Showing papers by "Herbert Edelsbrunner published in 2001"

Geometry and Topology for Mesh Generation

Abstract: 1. Delaunay triangulations 2. Triangle meshes 3. Combinatorial topology 4. Surface simplification 5. Delaunay tetrahedrizations 6. Tetrahedron meshes 7. Open problems.

Hierarchical morse complexes for piecewise linear 2-manifolds

Abstract: We present algorithms for constructing a hierarchy of increasingly coa rse Morse complexes that decompose a piecewise linear 2-manifold While Morse complexes are defined only in the smooth category, we extend the construction to the piecewise linear category by ensuring structural integrity and simulating differentiability We then simplify Morse complexes by cancelling pairs of critical points in order of increasing persistence

Dynamic skin triangulation

Abstract: This paper describes an algorithm for maintaining an approximating triangulation of a deforming surface in R3. The triangulation adapts dynamically to changing shape, curvature, and topology of the surface.

Manufacturing methods and systems for rapid production of hearing-aid shells

Abstract: Methods, apparatus and computer program products provide efficient techniques for designing and printing shells of hearing-aid devices with a high degree of quality assurance and reliability and with a reduced number of manual and time consuming production steps and operations. These techniques also preferably provide hearing-aid shells having internal volumes that can approach a maximum allowable ratio of internal volume relative to external volume. These high internal volumes facilitate the inclusion of hearing-aid electrical components having higher degrees of functionality and/or the use of smaller and less conspicuous hearing-aid shells. A preferred method includes operations to generate a watertight digital model of a hearing-aid shell by thickening a three-dimensional digital model of a shell surface in a manner that eliminates self-intersections and results in a thickened model having an internal volume that is a high percentage of an external volume of the model. This thickening operation preferably includes nonuniformly thickening the digital model of a shell surface about a directed path that identifies a location of an undersurface hearing-aid vent. This directed path may be drawn on the shell surface by a technician (e.g., audiologist) or computer-aided design operator, for example. Operations are then preferably performed to generate a digital model of an undersurface hearing-aid vent in the thickened model of the shell surface, at a location proximate the directed path.

An Experimental Study of Sliver Exudation.

Abstract: We present results on a two-step improvement of mesh quality in three-dimensional Delaunay triangulations. The first step refines the triangulation by inserting sinks and eliminates tetrahedra with large circumradius over shortest edge length ratio. The second step assigns weights to the vertices to eliminate slivers. Our experimental findings pro- vide evidence for the practical effectiveness of sliver exu- dation.

Dynamic Skin Triangulation

Abstract: This paper describes an algorithm for maintaining an approximating triangulation of a deforming surface in R 3 . The surface is the envelope of an infinite family of spheres defined and controlled by a finite collection of weighted points. The triangulation adapts dynamically to changing shape, curvature, and topology of the surface.

Computing and comprehending topology: persistence and hierarchical morse complexes

Abstract: algebra is based on abstracting from algebra its core properties, and studying algebra in terms of those properties. 3.1.1 Binary Operations We begin by extending the concept of addition. For a review of sets, see section 2.1.1. Definition 3.1 (Binary Operation) A binary operation ∗ on a set S is a rule that assigns to each ordered pair (a, b) of elements of S some element in S. It must assign a single element to each pair (otherwise it’s not defined or not well-defined, for assigning zero or more than one elements, respectively), and it must assign an element in S for the operation to be closed.

Sink-insertion for mesh improvement

Abstract: We propose sink-insertion as a new technique to improve the mesh quali ty of Delaunay triangulations. We compare it with the conventional circumcenter-insertion technique under three scheduling regimes: incremental, in blocks, and in parallel. Justification for sink-insertion is given in terms of mesh quality, numerical robustness, running time, and ease of parallelization.

Methods, apparatus and computer program products for modeling three-dimensional colored objects

Abstract: Methods, apparatus and computer program products can generate light weight but highly realistic and accurate colored models of three-dimensional colored objects. The colored model may be generated from a second plurality of points that define a coarse digital representation of the surface and at least one texture map containing information derived from a first plurality of colored points that define a fine digital representation of the surface. This derivation is achieved by mapping points within the texture map to the fine digital representation of the three-dimensional surface. Colored scan data may be used to construct the fine digital representation as a triangulated surface (i.e., triangulation) using a wrapping operation. This triangulated surface may be a two-manifold with or without nonzero boundary and the colored scan data may constitute raw point data with each datum comprising three real numbers (x-,y-, z-coordinates) providing geometric information and three integer numbers (r-,g-,b-values) providing color information. Operations are then performed to create the coarse digital representation from the fine digital representation and also preferably create a plurality a texture maps from the fine and coarse digital representations. One map may contain color information and another map may recover geometric detail lost in the simplification process associated with generating the coarse digital representation from the fine digital representation. An additional map may also be generated that corrects for differences in directions of normal vectors associated with the coarse and fine digital representations.

Shape space from deformation

Abstract: The construction of shape spaces is studied from a mathematical and a computational viewpoint. A program is outlined reducing the problem to four tasks: the representation of geometry, the canonical deformation of geometry, the measuring of distance in shape space, and the selection of base shapes. The technical part of this paper focuses on the second task: the specification of a deformation mixing two or more shapes in continuously changing proportions.

Computing Linking Numbers of a Filtration

Abstract: We develop fast algorithms for computing the linking number of a simplicial complex within a filtration. We give experimental results in applying our work toward the detection of non-trivial tangling in biomolecules, modeled as alpha complexes.

Design and analysis of planar shape deformation

Abstract: Shape deformation refers to the continuous change of one geometric object to another. We develop a software tool for planning, analyzing and visualizing deformations between two shapes in R^2. The deformation is generated automatically without any user intervention or specification of feature correspondences. A unique property of the tool is the explicit availability of a two-dimensional shape space, which can be used for designing the deformation either automatically by following constraints and objectives or manually by drawing deformation paths.

180 Wrapped Tubes

Abstract: The 180 models collected in this paper are produced by sampling and wrapping point sets on tubes. The surfaces are represented as triangulated 2-manifolds and available as stl- files from the author's web site atwww.cs.duke.edu/ edels. Each tube is obtained by thickening a circle or a smooth torus knot, and for some we use the degrees of freedom in the thickening process to encode meaningful information, such as curvature or torsion.

Tetrahedral mesh improvement, algorithms and experiments

Abstract: We propose a two-step procedure to improve the quality of a three-dimensional Delaunay mesh. The first step refines the mesh by inserting sinks to remove skinny tetrahedra with large ratio between circumradius and shortest edge length. The second step eliminates remaining slivers by replacing Euclidean distance with weighted distance. We strive to maintain a balance between rigorous proofs of theoretical ideas, robust implementations of algorithms, and their practical evaluations by computational experiments. Sink insertion is a variant of Delaunay refinement, which improves mesh quality by inserting special circumcenters, called sinks, to remove poor quality tetrahedra. We perform computational experiments to compare sink insertion with circumcenter insertion in terms of mesh quality, running time, and ease of parallelization. The persistent appearance of slivers in large three-dimensional Delaunay meshes has been reported as early as 1985. They persist even after treatment with the Delaunay refinement algorithm. Cheng et al. proposed to remove slivers by assigning real weights to the points and change the Delaunay to the weighted Delaunay mesh. This is referred to as the sliver exudation algorithm. Their theoretical bound on the achieved minimum mesh quality is a constant that is positive but exceedingly small. We perform computational experiments to testify the practical effectiveness of sliver exudation.