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.

There are many imaging techniques that generate three-dimensional volume images today. With higher precision in the image acquisition equipment, storing and processing these images require increasing amount of data processing capacity. Traditionally, three-dimensional images are represented by cubic (or cuboid) picture elements on a cubic grid. The two-dimensional hexagonal grid has some advantages over the traditionally used square grid. For example, less samples are needed to get the same reconstruction quality, it is less rotational dependent, and each picture element has only one type of neighbor which simplifies many algorithms. The corresponding three-dimensional grids are the face-centered cubic (fcc) grid and the body-centered cubic (bcc) grids. In this thesis, image representations using non-standard grids is examined. The focus is on the fcc and bcc grids and tools for processing images on these grids, but distance functions and related algorithms (distance transforms and various representations of objects) are defined in a general framework allowing any point-lattice in any dimension. Formulas for point-to-point distance and conditions for metricity are given in the general case and parameter optimization is presented for the fcc and bcc grids. Some image acquisition and visualization techniques for the fcc and bcc grids are also presented. More theoretical results define distance functions for grids of arbitrary dimensions. Less samples are needed to represent images on non-standard grids. Thus, the huge amount of data generated by for example computerized tomography can be reduced by representating the images on non-standard grids such as the fcc or bcc grids. The thesis gives a tool-box that can be used to acquire, process, and visualize images on high-dimensional, non-standard grids.

In this paper we introduce an algorithm for the creation of polyhedral approximations for objects represented as strongly connected sets of voxels in three-dimensional binary images. The algorithm generates the convex hull of a given object and modifies the hull afterwards by recursive repetitions of generating convex hulls of subsets of the given voxel set or subsets of the background voxels. The result of this method is a polyhedron which separates object voxels from background voxels. The objects processed by this algorithm and also the background voxel components inside the convex hull of the objects are restricted to have genus 0. The second aim of this paper is to present some improvements to our convex hull algorithm to reduce computation time.

This thesis presents a study on rotation in 2 dimensional and 3 dimensional discrete spaces. In computer science, using floating numbers is problematic due to computation errors. Thus we chose during this thesis to work only in discrete space. In the field of computer vision, the rotation is a transformation required for many applications. Using discretized Euclidean rotation gives bad results. Then, it is necessary to develop new rotation methods adapted to the discrete spaces. We mainly studied the hinge angles that represent the discontinuity of the rotation in the discrete space. Indeed, it is possible to perform two rotations of the same digital image with two angles that are slightly different and obtain the same result. This is captured by hinge angles. Using these angles allow us to describe a discrete rotation that gives the same results than the discretized Euclidean rotation without using floating numbers. They also allow describing an incremental rotation that performs all possible rotations of a given digital image. Using hinge angles can also be extended to the rotations in 3 dimensional discrete spaces. The extension requires the multi-grids that are rotation planes containing three sets of parallel lines. These parallel lines represent the discontinuities of the rotation in 3D discrete space. Thus they are useful to describe the hinge angles in rotation planes. Multi-grids allow obtaining the same results in 3D discrete rotations than the results obtained in 2D discrete rotations. This thesis presents a study on rotation in 2 dimensional and 3 dimensional discrete spaces. In computer science, using floating numbers is problematic due to computation errors. Thus we chose during this thesis to work only in discrete space. In the field of computer vision, the rotation is a transformation required for many applications. Using discretized Euclidean rotation gives bad results. Then, it is necessary to develop new rotation methods adapted to the discrete spaces. We mainly studied the hinge angles that represent the discontinuity of the rotation in the discrete space. Indeed, it is possible to perform two rotations of the same digital image with two angles that are slightly different and obtain the same result. This is captured by hinge angles. Using these angles allow us to describe a discrete rotation that gives the same results than the discretized Euclidean rotation without using floating numbers. They also allow describing an incremental rotation that performs all possible rotations of a given digital image. Using hinge angles can also be extended to the rotations in 3 dimensional discrete spaces. The extension requires the multi-grids that are rotation planes containing three sets of parallel lines. These parallel lines represent the discontinuities of the rotation in 3D discrete space. Thus they are useful to describe the hinge angles in rotation planes. Multi-grids allow obtaining the same results in 3D discrete rotations than the results obtained in 2D discrete rotations

In this paper we introduce an algorithm for the creation of polyhedral approximations for objects represented as strongly connected sets of voxels in three-dimensional binary images. The algorithm generates the convex hull of a given object and modifies the hull afterwards by recursive repetitions of generating convex hulls of subsets of the given voxel set or subsets of the background voxels. The result of this method is a polyhedron which separates object voxels from background voxels. The objects processed by this algorithm and also the background voxel components inside the convex hull of the objects are restricted to have genus 0. The second aim of this paper is to present some improvements to our convex hull algorithm to reduce computation time.

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

Cette thèse présente une étude sur les rotations dans les espaces discrets en 2 dimensions et en 3 dimensions. Dans le cadre de l'informatique, l'utilisation des nombres flottants n'est pas recommandée du fait des erreurs de calculs que cela implique. Nous avons donc fait le choix de nous concentrer sur les espaces discrets. Dans le domaine de la vision par ordinateur, la rotation est une transformation requise pour de nombreuses applications. L'utilisation de la rotation continue discrétisée donne des résultats de mauvaise qualité. Pour cette raison, il est nécessaire de développer de nouvelles méthodes de rotation adaptées aux espaces discrets. Nous nous sommes principalement intéressés aux angles charnières qui représentent la discontinuité de la rotation dans les espaces discrets. Dans ces espaces, deux rotations d'une image avec deux angles très proches peuvent donner le même résultat, ce qui est capturé par les angles charnières. L'utilisation de ces angles permet de décrire une rotation qui donne les mêmes résultats que la rotation continue discrétisée tout en n'utilisant que des nombres entiers. Ils permettent aussi de définir une rotation incrémentale qui décrit toutes les rotations possibles d'une image digitale donnée. Les angles charnières peuvent être étendus dans les espaces discrets en trois dimensions. Pour cela, on définit les multi-grilles qui sont des plans de rotations contenant trois ensembles de droites parallèles. Elles représentent les discontinuités de la rotation en 3D. Les multi-grilles permettent d'obtenir les mêmes résultats en 3D que ceux obtenus en 2D / This thesis presents a study on rotation in 2 dimensional and 3 dimensional discrete spaces. In computer science, using floating numbers is problematic due to computation errors. Thus we chose during this thesis to work only in discrete space. In the field of computer vision, the rotation is a transformation required for many applications. Using discretized Euclidean rotation gives bad results. Then, it is necessary to develop new rotation methods adapted to the discrete spaces. We mainly studied the hinge angles that represent the discontinuity of the rotation in the discrete space. Indeed, it is possible to perform two rotations of the same digital image with two angles that are slightly different and obtain the same result. This is captured by hinge angles. Using these angles allow us to describe a discrete rotation that gives the same results than the discretized Euclidean rotation without using floating numbers. They also allow describing an incremental rotation that performs all possible rotations of a given digital image. Using hinge angles can also be extended to the rotations in 3 dimensional discrete spaces. The extension requires the multi-grids that are rotation planes containing three sets of parallel lines. These parallel lines represent the discontinuities of the rotation in 3D discrete space. Thus they are useful to describe the hinge angles in rotation planes. Multi-grids allow obtaining the same results in 3D discrete rotations than the results obtained in 2D discrete rotations. This thesis presents a study on rotation in 2 dimensional and 3 dimensional discrete spaces. In computer science, using floating numbers is problematic due to computation errors. Thus we chose during this thesis to work only in discrete space. In the field of computer vision, the rotation is a transformation required for many applications. Using discretized Euclidean rotation gives bad results. Then, it is necessary to develop new rotation methods adapted to the discrete spaces. We mainly studied the hinge angles that represent the discontinuity of the rotation in the discrete space. Indeed, it is possible to perform two rotations of the same digital image with two angles that are slightly different and obtain the same result. This is captured by hinge angles. Using these angles allow us to describe a discrete rotation that gives the same results than the discretized Euclidean rotation without using floating numbers. They also allow describing an incremental rotation that performs all possible rotations of a given digital image. Using hinge angles can also be extended to the rotations in 3 dimensional discrete spaces. The extension requires the multi-grids that are rotation planes containing three sets of parallel lines. These parallel lines represent the discontinuities of the rotation in 3D discrete space. Thus they are useful to describe the hinge angles in rotation planes. Multi-grids allow obtaining the same results in 3D discrete rotations than the results obtained in 2D discrete rotations

La problématique centrale de cette thèse est l'élaboration d'un opérateur de Laplace--Beltrami discret sur les surfaces digitales. Ces surfaces proviennent de la théorie de la géométrie discrète, c’est-à-dire la géométrie qui s'intéresse à des sous-ensembles des entiers relatifs. Nous nous plaçons ici dans un cadre théorique où les surfaces digitales sont le résultat d'une approximation, ou processus de discrétisation, d'une surface continue sous-jacente. Cette méthode permet à la fois de prouver des théorèmes de convergence des quantités discrètes vers les quantités continues, mais aussi, par des analyses numériques, de confirmer expérimentalement ces résultats. Pour la discrétisation de l’opérateur, nous faisons face à deux problèmes : d'un côté, notre surface n'est qu'une approximation de la surface continue sous-jacente, et de l'autre côté, l'estimation triviale de quantités géométriques sur la surface digitale ne nous apporte pas en général une bonne estimation de cette quantité. Nous possédons déjà des réponses au second problème : ces dernières années, de nombreux articles se sont attachés à développer des méthodes pour approximer certaines quantités géométriques sur les surfaces digitales (comme par exemple les normales ou bien la courbure), méthodes que nous décrirons dans cette thèse. Ces nouvelles techniques d'approximation nous permettent d'injecter des informations de mesure sur les éléments de notre surface. Nous utilisons donc l'estimation de normales pour répondre au premier problème, qui nous permet en fait d'approximer de façon précise le plan tangent en un point de la surface et, via une méthode d'intégration, palier à des problèmes topologiques liées à la surface discrète. Nous présentons un résultat théorique de convergence du nouvel opérateur discrétisé, puis nous illustrons ensuite ses propriétés à l’aide d’une analyse numérique de l’opérateur. Nous effectuons une comparaison détaillée du nouvel opérateur par rapport à ceux de la littérature adaptés sur les surfaces digitales, ce qui nous permet, au moins pour la convergence, de montrer que seul notre opérateur possède cette propriété. Nous illustrons également l’opérateur via quelques unes de ces applications comme sa décomposition spectrale ou bien encore le flot de courbure moyenne / The central issue of this thesis is the development of a discrete Laplace--Beltrami operator on digital surfaces. These surfaces come from the theory of discrete geometry, i.e. geometry that focuses on subsets of relative integers. We place ourselves here in a theoretical framework where digital surfaces are the result of an approximation, or discretization process, of an underlying smooth surface. This method makes it possible both to prove theorems of convergence of discrete quantities towards continuous quantities, but also, through numerical analyses, to experimentally confirm these results. For the discretization of the operator, we face two problems: on the one hand, our surface is only an approximation of the underlying continuous surface, and on the other hand, the trivial estimation of geometric quantities on the digital surface does not generally give us a good estimate of this quantity. We already have answers to the second problem: in recent years, many articles have focused on developing methods to approximate certain geometric quantities on digital surfaces (such as normals or curvature), methods that we will describe in this thesis. These new approximation techniques allow us to inject measurement information into the elements of our surface. We therefore use the estimation of normals to answer the first problem, which in fact allows us to accurately approximate the tangent plane at a point on the surface and, through an integration method, to overcome topological problems related to the discrete surface. We present a theoretical convergence result of the discretized new operator, then we illustrate its properties using a numerical analysis of it. We carry out a detailed comparison of the new operator with those in the literature adapted on digital surfaces, which allows, at least for convergence, to show that only our operator has this property. We also illustrate the operator via some of these applications such as its spectral decomposition or the mean curvature flow

Für die Visualisierung dreidimensionaler digitaler Objekte ist im Allgemeinen nur ihre Oberfläche von Interesse. Da von den bildgebenden Verfahren das gesamte räumliche Objekt in Form einer Volumenstruktur digitalisiert wird, muss aus den Daten die Oberfläche berechnet werden. In dieser Arbeit wird ein Algorithmus vorgestellt, der die Oberfläche dreidimensionaler digitaler Objekte, die als Menge von Voxeln gegeben sind, approximiert und dabei Polyeder erzeugt, die die Eigenschaft besitzen, die Voxel des Objektes von den Voxeln des Hintergrundes zu trennen. Weiterhin werden nicht-konvexe Objekte klassifiziert und es wird untersucht, für welche Klassen von Objekten die erzeugten Polyeder die minimale Flächenanzahl und den minimalen Oberflächeninhalt besitzen.