification task without losing the properties of the first layers of the LGN-CNN. Since our focus was to analyze the structures that arises in the first two layers, we have not further investigated the performance of the CNN.

( a )

 a Filter Ψ 0 of first layer of LGN-CNN. (b) Minus the Laplacian of Gaussian (LoG).

Figure 4

 4 Figure 4.3 -Comparison between the filter Ψ 0 and minus the LoG. The two have a high correlation of 95.21% computed using the built-in function of MatLab corr2.

Figure 4 .

 4 Figure 4.3 shows the 3D filter Ψ 0 obtained after the training phase. Then, figure 4.3b shows its approximation with minus the LoG.The two have a high correlation of 95.21% computed using the built-in function of MatLab corr2. Thus, Ψ 0 spontaneously attains a LoG shape, modeling the RPs of LGN cells. Moreover, figure 4.4a shows the 2D image of Ψ 0 obtained after the training phase. Then, in figures 4.4b and 4.4c we plot a 2D approximation of Ψ 0 as a 280 × 280 filter and minus the LoG which shows the rotational symmetric pattern of Ψ 0 .

Figure 4

 4 Figure 4.4 -The first figure shows the 7 × 7 filter Ψ 0 of the LGN-CNN. To better visualize the filter Ψ 0 , we provide an approximating 280 × 280 filter and minus the LoG.

Figure 4

 4 Figure 4.5 -On-center/off-surround and off-center/on-surround filters of 0 with 2 filters.

2 CRemark 2 .

 22 dz -y(I)|dµ(I) = Finally, since Γ is close under rotation and y(I) = y(R -θ (I)) by hypothesis = ((I) * Ψ 0 )(z) dz -y(I)|dµ(I)Thus, R θ (Ψ 0 ) attains the same value of Ψ 0 for every choice of θ. But thanks to the uniqueness of the solution of this problem because of the compactness of Γ and convexity of the functional (4.5), Ψ 0 = R θ (Ψ 0 ) ∀θ ∈ [0, 2π] and this concludes the proof. Let us note that the first part of the proof of theorem 4.3.1 is valid for a general LGN-CNN architecture. In particular, if we have a solution Ψ 0 to the minimization problem (4.5), then every rotation R θ (Ψ 0 ) of an angle θ ∈ [0, 2π] is still a solution. The uniqueness of solution in theorem 4.3.1 guarantees that Ψ 0 is rotational invariant whereas for a general LGN-CNN this does not hold.

Figure 4

 4 Figure 4.6 -Filter Ψ 0 of LGN-CNN obtains after training.

( a )

 a Filter Ψ 0 of first layer of LGN-CNN. (b) Gaussian function.

Figure 4

 4 Figure 4.7 -Comparison between the filter Ψ 0 and a Gaussian. We can see that Ψ 0 is really close to a discrete approximation of a Gaussian fitted to the data.

Figure 4 . 8 -

 48 Figure 4.8 -On the left: filters from LGN-CNN. On the right: their approximation with the function (1.2).

Figure 4 . 9 -

 49 Figure 4.9 -On the left: filters from classical CNN. On the right: their approximation with the function (1.2).

Figure 4 .

 4 Figure 4.10 -Comparison between the statistical distribution on (n x , n y ) plane of filters of a classical CNN, of the LGN-CNN architecture and of RPs of real data.

Figure 4 .

 4 Figure 4.10 shows the three plots. As a first result, the introduction of 0 modifies the elongation of Gabor filters in 1 . In particular, in classical CNN the filters are often more elongated in the x direction as we can see from the slope of the interpolating line in figure 4.10a. In figure 4.10b we can see that the slope changes greatly and that the filters become much more elongated in the y direction. This behavior is similar to the RPs of simple cells in V1 as can be observed from the distribution of Ringach in figure 4.10c. Indeed, the slopes of the interpolating lines of Ringach distribution and of LGN-CNN 1 distribution are quite close. This enforces even more the link between the LGN-CNN and the structure of the visual system motivating us to pursue in this direction.

Figure 5

 5 Figure 5.1 -Scheme of the CNN architecture. Convolutional layers are numbered from 0 to 10 , fully connected classification layers are denoted as F C 1 , F C 2 , F C 3 . Horizontal connections governed by the connectivity kernel K 1 are represented as an operator acting on the feature space associated with layer 1 .

Figure 5 . 2 -

 52 Figure 5.2 -Behavior of the mean testing accuracy w.r.t. the number of layers (A) and units (B), for both classical CNNs and LGN-CNNs, with and without the lateral connectivity. Error bars represent the standard error of the mean over 20 trainings. The selected LGN-CNN with transition kernel is the one marked by a red star in both plots. (A) The x-axis represents the number of "standard" convolutional layers. (B) The x-axis represents the number of filters in each convolutional layer, expressed as a percentage w.r.t. the number of filters in the selected model.

Figure 5 .

 5 Figure 5.3B shows its approximation with σ = 0.184. The two functions have a high correlation of 93.67%, obtained by applying the built-in function corrcoef from the Python library NumPy.

Figure 5

 5 Figure 5.3 -(A) The learned filter Ψ 0 of the LGN-CNN architecture with transition kernel. (B) Its approximation as a LoG, with optimal σ = 0.184, yielding a correlation of 93.67% with the learned filter.

Figure 5

 5 Figure 5.4 -(A) Learned filters of 1 of our CNN architecture. (B) Learned filters of 1 of the same CNN architecture, but without 0 .

Figure 5

 5 Figure 5.5 -(A): the learned filters of 1 with odd parity, ordered w.r.t. the orientation θ obtained from the Gabor approximation. (B): the approximating odd Gabor filters, labelled by their orientation.

Figure 5

 5 Figure 5.6 -(A): the learned filters of 1 with even parity, ordered w.r.t. the orientation θ obtained from the Gabor approximation. (B): the approximating even Gabor filters, labelled by their orientation.

Figure 5

 5 Figure 5.7 -Behavior of learned short-range intracortical connections w.r.t. the relative orientation. The curve displays the strength of interaction between the filter Ψ f with orientation θ f = 2π5 and the other filters Ψ g centered at the same point (i.e. with relative displacement = (0, 0)) as a function of their orientation θ g .The curve has been smoothed using the MatLab built-in function smoothdata.

Figure 5

 5 Figure 5.8 -(A) The 7 × 7 filter Ψ f , with orientation θ f = 3π 10 . (B) The vector field V and its integral curves obtained from the kernel K 1 computed around Ψ f , with θ f = 3π 10 . (C)The association field of the transition kernel K 1 (blue), and its approximation using the integral curves defined in (1.7) (red). For better

Figure 5

 5 Figure 5.9 -(A) In blue: the vector field of the transition kernel K 1 around Ψ f . In red: the vector field of the best-fitting fundamental solution of the sub-Riemannian heat equation, with α = 4.40. (B) In green: the vector field of the best-fitting fundamental solution of the sub-Riemannian heat equation over a wider spatial region. In red: the local vector field fitted to the learned kernel.

  Retinex effects of LGN-type learned filtersWe assume that the perceived image is obtained via the application of another operator associated to a kernelM : Γ ⊂ R 2 → R (6.4)via the following equationI = M * (M * I). (6.5)We also impose that M is the kernel associated to the inverse operator to O M , so thatM * M = δ. (6.6)ConsequentlyM * I = M * I. (6.7)We observe that one main difference between our approach and the one proposed by Morel regards the Neumann boundary conditions. Indeed, Morel imposes directly the Neumann boundary conditions to the Poisson equation whereas, in our algorithm, the Neumann boundary conditions imposed to the filter are inherited

( a )

 a Exact inverse of a Laplacian log( x 2 + y 2 ) in the interval [-5, 5]. (b) Inverse of discrete Laplacian. (c) Inverse of LoG. (d) Inverse of the first layer of an LGN-CNN.

Figure 6 . 1 -

 61 Figure 6.1 -Comparison between inverse operators. First row: inverse of the discrete Laplacian and its exact inverse operator. Second row: inverse operators of a discrete LoG and the first filter Ψ 0 of an LGN-CNN.

( a )

 a Starting image with two gray dots on a gradient background. (b) Retinex effects of exact inverse of Laplacian. (c) Retinex effects of discrete Laplacian. (d) Retinex effects of LoG.(e) Retinex effects of Ψ 0 of LGN-CNN.

Figure 6 . 2 -

 62 Figure 6.2 -Retinex effects of some inverse operators on starting image 6.2a.

Figure 6 . 4 -

 64 Figure 6.4 -On the left: a gray-scale image I, the convolved image I and the reconstructed image I via eq. (6.7). On the right: the corresponding histograms of the gray-scale values.

Figure 7 . 1 -

 71 Figure 7.1 -Bank of 31 11 × 11 Gabor filters with σ = 1 and λ = 1.

  as defined in equation (1.2), where p = (x, y, θ) ∈ G = R 2 × S 1 . In the present case we consider Gabor filters defined in the interval [-1.5, 1.5] × [-1.5, 1.5] × [-1.5, 1.5] with σ = 1 and λ = 1. The step in each spatial dimension is 0.3 and in the third dimension is 0.1; in this way our bank of filters is composed of 31 filters of size 11 × 11 (see figure7.1).

Figure 7 . 2 - 2 iterations

 722 Figure 7.2 -2 iterations Kernel with respect to the filter with θ = 0.

Figure 7

 7 Figure 7.3 -(a) Solution of the heat equation. From: [Favali, 2017]. (b) Isosurface computed on the connectivity kernel iteratively generated from equation (7.1).

Figure 7

 7 Figure 7.4 -On the left image containing collinear and cocircular oriented elements.On the right the grouping operated via a connectivity kernel generated by a bank of Gabor filters. In order: red and green eigenvectors.

  Figure 7.5 -On the left image containing three clouds of points. On the right its grouping operated via a connectivity kernel generated by a bank of Gabor filters.

Figure 7

 7 Figure 7.6 -On the left image containing two clouds of points and a set of collinear segments. On the right its grouping operated via a kernel generated from a bank of Gabor filters. In order: red, green and blue eigenvectors.

Figure 7 . 7 -

 77 Figure 7.7 -Bank of 64 [7 × 7] filters of the red canal of the first convolutional layer of GoogLeNet.

  .3) which expresses the response of a filter to a visual stimulus. The second layer receives an input of sizes [112 × 112 × 64] after I has passed through the first convolutional layer and a POOLING. Analogously to the first layer we can define the function z 2I and the maxima suppression principle as follows:

Figure 7

 7 Figure 7.8 -On the left image containing collinear and cocircular oriented elements.

Figure 7

 7 Figure 7.9 -On the left image containing three clouds of points. On the right its grouping operated via a weghted combination of two layers. In order: red, green and blue eigenvectors.

Figure 7 .

 7 Figure 7.10 -On the left image containing two clouds of points and a set of collinear segments. On the right its grouping operated via a weighted combination of two layers of GoogLeNet. In order: red, green and blue eigenvectors.

7. 3 . 2

 32 Perceptual grouping with neural networks trained on MNIST We can now study the perceptual grouping ability of two trained CNNs. Since they have several layers we can analyze their behaviors varying the number of layers involved in the task.Let us consider the first architecture trained on MNIST. We analyze just the behaviour of the first two layers since the third one is extremely local (recall that the spatial dimension of the input is 4 as the spatial dimension of the filters).Therefore, we can consider the first bank of filters(Ψ 1 p ) = (Ψ 1 x,y,f 1 ) defined on G 1 = R 2 × F 1where F 1 = 1 . . . 128 labels the filters and the second bank of filters(Ψ 2 p ) = (Ψ 2 x,y,f 1 ,f 2 ) defined on G 2 = R 2 × F 1 × F 2 where F 2 = 1 . . . 128.Then, given an input I y)I(x, y)dxdy, and the principle of maxima suppression as z 1

  Figure 7.11 -On the left image containing two clouds of points and a set of collinear segments. On the right its grouping operated via a weighted combination of two layers. In order: red, green, blue and yellow eigenvectors.

  Figure 7.11 displays the grouping result. The first and third eigenvectors represent the bigger clouds of points split into two parts.

2. 2 .

 2 2 to every layer obtaining three affinity matrices A 1 I , A 2 I and A 3 I . Then we can define a linear combination of them as A

Figure 7 .

 7 Figure 7.12 -On the left image containing two clouds of points and a set of collinear segments. On the right its grouping operated via a weighted combination of three layers. In order: red, green and blue eigenvectors.

Figure 7 .

 7 Figure 7.13 -First column: starting images with 2 segments at different distances. Second column: first eigenvector of first layer. Third column: first eigenvector of second layer. Fourth column: first eigenvector of first and second layers applied together.

Figure 7 .

 7 Figure 7.14 -First column: starting images with corners of a square at different distances. Second column: first eigenvector of first layer. Third column: first eigenvector of second layer. Fourth column: first eigenvector of first and second layers applied together.

Figure 7 .

 7 Figure 7.15 -First column: starting images with corners of a square at different distances. Second column: first eigenvector of first layer. Third column: first eigenvector of second layer. Fourth column: first eigenvector of third layers. Fifth column: first eigenvector of first and second layers applied together. Sixth column: first eigenvector of first, second and third layers applied together.

  forme d'un faisceau de fibres équipé d'une structure de contact. Dans la même période, Steven Zucker a observé expérimentalement la relation entre la mesure de la courbure euclidienne et le rôle des cellules terminales

	x	RESUM	É
	a développé un modèle du cortex visuel sous la
	ix

  ont observé que la structure corticale est en fait définie

	RESUM É	xi
	and Cowan, 2003] ont étudié la dynamique de la population neuronale, modélisant
	le cortex dans le même groupe SE(2). Un résultat impressionnant a été obtenu
	lorsque la carte d'activation a été calculée en l'absence d'une entrée externe ; en
	effet, ils ont simulé les effets de drogues chimiques en obtenant une distribution
	d'activation correspondant à des hallucinations visuelles, comme on a rapporté
	dans la littérature classique. Dans [Duits and Franken, 2010a,b] Remco Duits a
	déplacé l'attention travaillant dans le groupe SE(2) du point de vue du traitement
	d'image. Il a soulevé et propagé le signal visuel dans l'espace R 2 × S 1 et, appliquant
	des noyaux inversibles, a pu reconstruire le stimulus sans perte d'information dans
	le plan image 2D. Une approche phénoménologique se retrouve dans l'ensemble de
	dans le groupe de rotation et de translation euclidien SE(2) doté d'une métrique
	sous-riemannienne. Dans cette structure de faisceaux de fibres, ils ont exprimé
	la propagation corticale en termes de diffusion sous-riemannienne d'EDP afin de
	modéliser le phénomène de complétion d'image. Bressloff et Cowan dans [Bressloff

l'oeuvre de Jean-Michel Morel et de son groupe computationnel de Gestalt [voir par exemple

  y)dxdy ,

	RESUM É	xiii
	une structure hiérarchique, où chaque couche reçoit l'entrée de la précédente et
	fournit la sortie à la suivante. Malgré cette simplification, ils ont atteint des
	valeurs optimales dans les processus typiques du système visuel naturel, comme
	par exemple la détection d'objets
	Des nombreux efforts ont été faits pour relier les architectures corticales avec les
	réseaux neuronaux convolutifs (CNNs). En effet, les premiers réseaux neuronaux
	ont été inspirés par une simplification de la structure du système visuel, présentant

  In Table4.2 we report the correlations with different values of L 2 regularization term and for two LGN-CNN architectures, the one introduced in section 4.2 which is indicated by 'LGN + 4 layers' and the deeper one described above indicated by'LGN + 8 layers'. All the other training parameters are provided in section 4.2.

		No DA Mild DA Hard DA
	LoG corr 93.77 % 93.68 %	95.21 %
	Ψ 0 S corr 93.92 % 94.16 %	97.15 %
	Table 4.3 -Correlation between Ψ 0 and its LoG approximation and between Ψ 0
	and Ψ 0 S varying the data augmentation (DA) applied to the dataset.
		LGN + 4 layers LGN + 8 layers
	0.01	88.3 %	85.40 %
	0.02	97.15 %	90.79 %
	0.03	93.06 %	91.41 %
	0.04	95.61%	92.58 %
	0.045	93.55 %	94.79 %
	0.05	95.44 %	94.11 %
	Table 4.2 -Correlation between Ψ 0 and Ψ 0 S varying the L 2 regularization term and
	the number of layers of the LGN-CNN architecture. Best rotational symmetric
	filters are selected in cyan for both architectures.

As we can see for

'

LGN + 4 layers' architecture the L 2 regularization term that let Ψ 0 to be the closest rotational symmetric filter is 0.02. This is the filter we have shown in figures 4.3 and 4.4. On the other hand, in the case of 'LGN + 8 layers' architecture the more rotationally symmetric filter is the one with 0.045 as the L 2 regularization term. This behavior suggests that the architecture of

  CNN is to classify the input as a digit or a cloth, indeed if the image belongs to MNIST or Fashion-MNIST dataset. The new training set is built by taking the

	first half of MNIST dataset (30000 images) and the first half of Fashion-MNIST
	dataset (30000 images), for a total of 60000 images. We follow the same steps for
	the test set (for a total of 10000 images) and we randomly sort the training and
	test sets. Then each image is labeled by 0 if it is a digit and by 1 if it is a cloth.
	The LGN-CNN is a really simple architecture that contains only a first layer

perform this test we build a new dataset of images starting from the dataset MNIST [a set of digits images, see LeCun and Cortes, 2010] and the dataset Fashion-MNIST [a set of cloths images, see

[START_REF] Xiao | Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms[END_REF]

. They are two similar datasets, composed by gray-scale images of size 28 × 28. The purpose of our

LGN-

  × 4 × 128] that gives an output of sizes [1 × 1 × 500]. At this point a RELU is used and a fully connected layer is applied, modifying the length of the output vector to 10. Finally a softmax loss is used to calculate the error done to each batch during the training session. The output represents the probabilities of the image to be a number from 0 to 9 and the CNN selects the label with the highest probability.The main difference between the first architecture and the second one is that the latter has a higher number of layers and receives larger images as input. In particular, the first convolutional layer, composed by 32 9 × 9 filters, receives a 56 × 56 gray-scale image. Then, a POOLING and a convolutional layer with 64 [5 × 5 × 32] filters are applied; after them another POOLING and another convolutional layer with 64 [5 × 5 × 64] filters are used. After the last POOLING and a convolutional layer composed by 500 [3 × 3 × 64] filters a RELU is applied.

	of 500 layers of sizes [4
	The
	output of the first convolutional layer is indeed a tensor of size [24 × 24 × 128] and
	after a POOLING it becomes a [12 × 12 × 128] input for the second convolutional
	layer. The latter is composed again of 128 filters but with sizes [5 × 5 × 128] and
	therefore its output is a [8 × 8 × 128] tensor. Then, after another POOLING, there
	is a third convolutional layer that, receiving a [4 × 4 × 128] input, is composed

Convolutional Neural Networks y(I i ), i = 1, . . . , N in H. Thus, we should define a loss function that estimates the error between F and y. Various loss functions, depending on the specific task, can be used. For example, in the case of a classification task, the softmax loss

Convolutional Neural Networks

This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Sk lodowska-Curie grant agreement No 754362.

Thèse pour obtenir le grade de

The LGN-CNN architecture with the transition kernel

The architecture is composed by 11 convolutional layers, each one followed by a ReLU function and a batch normalization layer; at the end of the architecture 3 fully-connected layers are present. Appropriate zero padding are applied in order to keep the spatial dimensions unchanged after each convolutional layer.

Figure 5.1 shows the network architecture whose layers are listed below. Convolutional layers are denoted by 0 , . . . , 10 , whereas fully connected layers are denoted by F C 1 , F C 2 , F C 3 . For simplicity we omit the ReLU function and the batch normalization layer. 0 LGN layer: a single filter Ψ 0 of size 11 × 11;

1 64 filters of size 7 × 7; lateral connectivity kernel K 1 of size 13 × 13 × 64 × 64; max pooling of square size 2;

2 64 filters of size 5 × 5 × 64;

3 , 4 64 filters of size 3 × 3 × 64; max pooling of square size 2; 5 , 6 64 filters of size 3 × 3 × 64;

7 128 filters of size 3 × 3 × 64; max pooling of square size 2;

8 , 9 , 10 128 filters of size 3 × 3 × 128; max pooling of square size 2; F C 1 , F C 2 , F C 3 1000, 200 and 10 output units respectively.

Since the neural network should receive as input a gray scale image, we transform the CIFAR-10 dataset into gray scale images. All the parameters except the kernels K 1 were first pre-trained in the absence of lateral connections for a maximum of 800 epochs, with early stopping when validation accuracy failed to increase for 80 consecutive epochs. After the pre-training we add the lateral connections in layer 1 , thus employing the full update rule in Eq. (5.1), and implement a second Finally, we can compare the image I with the reconstructed one I and see if any Retinex effects occur.

We show the inverse operators obtained through the algorithm described by equation (6.8). We start from the classical discrete Laplacian operator and then we move to some convolutional operators, in particular a discrete LoG and the filter Ψ 0 of the LGN-CNN. In order to compare the inverse operators, we show the 2D plots obtained by selecting a slice of the 3D inverse operator itself.

We first compare the inverse of the discrete Laplacian w.r.t. its exact inverse operator given by the function log( x 2 + y 2 ). Figure 6.1b shows the approximation of the inverse of the discrete Laplacian and 6.1a shows the exact inverse in the interval [-5, 5]. We observe that the approximation is close to the exact one.

In the second row of figure 6.1 we compare the inverse operators of the LoG and Ψ 0 where figure 6.1c shows the LoG inverse operator and figure 6.1d shows the Ψ 0 inverse operator. Indeed, since we have shown in chapter 4 that Ψ 0 approximates the LoG we should expect similar inverse operators. In the next section we will see if also the Retinex effects are similar.

Circles on a gradient background

We start with a simple gray-scale image (see figure 6.2a) in order to see how the different operators work. We use the same image as in [START_REF] Morel | A pde formalization of retinex theory[END_REF][START_REF] Limare | Retinex poisson equation: a model for color perception[END_REF] in which they obtain remarkable Retinex effects. It has a background that shifts from white (value 1) to black (value -1) maintaining the gradient constant from left to right. There are also two gray dots (same value 0), the left one with a brighter background, the right one with a darker one. Because of the different backgrounds, our visual system perceives the two dots differently: the left one is perceived as darker with respect to its true color; the right one is perceived as brighter.

We first consider the discrete Laplacian operator since we expect to obtain the same Retinex effects as in [START_REF] Morel | A pde formalization of retinex theory[END_REF]. Figure 6.2c shows the Then, we perform our experiment with the inverse of a LoG. We can note from figure 6.2d that the Retinex effects occur also in this case (with values close to -0.9 and 0.9), similarly to the Retinex effects of the discrete Laplacian. Also in this case the contours of the dots are not completely clear.

Finally, we test the inverse of the convolutional operator Ψ 0 of the LGN-CNN introduced in chapter 4.2. In figure 6.2e we see that this operator shows Retinex effects where the values of the dots are close to -0.9 and 0.9. In this case the contours of the dots are even clearer than the ones obtained with the LoG and the discrete Laplacian.

To summarize, we have shown that our algorithm reproduces the same results of [START_REF] Morel | A pde formalization of retinex theory[END_REF] in the case of the Laplacian operator. Furthermore, we have tested it on other operators obtaining remarkable results. It is particularly interested the case of the convolutional operator Ψ 0 since it is able to show Retinex effects even if it is a learned filter with no a-priori structure, enforcing the link between the LGN-CNN architecture and the visual system.

Adelson's checker

We can also test our algorithm on the Adelson's checker shadow illusion as in [START_REF] Morel | A pde formalization of retinex theory[END_REF][START_REF] Limare | Retinex poisson equation: a model for color perception[END_REF]. Since we are more interested in the Retinex effects of LoG and Ψ 0 operators we analyze their behaviors on the gray-scale image (see figure 6.3a). It shows a checkerboard with light gray and dark gray squares with a cylinder on it that shadows a part of the squares. In particular, the square labeled 'A' and the square labeled 'B' have the same gray-scale value (in our case, since -1 is black and 1 is white, they have -0,4953 value). The illusion is built in such a way that, even if they have the same value, they are perceived in a completely different way. Indeed square 'A', which is outside the shadow and surrounded by light gray squares, is perceived as a dark gray square. On the other hand, square 'B', which is inside the shadow and is surrounded by dark gray squares, is perceived as lighter.

We expect that the two operators should reproduce the same behavior of our perception, in particular square 'A' should have a smaller value whereas square 'B' should have a bigger value. Figure 6.3c shows the Retinex effects obtained with