Dans cette thèse, nous nous intéressons à l'étude des courbes discrètes et ses applications en analyse d'images. Nous avons proposé une amélioration de l'estimation de courbure reposant sur le cercle circonscrit. Celle-ci repose sur la notion de segment flou maximal d'épaisseur [nu] et sur la décomposition d'une courbe discrète en sa séquence de segments flous maximaux. Par la suite, nousavons appliqué cette idée en 3D afin d'estimer la courbure et la torsion discrète en chaque point d'une courbe 3D. Au niveau de l'application, nous avons développé une méthode rapide et fiable pour détecter les points dominants dans une courbe 2D. Un point dominant est un point dont la courbure est localement maximale. Les points dominants jouent un rôle très important dans la reconnaissance de formes. Notre méthode utilise un paramètre qui est l'épaisseur des segments flous maximaux. Reposant sur cette nouvelle méthode de détection des points dominants, nous avons développé des méthodes sans paramètres de détection des points dominants. Celles-ci se basent sur une approche multi-épaisseur. D'autre part, nous nous intéressons particulièrement au cercles et arcs discrets. Une méthode linéaire a été développé pour reconnaître des cercles et arcs discrets. Puisnous avons fait évoluer cette méthode afin de travailler avec des courbes bruitées en utilisant une méthode de détection du bruit. Nous proposons aussi une mesure de circularité. Une méthode linéaire qui utilise cette mesure a été aussi développée pour mesurer la circularité des courbes fermées. Par ailleurs, nous avons proposé une méthode rapide pour décomposer des courbes discrètes en arcs et en segments de droite. / In this thesis, we are interested in the study of discrete curves and its applications in image analysis. We have proposed an amelioration of curvature estimation based on circumcircle. This method is based on the notion of blurred segment of width [nu] and on the decomposition of a curve into the sequence of maximal blurred segment of width [nu]. Afterwards, we have applied this idea in 3D to estimate the discrete curvature and torsion at each point of a 3D curve. Concerning the applications, we have developed a rapid et reliable method to detect dominant points of a 2D curve. A dominant point is a point whose the curvature value is locally maximum. The dominant points play an important role in pattern recognition. Our method uses a parameter: the width of maximal blurred segments. Based on this novel method of dominant point detection, we proposed free-parameter methods for polygonal representation. They are based on a multi-width approach. Otherwise, we are interested in discrete arcs and circles. A linear method has been proposed for the recognition of arcs and circles. We then develop a new method for segmentation of noisy curves into arcs based on a method of noise detection. We also proposed a linear method to measure the circularity of closed curves. In addition, we have proposed a robust method to decompose a curve into arcs and line segments

Starting from the observation that substances and reactions are the central entities of chemistry, I have structured chemical knowledge into a formal space called a directed hypergraph, which arises when substances are connected by their reactions. I call this hypernet chemical space. In this thesis, I explore different levels of description of this space: its evolution over time, its curvature, and categorical models of its compositionality. The vast majority of the chemical literature focuses on investigations of particular aspects of some substances or reactions, which have been systematically recorded in comprehensive databases such as Reaxys for the last 200 years. While complexity science has made important advances in physics, biology, economics, and many other fields, it has somewhat neglected chemistry. In this work, I propose to take a global view of chemistry and to combine complexity science tools, modern data analysis techniques, and geometric and compositional theories to explore chemical space. This provides a novel view of chemistry, its history, and its current status. We argue that a large directed hypergraph, that is, a model of directed relations between sets, underlies chemical space and that a systematic study of this structure is a major challenge for chemistry. Using the Reaxys database as a proxy for chemical space, we search for large-scale changes in a directed hypergraph model of chemical knowledge and present a data-driven approach to navigate through its history and evolution. These investigations focus on the mechanistic features by which this space has been expanding: the role of synthesis and extraction in the production of new substances, patterns in the selection of starting materials, and the frequency with which reactions reach new regions of chemical space. Large-scale patterns that emerged in the last two centuries of chemical history are detected, in particular, in the growth of chemical knowledge, the use of reagents, and the synthesis of products, which reveal both conservatism and sharp transitions in the exploration of the space. Furthermore, since chemical similarity of substances arises from affinity patterns in chemical reactions, we quantify the impact of changes in the diversity of the space on the formulation of the system of chemical elements. In addition, we develop formal tools to probe the local geometry of the resulting directed hypergraph and introduce the Forman-Ricci curvature for directed and undirected hypergraphs. This notion of curvature is characterized by applying it to social and chemical networks with higher order interactions, and then used for the investigation of the structure and dynamics of chemical space. The network model of chemistry is strongly motivated by the observation that the compositional nature of chemical reactions must be captured in order to build a model of chemical reasoning. A step forward towards categorical chemistry, that is, a formalization of all the flavors of compositionality in chemistry, is taken by the construction of a categorical model of directed hypergraphs. We lifted the structure from a lineale (a poset version of a symmetric monoidal closed category) to a category of Petri nets, whose wiring is a bipartite directed graph equivalent to a directed hypergraph. The resulting construction, based on the Dialectica categories introduced by Valeria De Paiva, is a symmetric monoidal closed category with finite products and coproducts, which provides a formal way of composing smaller networks into larger in such a way that the algebraic properties of the components are preserved in the resulting network. Several sets of labels, often used in empirical data modeling, can be given the structure of a lineale, including: stoichiometric coefficients in chemical reaction networks, reaction rates, inhibitor arcs, Boolean interactions, unknown or incomplete data, and probabilities. Therefore, a wide range of empirical data types for chemical substances and reactions can be included in our model.

This thesis explores and explains a method using discrete curvature as a feature to find regions of vertices that can be classified as being likely to indicate the presence of an underlying tumor on a kidney surface mesh. Vertices are tagged based on curvature type and mathematical morphology is used to form regions on the mesh. The size and location of the tumor is approximated by fitting a sphere to this region. The method is intended to be employed in noninvasive radiotherapy with a dynamic soft tissue model. It could also provide an alternative to volumetric methods used to segment tumors. A validation is made using the images from which the kidney mesh was constructed, the tumor is visible as a comparison to the method result. The dynamic kidney model is validated using the Hausdorff distance and it is explained how this can be computed in an effective way using bounding volume hierarchies. Both the tumor finding method and the dynamic model show promising results since they lie within the limit used by practitioners during therapy.

Ce travail de thèse porte sur la modélisation dynamique 3D du rein et le suivi d’une tumeur de cet organe. Il s’inscrit dans le projet KiTT (Kidney Tumor Tracking) qui regroupe des chercheurs issus de plusieurs domaines : la modélisation géométrique, la radiologie et l’urologie. Le cadre de cette thèse suit une tendance de mini-invasivité des gestes chirurgicaux observée ces dernières années (HIFU, coelioscopie). Sa finalité est d’aboutir à un nouveau protocole de destruction de tumeurs rénales totalement non-invasif, par la diffusion d’agents physiques (ondes d’ultrasons) à travers la peau et focalisés sur la tumeur. Devant le mouvement et la déformation que le rein présente au cours du cycle respiratoire, la problématique de ces travaux de recherche est de connaître en permanence la position de la tumeur afin d’ajuster à moyen terme la diffusion des ondes en conséquence. / This Ph.D. thesis deals with the 3D dynamic modeling of the kidney and tracking a tumor of this organ. It is in line with the KiTT project (Kidney Tumor Tracking) which gathers researchers from different fileds: geometric modeling, radiology and urology. This work arised from the tendency of nowadays surgical gestures to be less and less invasive (HIFU, coelioscopy). Its goal is to result in a totally non-invasive protocol of kidney tumors eradication by transmitting ultrasound waves through the skin without breaking in it. As the kidney presents motions and deformations during the breathing phase, the main issue is to know the kidney and tumor positions at any time in order to adjust the waves accordingly.