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38
Delaunay Refinement Algorithms for Triangular Mesh Generation
 Computational Geometry: Theory and Applications
, 2001
"... Delaunay refinement is a technique for generating unstructured meshes of triangles for use in interpolation, the finite element method, and the finite volume method. In theory and practice, meshes produced by Delaunay refinement satisfy guaranteed bounds on angles, edge lengths, the number of tria ..."
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Cited by 100 (0 self)
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Delaunay refinement is a technique for generating unstructured meshes of triangles for use in interpolation, the finite element method, and the finite volume method. In theory and practice, meshes produced by Delaunay refinement satisfy guaranteed bounds on angles, edge lengths, the number of triangles, and the grading of triangles from small to large sizes. This article presents an intuitive framework for analyzing Delaunay refinement algorithms that unifies the pioneering mesh generation algorithms of L. Paul Chew and Jim Ruppert, improves the algorithms in several minor ways, and most importantly, helps to solve the difficult problem of meshing nonmanifold domains with small angles.
Representation and Detection of Deformable Shapes
 PAMI
, 2004
"... We describe some techniques that can be used to represent and detect deformable shapes in images. The main di#culty with deformable template models is the very large or infinite number of possible nonrigid transformations of the templates. This makes the problem of finding an optimal match of a ..."
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Cited by 78 (4 self)
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We describe some techniques that can be used to represent and detect deformable shapes in images. The main di#culty with deformable template models is the very large or infinite number of possible nonrigid transformations of the templates. This makes the problem of finding an optimal match of a deformable template to an image incredibly hard. Using a new representation for deformable shapes we show how to e#ciently find a global optimal solution to the nonrigid matching problem. The representation is based on the description of objects using triangulated polygons. Our matching algorithm can minimize a large class of energy functions, making it applicable to a wide range of problems. We present experimental results of detecting shapes in medical images and images of natural scenes. Our method does not depend on initialization and is very robust, yielding good matches even in images with high clutter.
Multiresolution shape deformations for meshes with dynamic connectivity
 In Computer Graphics Forum (Proc. Eurographics 2000
"... Multiresolution shape representation is a very effective way to decompose surface geometry into several levels of detail. Geometric modeling with such representations enables flexible modifications of the global shape while preserving the detail information. Many schemes for modeling with multiresol ..."
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Cited by 76 (10 self)
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Multiresolution shape representation is a very effective way to decompose surface geometry into several levels of detail. Geometric modeling with such representations enables flexible modifications of the global shape while preserving the detail information. Many schemes for modeling with multiresolution decompositions based on splines, polygonal meshes and subdivision surfaces have been proposed recently. In this paper we modify the classical concept of multiresolution representation by no longer requiring a global hierarchical structure that links the different levels of detail. Instead we represent the detail information implicitly by the geometric difference between independent meshes. The detail function is evaluated by shooting rays in normal direction from one surface to the other without assuming a consistent tesselation. In the context of multiresolution shape deformation, we propose a dynamic mesh representation which adapts the connectivity during the modification in order to maintain a prescribed mesh quality. Combining the two techniques leads to an efficient mechanism which enables extreme deformations of the global shape while preventing the mesh from degenerating. During the deformation, the detail is reconstructed in a natural and robust way. The key to the intuitive detail preservation is a transformation map which associates points on the original and the modified geometry with minimum distortion. We show several examples which demonstrate the effectiveness and robustness of our approach including the editing of multiresolution models and models with texture. 1.
2000, ‘Delaunay triangulations and Voronoi diagrams for Riemannian manifolds
 ACM Symposium on Computational Geometry
"... For a sufficiently dense set of points in any closed Riemannian manifold, we prove that a unique Delannay triangulation exists. This triangulation has the same properties as in Euclidean space. Algorithms for constructing these triangulations will also be described. 1. ..."
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Cited by 59 (2 self)
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For a sufficiently dense set of points in any closed Riemannian manifold, we prove that a unique Delannay triangulation exists. This triangulation has the same properties as in Euclidean space. Algorithms for constructing these triangulations will also be described. 1.
Isotropic remeshing of surfaces: A local parameterization approach
 In Proceedings of 12th International Meshing Roundtable
, 2003
"... We present a method for isotropic remeshing of arbitrary genus surfaces. The method is based on a mesh adaptation process, namely, a sequence of local modifications performed on a copy of the original mesh, while referring to the original mesh geometry. The algorithm has three stages. In the first s ..."
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Cited by 39 (4 self)
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We present a method for isotropic remeshing of arbitrary genus surfaces. The method is based on a mesh adaptation process, namely, a sequence of local modifications performed on a copy of the original mesh, while referring to the original mesh geometry. The algorithm has three stages. In the first stage the required number or vertices are generated by iterative simplification or refinement. The second stage performs an initial vertex partition using an areabased relaxation method. The third stage achieves precise isotropic vertex sampling prescribed by a given density function on the mesh. We use a modification of Lloyd’s relaxation method to construct a weighted centroidal Voronoi tessellation of the mesh. We apply these iterations locally on small patches of the mesh that are parameterized into the 2D plane. This allows us to handle arbitrary complex meshes with any genus and any number of boundaries. The efficiency and the accuracy of the remeshing process is achieved using a patchwise parameterization technique.
What Is a Good Linear Finite Element?  Interpolation, Conditioning, Anisotropy, and Quality Measures
 In Proc. of the 11th International Meshing Roundtable
, 2002
"... When a mesh of simplicial elements (triangles or tetrahedra) is used to form a piecewise linear approximation of a function, the accuracy of the approximation depends on the sizes and shapes of the elements. In finite element methods, the conditioning of the stiffness matrices also depends on the si ..."
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Cited by 33 (0 self)
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When a mesh of simplicial elements (triangles or tetrahedra) is used to form a piecewise linear approximation of a function, the accuracy of the approximation depends on the sizes and shapes of the elements. In finite element methods, the conditioning of the stiffness matrices also depends on the sizes and shapes of the elements. This article explains the mathematical connections between mesh geometry, interpolation errors, discretization errors, and stiffness matrix conditioning. These relationships are expressed by error bounds and element quality measures that determine the fitness of a triangle or tetrahedron for interpolation or for achieving low condition numbers. Unfortunately, the quality measures for these purposes do not fully agree with each other; for instance, small angles are bad for matrix conditioning but not for interpolation or discretization. The upper and lower bounds on interpolation error and element stiffness matrix conditioning given here are tighter than those usually seen in the literature, so the quality measures are likely to be unusually precise indicators of element fitness. Bounds are included for anisotropic cases, wherein long, thin elements perform better than equilateral ones. Surprisingly, there are circumstances wherein interpolation, conditioning, and discretization error are each best served by elements of different aspect ratios or orientations.
Isotropic Surface Remeshing
, 2003
"... This paper proposes a new method for isotropic remeshing of triangulated surface meshes. Given a triangulated surface mesh to be resampled and a userspecified density function defined over it, we first distribute the desired number of samples by generalizing error diffusion, commonly used in image ..."
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Cited by 31 (3 self)
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This paper proposes a new method for isotropic remeshing of triangulated surface meshes. Given a triangulated surface mesh to be resampled and a userspecified density function defined over it, we first distribute the desired number of samples by generalizing error diffusion, commonly used in image halftoning, to work directly on mesh triangles and feature edges. We then use the resulting sampling as an initial configuration for building a weighted centroidal Voronoi tessellation in a conformal parameter space, where the specified density function is used for weighting. We finally create the mesh by lifting the corresponding constrained Delaunay triangulation from parameter space. A precise control over the sampling is obtained through a flexible design of the density function, the latter being possibly lowpass filtered to obtain a smoother gradation. We demonstrate the versatility of our approach through various remeshing examples.
Numerical Conformal Mapping Using CrossRatios And Delaunay Triangulation
 SIAM J. Sci. Comput
"... We propose a new algorithm for computing the Riemann mapping of the unit disk to a polygon, also known as the SchwarzChristoffel transformation. The new algorithm, CRDT (for crossratios of the Delaunay triangulation), is based on crossratios of the prevertices, and also on crossratios of quadril ..."
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Cited by 24 (5 self)
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We propose a new algorithm for computing the Riemann mapping of the unit disk to a polygon, also known as the SchwarzChristoffel transformation. The new algorithm, CRDT (for crossratios of the Delaunay triangulation), is based on crossratios of the prevertices, and also on crossratios of quadrilaterals in a Delaunay triangulation of the polygon. The CRDT algorithm produces an accurate representation of the Riemann mapping even in the presence of arbitrary long, thin regions in the polygon, unlike any previous conformal mapping algorithm. We believe that CRDT solves all difficulties with crowding and global convergence, although these facts depend on conjectures that we have so far not been able to prove. We demonstrate convergence with computational experiments. The Riemann mapping has applications in twodimensional potential theory and mesh generation. We demonstrate CRDT on problems in long, thin regions in which no other known algorithm can perform comparably.