Modélisation de textures anisotropes par la transformée en ondelettes monogènes, et super-résolution de lignes 2-D

Keywords :

• anisotropie et régularité
• champs tangents
• synthèse de textures
• caractérisation de l'orientation
• ondelettes monogènes
• statistiques directionnelles
• déconvolution
• super-résolution
• algorithmes proximaux

Texture analysis is a component of image processing which holds 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 Mandelbrot's 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. This inverse problem modeling, and its resolution, provide a proof of concept opening perspectives to the development of a revised Hough transform for the continuous detection of lines in images.

Directors:

• Mme Valérie Perrier (Professeure - Grenoble INP )

Raporteurs:

• Mr Frédéric Richard (Professeur - Université Aix-Marseille )
• Mr Gabriel Peyré (Directeur de recherche - CNRS )

Examinators:

• Mr Pierre Weiss (chercheur - CNRS )
• Mr Annick Montanvert (Professeure - Université Grenoble Alpes )
• Mme Anne Estrade (Professeure - Université Paris-Descartes )