<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.

In this thesis we present and solve selected problems arising from digital geometry and combinatorics on words. We consider digital straight lines and, equivalently, upper mechanical words with positive irrational slopes a&lt;1 and intercept 0. We formulate a continued fraction (CF) based description of their run-hierarchical structure. Paper I gives a theoretical basis for the CF-description of digital lines. We define for each irrational positive slope less than 1 a sequence of digitization parameters which fully specifies the run-hierarchical construction. In Paper II we use the digitization parameters in order to get a description of runs using only integers. We show that the CF-elements of the slopes contain the complete information about the run-hierarchical structure of the line. The index jump function introduced by the author indicates for each positive integer k the index of the CF-element which determines the shape of the digitization runs on level k. In Paper III we present the results for upper mechanical words and compare our CF-based formula with two well-known methods, one of which was formulated by Johann III Bernoulli and proven by Markov, while the second one is known as the standard sequences method. Due to the special treatment of some CF-elements equal to 1 (essential 1's in Paper IV), our method is currently the only one which reflects the run-hierarchical structure of upper mechanical words by analogy to digital lines. In Paper IV we define two equivalence relations on the set of all digital lines with positive irrational slopes a&lt;1. One of them groups into classes all the lines with the same run length on all digitization levels, the second one groups the lines according to the run construction in terms of long and short runs on all levels. We analyse the equivalence classes with respect to minimal and maximal elements. In Paper V we take another look at the equivalence relation defined by run construction, this time independently of the context, which makes the results more general. In Paper VI we define a run-construction encoding operator, by analogy with the well-known run-length encoding operator. We formulate and present a proof of a fixed-point theorem for Sturmian words. We show that in each equivalence class under the relation based on run length on all digitization levels (as defined in Paper IV), there exists exactly one fixed point of the run-construction encoding operator.

This thesis consists of two parts: digital geometry and discrete optimization. In the first part we study the structure of digital straight line segments. We also study digital curves from a combinatorial point of view. In Paper I we study the straightness in the 8-connected plane and in the Khalimsky plane by considering vertical distances and unions of two segments. We show that we can investigate the straightness of Khalimsky arcs by using our knowledge from the 8-connected plane. In Paper II we determine the number of Khalimsky-continuous functions with 2, 3 and 4 points in their codomain. These enumerations yield examples of known sequences as well as new ones. We also study the asymptotic behavior of each of them. In Paper III we study the number of Khalimsky-continuous functions with codomain Z and N. This gives us examples of Schröder and Delannoy numbers. As a byproduct we get some relations between these numbers. In Paper IV we study the number of Khalimsky-continuous functions between two points in a rectangle. Using a generating function we get a recurrence formula yielding this numbers.   In the second part we study an analogue of discrete convexity, namely lateral convexity. In Paper V we define by means of difference operators the class of lateral convexity. The functions have plus infinity in their codomain. For the real-valued functions we need to check the difference operators for a smaller number of points. We study the relation between this class and integral convexity. In Paper VI we study the marginal function of real-valued functions in this class and its generalization. We show that for two points with a certain distance we have a Lipschitz property for the points where the infimum is attained. We show that if a function is in this class, the marginal function is also in the same class. / At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 4: Submitted. Paper 5: Manuscript. Paper 6: Manuscript.

La tomodensitométrie est une technique très utile qui permet de mener avec succès des analyses non-invasives dans plusieurs types d'applications, par exemple médicales ou industrielles. L'analyse manuelle des structures d'intérêt présentes dans une image peut prendre beaucoup de temps, être laborieuse et parfois même impossible à faire en raison de sa complexité. C'est pour cela que dans cette thèse, nous proposons et développons des algorithmes nécessaires à cette analyse, basés sur la géométrie discrète et la topologie. Ces algorithmes peuvent servir dans de nombreuses applications, et en particulier au niveau de l'analyse quantitative automatique de l'arbre bronchique humain, sur la base d'images de tomodensitométrie. La première partie introduit les notions fondamentales de la topologie et de la géométrie discrètes utiles dans cette thèse. Ensuite, nous présentons le principe de méthodes utilisées dans de nombreuses applications : les algorithmes de squelettisation, de calcul de l'axe médian, les algorithmes de fermeture de tunnels et les estimateurs de tangentes. La deuxième partie présente les nouvelles méthodes que nous proposons et qui permettent de résoudre des problèmes particuliers. Nous avons introduit deux méthodes nouvelles de filtrage d'axe médian. La première, que nous appelons "hierarchical scale medial axis", est inspirée du "scale axis transform", sans les inconvénients qui sont propres à la méthode originale. La deuxième est une méthode nommée "discrete adaptive medial axis", où le paramètre de filtrage est adapté dynamiquement aux dimensions locales de l'objet. Dans cette partie, nous introduisons également des estimateurs de tangente nouveaux et efficaces, agissant sur des courbes discrètes tridimensionnelles, et que nous appelons "3Dlambda maximal segment tangent direction". Enfin, nous avons montré que la géométrie discrète et les algorithmes topologiques pouvaient être utiles dans le problème de l'analyse quantitative de l'arbre bronchique humain à partir d'images tomodensitométriques. Dans une chaîne de traitements de structure classique par rapport à l'état de l'art, nous avons appliqué des méthodes de topologie et de géométrie discrète afin de résoudre des problèmes particuliers dans chaque étape du processus de l'analyse quantitative. Nous proposons une méthode robuste pour segmenter l'arbre bronchique à partir d'un ensemble de données tomographiques (CT). La méthode est basée sur un algorithme de fermeture de tunnels qui est utilisé comme outil pour réparer des images CT abîmées par les erreurs d'acquisition. Nous avons aussi proposé un algorithme qui sert à créer un modèle artificiel d'arbre bronchique. Ce modèle est utilisé pour la validation des algorithmes présentés dans cette thèse. Ensuite nous comparons la qualité des différents algorithmes en utilisant un ensemble de test constitué de fantômes (informatiques) et d'un ensemble de données CT réelles. Nous montrons que les méthodes récemment présentées dans le cadre des complexes cubiques, combinées avec les méthodes présentées dans cette thèse, permettent de surmonter des problèmes indiqués par la littérature et peuvent être un bon fondement pour l'implémentation future des systèmes de quantification automatique des particularités de l'arbre bronchique / 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

En informatique, les images sont numériques et donc composées de pixels en 2D et de voxels en 3D. Dans une scène virtuelle 3D, il est impossible de manipuler directement les objets comme des ensembles de voxels en raison du trop gros volume de données. Les objets sont alors polyédrisés, c’est-à-dire remplacés par une collection de facettes. Pour ce faire, il est primordial de savoir décider si un sous-ensemble de voxels peut être transformé en une facette dans la représentation polyédrique. Ce problème est appelé reconnaissance de plans discrets. Pour le résoudre, nous mettons en place un nouvel algorithme spécialement adapté pour les ensembles de voxels denses dans une boite englobante. Notre méthode atteint une complexité quasi-linéaire dans ce cas et s’avère efficace en pratique. En parallèle, nous nous intéressons à un problème algorithmique annexe intervenant dans notre méthode de reconnaissance de plans discrets. Il s’agit de calculer les deux enveloppes convexes des points de Z2 contenus dans un domaine vertical borné et situés de part et d’autre d’une droite quelconque. Nous proposons une méthode de complexité optimale et adaptative pour calculer ces enveloppes convexes. Nous présentons le problème de manière détournée : déterminer le nombre rationnel à dénominateur borné qui approxime au mieux un nombre réel donné. Nous établissons le lien entre ce problème numérique et son interprétation géométrique dans le plan. Enfin, nous proposons indépendamment un nouvel algorithme pour calculer l’épaisseur d’un ensemble de points dans le réseau Zd. Notre méthode est optimale en 2D et gloutonne mais efficace en dimension supérieure / In computer science, pictures are digital and so, they are composed of pixels in 2D or of voxels in 3D. In 3D virtual scenes, we cannot directly manipulate objects as sets of voxels because the data are too huge. As a result, the objects are transformed into polyhedra, i.e. collections of facets. For this, we must be able to decide if a subset of voxels can be replaced by a facet in the polyhedrisation. This problem is called digital plane recognition. To solve it, we design a new algorithm especially adapted for sets of voxels which are dense in a bounding box. Our method achieves a quasi-linear worst-case time complexity in this case and it is efficient in practice. In parallel, we study another algorithmic problem which occures in our digital plane recognition algorithm. It is computing the two convex hulls of grid points lying in a bounded vertical domain and located on either side of a straight line. We propose an optimal time complexity method to compute these convex hulls and which is also output sensitive. We present the problem in a different way : find the rational number of bounded denominator that best approximates a given real number. We establish the link between this numerical problem and geometry. Finally, we independently propose a new algorithm to compute the lattice width of a set of points in Zd. Our method is optimal in 2D and is greedy but efficent in higher dimension

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.

Représenter un objet géométrique complexe par un ensemble de primitives simples est une tâche souvent fondamentale, que ce soit pour la reconstruction et la réparation de données, ou encore pour faciliter la visualisation ou la manipulation des données. Le choix de la ou les primitives, ainsi que celui de la méthode d'approximation, impactent fortement les propriétés de la représentation de forme qui sera obtenue.Dans cette thèse, nous utilisons les boules comme seule primitive. Nous prenons ainsi un grand soin à décrire les unions finies de boules et leur structure. Pour cela, nous nous reposons sur les faisceaux de boules. En particulier, nous aboutissons à une description valide en toute dimension, sans hypothèse de position générale. En chemin, nous obtenons également plusieurs résultats portant sur les tests d'inclusion locale et globale dans une union de boules.Nous proposons également une nouvelle méthode d'approximation par union finie de boules, l'approximation par boules à (delta,epsilon)-près. Cette approche contraint l'union de boules à couvrir un sous-ensemble de la forme d'origine (précisément, un epsilon-érodé), tout en étant contenu dans un sur-ensemble de la forme (un delta-dilaté). En nous appuyant sur nos précédents résultats portant sur les unions de boules, nous démontrons plusieurs propriétés de ces approximations. Nous verrons ainsi que calculer une approximation par boules à (delta,epsilon)-près qui soit de cardinal minimum est un problème NP-complet. Pour des formes simples dans le plan, nous présentons un algorithme polynomial en temps et en espace qui permet de calculer ces approximations de cardinal minimum. Nous concluons par une généralisation de notre méthode d'approximation pour une plus large variété de sous-ensembles et sur-ensembles. / Describing a complex geometric shape with a set of simple primitives is often a fundamental task for shape reconstruction, visualization, analysis and manipulation. The type of primitives, as well as the choice of approximation scheme, both greatly impact the properties of the resulting shape representation.In this PhD, we focus on balls as primitives. Using pencils of balls, we carefully describe finite unions of balls and their structure. In particular, our description holds in all dimension without assuming general position. On our way, we also establish various results and tools to test local and global inclusions within these unions.We also propose a new approximation scheme by union of balls, the (delta,epsilon)-ball approximation. This scheme constrains the approximation to cover a core subset of the original shape (specifically, an epsilon-erosion), while being contained within a superset of the shape (a delta-dilation). Using our earlier results regarding finite unions of balls, we prove several properties of these approximations. We show that computing a cardinal minimum (delta,epsilon)-ball approximation is an NP-complete problem. For simple planar shapes however, we present a polynomial time and space algorithm that outputs a cardinal minimum approximation. We then conclude by generalizing the approximation scheme to a wider range of core subsets and bounding supersets.