Anisotropic Triangulations via Discrete Riemannian Voronoi Diagrams

The construction of anisotropic triangulations is desirable for various applications, such as the numerical solving of partial differential equations and the representation of surfaces in graphics. To solve this notoriously difficult problem in a practical way, we introduce the discrete Riemannian Voronoi diagram, a discrete structure that approximates the Riemannian Voronoi diagram. This structure has been implemented and was shown to lead to good triangulations in R^2 and on surfaces embedded in R^3 as detailed in our experimental companion paper. In this paper, we study theoretical aspects of our structure. Given a finite set of points P in a domain Omega equipped with a Riemannian metric, we compare the discrete Riemannian Voronoi diagram of P to its Riemannian Voronoi diagram. Both diagrams have dual structures called the discrete Riemannian Delaunay and the Riemannian Delaunay complex. We provide conditions that guarantee that these dual structures are identical. It then follows from previous results that the discrete Riemannian Delaunay complex can be embedded in Omega under sufficient conditions, leading to an anisotropic triangulation with curved simplices. Furthermore, we show that, under similar conditions, the simplices of this triangulation can be straightened

An Analytical Representation of the 2d Generalized Balanced Power Diagram

Tessellations are an important tool to model the microstructure of cellular and polycrystalline materials. Classical tessellation models include the Voronoi diagram and Laguerre tessellation whose cells are polyhedra. Due to the convexity of their cells, those models may be too restrictive to describe data that includes possibly anisotropic grains with curved boundaries. Several generalizations exist. The cells of the generalized balanced power diagram are induced by elliptic distances leading to more diverse structures. So far, methods for computing the generalized balanced power diagram are restricted to discretized versions in the form of label images. In this work, we derive an analytic representation of the vertices and edges of the generalized balanced power diagram in 2d. Based on that, we propose a novel algorithm to compute the whole diagram

An obstruction to Delaunay triangulations in Riemannian manifolds

Delaunay has shown that the Delaunay complex of a finite set of points $P$ of Euclidean space $\mathbb{R}^m$ triangulates the convex hull of $P$, provided that $P$ satisfies a mild genericity property. Voronoi diagrams and Delaunay complexes can be defined for arbitrary Riemannian manifolds. However, Delaunay's genericity assumption no longer guarantees that the Delaunay complex will yield a triangulation; stronger assumptions on $P$ are required. A natural one is to assume that $P$ is sufficiently dense. Although results in this direction have been claimed, we show that sample density alone is insufficient to ensure that the Delaunay complex triangulates a manifold of dimension greater than 2.Comment: This is a revision and extension of a note that appeared as an appendix in the (otherwise unpublished) report arXiv:1303.649

Discretized Riemannian Delaunay Triangulations

Anisotropic meshes are desirable for various applications, such as the numerical solving of partial differential equations and graphics.In this report, we introduce an algorithm to compute discrete approximations of Riemannian Voronoi diagrams on 2-manifolds.This is not straightforward because geodesics, shortest paths between points, and therefore distances cannot, in general, be computed exactly.We give conditions that guarantee that our discrete Riemannian Voronoi diagram is combinatorially equivalent to the exact Riemannian Voronoi diagram.This allows us to build upon recent theoretical results on Riemannian Delaunay triangulations, and guarantee that the dual of our discrete Riemannian Voronoi diagram is an embedded triangulation using both approximate geodesics and straight edges.Our implementation employs recent developments in the numerical computation of geodesic distances.We observe that, in practice, our discrete Voronoi Diagram is correct in a far wider range of settings than our theoretical bounds imply.Both the theoretical guarantees on the approximation of the Voronoi diagram and the implementation are new and provides a step towards the practical application of Riemannian Delaunay triangulations.Les maillages anisotropes sont désirables pour de nombreuses applications, telles que la résolution numérique d'équations aux dérivées partielles ou la visualisation.Dans ce rapport, nous présentons un algorithme qui permet de calculer une approximation discrète d'un diagramme de Voronoi Riemannien sur une 2-variété.Il s'agit d'une tache complexe car ce diagramme est basé sur la notion de courbe géodésique, qui ne peut en général pas être calculée de manière exacte.Nous donnons dans ce rapport des conditions qui garantissent que notre diagramme de Voronoi Riemannien discret est combinatoirement équivalent au diagramme de Voronoi Riemannien exact.Ceci nous permet ensuite d'utiliser des résultats récents sur les triangulations de Delaunay Riemanniennes pour garantir le fait que le dual de notre diagramme de Voronoi Riemannien discret est une triangulation plongée, à la fois en utilisant des arêtes géodésiques et des arêtes droites.Notre implémentation est basée sur de récentes avancées dans le calcul numérique des distances géodésiques.Nous observons en pratique que notre diagramme de Voronoi Riemannien discret est correct dans des conditions beaucoup moins contraignantes que ce que notre théorie implique.Les garanties théoriques et l'approximation du diagramme de Voronoi sont nouvelles et sont une étape de plus vers une utilisation pratique des triangulations de Delaunay Riemanniennes

Anisotropic triangulations via discrete Riemannian Voronoi diagrams

International audienceThe construction of anisotropic triangulations is desirable for various applications, such as the numerical solving of partial differential equations and the representation of surfaces in graphics. To solve this notoriously difficult problem in a practical way, we introduce the discrete Riemannian Voronoi diagram, a discrete structure that approximates the Riemannian Voronoi diagram. This structure has been implemented and was shown to lead to good triangulations in R2 and on surfaces embedded in R3 as detailed in our experimental companion paper. In this paper, we study theoretical aspects of our structure. Given a finite set of points P in a domain Ω equipped with a Riemannian metric, we compare the discrete Riemannian Voronoi diagram of P to its Riemannian Voronoi diagram. Both diagrams have dual structures called the discrete Riemannian Delaunay and the Riemannian Delaunay complex. We provide conditions that guarantee that these dual structures are identical. It then follows from previous results that the discrete Riemannian Delaunay complex can be embedded in Ω under sufficient conditions, leading to an anisotropic triangulation with curved simplices. Furthermore, we show that, under similar conditions, the simplices of this triangulation can be straightened

Um novo arcabouço para análise de qualidade de imagens de impressões digitais de alta resolução

Orientador: Neucimar Jerônimo LeiteTese (doutorado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: A falta de robustez referente à degradação de qualidade de conjuntos de características extraídas de padrões de cristas-e-vales, contidos na epiderme dos dedos humanos, é uma das questões em aberto na análise de imagens de impressões digitais, com implicações importantes em problemas de segurança, privacidade e fraude de identificação. Neste trabalho, introduzimos uma nova metodologia para analisar a qualidade de conjuntos de características de terceiro nível em imagens de impressões digitais representados, aqui, por poros de transpiração. A abordagem sugerida leva em conta a interdependência espacial entre as características consideradas e algumas transformações básicas envolvendo a manipulação de processos pontuais e sua análise a partir de ferramentas anisotrópicas. Foram propostos dois novos algoritmos para o cálculo de índices de qualidade que se mostraram eficazes na previsão da qualidade da correspondência entre as impressões e na definição de pesos de filtragem de características de baixa qualidade a ser empregado num processo de identificação. Para avaliar experimentalmente o desempenho destes algoritmos e suprir a ausência de uma base de dados com níveis de qualidade controlados, criamos uma base de dados com diferentes recursos de configuração e níveis de qualidade. Neste trabalho, propusemos ainda um método para reconstruir imagens de fase da impressão digital a partir de um dado conjunto de coordenadas de poros. Para validar esta idéia sob uma perspectiva de identificação, consideramos conjuntos de minúcias presentes nas imagens reconstruídas, inferidas a partir das configurações de poros, e associamos este resultado ao problema típico de casamento de impressões digitaisAbstract: The lack of robustness against the quality degradation affecting sets of features extracted from patterns of epidermal ridges on our fingers is one of the open issues in fingerprint image analysis, with implications for security, privacy, and identity fraud. In this doctorate work we introduce a new methodology to analyze the quality of sets of level-3 fingerprint features represented by pores. Our approach takes into account the spatial interrelationship between the considered features and some basic transformations involving point process and anisotropic analysis. We propose two new quality index algorithms, which have proved to be effective as a matcher predictor and in the definition of weights filtering out low-quality features from an identification process. To experimentally assess the performance of these algorithms and supply the absence of a feature-based controlled quality database in the biometric community, we created a dataset with features configurations containing different levels of quality. In this work, we also proposed a method for reconstructing phase images from a given set of pores coordinates. To validate this idea from an identification perspective, we considered the set of minutia present in the reconstructed images and inferred from the pores configurations and used this result in fingerprint matchingsDoutoradoCiência da ComputaçãoDoutor em Ciência da Computação01-P-3951/2011147050/2012-0CAPESCNP

Delaunay Triangulation of Manifolds

International audienceWe present an algorithm for producing Delaunay triangulations of manifolds. The algorithm can accommodate abstract manifolds that are not presented as submanifolds of Euclidean space. Given a set of sample points and an atlas on a compact manifold, a manifold Delaunay complex is produced for a perturbed point set provided the transition functions are bi-Lipschitz with a constant close to 1, and the original sample points meet a local density requirement; no smoothness assumptions are required. If the transition functions are smooth, the output is a triangulation of the manifold. The output complex is naturally endowed with a piecewise flat metric which, when the original manifold is Riemannian, is a close approximation of the original Riemannian metric. In this case the output complex is also a Delaunay triangulation of its vertices with respect to this piecewise flat metric