12 research outputs found
Texture Modeling by Gaussian fields with prescribed local orientation
International audienceThis paper presents a new framework for oriented texture modeling. We introduce a new class of Gaussian fields, called Locally Anisotropic Fractional Brownian Fields, with prescribed local orientation at any point. These fields are a local version of a specific class of anisotropic self-similar Gaussian fields with stationary increments. The simulation of such textures is obtained using a new algorithm mixing the tangent field formulation and a turning band method, this latter method having proved its efficiency for generating stationary anisotropic textures. Numerical experiments show the ability of the method for synthesis of textures with prescribed local orientation
Convex Super-Resolution Detection of Lines in Images
International audienceIn this paper, we present a new convex formulation for the problem of recovering lines in degraded images. Following the recent paradigm of super-resolution, we formulate a dedicated atomic norm penalty and we solve this optimization problem by means of a primalâdual algorithm. This parsimonious model enables the reconstruction of lines from lowpass measurements, even in presence of a large amount of noise or blur. Furthermore, a Prony method performed on rows and columns of the restored image, provides a spectral estimation of the line parameters, with subpixel accuracy
Modélisation de textures anisotropes par la transformée en ondelettes monogéniques
Texture analysis is a component of image processing which hold the interest in the various applications it covers. In medical imaging, the images recorded such as bone X-rays or mammograms show a highly irregular micro-architecture, which invites to consider these textures formation as a realization of a random field. Following Benoit Mandelbrotâs pioneer work, many models derived from the fractional Brownian field have been proposed to characterize the fractal behavior of images and to synthesize textures with prescribed roughness. Thus, the parameters estimation of these models has made possible to link the fractal dimension of these images to the detection of bone structure alteration as it is observed in the case of osteoporosis. More recently, other models known as anisotropic random fields have been used to describe phenomena with preferred directions, for example for detecting abnormalities in the mammary tissues.This thesis deals with the development of new models of anisotropic fields, allowing to locally control the anisotropy of the textures. A first contribution was to define a generalized anisotropic fractional Brownian field (GAFBF), and a second model based on an elementary field deformation (WAFBF), both allowing to prescribe the local orientation of the texture. The study of the local structure of these fields is carried out using the tangent fields formalism. Simulation procedures are implemented to concretely observe the behavior, and serve as a benchmark for the validation of anisotropy detection tools. Indeed, the investigation of local orientation and anisotropy in the context of textures still raises many mathematical problems, starting with the rigorous definition of this orientation. Our second contribution is in this perspective. By transposing the orientation detection methods based on the monogenic wavelet transform, we have been able, for a wide class of random fields, to define an intrinsic notion of orientation. In particular, the study of the two new models of anisotropic fields introduced previously allowed to formally link this notion of orientation with the anisotropy parameters of these models. Connections with directional statistics are also established, in order to characterize the probability distribution of orientation estimators.Finally, a third part of this thesis was devoted to the problem of the lines detection in images. The underlying model is that of a superposition of diffracted lines (i.e, convoluted by a blur kernel) with presence of noise, whose position and intensity parameters must be recovered with sub-pixel precision. We have developed a method based on the super-resolution paradigm. The reformulation of the problem in the framework of 1-D atoms lead to an optimization problem under constraints, and enables to reconstruct these lines by reaching this precision. The algorithms used to perform the minimization belong to the family of algorithms known as proximal algorithms. The modelization and the resolution of this inverse problem, provides a proof of concept opening perspectives to the development of a revised Hough transform for the continuous detection of lines in images.Lâanalyse de texture est une composante du traitement dâimage qui suscite beaucoup dâintĂ©rĂȘt tant les applications quâelle recouvre sont diverses. En imagerie mĂ©dicale, les signaux enregistrĂ©s sous forme dâimages telles que les radiographies de lâos ou les mammographies, prĂ©sentent une micro-architecture fortement irrĂ©guliĂšre qui invite Ă considĂ©rer la formation de ces textures comme la rĂ©alisation dâun champ alĂ©atoire. Suite aux travaux prĂ©curseurs de Benoit Mandelbrot, de nombreux modĂšles dĂ©rivĂ©s du champ brownien fractionnaire ont Ă©tĂ© proposĂ©s pour caractĂ©riser lâaspect fractal des images et synthĂ©tiser des textures Ă rugositĂ© prescrite. Ainsi lâestimation des paramĂštres de ces modĂšles, a notamment permis de relier la dimension fractale des images Ă la dĂ©tection de modifications de la structure osseuse telle quâon lâobserve en cas dâostĂ©oporose. Plus rĂ©cemment, dâautres modĂšles de champs alĂ©atoires, dits anisotropes, ont Ă©tĂ© utilisĂ©s pour dĂ©crire des phĂ©nomĂšnes prĂ©sentant des directions privilĂ©giĂ©es, et dĂ©tecter par exemple des anomalies dans les tissus mammaires.Cette thĂšse porte sur lâĂ©laboration de nouveaux modĂšles de champs anisotropes, permettant de contrĂŽler localement lâanisotropie des textures. Une premiĂšre contribution a consistĂ© Ă dĂ©finir un champ brownien fractionnaire anisotrope gĂ©nĂ©ralisĂ© (GAFBF), et un second modĂšle basĂ© sur une dĂ©formation de champs Ă©lĂ©mentaires (WAFBF), permettant tous deux de prescrire lâorientation locale de la texture. LâĂ©tude de la structure locale de ces champs est menĂ©e Ă lâaide du formalisme des champs tangents. Des procĂ©dures de simulation sont mises en oeuvres pour en observer concrĂštement le comportement, et servir de benchmark Ă la validation dâoutils de dĂ©tection de lâanisotropie. En effet lâĂ©tude de lâorientation locale et de lâanisotropie dans le cadre des textures soulĂšve encore de nombreux problĂšmes mathĂ©matiques, Ă commencer par la dĂ©finition rigoureuse de cette orientation. Notre seconde contribution sâinscrit dans cette perspective. En transposant les mĂ©thodes de dĂ©tection de lâorientation basĂ©es sur la transformĂ©e en ondelettes monogĂ©niques, nous avons Ă©tĂ© en mesure, pour une vaste classe de champs alĂ©atoires, de dĂ©finir une notion dâorientation intrinsĂšque. En particulier lâĂ©tude des deux nouveaux modĂšles de champs anisotropes introduits prĂ©cĂ©demment, a permis de relier formellement cette notion dâorientation aux paramĂštres dâanisotropie de ces modĂšles. Des connexions avec les statistiques directionnelles sont Ă©galement Ă©tablies, de façon Ă caractĂ©riser la loi de probabilitĂ© des estimateurs dâorientation.Enfin une troisiĂšme partie de la thĂšse est consacrĂ©e au problĂšme de la dĂ©tection de lignes dans les images. Le modĂšle sous jacent est celui dâune superposition de lignes diffractĂ©es (c-a-d convoluĂ©es par un noyau de flou) et bruitĂ©es, dont il sâagit de retrouver les paramĂštres de position et dâintensitĂ© avec une prĂ©cision sub-pixel. Nous avons dĂ©veloppĂ© dans cet objectif une mĂ©thode basĂ©e sur le paradigme de la super-rĂ©solution. La reformulation du problĂšme en termes dâatomes 1-D a permis de dĂ©gager un problĂšme dâoptimisation sous contraintes, et de reconstruire ces lignes en atteignant cette prĂ©cision. Les algorithmes employĂ©s pour effectuer la minimisation appartiennent Ă la famille des algorithmes dits proximaux. La formalisation de ce problĂšme inverse et sa rĂ©solution, constituent une preuve de concept ouvrant des perspectives Ă lâĂ©laboration dâune transformĂ©e de Hough revisitĂ©e pour la dĂ©tection âcontinueâ de lignes dans les images
Modelisation of anisotropic textures by the monogenic wavelet transform
Lâanalyse de texture est une composante du traitement dâimage qui suscite beaucoup dâintĂ©rĂȘt tant les applications quâelle recouvre sont diverses. En imagerie mĂ©dicale, les signaux enregistrĂ©s sous forme dâimages telles que les radiographies de lâos ou les mammographies, prĂ©sentent une micro-architecture fortement irrĂ©guliĂšre qui invite Ă considĂ©rer la formation de ces textures comme la rĂ©alisation dâun champ alĂ©atoire. Suite aux travaux prĂ©curseurs de Benoit Mandelbrot, de nombreux modĂšles dĂ©rivĂ©s du champ brownien fractionnaire ont Ă©tĂ© proposĂ©s pour caractĂ©riser lâaspect fractal des images et synthĂ©tiser des textures Ă rugositĂ© prescrite. Ainsi lâestimation des paramĂštres de ces modĂšles, a notamment permis de relier la dimension fractale des images Ă la dĂ©tection de modifications de la structure osseuse telle quâon lâobserve en cas dâostĂ©oporose. Plus rĂ©cemment, dâautres modĂšles de champs alĂ©atoires, dits anisotropes, ont Ă©tĂ© utilisĂ©s pour dĂ©crire des phĂ©nomĂšnes prĂ©sentant des directions privilĂ©giĂ©es, et dĂ©tecter par exemple des anomalies dans les tissus mammaires.Cette thĂšse porte sur lâĂ©laboration de nouveaux modĂšles de champs anisotropes, permettant de contrĂŽler localement lâanisotropie des textures. Une premiĂšre contribution a consistĂ© Ă dĂ©finir un champ brownien fractionnaire anisotrope gĂ©nĂ©ralisĂ© (GAFBF), et un second modĂšle basĂ© sur une dĂ©formation de champs Ă©lĂ©mentaires (WAFBF), permettant tous deux de prescrire lâorientation locale de la texture. LâĂ©tude de la structure locale de ces champs est menĂ©e Ă lâaide du formalisme des champs tangents. Des procĂ©dures de simulation sont mises en oeuvres pour en observer concrĂštement le comportement, et servir de benchmark Ă la validation dâoutils de dĂ©tection de lâanisotropie. En effet lâĂ©tude de lâorientation locale et de lâanisotropie dans le cadre des textures soulĂšve encore de nombreux problĂšmes mathĂ©matiques, Ă commencer par la dĂ©finition rigoureuse de cette orientation. Notre seconde contribution sâinscrit dans cette perspective. En transposant les mĂ©thodes de dĂ©tection de lâorientation basĂ©es sur la transformĂ©e en ondelettes monogĂ©niques, nous avons Ă©tĂ© en mesure, pour une vaste classe de champs alĂ©atoires, de dĂ©finir une notion dâorientation intrinsĂšque. En particulier lâĂ©tude des deux nouveaux modĂšles de champs anisotropes introduits prĂ©cĂ©demment, a permis de relier formellement cette notion dâorientation aux paramĂštres dâanisotropie de ces modĂšles. Des connexions avec les statistiques directionnelles sont Ă©galement Ă©tablies, de façon Ă caractĂ©riser la loi de probabilitĂ© des estimateurs dâorientation.Enfin une troisiĂšme partie de la thĂšse est consacrĂ©e au problĂšme de la dĂ©tection de lignes dans les images. Le modĂšle sous jacent est celui dâune superposition de lignes diffractĂ©es (c-a-d convoluĂ©es par un noyau de flou) et bruitĂ©es, dont il sâagit de retrouver les paramĂštres de position et dâintensitĂ© avec une prĂ©cision sub-pixel. Nous avons dĂ©veloppĂ© dans cet objectif une mĂ©thode basĂ©e sur le paradigme de la super-rĂ©solution. La reformulation du problĂšme en termes dâatomes 1-D a permis de dĂ©gager un problĂšme dâoptimisation sous contraintes, et de reconstruire ces lignes en atteignant cette prĂ©cision. Les algorithmes employĂ©s pour effectuer la minimisation appartiennent Ă la famille des algorithmes dits proximaux. La formalisation de ce problĂšme inverse et sa rĂ©solution, constituent une preuve de concept ouvrant des perspectives Ă lâĂ©laboration dâune transformĂ©e de Hough revisitĂ©e pour la dĂ©tection âcontinueâ de lignes dans les images.Texture analysis is a component of image processing which hold the interest in the various applications it covers. In medical imaging, the images recorded such as bone X-rays or mammograms show a highly irregular micro-architecture, which invites to consider these textures formation as a realization of a random field. Following Benoit Mandelbrotâs pioneer work, many models derived from the fractional Brownian field have been proposed to characterize the fractal behavior of images and to synthesize textures with prescribed roughness. Thus, the parameters estimation of these models has made possible to link the fractal dimension of these images to the detection of bone structure alteration as it is observed in the case of osteoporosis. More recently, other models known as anisotropic random fields have been used to describe phenomena with preferred directions, for example for detecting abnormalities in the mammary tissues.This thesis deals with the development of new models of anisotropic fields, allowing to locally control the anisotropy of the textures. A first contribution was to define a generalized anisotropic fractional Brownian field (GAFBF), and a second model based on an elementary field deformation (WAFBF), both allowing to prescribe the local orientation of the texture. The study of the local structure of these fields is carried out using the tangent fields formalism. Simulation procedures are implemented to concretely observe the behavior, and serve as a benchmark for the validation of anisotropy detection tools. Indeed, the investigation of local orientation and anisotropy in the context of textures still raises many mathematical problems, starting with the rigorous definition of this orientation. Our second contribution is in this perspective. By transposing the orientation detection methods based on the monogenic wavelet transform, we have been able, for a wide class of random fields, to define an intrinsic notion of orientation. In particular, the study of the two new models of anisotropic fields introduced previously allowed to formally link this notion of orientation with the anisotropy parameters of these models. Connections with directional statistics are also established, in order to characterize the probability distribution of orientation estimators.Finally, a third part of this thesis was devoted to the problem of the lines detection in images. The underlying model is that of a superposition of diffracted lines (i.e, convoluted by a blur kernel) with presence of noise, whose position and intensity parameters must be recovered with sub-pixel precision. We have developed a method based on the super-resolution paradigm. The reformulation of the problem in the framework of 1-D atoms lead to an optimization problem under constraints, and enables to reconstruct these lines by reaching this precision. The algorithms used to perform the minimization belong to the family of algorithms known as proximal algorithms. The modelization and the resolution of this inverse problem, provides a proof of concept opening perspectives to the development of a revised Hough transform for the continuous detection of lines in images
Génération de modÚles graphiques
International audienceNous étudions la répartition des valeurs de corrélation obtenues à partir de l'algorithme de Córdoba et al. pour différentes structures de graphes. Nous mettons en lumiÚre un biais possible de l'algorithme et la difficulté à simuler des valeurs similaires à des données réelles
Génération de modÚles graphiques
International audienceNous étudions la répartition des valeurs de corrélation obtenues à partir de l'algorithme de Córdoba et al. pour différentes structures de graphes. Nous mettons en lumiÚre un biais possible de l'algorithme et la difficulté à simuler des valeurs similaires à des données réelles
Génération de modÚles graphiques
International audienceNous étudions la répartition des valeurs de corrélation obtenues à partir de l'algorithme de Córdoba et al. pour différentes structures de graphes. Nous mettons en lumiÚre un biais possible de l'algorithme et la difficulté à simuler des valeurs similaires à des données réelles
Modélisation Parcimonieuse de CNNs avec des Paquets d'Ondelettes Dual-Tree
International audienceWe propose to improve the mathematical interpretability of convolutional neural networks (CNNs) for image classification. In this purpose, we replace the first layers of existing models such as AlexNet or ResNet by an operator containing the dual-tree wavelet packet transform, i.e., a redundant decomposition using complex and oriented waveforms. Our experiments show that these modified networks behave very similarly to the original models once trained. The goal is then to study this operator from a theoretical point of view and to identify potential optimizations. We want to analyze its main properties such as directional selectivity, stability with respect to small shifts and rotations, thus retaining discriminant information while decreasing intra-class variability. This work is a step toward a more complete description of CNNs using well-defined mathematical operators, characterized by a small number of arbitrary parameters, making them easier to interpret.Nous proposons dâamĂ©liorer lâinterprĂ©tabilitĂ© mathĂ©matique des rĂ©seaux neuronaux convolutifs (CNNs) pour la classification dâimages. Pour cela, nous remplaçons les premiĂšres couches de rĂ©seaux tels quâAlexNet ou ResNet par un opĂ©rateur faisant intervenir une version complexe,orientĂ©e et redondante de la transformĂ©e en paquets dâondelettes discrĂšte, appelĂ©e en anglais dual-tree wavelet packet transform. Nous montrons expĂ©rimentalement que ces rĂ©seaux modifiĂ©s se comportent de maniĂšre trĂšs similaire aux modĂšles originaux une fois entraĂźnĂ©s. Lâobjectif est ensuite dâĂ©tudier, dâun point de vue thĂ©orique, lâopĂ©rateur mathĂ©matique ainsi introduit, et dâidentifier des leviers dâoptimisation. Nous souhaitons analyser ses principales propriĂ©tĂ©s telles que la sĂ©lectivitĂ© directionnelle, la stabilitĂ© par translation et rotation, qui permettent de discriminer des images de nature diffĂ©rente tout en attĂ©nuant les sources de variabilitĂ© au sein dâune mĂȘme classe dâimages. Ce travail est un pas vers une description plus complĂšte des CNNs existants Ă lâaide dâopĂ©rateurs mathĂ©matiques bien dĂ©finis, caractĂ©risĂ©s par un faible nombre de paramĂštres arbitraires, les rendant de fait plus aisĂ©s Ă interprĂ©ter
A convex approach to super-resolution and regularization of lines in images
International audienceWe present a new convex formulation for the problem of recovering lines in degraded images. Following the recent paradigm of super-resolution, we formulate a dedicated atomic norm penalty and solve this optimization problem by a primal-dual algorithm. Then, a spectral estimation method recovers the line parameters, with subpixel accuracy