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71
Parallel Delaunay Refinement: Algorithms and Analyses
 In Proceedings, 11th International Meshing Roundtable
, 2002
"... In this paper, we analyze the complexity of natural parallelizations of Delaunay refinement methods for mesh generation. The parallelizations employ a simple strategy: at each iteration, they choose a set of "independent" points to insert into the domain, and then update the Delaunay triangulation. ..."
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Cited by 23 (3 self)
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In this paper, we analyze the complexity of natural parallelizations of Delaunay refinement methods for mesh generation. The parallelizations employ a simple strategy: at each iteration, they choose a set of "independent" points to insert into the domain, and then update the Delaunay triangulation. We show that such a set of independent points can be constructed efficiently in parallel and that the number of iterations needed is O(log²(L/s)), where L is the diameter of the domain, and s is the smallest edge in the output mesh. In addition, we show that the insertion of each independent set of points can be realized sequentially by Ruppert's method in two dimensions and Shewchuk's in three dimensions. Therefore, our parallel Delaunay refinement methods provide the same element quality and mesh size guarantees as the sequential algorithms in both two and three dimensions. For quasiuniform meshes, such as those produced by Chew's method, we show that the number of iterations can be reduced to O(log(L/s)). To the best of our knowledge, these are the first provably polylog(L/s) parallel time Delaunay meshing algorithms that generate wellshaped meshes of size optimal to within a constant.
Exposing Resource Tradeoffs in RegionBased Communication Abstractions for Sensor Networks
, 2003
"... We argue that communication abstractions for wireless sensor networks should expose the tradeoff between accuracy and resource usage, allowing applications to adapt to changing network conditions and tune energy and bandwidth requirements. We describe abstract regions, a family of spatial operators ..."
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Cited by 20 (0 self)
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We argue that communication abstractions for wireless sensor networks should expose the tradeoff between accuracy and resource usage, allowing applications to adapt to changing network conditions and tune energy and bandwidth requirements. We describe abstract regions, a family of spatial operators that capture local communication within regions of the network, which may be defined in terms of radio connectivity, geographic location, or other properties of nodes. Abstract regions provide feedback on the quality of collective operations, and expose an interface for tuning resource consumption. We present the implementation of abstract regions in the TinyOS programming environment, as well as preliminary results demonstrating their use for building adaptive sensor network applications.
Delaunay Triangulations of Imprecise Points in Linear Time after Preprocessing
, 2008
"... An assumption of nearly all algorithms in computational geometry is that the input points are given precisely, so it is interesting to ask what is the value of imprecise information about points. We show how to preprocess a set of n disjoint unit disks in the plane in O(n log n) time so that if one ..."
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Cited by 20 (5 self)
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An assumption of nearly all algorithms in computational geometry is that the input points are given precisely, so it is interesting to ask what is the value of imprecise information about points. We show how to preprocess a set of n disjoint unit disks in the plane in O(n log n) time so that if one point per disk is specified with precise coordinates, the Delaunay triangulation can be computed in linear time. From the Delaunay, one can obtain the Gabriel graph and a Euclidean minimum spanning tree; it is interesting to note the roles that these two structures play in our algorithm to quickly compute the Delaunay.
Meshing 3D domains bounded by piecewise smooth surfaces
"... This report provides an algorithm to mesh 3D domains bounded by piecewise smooth surfaces. The algorithm may handle as well subdivisions of the domain forming non manifold surfaces. The boundaries and constraints are assumed to be described as a complex formed by a set of vertices, a set of curved ..."
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Cited by 18 (1 self)
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This report provides an algorithm to mesh 3D domains bounded by piecewise smooth surfaces. The algorithm may handle as well subdivisions of the domain forming non manifold surfaces. The boundaries and constraints are assumed to be described as a complex formed by a set of vertices, a set of curved segments and a set of surface patches. Each curve segment is assumed to be a piece of a closed smooth curves and each surface patch is assumed to be included in a smooth surface without boundary. The meshing algorithm is a Delaunay refinement and it uses the notion of restricted Delaunay triangulation to approximate the input curved segments and surfaces patches. The algorithm is shown to end up with a set of vertices whose restricted Delaunay triangulation to any input feature forms an homeomorphic and accurate approximation of this feature. The algorithm also provides guarantees on the size and shape of facets approximating the input surface patches and on the size and shape of the tetrahedra in the domain. In its actual state the algorithm suffers from a severe angular restriction on input constraints. It basically assumes that linear subspaces which are tangent to distinct input features on a common point form angles measuring at least 90 degrees.
SUKUMAR N.: Maximum entropy coordinates for arbitrary polytopes
 Proceedings of SGP 2008
, 2008
"... Barycentric coordinates can be used to express any point inside a triangle as a unique convex combination of the triangle’s vertices, and they provide a convenient way to linearly interpolate data that is given at the vertices of a triangle. In recent years, the ideas of barycentric coordinates and ..."
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Cited by 12 (3 self)
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Barycentric coordinates can be used to express any point inside a triangle as a unique convex combination of the triangle’s vertices, and they provide a convenient way to linearly interpolate data that is given at the vertices of a triangle. In recent years, the ideas of barycentric coordinates and barycentric interpolation have been extended to arbitrary polygons in the plane and general polytopes in higher dimensions, which in turn has led to novel solutions in applications like mesh parameterization, image warping, and mesh deformation. In this paper we introduce a new generalization of barycentric coordinates that stems from the maximum entropy principle. The coordinates are guaranteed to be positive inside any planar polygon, can be evaluated efficiently by solving a convex optimization problem with Newton’s method, and experimental evidence indicates that they are smooth inside the domain. Moreover, the construction of these coordinates can be extended to arbitrary polyhedra and higherdimensional polytopes.
Geometry completion and detail generation by texture synthesis. The Visual Computer 21
, 2005
"... We present a novel method for patching holes in polygonal meshes and synthesizing surface with details based on existing geometry. The most novel feature of our proposed method is to transform the 3D geometry synthesis problem into a 2D domain by parameterizing surfaces and solve this problem in tha ..."
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Cited by 11 (0 self)
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We present a novel method for patching holes in polygonal meshes and synthesizing surface with details based on existing geometry. The most novel feature of our proposed method is to transform the 3D geometry synthesis problem into a 2D domain by parameterizing surfaces and solve this problem in that domain. We then derive local geometry gradient images that encode intrinsic local geometry properties, which are invariant to object translation and rotation. The 3D geometry of holes is then reconstructed from synthesized local gradient images. This method can be extended to execute other mesh editing operations such as geometry detail transfer or synthesis. The resulting major benefits of performing geometry synthesis in 2D are more flexible and robust control, better leverage of the wealth of current 2D image completion methods and greater efficiency. 1.
Generalized delaunay mesh refinement: From scalar to parallel
 IN PROCEEDINGS OF THE 15TH INTERNATIONAL MESHING ROUNDTABLE
, 2006
"... The contribution of the current paper is threefold. First, we generalize the existing sequential point placement strategies for guaranteed quality Delaunay refinement: instead of a specific position for a new point, we derive a selection disk inside the circumdisk of a poor quality triangle. We pr ..."
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Cited by 9 (3 self)
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The contribution of the current paper is threefold. First, we generalize the existing sequential point placement strategies for guaranteed quality Delaunay refinement: instead of a specific position for a new point, we derive a selection disk inside the circumdisk of a poor quality triangle. We prove that any point placement algorithm that inserts a point inside the selection disk of a poor quality triangle will terminate and produce a sizeoptimal mesh. Second, we extend our theoretical foundation for the parallel Delaunay refinement. Our new parallel algorithm can be used in conjunction with any sequential point placement strategy that chooses a point within the selection disk. Third, we implemented our algorithm in C++ for shared memory architectures and present the experimental results. Our data show that even on workstations with a few cores, which are now in common use, our implementation is significantly faster the best sequential counterpart.
Motion planning with dynamics by a synergistic combination of layers of planning
 IEEE Trans. on Robotics
, 2010
"... Abstract—Effectively incorporating robot dynamics into motion planning has been an active area of research in robotics over the last decade. Toward this goal, this work proposes a novel multilayered approach, termed Synergistic Combination of Layers of Planning (SyCLoP), that synergistically combin ..."
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Cited by 9 (5 self)
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Abstract—Effectively incorporating robot dynamics into motion planning has been an active area of research in robotics over the last decade. Toward this goal, this work proposes a novel multilayered approach, termed Synergistic Combination of Layers of Planning (SyCLoP), that synergistically combines highlevel discrete planning and samplingbased motion planning. Initially, SyCLoP uses a workspace decomposition to construct a discrete model of the motionplanning problem. At each iteration, highlevel planning, which draws from research in AI and logic, searches the discrete model for a feasible plan, i.e., a sequence of decomposition regions that can effectively guide motion planning as it extends a tree during the search for a solution trajectory. In return, information gathered by motion planning, such as progress made in connecting decomposition regions, is fed back to highlevel planning. In this way, the planning layers in SyCLoP are not independent, but work in tandem to compute in future iterations increasingly feasible highlevel plans and quickly grow the tree toward the goal. The synergistic combination of highlevel discrete planning and samplingbased motion planning allows SyCLoP to effectively solve challenging motionplanning problems with dynamics. Simulation experiments with highdimensional dynamical models of ground and flying vehicles demonstrate computational speedups of up to two orders of magnitude over stateoftheart motion planners. In addition, SyCLoP is wellsuited for hybrid systems, which move beyond continuous models by employing discrete logic to instantaneously modify the dynamics. I.
Meshing of Surfaces
 EFFECTIVE COMPUTATIONAL GEOMETRY FOR CURVES AND SURFACES, (JEANDANIEL BOISSONNAT, MONIQUE TEILLAUD, EDITORS)
, 2007
"... Meshing is the process of computing, for a given surface, a representation consisting of pieces of simple surface patches, like triangles. This survey discusses all currently known surface (and curve) meshing algorithms that come with correctness and quality guarantees. ..."
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Cited by 8 (1 self)
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Meshing is the process of computing, for a given surface, a representation consisting of pieces of simple surface patches, like triangles. This survey discusses all currently known surface (and curve) meshing algorithms that come with correctness and quality guarantees.
A.W.: Anisotropic LaplaceBeltrami eigenmaps: Bridging Reeb graphs and skeletons
 In: Proceedings of Mathematical Methods in Biomedical Image Analysis (MMBIA
, 2008
"... In this paper we propose a novel approach of computing skeletons of robust topology for simply connected surfaces with boundary by constructing Reeb graphs from the eigenfunctions of an anisotropic LaplaceBeltrami operator. Our work brings together the idea of Reeb graphs and skeletons by incorpora ..."
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Cited by 8 (2 self)
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In this paper we propose a novel approach of computing skeletons of robust topology for simply connected surfaces with boundary by constructing Reeb graphs from the eigenfunctions of an anisotropic LaplaceBeltrami operator. Our work brings together the idea of Reeb graphs and skeletons by incorporating a fluxbased weight function into the LaplaceBeltrami operator. Based on the intrinsic geometry of the surface, the resulting Reeb graph is pose independent and captures the global profile of surface geometry. Our algorithm is very efficient and it only takes several seconds to compute on neuroanatomical structures such as the cingulate gyrus and corpus callosum. In our experiments, we show that the Reeb graphs serve well as an approximate skeleton with consistent topology while following the main body of conventional skeletons quite accurately. 1.