Results 1  10
of
65
Computing Contour Trees in All Dimensions
, 1999
"... We show that contour trees can be computed in all dimensions by a simple algorithm that merges two trees. Our algorithm extends, simplifies, and improves work of Tarasov and Vyalyi and of van Kreveld et al. ..."
Abstract

Cited by 131 (8 self)
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We show that contour trees can be computed in all dimensions by a simple algorithm that merges two trees. Our algorithm extends, simplifies, and improves work of Tarasov and Vyalyi and of van Kreveld et al.
Restricted Delaunay triangulations and normal cycle
 In: ACM Symposium on Computational Geometry
, 2003
"... We address the problem of curvature estimation from sampled smooth surfaces. Building upon the theory of normal cycles, we derive a de�nition of the curvature tensor for polyhedral surfaces. This de�nition consists in a very simple and new formula. When applied to a polyhedral approximation of a smo ..."
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Cited by 113 (2 self)
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We address the problem of curvature estimation from sampled smooth surfaces. Building upon the theory of normal cycles, we derive a de�nition of the curvature tensor for polyhedral surfaces. This de�nition consists in a very simple and new formula. When applied to a polyhedral approximation of a smooth surface, it yields an ef�cient and reliable curvature estimation algorithm. Moreover, we bound the difference between the estimated curvature and the one of the smooth surface in the case of restricted Delaunay triangulations. Categories and Subject Descriptors F.2.2 [Analysis of algorithms and problem complexity]: [Geometrical problems and computations, Computations on discrete
Featurebased surface parameterization and texture mapping
 ACM Transactions on Graphics
, 2005
"... and precomputation of solid textures. The stretch caused by a given parameterization determines the sampling rate on the surface. In this article, we present an automatic parameterization method for segmenting a surface into patches that are then flattened with little stretch. Many objects consist o ..."
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Cited by 71 (5 self)
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and precomputation of solid textures. The stretch caused by a given parameterization determines the sampling rate on the surface. In this article, we present an automatic parameterization method for segmenting a surface into patches that are then flattened with little stretch. Many objects consist of regions of relatively simple shapes, each of which has a natural parameterization. Based on this observation, we describe a threestage featurebased patch creation method for manifold surfaces. The first two stages, genus reduction and feature identification, are performed with the help of distancebased surface functions. In the last stage, we create one or two patches for each feature region based on a covariance matrix of the feature’s surface points. To reduce stretch during patch unfolding, we notice that stretch is a 2 × 2 tensor, which in ideal situations is the identity. Therefore, we use the GreenLagrange tensor to measure and to guide the optimization process. Furthermore, we allow the boundary vertices of a patch to be optimized by adding scaffold triangles. We demonstrate our featurebased patch creation and patch unfolding methods for several textured models. Finally, to evaluate the quality of a given parameterization, we describe an imagebased error measure that takes into account stretch, seams, smoothness, packing efficiency, and surface visibility.
Hierarchical Morse Complexes for Piecewise Linear 2Manifolds
, 2001
"... We present algorithms for constructing a hierarchy of increasingly coarse Morse complexes that decompose a piecewise linear 2manifold. While Morse complexes are defined only in the smooth category, we extend the construction to the piecewise linear category by ensuring structural integrity and simu ..."
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Cited by 47 (5 self)
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We present algorithms for constructing a hierarchy of increasingly coarse Morse complexes that decompose a piecewise linear 2manifold. 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. Keywords Computational topology, PL manifolds, Morse theory, topological persistence, hierarchy, algorithms, implementation, terrains 1. INTRODUCTION In this paper, we define the Morse complex decomposing a piecewise linear 2manifold and present algorithms for constructing and simplifying this complex. 1.1 Motivation Physical simulation problems often start with a space and measurements over this space. If the measurements are scalar values, we talk about a height function of that space. We use this name throughout the paper, although the functions can ...
On the Curvature of Piecewise Flat Spaces
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 1984
"... We consider analogs of the LipschitzKilling curvatures of smooth Riemannian manifolds for piecewise flat spaces. In the special case of scalar curvature, the definition is due to T. Regge considerations in this spirit date back to J. Steiner. We show that if a piecewise flat space approximates a s ..."
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Cited by 47 (2 self)
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We consider analogs of the LipschitzKilling curvatures of smooth Riemannian manifolds for piecewise flat spaces. In the special case of scalar curvature, the definition is due to T. Regge considerations in this spirit date back to J. Steiner. We show that if a piecewise flat space approximates a smooth space in a suitable sense, then the corresponding curvatures are close in the sense of measures.
On the convergence of metric and geometric properties of polyhedral surfaces
 GEOMETRIAE DEDICATA
, 2005
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Vines and vineyards by updating persistence in linear time
 In “Proc. 22nd
, 2006
"... Persistent homology is the mathematical core of recent work on shape, including reconstruction, recognition, and matching. Its pertinent information is encapsulated by a pairing of the critical values of a function, visualized by points forming a diagram in the plane. The original algorithm in [10] ..."
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Cited by 29 (9 self)
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Persistent homology is the mathematical core of recent work on shape, including reconstruction, recognition, and matching. Its pertinent information is encapsulated by a pairing of the critical values of a function, visualized by points forming a diagram in the plane. The original algorithm in [10] computes the pairs from an ordering of the simplices in a triangulation and takes worstcase time cubic in the number of simplices. The main result of this paper is an algorithm that maintains the pairing in worstcase linear time per transposition in the ordering. A sideeffect of the algorithm’s analysis is an elementary proof of the stability of persistence diagrams [7] in the special case of piecewiselinear functions. We use the algorithm to compute 1parameter families of diagrams which we apply to the study of protein folding trajectories. Categories and Subject Descriptors F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems—Geometrical problems and
Progressive Simplification of Tetrahedral Meshes Preserving All Isosurface Topologies
 Computer Graphics Forum
, 2003
"... In this paper, we propose a novel technique for constructing multiple levels of a tetrahedral volume dataset while preserving the topologies of all isosurfaces embedded in the data. Our simplification technique has two major phases. In the segmentation phase, we segment the volume data into topolo ..."
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Cited by 28 (2 self)
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In this paper, we propose a novel technique for constructing multiple levels of a tetrahedral volume dataset while preserving the topologies of all isosurfaces embedded in the data. Our simplification technique has two major phases. In the segmentation phase, we segment the volume data into topologicalequivalence regions, that is, the subvolumes within each of which all isosurfaces have the same topology. In the simplification phase, we simplify each topologicalequivalence region independently, one by one, by collapsing edges from the smallest to the largest errors (within the userspecified error tolerance, for a given error metrics), and ensure that we do not collapse edges that may cause an isosurfacetopology change. We also avoid creating a tetrahedral cell of negative volume (i.e., avoid the foldover problem). In this way, we guarantee to preserve all isosurface topologies in the entire simplification process, with a controlled geometric error bound. Our method also involves several additional novel ideas, including using the Morse theory and the implicit fully augmented contour tree, identifying types of edges that are not allowed to be collapsed, and developing efficient techniques to avoid many unnecessary or expensive checkings, all in an integrated manner. The experiments show that all the resulting isosurfaces preserve the topologies, and have good accuracies in their geometric shapes. Moreover, we obtain nice datareduction rates, with competitively fast running times.
Discrete OneForms on Meshes and Applications to 3D Mesh Parameterization
 Journal of CAGD
, 2006
"... We describe how some simple properties of discrete oneforms directly relate to some old and new results concerning the parameterization of 3D mesh data. Our first result is an easy proof of Tutte's celebrated "springembedding" theorem for planar graphs, which is widely used for parameterizing mesh ..."
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Cited by 25 (1 self)
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We describe how some simple properties of discrete oneforms directly relate to some old and new results concerning the parameterization of 3D mesh data. Our first result is an easy proof of Tutte's celebrated "springembedding" theorem for planar graphs, which is widely used for parameterizing meshes with the topology of a disk as a planar embedding with a convex boundary. Our second result generalizes the first, dealing with the case where the mesh contains multiple boundaries, which are free to be nonconvex in the embedding. We characterize when it is still possible to achieve an embedding, despite these boundaries being nonconvex. The third result is an analogous embedding theorem for meshes with genus 1 (topologically equivalent to the torus). Applications of these results to the parameterization of meshes with disk and toroidal topologies are demonstrated. Extensions to higher genus meshes are discussed.