Morphologie, Géométrie et Statistiques en imagerie non-standard

Digital image processing has followed the evolution of electronic and computer science. It is now current to deal with images valued not in {0,1} or in gray-scale, but in manifolds or probability distributions. This is for instance the case for color images or in diffusion tensor imaging (DTI). Each kind of images has its own algebraic, topological and geometric properties. Thus, existing image processing techniques have to be adapted when applied to new imaging modalities. When dealing with new kind of value spaces, former operators can rarely be used as they are. Even if the underlying notion has still a meaning, a work must be carried out in order to express it in the new context.The thesis is composed of two independent parts. The first one, "Mathematical morphology on non-standard images", concerns the extension of mathematical morphology to specific cases where the value space of the image does not have a canonical order structure. Chapter 2 formalizes and demonstrates the irregularity issue of total orders in metric spaces. The main results states that for any total order in a multidimensional vector space, there are images for which the morphological dilations and erosions are irregular and inconsistent. Chapter 3 is an attempt to generalize morphology to images valued in a set of unordered labels.The second part "Probability density estimation on Riemannian spaces" concerns the adaptation of standard density estimation techniques to specific Riemannian manifolds. Chapter 5 is a work on color image histograms under perceptual metrics. The main idea of this chapter consists in computing histograms using local Euclidean approximations of the perceptual metric, and not a global Euclidean approximation as in standard perceptual color spaces. Chapter 6 addresses the problem of non parametric density estimation when data lay in spaces of Gaussian laws. Different techniques are studied, an expression of kernels is provided for the Wasserstein metric.Le traitement d'images numériques a suivi l'évolution de l'électronique et de l'informatique. Il est maintenant courant de manipuler des images à valeur non pas dans {0,1}, mais dans des variétés ou des distributions de probabilités. C'est le cas par exemple des images couleurs où de l'imagerie du tenseur de diffusion (DTI). Chaque type d'image possède ses propres structures algébriques, topologiques et géométriques. Ainsi, les techniques existantes de traitement d'image doivent être adaptés lorsqu'elles sont appliquées à de nouvelles modalités d'imagerie. Lorsque l'on manipule de nouveaux types d'espaces de valeurs, les précédents opérateurs peuvent rarement être utilisés tel quel. Même si les notions sous-jacentes ont encore un sens, un travail doit être mené afin de les exprimer dans le nouveau contexte. Cette thèse est composée de deux parties indépendantes. La première, « Morphologie mathématiques pour les images non standards », concerne l'extension de la morphologie mathématique à des cas particuliers où l'espace des valeurs de l'image ne possède pas de structure d'ordre canonique. Le chapitre 2 formalise et démontre le problème de l'irrégularité des ordres totaux dans les espaces métriques. Le résultat principal de ce chapitre montre qu'étant donné un ordre total dans un espace vectoriel multidimensionnel, il existe toujours des images à valeur dans cet espace tel que les dilatations et les érosions morphologiques soient irrégulières et incohérentes. Le chapitre 3 est une tentative d'extension de la morphologie mathématique aux images à valeur dans un ensemble de labels non ordonnés.La deuxième partie de la thèse, « Estimation de densités de probabilités dans les espaces de Riemann » concerne l'adaptation des techniques classiques d'estimation de densités non paramétriques à certaines variétés Riemanniennes. Le chapitre 5 est un travail sur les histogrammes d'images couleurs dans le cadre de métriques perceptuelles. L'idée principale de ce chapitre consiste à calculer les histogrammes suivant une approximation euclidienne local de la métrique perceptuelle, et non une approximation globale comme dans les espaces perceptuels standards. Le chapitre 6 est une étude sur l'estimation de densité lorsque les données sont des lois Gaussiennes. Différentes techniques y sont analysées. Le résultat principal est l'expression de noyaux pour la métrique de Wasserstein

Generalized Morphology using Sponges

Mathematical morphology has traditionally been grounded in lattice theory. For non-scalar data lattices often prove too restrictive, however. In this paper we present a more general alternative, sponges, that still allows useful definitions of various properties and concepts from morphological theory. It turns out that some of the existing work on “pseudo-morphology” for non-scalar data can in fact be considered “proper” mathematical morphology in this new framework, while other work cannot, and that this correlates with how useful/intuitive some of the resulting operators are

Notes in Pure Mathematics & Mathematical Structures in Physics

These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.Comment: Small improvements and addition

Exploiting Numerical Features in Computer Tomography Data Processing and Management by CICT

The impact of high resolution Computer Tomography (HRCT) technology is to generate new challenges associated with the problem of formation, acquisition, compression, transmission, and analysis of enormous amount of data. In the past, computational information conservation theory (CICT) has shown potentiality to provide us with new techniques to face this challenge conveniently. CICT explores, at elementary level, the basic properties and relationships of Q arithmetic to achieve full numeric algorithmic information conservation with strong connections to modular group theory and combinatorial optimization. Traditional rational number system can be regarded as a highly sophisticated open logic, powerful and flexible LTR and RTL formal language of languages, with self-defining consistent words and rules, starting from elementary generators and relations. CICT supply us with optimized exponential cyclic sequences (OECS) which inherently capture the hidden symmetries and asymmetries of the hyperbolic space encoded by rational numbers. Symmetry and asymmetry relations can be seen as the operational manifestation of universal categorical irreducible arithmetic dichotomy (”correspondence” and ”incidence”) at the innermost logical data structure level. These two components are inseparable from each other, and in continuous reciprocal interaction. According to Pierre Curie, symmetry breaking has the following role: for the occurrence of a phenomenon in a medium, the original symmetry group of the medium must be lowered (broken, in today's terminology) to the symmetry group of the phenomenon (or to a subgroup of the phenomenons symmetry group) by the action of some cause. In this sense symmetry breaking, or asymmetry, is what creates a phenomenon. The same dichotomy generates ”pairing” and ”fixed point” properties for digit group with the same word length, in word combinatorics. Correspondence and Incidence manifest themselves even into single digit fundamental property (i.e. ”evenness” and ”oddness”), till binary elementary symbols (”0”, ”1”). This new awareness can be exploited into the development of competitive optimized algorithm and application. Practical examples will be presented

Courbure discrète : théorie et applications

International audienceThe present volume contains the proceedings of the 2013 Meeting on discrete curvature, held at CIRM, Luminy, France. The aim of this meeting was to bring together researchers from various backgrounds, ranging from mathematics to computer science, with a focus on both theory and applications. With 27 invited talks and 8 posters, the conference attracted 70 researchers from all over the world. The challenge of finding a common ground on the topic of discrete curvature was met with success, and these proceedings are a testimony of this wor

Recent Advances in Industrial and Applied Mathematics

This open access book contains review papers authored by thirteen plenary invited speakers to the 9th International Congress on Industrial and Applied Mathematics (Valencia, July 15-19, 2019). Written by top-level scientists recognized worldwide, the scientific contributions cover a wide range of cutting-edge topics of industrial and applied mathematics: mathematical modeling, industrial and environmental mathematics, mathematical biology and medicine, reduced-order modeling and cryptography. The book also includes an introductory chapter summarizing the main features of the congress. This is the first volume of a thematic series dedicated to research results presented at ICIAM 2019-Valencia Congress

Recent Advances in Industrial and Applied Mathematics

This open access book contains review papers authored by thirteen plenary invited speakers to the 9th International Congress on Industrial and Applied Mathematics (Valencia, July 15-19, 2019). Written by top-level scientists recognized worldwide, the scientific contributions cover a wide range of cutting-edge topics of industrial and applied mathematics: mathematical modeling, industrial and environmental mathematics, mathematical biology and medicine, reduced-order modeling and cryptography. The book also includes an introductory chapter summarizing the main features of the congress. This is the first volume of a thematic series dedicated to research results presented at ICIAM 2019-Valencia Congress

Birefringent properties of the human cornea in vivo : towards a new model of corneal structure

The fundamental corneal properties of mechanical rigidity, maintenance of curvature and optical transparency result from the specific organisation of collagen fibrils in the corneal stroma. The exact arrangement of stromal collagen is currently unknown but several structural models have been proposed. The purpose of the present study is to investigate inconsistencies between current x‐ray derived structural models of the cornea and optically derived birefringence data. Firstly, the thesis reviews the current understanding of corneal structure, particularly in relation to corneal birefringence. It also reviews and develops the different analytical approaches used to model optical biaxial behaviour, particularly as applied to predict corneal optical phase retardation. The second part develops a novel technique of elliptic polarization biomicroscopy (EPB), enabling study of corneal birefringence in vivo. Using EPB, the pattern of corneal retardation is recorded for a range of human subjects. This dataset is then used to investigate both central and peripheral corneal birefringence as well as the corneal microstructure. A key finding is that the central parts of the cornea exhibit a retardation pattern compatible with a negative biaxial crystal, whereas the peripheral corneal regions do not. Furthermore, within the central regions of the cornea, orthogonal confocal conic fibrillar structures are identified which resemble the analytically derived contours of equal refractive index of an ideal negative biaxial crystal. The third part of this work presents a synthesis of previous published experimental, anatomical and theoretical findings and the experimental results presented in this thesis. Based on these findings, a novel corneal structural model is proposed that comprises overlapping spherical elliptic structural units. Finally, ensuing biomechanical and clinical consequences of the spherical elliptic structural model and of the EPB technique are discussed including their potential diagnostic and surgical applications