This dissertation is a uniïfication of an analysis-based approach and the traditional topological-based approach to Poincaré duality. We examine the role of the discrete Hodge star operator in proving and in realizing the Poincaré duality isomorphism (between cohomology and homology in complementary degrees) in a cellular setting without reference to a dual cell complex. More specifically, we provide a proof of this version of Poincaré duality over R via the simplicial discrete Hodge star defined by Scott Wilson in [19] without referencing a dual cell complex. We also express the Poincaré duality isomorphism over both R and Z in terms of this discrete operator. Much of this work is dedicated to extending these results to a cubical setting, via the introduction of a cubical version of Whitney forms. A cubical setting provides a place for Robin Forman's complex of nontraditional differential forms, defined in [7], in the uniïfication of analytic and topological perspectives discussed in this dissertation. In particular, we establish a ring isomorphism (on the cohomology level) between Forman's complex of differential forms with his exterior derivative and product and a complex of cubical cochains with the discrete coboundary operator and the standard cubical cup product. / Ph. D.

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.

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.

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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.

Orientador: Hélio Pedrini / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Computação / Made available in DSpace on 2018-08-24T00:14:20Z (GMT). No. of bitstreams: 1 Silva_RicardoDutrada_D.pdf: 13549105 bytes, checksum: 3d49e5116a70a72601ba4cc3b3c85762 (MD5) Previous issue date: 2013 / Resumo: A Teoria de Morse é importante para o estudo da topologia em funções escalares como elevação de terrenos e dados provenientes de simulações físicas, a qual relaciona a topologia de uma função com seus pontos críticos. A teoria contínua foi adaptada para dados discretos através de construções como os complexos de Morse-Smale e o complexo discreto de Morse. Complexos de Morse têm sido aplicados em processamento de imagens, no entanto, ainda existem desafios envolvendo algoritmos e considerações práticas para a computação e modelagem dos complexos para imagens. Complexos de Morse podem ser usados como um meio de definir a conexão entre pontos de interesse em imagens. Normalmente, pontos de interesse são considerados como elementos independentes descritos por informação local. Tal abordagem apresenta limitações uma vez que informação local pode não ser suficiente para descrever certas regiões da imagem. Pontos de mínimo e máximo são comumente utilizados como pontos de interesse em imagens, os quais podem ser obtidos a partir dos complexos de Morse, bem como sua conectividade no espaço de imagem. Esta tese apresenta uma abordagem dirigida por algoritmos e estruturas de dados para computar o complexo de Morse discreto em imagens bidimensionais. A construção é ótima e permite fácil manipulação do complexo. Resultados teóricos e experimentais são apresentados para mostrar que o método é eficaz. Experimentos realizados incluem a computação de homologia persistente e hierarquias de complexos sobre dados de elevação de terrenos. Outra contribuição é a proposição de um operador topológico, chamado Contexto Local de Morse, computado sobre complexos de Morse, para extrair vizinhanças de pontos de interesse para explorar a informação estrutural de imagens. O contexto local de Morse é usado no desenvolvimento de um algoritmo que auxilia a redução do número de casamentos incorretos entre pontos de interesse e na obtenção de uma medida de confiança para tais correspondências. A abordagem proposta é testada em pares de imagens sintéticas e de imagens subaquáticas, para as quais métodos existentes podem obter muitas correspondências incorretas / Abstract: The Morse theory is important for studying the topology of scalar functions such as elevation of terrains and data from physical simulations, which relates the topology of a function to critical points. The smooth theory has been adapted to discrete data through constructions such as the Morse-Smale complexes and the discrete Morse complex. Morse complexes have been applied to image processing, however, there are still challenges involving algorithms and practical considerations for computation and modeling of the complexes. Morse complexes can be used as means of defining the connectedness of interest points in images. Usually, interest points are considered as independent elements described by local information. Such an approach has its limitations since local information may not suffice for describing certain image regions. Minimum and maximum points are widely used as interest points in images, which can be obtained from Morse complexes, as well as their connectivity in the image space. This thesis presents an algorithmic and data structure driven approach to computing the discrete Morse complex of 2-dimensional images. The construction is optimal and allows easy manipulation of the complex. Theoretical and applied results are presented to show the effectiveness of the method. Applied experiments include the computation of persistent homology and hierarchies of complexes over elevation terrain data. Another contribution is the proposition of a topological operator, called Local Morse Context (LMC), computed over Morse complexes, for extracting neighborhoods of interest points to explore the structural information in images. The LMC is used in the development of a matching algorithm, which helps reducing the number of incorrect matches between images and obtaining a confidence measure of whether a correspondence is correct or incorrect. The approach is tested in synthetic and challenging underwater stereo pairs of images, for which available methods may obtain many incorrect correspondences / Doutorado / Ciência da Computação / Doutor em Ciência da Computação

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.

The thesis deals with the concept of level of detail and its use in 3D GIS. The aim of this work is to design a data structure that will allow continuous rendering of discrete 3D building models with different levels of geometric detail, which were created by generalization method based on mathematical morphology approach. The proposed solution creates links of corresponding geometric primitives of models at different level of detail at first and then reconstructs them using the extrusion method. The data structure created in this way is able to generate and render any model, including intermediate models, which are represented as slices through the data structure across the axis of geometric detail.

Dans cette thèse, nous étudions les problèmes de reconstruction statique et dynamique à partir de vues multiples et texturation, en s'appuyant sur des applications réelles et pratiques. Nous proposons trois méthodes de reconstruction destinées à l'estimation d'une représentation d'une scène statique/dynamique à partir d'un ensemble d'images/vidéos. Nous considérons ensuite le problème de texturation multi-vues en se concentrant sur la qualité visuelle de rendu..