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An Optimal Transport Approach to Robust Reconstruction and Simplification of 2D Shapes
, 2011
"... We propose a robust 2D shape reconstruction and simplification algorithm which takes as input a defectladen point set with noise and outliers. We introduce an optimaltransport driven approach where the input point set, considered as a sum of Dirac measures, is approximated by a simplicial complex ..."
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Cited by 12 (3 self)
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We propose a robust 2D shape reconstruction and simplification algorithm which takes as input a defectladen point set with noise and outliers. We introduce an optimaltransport driven approach where the input point set, considered as a sum of Dirac measures, is approximated by a simplicial complex considered as a sum of uniform measures on 0 and 1simplices. A finetocoarse scheme is devised to construct the resulting simplicial complex through greedy decimation of a Delaunay triangulation of the input point set. Our method performs well on a variety of examples ranging from line drawings to grayscale images, with or without noise, features, and boundaries.
Computing SelfSupporting Surfaces by Regular Triangulation
"... and their corresponding power cells are colored in orange. Top right: initial selfsupporting mesh. Spikes appear due to extremely small reciprocal areas. Bottom right: applying our smoothing scheme (5 iterations) improves mesh quality. The power diagrams (black) show how power cell area is distribu ..."
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Cited by 10 (1 self)
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and their corresponding power cells are colored in orange. Top right: initial selfsupporting mesh. Spikes appear due to extremely small reciprocal areas. Bottom right: applying our smoothing scheme (5 iterations) improves mesh quality. The power diagrams (black) show how power cell area is distributed more evenly. Masonry structures must be compressively selfsupporting; designing such surfaces forms an important topic in architecture as well as a challenging problem in geometric modeling. Under certain conditions, a surjective mapping exists between a power diagram, defined by a set of 2D vertices and associated weights, and the reciprocal diagram that characterizes the force diagram of a discrete selfsupporting network. This observation lets us define a new and convenient parameterization for the space of selfsupporting networks. Based on it and the discrete geometry of this design space, we present novel geometry processing methods including surface smoothing and remeshing which significantly reduce the magnitude of force densities and homogenize their distribution.
New Bounds on the Size of Optimal Meshes
"... The theory of optimal size meshes gives a method for analyzing the output size (number of simplices) of a Delaunay refinement mesh in terms of the integral of a sizing function over the input domain. The input points define a maximal such sizing function called the feature size. This paper presents ..."
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Cited by 4 (1 self)
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The theory of optimal size meshes gives a method for analyzing the output size (number of simplices) of a Delaunay refinement mesh in terms of the integral of a sizing function over the input domain. The input points define a maximal such sizing function called the feature size. This paper presents a way to bound the feature size integral in terms of an easy to compute property of a suitable ordering of the point set. The key idea is to consider the pacing of an ordered point set, a measure of the rate of change in the feature size as points are added one at a time. In previous work, Miller et al. showed that if an ordered point set has pacing φ, then the number of vertices in an optimal mesh will be O(φ d n), where d is the input dimension. We give a new analysis of this integral showing that the output size is only Θ(n + nlogφ). The new analysis tightens bounds from several previous results and provides matching lower bounds. Moreover, it precisely characterizes inputs that yield outputs of size O(n).
Weighted triangulations for geometry processing
"... In this paper, we investigate the use of weighted triangulations as discrete, augmented approximations of surfaces for digital geometry processing. By incorporating a scalar weight per mesh vertex, we introduce a new notion of discrete metric that defines an orthogonal dual structure for arbitrary t ..."
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Cited by 3 (0 self)
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In this paper, we investigate the use of weighted triangulations as discrete, augmented approximations of surfaces for digital geometry processing. By incorporating a scalar weight per mesh vertex, we introduce a new notion of discrete metric that defines an orthogonal dual structure for arbitrary triangle meshes and thus extends weighted Delaunay triangulations to surface meshes. We also present alternative characterizations of this primaldual structure (through combinations of angles, areas, and lengths) and, in the process, uncover closedform expressions of mesh energies that were previously known in implicit form only. Finally, we demonstrate how weighted triangulations provide a faster and more robust approach to a series of geometry processing applications, including the generation of wellcentered meshes, selfsupporting surfaces, and sphere packing.
Frame Field Singularity Correction for Automatic Hexahedralization
"... Abstract—We present an automatic hexahedralization tool, based on a systematic treatment that removes some of the singularities that would lead to degenerate volumetric parameterization. Such singularities could be abundant in automatically generated frame fields guiding the interior and boundary la ..."
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Abstract—We present an automatic hexahedralization tool, based on a systematic treatment that removes some of the singularities that would lead to degenerate volumetric parameterization. Such singularities could be abundant in automatically generated frame fields guiding the interior and boundary layouts of the hexahedra in an all hexahedral mesh. We first give the mathematical definitions of the inadmissible singularities prevalent in frame fields, including newly introduced surface singularity types. We then give a practical framework for adjusting singularity graphs by automatically modifying the rotational transition of frames between charts (cells of a tetrahedral mesh for the volume) to resolve the issues detected in the internal and boundary singularity graph. After applying an additional resmoothing of the frame field with the modified transition conditions, we cut the volume into a topologically trivial domain, with the original topology encoded by the selfintersections of the boundary of the domain, and solve a mixed integer problem on this domain for a global parameterization. Finally, a properly connected hexahedral mesh is constructed from the integer isosurfaces of (u, v, w) in the parameterization. We demonstrate the applicability of the method on complex shapes, and discuss its limitations. Index Terms—automatic hexahedral meshing, frame field, field singularity, volumetric parameterization 1
TABLE OF CONTENTS
, 2001
"... This work was supported by the U.S. Army Medical Research and Materiel Command under Contract No. DAMD 1799C9001. The views, opinions and/or findings contained in this report are those of the authors and should not be construed as an official Department of the Army position, policy, or decision u ..."
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Cited by 1 (0 self)
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This work was supported by the U.S. Army Medical Research and Materiel Command under Contract No. DAMD 1799C9001. The views, opinions and/or findings contained in this report are those of the authors and should not be construed as an official Department of the Army position, policy, or decision unless so designated by other documentation. General Methodology for Business Case Analysis
Earth Mover’s Distances on Discrete Surfaces
"... We introduce a novel method for computing the earth mover’s distance (EMD) between probability distributions on a discrete surface. Rather than using a large linear program with a quadratic number of variables, we apply the theory of optimal transportation and pass to a dual differential formulati ..."
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Cited by 1 (0 self)
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We introduce a novel method for computing the earth mover’s distance (EMD) between probability distributions on a discrete surface. Rather than using a large linear program with a quadratic number of variables, we apply the theory of optimal transportation and pass to a dual differential formulation with linear scaling. After discretization using finite elements (FEM) and development of an accompanying optimization method, we apply our new EMD to problems in graphics and geometry processing. In particular, we uncover a class of smooth distances on a surface transitioning from a purely spectral distance to the geodesic distance between points; these distances also can be extended to the volume inside and outside the surface. A number of additional applications of our machinery to geometry problems in graphics are presented.
Automatic Frame Field Guided Hexahedral Mesh Generation Technique Report
"... We present a hexahedralization method based on a systematic treatment that eradicates the singularities that would lead to degeneration in the volumetric parameterization. Such singularities could be abundant in automatically generated frame fields guiding the interior and boundary layouts of the ..."
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Cited by 1 (0 self)
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We present a hexahedralization method based on a systematic treatment that eradicates the singularities that would lead to degeneration in the volumetric parameterization. Such singularities could be abundant in automatically generated frame fields guiding the interior and boundary layouts of the hexahedra in a purely hexahedral mesh. We first give the mathematical definitions of different types of surface singularities, which are involved in making the surface of the output hexahedral mesh conform to the input model while optimizing mesh quality. We then give a practical framework for adjusting singularity graphs by automatically modifying the rotational transition of frames between charts (cells of a tetrahedral mesh for the volume) to resolve the issues detected in the internal and boundary singularity graph. After applying an additional resmoothing of the frame field with the modified transition conditions, we cut the volume into a domain with a spherelike topology automatically, by combining the tetrahedra. Finally, we efficiently solve a mixed integer problem on this domain for a global parameterization, which follows the frame field and respects the transitions across the chart boundaries, and creates a properly connected hexahedral mesh using integer grids of the parameterization. 1
Caltech Mathieu Desbrun Caltech
"... We present a novel approach for the analysis and design of selfsupporting simplicial masonry structures. A finitedimensional formulation of their compressive stress field is derived, offering a new interpretation of thrust networks through numerical homogenization theory. We further leverage geomet ..."
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We present a novel approach for the analysis and design of selfsupporting simplicial masonry structures. A finitedimensional formulation of their compressive stress field is derived, offering a new interpretation of thrust networks through numerical homogenization theory. We further leverage geometric properties of the resulting force diagram to identify a set of reduced coordinates characterizing the equilibrium of simplicial masonry. We finally derive computational formfinding tools that improve over previous work in efficiency, accuracy, and scalability.
Parametrization of Generalized PrimalDual Triangulations
"... Motivated by practical numerical issues in a number of modeling and simulation problems, we introduce the notion of a compatible dual complex to a primal triangulation, such that a simplicial mesh and its compatible dual complex (made out of convex cells) form what we call a primaldual triangulatio ..."
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Motivated by practical numerical issues in a number of modeling and simulation problems, we introduce the notion of a compatible dual complex to a primal triangulation, such that a simplicial mesh and its compatible dual complex (made out of convex cells) form what we call a primaldual triangulation. Using algebraic and computational geometry results, we show that compatible dual complexes exist only for a particular type of triangulation known as weakly regular. We also demonstrate that the entire space of primaldual triangulations, which extends the well known (weighted) Delaunay/Voronoi duality, has a convenient, geometric parametrization. We finally discuss how this parametrization may play an important role in discrete optimization problems such as optimal mesh generation, as it allows us to easily explore the space of primaldual structures along with some important subspaces. 1