Topic
Delaunay triangulation
About: Delaunay triangulation is a research topic. Over the lifetime, 5816 publications have been published within this topic receiving 126615 citations. The topic is also known as: Delone triangulation.
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TL;DR: In this paper, a survey of acute and nonobtuse simplices and associated spatial partitions is presented, including path-simplices, the generalization of right triangles to higher dimensions.
Abstract: This paper surveys some results on acute and nonobtuse simplices and associated spatial partitions. These partitions are relevant in numerical mathematics, including piecewise polynomial approximation theory and the finite element method. Special attention is paid to a basic type of nonobtuse simplices called path-simplices, the generalization of right triangles to higher dimensions. In addition to applications in numerical mathematics, we give examples of the appearance of acute and nonobtuse simplices in other areas of mathematics.
92 citations
TL;DR: A hybrid approach to accurate quantification of vascular structures from magnetic resonance angiography (MRA) images using level set methods and deformable geometric models constructed with 3-D Delaunay triangulation is presented.
Abstract: The aim of this paper is to present a hybrid approach to accurate quantification of vascular structures from magnetic resonance angiography (MRA) images using level set methods and deformable geometric models constructed with 3-D Delaunay triangulation. Multiple scale filtering based on the analysis of local intensity structure using the Hessian matrix is used to effectively enhance vessel structures with various diameters. The level set method is then applied to automatically segment vessels enhanced by the filtering with a speed function derived from enhanced MRA images. Since the goal of this paper is to obtain highly accurate vessel borders, suitable for use in fluid flow simulations, in a subsequent step, the vessel surface determined by the level set method is triangulated using 3-D Delaunay triangulation and the resulting surface is used as a parametric deformable model. Energy minimization is then performed within a variational setting with a first-order internal energy; the external energy is derived from 3-D image gradients. Using the proposed method, vessels are accurately segmented from MRA data.
92 citations
09 Jan 2001
TL;DR: The main ingredient of the algorithm is a novel refinement technique which systematically forbids the formation of shivers, a family of bad elements that none of the previous known algorithms can cleanly remove, especially near the domain boundary.
Abstract: A triangular mesh in 3D is a decomposition of a given geometric domain into tetrahedra. The mesh is well-shaped if the aspect ratio of every of its tetrahedra is bounded from above by a constant. It is Delaunay if the interior of the circum-sphere of each of its tetrahedra does not contain any other mesh vertices. Generating a well-shaped Delaunay mesh for any 3D domain has been a long term outstanding problem. In this paper, we present an efficient 3D Delaunay meshing algorithm that mathematically guarantees the well-shape quality of the mesh, if the domain does not have acute angles. The main ingredient of our algorithm is a novel refinement technique which systematically forbids the formation of shivers, a family of bad elements that none of the previous known algorithms can cleanly remove, especially near the domain boundary — needless to say, that our algorithm ensure that there is no sliver near the boundary of the domain.
91 citations
TL;DR: Bregman divergences as discussed by the authors allow one to define information-theoretic Voronoi diagrams in statistical parametric spaces based on the relative entropy of distributions and establish correspondences between those diagrams, and show how to compute them efficiently.
Abstract: The Voronoi diagram of a finite set of objects is a fundamental geometric structure that subdivides the embedding space into regions, each region consisting of the points that are closer to a given object than to the others. We may define various variants of Voronoi diagrams depending on the class of objects, the distance function and the embedding space. In this paper, we investigate a framework for defining and building Voronoi diagrams for a broad class of distance functions called Bregman divergences. Bregman divergences include not only the traditional (squared) Euclidean distance but also various divergence measures based on entropic functions. Accordingly, Bregman Voronoi diagrams allow one to define information-theoretic Voronoi diagrams in statistical parametric spaces based on the relative entropy of distributions. We define several types of Bregman diagrams, establish correspondences between those diagrams (using the Legendre transformation), and show how to compute them efficiently. We also introduce extensions of these diagrams, e.g., k-order and k-bag Bregman Voronoi diagrams, and introduce Bregman triangulations of a set of points and their connection with Bregman Voronoi diagrams. We show that these triangulations capture many of the properties of the celebrated Delaunay triangulation.
91 citations
TL;DR: The proposed method connects minutiae using a Delaunay triangulation and analyzes the relative position and orientation of each minutia with respect to its neighbors obtained by the triangle structure.
Abstract: We present a new technique for fingerprint minutiae matching. The proposed method connects minutiae using a Delaunay triangulation and analyzes the relative position and orientation of each minutia with respect to its neighbors obtained by the triangle structure. Due to non-linear deformations, we admit a certain degree of triangle deformation. If rotations and translations are present, the triangle structure does not change consistently. Two fingerprints are considered matching, if their triangle structures are similar according the neighbor relationship. The algorithm performance are evaluated on a public domain database.
91 citations