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

Made available in DSpace on 2014-12-17T15:47:49Z (GMT). No. of bitstreams: 1 IriaCSC.pdf: 926529 bytes, checksum: aa23848c0d07c85faded67f0781041fc (MD5) Previous issue date: 2008-08-04 / A 3D binary image is considered well-composed if, and only if, the union of the faces shared by the foreground and background voxels of the image is a surface in R3. Wellcomposed images have some desirable topological properties, which allow us to simplify and optimize algorithms that are widely used in computer graphics, computer vision and image processing. These advantages have fostered the development of algorithms to repair bi-dimensional (2D) and three-dimensional (3D) images that are not well-composed. These algorithms are known as repairing algorithms. In this dissertation, we propose two repairing algorithms, one randomized and one deterministic. Both algorithms are capable of making topological repairs in 3D binary images, producing well-composed images similar to the original images. The key idea behind both algorithms is to iteratively change the assigned color of some points in the input image from 0 (background)to 1 (foreground) until the image becomes well-composed. The points whose colors are changed by the algorithms are chosen according to their values in the fuzzy connectivity map resulting from the image segmentation process. The use of the fuzzy connectivity map ensures that a subset of points chosen by the algorithm at any given iteration is the one with the least affinity with the background among all possible choices / Uma imagem bin?ria 3D ? considerada bem-composta se, e somente se, a uni?o das faces compartilhadas pelos voxels do foreground e do background da referida imagem ? uma superf?cie em R3 . Imagens bem-compostas se beneficiam de propriedades topol?gicas desej?veis, as quais nos permitem simplificar e otimizar algoritmos amplamente usados na computa??o gr?fica, vis?o computacional e processamento de imagens. Estas vantagens t?m motivado o desenvolvimento de algoritmos para reparar imagens bi e tridimensionais que n?o sejam bem-compostas. Estes algoritmos s?o conhecidos como algoritmos reparadores. Nesta disserta??o, propomos dois algoritmos reparadores, um aleat?rio e um determin?stico. Ambos s?o capazes de fazer reparos topol?gicos em imagens bin?rias 3D, produzindo imagens bem-compostas similares ?s imagens originais. A id?ia fundamental por tr?s de ambos algoritmos ? mudar iterativamente a cor atribu?da de alguns pontos da imagem de entrada de 0 (background) para 1 (foreground) at? a imagem se tornar bem-composta. Os pontos cujas cores s?o mudadas pelos algoritmos s?o escolhidos de acordo com seus valores no mapa de conectividade fuzzy, resultante do processo de segmenta??o da imagem. O uso do mapa de conectividade fuzzy garante que um subconjunto dos pontos escolhidos pelo algoritmo em qualquer itera??o seja um com a menor afinidade com o background dentre todas as escolhas poss?veis