Discrete Laplace operators are ubiquitous in applications spanning geometric modeling to simulation. For robustness and efficiency, many applications require discrete operators that retain key structural properties inherent to the continuous setting. Building on the smooth setting, we present a set of natural properties for discrete Laplace operators for triangular surface meshes. We prove an important theoretical limitation: discrete Laplacians cannot satisfy all natural properties; retroactively, this explains the diversity of existing discrete Laplace operators. Finally, we present a family of operators that includes and extends well-known and widely-used operators.

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We present algorithms to produce Delaunay meshes from arbitrary triangle meshes by edge flipping and geometry-preserving refinement and prove their correctness. In particular we show that edge flipping serves to reduce mesh surface area, and that a poorly sampled input mesh may yield unflippable edges necessitating refinement to ensure a Delaunay mesh output. Multiresolution Delaunay meshes can be obtained via constrained mesh decimation. We further examine the usefulness of trading off the geometry-preserving feature of our algorithm with the ability to create fewer triangles. We demonstrate the performance of our algorithms through several experiments.

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In this paper, we are concerned with Delaunay triangulations of the vertex set of a piecewise flat (pwf) surface. We first propose the notion of well-formed Voronoi diagrams and establish a precise dual relationship between them and proper Delaunay triangulations on pwf surfaces. Then we provide an algorithm which, given any input manifold triangle mesh, constructs a Delaunay mesh: a manifold triangle mesh whose edges form an intrinsic Delaunay triangulation of its vertex set. Rather than relying on a geodesic Delaunay triangulation on the input mesh, our algorithm swaps the physical mesh edges based on the locally Delaunay criterion. We prove that when a physical edge that is not locally Delaunay is swapped, the surface area of the mesh is reduced. In order to ensure a proper Delaunay triangulation, some new vertices may need to be introduced, leading to a refinement scheme, and we detail the cases involved.

We introduce Hodge-optimized triangulations (HOT), a family of well-shaped primal-dual pairs of complexes designed for fast and accurate computations in computer graphics. Previous work most commonly employs barycentric or circumcentric duals; while barycentric duals guarantee that the dual of each simplex lies within the simplex, circumcentric duals are often preferred due to the induced orthogonality between primal and dual complexes. We instead promote the use of weighted duals (“power diagrams”). They allow greater flexibility in the location of dual vertices while keeping primal-dual orthogonality, thus providing a valuable extension to the usual choices of dual by only adding one additional scalar per primal vertex. Furthermore, we introduce a family of functionals on pairs of complexes that we derive from bounds on the errors induced by diagonal Hodge stars, commonly used in discrete computations. The minimizers of these functionals, called HOT meshes, are shown to be generalizations of Centroidal Voronoi Tesselations and Optimal Delaunay Triangulations, and to provide increased accuracy and flexibility for a variety of computational purposes.

Abstract. The exterior surface of the brain is characterized by a juxtaposition of crests and troughs that together form a folding pattern. The majority of the deformations that occur in the normal course of adult human development result in folds changing their length or width. Current statistical shape analysis methods cannot easily discriminate between these two cases. Using discrete exterior calculus and Tikhonov regularization, we develop a method to estimate a dense orientation field in the tangent space of a surface described by a triangulated mesh, in the direction of its folds. We then use this orientation field to distinguish between shape differences in the direction parallel to folds and those in the direction across them. We test the method quantitatively on synthetic data and qualitatively on a database consisting of segmented cortical surfaces of 92 healthy subjects and 97 subjects with Alzheimer’s disease. The method estimates the correct fold directions and also indicates that the healthy and diseased subjects are distinguished by shape differences that are in the direction perpendicular to the underlying hippocampi, a finding which is consistent with the neuroscientific literature. These results demonstrate the importance of direction specific computational methods for shape analysis. 1

We undertake a study of the local properties of 2-Gabriel meshes: manifold triangle meshes each of whose faces has an open Euclidean diametric ball that contains no mesh vertices. We show that, under mild constraints on the dihedral angles, such meshes are Delaunay meshes: the open geodesic circumdisk of each face contains no mesh vertex. The analysis is done by means of the Delaunay edge flipping algorithm and it reveals the details of the distinction between these two mesh structures. In particular we observe that the obstructions which prohibit the existence of Gabriel meshes as homeomorphic representatives of smooth surfaces do not hinder the construction of Delaunay meshes. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry