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We present a novel framework to exert a topology control over a level set evolution. Level set methods offer several advantages over parametric active contours, in particular automated topological changes. In some applications, where some a priori knowledge of the target topology is available, topological changes may not be desirable. A method, based on the concept of simple point borrowed from digital topology, was recently proposed to achieve a strict topology preservation during a level set evolution. However, topologically constrained evolutions often generate topological barriers that lead to large geometric inconsistencies. We introduce a topologically controlled level set framework that greatly alleviates this problem. Unlike existing work, our method allows connected components to merge, split or vanish under some specific conditions that ensure that no topological defects are generated. We demonstrate the strength of our method on a wide range of numerical experiments.

Durant cette thèse, nous nous sommes intéressés à la problématique de la captation de mouvement sans marqueurs. Une approche classique est basée sur l'utilisation d'un modèle prédéfini du sujet, et est divisée en deux phases : celle d'initialisation, où la pose initiale du sujet est estimée, et celle de suivi, où la pose actuelle du sujet est estimée à partir des précédentes. Souvent, la phase d'initialisation est faite manuellement, rendant impossible l'utilisation en direct, ou nécessite des actions spécifiques du sujet. Nous proposons une phase d'initialisation automatique et temps-réel, utilisant l'information topologique extraite par squelettisation d'une reconstruction 3D du sujet. Cette information est représentée sous forme d'arbre (arbre de données), qui est mis en correspondance avec un arbre utilisé comme modèle, afin d'identifier les différentes parties du sujet. Pour obtenir une telle méthode, nous apportons des contributions dans les domaines de la topologie discrète et de la théorie des graphes. Comme notre méthode requiert le temps réel, nous nous intéressons d'abord à l'optimisation du temps de calcul des méthodes de squelettisation, ainsi qu'à l'élaboration de nouveaux algorithmes rapides fournissant de bons résultats. Nous nous intéressons ensuite à la définition d'une mise en correspondance efficace entre l'arbre de données et celui décrivant le modèle. Enfin, nous améliorons la robustesse de notre méthode en ajoutant des contraintes novatrices au modèle. Nous terminons par l'application de notre méthode sur différents jeux de données, démontrantses propriétés : rapidité robustesse et adaptabilité à différents types de sujet / This manuscript deals with the problem of markerless motion capture. An approach to thisproblem is model-based and is divided into two steps : an initialization step in which the initialpose is estimated, and a tracking which computes the current pose of the subject using infor-mation of previous ones. Classically, the initialization step is done manually, for bidding the possibility to be used online, or requires constraining actions of the subject. We propose an automatic real-time markerless initialization step, that relies on topological information provided by skeletonization of a 3D reconstruction of the subject. This topological information is then represented as a tree, which is matched with another tree used as modeldescription, in order to identify the different parts of the subject. In order to provide such a method, we propose some contributions in both digital topology and graph theory researchfields. As our method requires real-time computation, we first focus on the speed optimization of skeletonization methods, and on the design of new fast skeletonization schemes providing good results. In order to efficiently match the tree representing the topological information with the tree describing the model, we propose new matching definitions and associated algorithms. Finally, we study how to improve the robustness of our method by the use of innovative con-straints in the model. This manuscript ends by a study of the application of our method on several data sets, demon-strating its interesting properties : fast computation, robustness, and adaptability to any kindof subjects

En géométrie discrète, les objets euclidiens sont représentés par leurs approximations discrètes, telles que des sous-ensembles du réseau des points à coordonnées entières. Les déplacements de ces ensembles doivent être définis comme des applications depuis et sur un espace discret donné. Une façon de concevoir de telles transformations est de combiner des déplacements continus définis sur un espace euclidien avec un opérateur de discrétisation. Cependant, les déplacements discrétisés ne satisfont souvent plus les propriétés de leurs équivalents continus. En effet, en raison de la discrétisation, de telles transformations ne préservent pas les distances, et la bijectivité et la connexité entre les points sont généralement perdues. Dans le contexte des espaces discrets 2D, nous étudions des déplacements discrétisés sur les réseaux d'entiers de Gauss et d'Eisenstein. Nous caractérisons les déplacements discrétisés bijectifs sur le réseau carré, et les rotations bijectives discrétisées sur le réseau hexagonal régulier. En outre, nous comparons les pertes d'information induites par des déplacements discrétisés non bijectifs définis sur ces deux réseaux. Toutefois, pour des applications pratiques, l'information pertinente n'est pas la bijectivité globale, mais celle d'un déplacement discrétisé restreint à un sous-ensemble fini donné d'un réseau. Nous proposons deux algorithmes testant cette condition pour les sous-ensembles du réseau entier, ainsi qu'un troisième algorithme fournissant des intervalles d'angles optimaux qui préservent cette bijectivité restreinte. Nous nous concentrons ensuite sur les déplacements discrétisés sur le réseau cubique 3D. Tout d'abord, nous étudions à l'échelle locale des défauts géométriques et topologiques induits par des déplacements discrétisés. Une telle analyse consiste à générer toutes les images d'un ensemble du réseau fini sous des déplacements discrétisés. Un tel problème revient à calculer un arrangement d'hypersurfaces dans un espace de paramètres de dimension six. La dimensionnalité et les cas dégénérés rendent le problème insoluble, en pratique, par les techniques usuelles. Nous proposons une solution ad hoc reposant sur un découplage des paramètres, et un algorithme pour calculer des points d'échantillonnage de composantes connexes 3D dans un arrangement de polynômes du second degré. Enfin, nous nous concentrons sur le problème ouvert de déterminer si une rotation discrétisée 3D est bijective ou non. Dans notre approche, nous explorons les propriétés arithmétiques des quaternions de Lipschitz. Ceci conduit à un algorithme qui détermine si une rotation discrétisée donnée, associée à un quaternion de Lipschitz, est bijective ou non / In digital geometry, Euclidean objects are represented by their discrete approximations, e.g. subsets of the lattice of integers. Rigid motions of such sets have to be defined as maps from and onto a given discrete space. One way to design such motions is to combine continuous rigid motions defined on Euclidean space with a digitization operator. However, digitized rigid motions often no longer satisfy properties of their continuous siblings. Indeed, due to digitization, such transformations do not preserve distances, while bijectivity and point connectivity are generally lost. In the context of 2D discrete spaces, we study digitized rigid motions on the lattices of Gaussian and Eisenstein integers. We characterize bijective digitized rigid motions on the integer lattice, and bijective digitized rotations on the regular hexagonal lattice. Also, we compare the information loss induced by non-bijective digitized rigid motions defined on both lattices. Yet, for practical applications, the relevant information is not global bijectivity, but bijectivity of a digitized rigid motion restricted to a given finite subset of a lattice. We propose two algorithms testing that condition for subsets of the integer lattice, and a third algorithm providing optimal angle intervals that preserve this restricted bijectivity. We then focus on digitized rigid motions on 3D integer lattice. First, we study at a local scale geometric and topological defects induced by digitized rigid motions. Such an analysis consists of generating all the images of a finite digital set under digitized rigid motions. This problem amounts to computing an arrangement of hypersurfaces in a 6D parameter space. The dimensionality and degenerate cases make the problem practically unsolvable for state-of-the-art techniques. We propose an ad hoc solution, which mainly relies on parameter uncoupling, and an algorithm for computing sample points of 3D connected components in an arrangement of second degree polynomials. Finally, we focus on the open problem of determining whether a 3D digitized rotation is bijective or not. In our approach, we explore arithmetic properties of Lipschitz quaternions. This leads to an algorithm which answers whether a given digitized rotation—related to a Lipschitz quaternion—is bijective or not

Chronic obstructive pulmonary disease (COPD) is a type of obstructive lung disease characterized by chronically poor airflow, which is the result of breakdown of lung tissue (known as emphysema) and small airways disease. It typically worsens over time. Most treatments are limited to the management of symptoms, which makes early detection more valuable to treat the disease etiology itself. With the advancement of computed tomography (CT), it is able to provide high resolution structural and functional imaging to distinguish the lung anatomic structures, as well as characterize their changes over time. Previously, the majority of CT-based measures have focused on quantifying the extent of airway and parenchymal damage. Recent studies suggests that pulmonary vascular dysfunction is an early lesion in COPD and associated with an emphysematous phenotype. Few studies attempted to quantify pulmonary vessel morphology and compared those measures across COPD groups. However, the scope of examined vascular structures in these studies was limited, majorly due to the lack of a standardized method to quantify a broad range of vascular structures. In this thesis, we propose to use anatomically defined airway branches as references to locate and morphologically quantify central pulmonary arteries in different lung regions. Although pulmonary vessel trees have complex topologic and geometric structures, airway tree possesses much simpler and consistent branching patterns and standardized anatomic nomenclatures are available up to sub-segmental levels. It is also well-known that airway and arterial branches have a unique pairing that is established by their spatial proximity and parallel configuration. Therefore, anatomically labeled airway tree provides a robust benchmark to locate consistent arterial segments for both intra- and inter-subjects. New methods have been developed for quantitative assessment of arterial morphology matched and standardized by associated airways at different anatomic branches. First, the skeletons of airway and vessel trees are generated to provide simple and hierarchical representations. Then, topologic and geometric properties of airways and arteries, such as distance, orientation and anatomic positon information, are explored to locate the target arterial segments. Finally, the morphologic properties, e.g. cross-sectional area, of target arterial segments are robustly computed. The developed methods in this thesis provides a standardized framework to assess and compare the vascular measurements in intra- and inter- subjects from a broad range of vessel branches in different lung regions. The work also serves as a practical tool for large longitudinal or cross-sectional studies to explore the pulmonary vessel roles played at the early stage of COPD. The major contribution of this thesis include: (1) developing two novel skeletonization methods that are applicable to airway and pulmonary vessel trees; (2) developing a semi-automatic method to locate and quantify central pulmonary arterial morphology associate to anatomic airway branches; (3) developing a fully automatic method to identify and reconstruct central pulmonary arterial segments associated to anatomic airway branches and quantify their morphology; (4) validating the methods using computerized phantoms, physical phantoms and human subjects; (5) applying the developed methods to two human lung disease studies.

Computed tomography is a very useful technic which allow non-invasive diagnosis in many applications for example is used with success in industry and medicine. However, manual analysis of the interesting structures can be tedious and extremely time consuming, or even impossible due its complexity. Therefore in this thesis we study and develop discrete geometry and topology algorithms suitable for use in many practical applications, especially, in the problem of automatic quantitative analysis of the human airway trees based on computed tomography images. In the first part, we define basic notions used in discrete topology and geometry then we showed that several class of discrete methods like skeletonisation algorithms, medial axes, tunnels closing algorithms and tangent estimators, are widely used in several different practical application. The second part consist of a proposition and theory of a new methods for solving particular problems. We introduced two new medial axis filtering method. The hierarchical scale medial axis which is based on previously proposed scale axis transform, however, is free of drawbacks introduced in the previously proposed method and the discrete adaptive medial axis where the filtering parameter is dynamically adapted to the local size of the object. In this part we also introduced an efficient and parameter less new tangent estimators along three-dimensional discrete curves, called 3D maximal segment tangent direction. Finally, we showed that discrete geometry and topology algorithms can be useful in the problem of quantitative analysis of the human airway trees based on computed tomography images. According to proposed in the literature design of such system we applied discrete topology and geometry algorithms to solve particular problems at each step of the quantitative analysis process. First, we propose a robust method for segmenting airway tree from CT datasets. The method is based on the tunnel closing algorithm and is used as a tool to repair, damaged by acquisition errors, CT images. We also proposed an algorithm for creation of an artificial model of the bronchial tree and we used such model to validate algorithms presented in this work. Then, we compare the quality of different algorithms using set of experiments conducted on computer phantoms and real CT dataset. We show that recently proposed methods which works in cubical complex framework, together with methods introduced in this work can overcome problems reported in the literature and can be a good basis for the further implementation of the system for automatic quantification of bronchial tree properties

This manuscript deals with the problem of markerless motion capture. An approach to thisproblem is model-based and is divided into two steps : an initialization step in which the initialpose is estimated, and a tracking which computes the current pose of the subject using infor-mation of previous ones. Classically, the initialization step is done manually, for bidding the possibility to be used online, or requires constraining actions of the subject. We propose an automatic real-time markerless initialization step, that relies on topological information provided by skeletonization of a 3D reconstruction of the subject. This topological information is then represented as a tree, which is matched with another tree used as modeldescription, in order to identify the different parts of the subject. In order to provide such a method, we propose some contributions in both digital topology and graph theory researchfields. As our method requires real-time computation, we first focus on the speed optimization of skeletonization methods, and on the design of new fast skeletonization schemes providing good results. In order to efficiently match the tree representing the topological information with the tree describing the model, we propose new matching definitions and associated algorithms. Finally, we study how to improve the robustness of our method by the use of innovative con-straints in the model. This manuscript ends by a study of the application of our method on several data sets, demon-strating its interesting properties : fast computation, robustness, and adaptability to any kindof subjects

Le processus de discrétisation faisant inévitablement appel à des capteurs, et ceux-ci étant limités de par leur nature, de nombreux effets secondaires apparaissent alors lors de ce processus; en particulier, nous perdons la propriété d'être "bien-composé" dans le sens où deux objects discrétisés peuvent être connectés ou non en fonction de la connexité utilisée dans l'image discrète, ce qui peut amener à des ambigüités. De plus, les images discrétisées sont des tableaux de valeurs numériques, et donc ne possèdent pas de topologie par nature, contrairement à notre modélisation usuelle du monde en mathématiques et en physique. Perdre toutes ces propriétés rend difficile l'élaboration d'algorithmes topologiquement corrects en traitement d'images: par exemple, le calcul de l'arbre des formes nécessite que la representation d'une image donnée soit continue et bien-composée; dans le cas contraire, nous risquons d'obtenir des anomalies dans le résultat final. Quelques representations continues et bien-composées existent déjà, mais elles ne sont pas simultanément n-dimensionnelles et auto-duales. La n-dimensionalité est cruciale sachant que les signaux usuels sont de plus en plus tridimensionnels (comme les vidéos 2D) ou 4-dimensionnels (comme les CT-scans). L'auto-dualité est nécéssaire lorsqu'une même image contient des objets a contrastes divers. Nous avons donc développé une nouvelle façon de rendre les images bien-composées par interpolation de façon auto-duale et en n-D; suivie d'une immersion par l'opérateur span, cette interpolation devient une représentation auto-duale continue et bien-composée du signal initial n-D. Cette représentation bénéficie de plusieurs fortes propriétés topologiques: elle vérifie le théorème de la valeur intermédiaire, les contours de chaque coupe de la représentation sont déterminés par une union disjointe de surfaces discrète, et ainsi de suite / Digitization of the real world using real sensors has many drawbacks; in particular, we loose ``well-composedness'' in the sense that two digitized objects can be connected or not depending on the connectivity we choose in the digital image, leading then to ambiguities. Furthermore, digitized images are arrays of numerical values, and then do not own any topology by nature, contrary to our usual modeling of the real world in mathematics and in physics. Loosing all these properties makes difficult the development of algorithms which are ``topologically correct'' in image processing: e.g., the computation of the tree of shapes needs the representation of a given image to be continuous and well-composed; in the contrary case, we can obtain abnormalities in the final result. Some well-composed continuous representations already exist, but they are not in the same time n-dimensional and self-dual. n-dimensionality is crucial since usual signals are more and more 3-dimensional (like 2D videos) or 4-dimensional (like 4D Computerized Tomography-scans), and self-duality is necessary when a same image can contain different objects with different contrasts. We developed then a new way to make images well-composed by interpolation in a self-dual way and in n-D; followed with a span-based immersion, this interpolation becomes a self-dual continuous well-composed representation of the initial n-D signal. This representation benefits from many strong topological properties: it verifies the intermediate value theorem, the boundaries of any threshold set of the representation are disjoint union of discrete surfaces, and so on

<p>Digital geometry is the geometry of digital images. Compared to Euclid’s geometry, which has been studied for more than two thousand years, this field is very young.</p><p>Efim Khalimsky’s topology on the integers, invented in the 1970s, is a digital counterpart of the Euclidean topology on the real line. The Khalimsky topology became widely known to researchers in digital geometry and computer imagery during the early 1990s.</p><p>Suppose that a continuous function is defined on a subspace of an <i>n-</i>dimensional Khalimsky space. One question to ask is whether this function can be extended to a continuous function defined on the whole space. We solve this problem. A related problem is to characterize the subspaces on which every continuous function can be extended. Also this problem is solved.</p><p>We generalize and solve the extension problem for integer-valued, Khalimsky-continuous functions defined on arbitrary smallest-neighborhood spaces, also called Alexandrov spaces.</p><p>The notion of a digital straight line was clarified in 1974 by Azriel Rosenfeld. We introduce another type of digital straight line, a line that respects the Khalimsky topology in the sense that a line is a topological embedding of the Khalimsky line into the Khalimsky plane.</p><p>In higher dimensions, we generalize this construction to digital Khalimsky hyperplanes, surfaces and curves by digitization of real objects. In particular we study approximation properties and topological separation properties. </p><p>The last paper is about Khalimsky manifolds, spaces that are locally homeomorphic to <i>n-</i>dimensional Khalimsky space. We study different definitions and address basic questions such as uniqueness of dimension and existence of certain manifolds.</p>

Digital geometry is the geometry of digital images. Compared to Euclid’s geometry, which has been studied for more than two thousand years, this field is very young. Efim Khalimsky’s topology on the integers, invented in the 1970s, is a digital counterpart of the Euclidean topology on the real line. The Khalimsky topology became widely known to researchers in digital geometry and computer imagery during the early 1990s. Suppose that a continuous function is defined on a subspace of an n-dimensional Khalimsky space. One question to ask is whether this function can be extended to a continuous function defined on the whole space. We solve this problem. A related problem is to characterize the subspaces on which every continuous function can be extended. Also this problem is solved. We generalize and solve the extension problem for integer-valued, Khalimsky-continuous functions defined on arbitrary smallest-neighborhood spaces, also called Alexandrov spaces. The notion of a digital straight line was clarified in 1974 by Azriel Rosenfeld. We introduce another type of digital straight line, a line that respects the Khalimsky topology in the sense that a line is a topological embedding of the Khalimsky line into the Khalimsky plane. In higher dimensions, we generalize this construction to digital Khalimsky hyperplanes, surfaces and curves by digitization of real objects. In particular we study approximation properties and topological separation properties. The last paper is about Khalimsky manifolds, spaces that are locally homeomorphic to n-dimensional Khalimsky space. We study different definitions and address basic questions such as uniqueness of dimension and existence of certain manifolds.