Results 1  10
of
10
Topological inference via meshing
 IN SOCG: PROCEEDINGS OF THE 26TH ACM SYMPOSIUM ON COMPUTATIONAL GEOMETRY
, 2009
"... ..."
Dynamic wellspaced point sets
 In SCG ’10: Proceedings of the 26th Annual Symposium on Computational Geometry
, 2010
"... In a wellspaced point set the Voronoi cells all have bounded aspect ratio, i.e., the distance from the Voronoi site to the farthest point in the Voronoi cell divided by the distance to the nearest neighbor in the set is bounded by a small constant. Wellspaced point sets satisfy some important geom ..."
Abstract

Cited by 9 (5 self)
 Add to MetaCart
(Show Context)
In a wellspaced point set the Voronoi cells all have bounded aspect ratio, i.e., the distance from the Voronoi site to the farthest point in the Voronoi cell divided by the distance to the nearest neighbor in the set is bounded by a small constant. Wellspaced point sets satisfy some important geometric properties and yield quality Voronoi or simplicial meshes that can be important in scientific computations. In this paper, we consider the dynamic wellspaced pointsets problem, which requires computing the wellspaced superset of a dynamically changing input set, e.g., as points are inserted or deleted. We present a dynamic algorithm that allows inserting/deleting points into/from the input in worstcase O(log ∆) time, where ∆ is the geometric spread, a natural measure that is bounded by O(log n) when input points are represented by logsize words. We show that the runtime of the dynamic update algorithm is optimal in the worst case by showing that there exists inputs and modifications that require Ω(log ∆) Steiner points to be inserted to the output. Our algorithm generates sizeoptimal outputs: the resulting output sets are never more than a constant factor larger than the minimum size necessary. A preliminary implementation indicates that the algorithm is indeed fast in practice. To the best of our knowledge, this is the first time and sizeoptimal dynamic algorithm for wellspaced point sets.
Beating the Spread: TimeOptimal Point Meshing
, 2011
"... We present NetMesh, a new algorithm that produces a conforming Delaunay mesh for point sets in any fixed dimension with guaranteed optimal mesh size and quality. Our comparison based algorithm runs in time O(n log n+m), where n is the input size and m is the output size, and with constants depending ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
We present NetMesh, a new algorithm that produces a conforming Delaunay mesh for point sets in any fixed dimension with guaranteed optimal mesh size and quality. Our comparison based algorithm runs in time O(n log n+m), where n is the input size and m is the output size, and with constants depending only on the dimension and the desired element quality bounds. It can terminate early in O(n logn) time returning a O(n) size Voronoi diagram of a superset of P with a relaxed quality bound, which again matches the known lower bounds. The previous best results in the comparison model depended on the log of the spread of the input, the ratio of the largest to smallest pairwise distance among input points. We reduce this dependence to O(log n) by using a sequence of εnets to determine input insertion order in an incremental Voronoi diagram. We generate a hierarchy of wellspaced meshes and use these to show that the complexity of the Voronoi diagram stays linear in the number of points throughout the construction.
A New Approach to OutputSensitive Voronoi Diagrams
"... We describe a new algorithm for computing the Voronoi diagram of a set of n points in constantdimensional Euclidean space. The running time of our algorithm is O(f log n log ∆) where f is the output complexity of the Voronoi diagram and ∆ is the spread of the input, the ratio of largest to smallest ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
We describe a new algorithm for computing the Voronoi diagram of a set of n points in constantdimensional Euclidean space. The running time of our algorithm is O(f log n log ∆) where f is the output complexity of the Voronoi diagram and ∆ is the spread of the input, the ratio of largest to smallest pairwise distances. Despite the simplicity of the algorithm and its analysis, it improves on the state of the art for all inputs with polynomial spread and nearlinear output size. The key idea is to first build the Voronoi diagram of a superset of the input points using ideas from Voronoi refinement mesh generation. Then, the extra points are removed in a straightforward way that allows the total work to be bounded in terms of the output complexity, yielding the output sensitive bound. The removal only involves local flips and is inspired by kinetic data structures. 1
Dynamic Mesh Refinement
, 2007
"... Mesh refinement is the problem to produce a triangulation (typically Delaunay) of an input set of points augmented by Steiner points, such that every triangle or tetrahedron has good quality (no small angles). The requirement arises from the applications: in scientific computing and in graphics, mes ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
Mesh refinement is the problem to produce a triangulation (typically Delaunay) of an input set of points augmented by Steiner points, such that every triangle or tetrahedron has good quality (no small angles). The requirement arises from the applications: in scientific computing and in graphics, meshes are often used to discretely represent the value of a function over space. In addition to the quality requirement, the user often has input segments or polygons (generally, a piecewise linear complex) they would like see retained in the mesh; the mesh must respect these constraints. Finally, the mesh should be sizeconforming: the size of mesh elements should be related to a particular sizing function based on the distance between input features. The static meshing problem is increasingly wellunderstood: one can download software with provable guarantees that on reasonable input, the meshes will have good quality, will respect the input, and will be sizeconforming; more recently, these algorithms have started to come with optimal runtimes of O(n lg(L/s) +m), where L/s is the spread of the input. As a first result, I
Mathieu Desbrun
"... We present a practical approach to isotropic tetrahedral meshing of 3D domains bounded by piecewise smooth surfaces. Building upon recent theoretical and practical advances, our algorithm interleaves Delaunay refinement and mesh optimization to generate quality meshes that satisfy a set of userdefi ..."
Abstract
 Add to MetaCart
We present a practical approach to isotropic tetrahedral meshing of 3D domains bounded by piecewise smooth surfaces. Building upon recent theoretical and practical advances, our algorithm interleaves Delaunay refinement and mesh optimization to generate quality meshes that satisfy a set of userdefined criteria. This interleaving is shown to be more conservative in number of Steiner point insertions than refinement alone, and to produce higher quality meshes than optimization alone. A careful treatment of boundaries and their features is presented, offering a versatile framework for designing smoothly graded tetrahedral meshes. as they provide a blazingly fast approach to meshing most of the domain. While smooth surface boundaries can be efficiently handled with guaranteed minimum dihedral angles [Labelle and Shewchuk
Efficient Mesh Generation for Piecewise Linear Complexes
, 2009
"... The mesh generation problem is to output a set of tetrahedra that discretize an input geometry. The input is given as a piecewise linear complex (PLC), a set of points, lines, and polygons to which the output tetrahedra must conform. Additionally, a mesh generation algorithm must make guarantees on ..."
Abstract
 Add to MetaCart
The mesh generation problem is to output a set of tetrahedra that discretize an input geometry. The input is given as a piecewise linear complex (PLC), a set of points, lines, and polygons to which the output tetrahedra must conform. Additionally, a mesh generation algorithm must make guarantees on the quality and number of output tetrahedra. Downstream applications in scientific computing and visualization necessitate these guarantees on the mesh. Recent advances have led to provably correct algorithms for a number of input classes. Particular difficulties arise when the input contains creases, regions where input segments or polygons meet at acute angles. When the input is without creases, the mesh generation problem is better understood. Algorithms for such inputs exist with nearoptimal runtimes of O(n log ∆+m), where n and m are the size of the input and output, and ∆ is the ratio of largesttosmallest distances in the input geometry. The principle result of this thesis is to extend this result to the general case of piecewise linear complexes with creases. Correct algorithms to handle inputs with creases involve explicitly constructing
Interleaving Delaunay Refinement and . . .
"... We present a practical approach to isotropic tetrahedral meshing of 3D domains bounded by piecewise smooth surfaces. Building upon recent theoretical and practical advances, our algorithm interleaves Delaunay refinement and mesh optimization to generate quality meshes that satisfy a set of userdefi ..."
Abstract
 Add to MetaCart
We present a practical approach to isotropic tetrahedral meshing of 3D domains bounded by piecewise smooth surfaces. Building upon recent theoretical and practical advances, our algorithm interleaves Delaunay refinement and mesh optimization to generate quality meshes that satisfy a set of userdefined criteria. This interleaving is shown to be more conservative in number of Steiner point insertions than refinement alone, and to produce higher quality meshes than optimization alone. A careful treatment of boundaries and their features is presented, offering a versatile framework for designing smoothly graded tetrahedral meshes.
Beating the Spread: TimeOptimal Point Meshing (Extended Abstract)
, 2011
"... We present NetMesh, a new algorithm that produces ac onforming Delaunay mesh for point sets in any fixed dimension with guaranteed optimal mesh size and quality. Our comparisonbased algorithm runs in O(nlogn + m) time, where n is the input size and m is the output size, and with constants depending ..."
Abstract
 Add to MetaCart
(Show Context)
We present NetMesh, a new algorithm that produces ac onforming Delaunay mesh for point sets in any fixed dimension with guaranteed optimal mesh size and quality. Our comparisonbased algorithm runs in O(nlogn + m) time, where n is the input size and m is the output size, and with constants depending only on the dimension and the desired element quality. It can terminate early in O(nlogn) time returning a O(n) size Voronoi diagram of a superset of P, which again matches the known lower bounds. The previous best results in the comparison model depended on the log of the spread of the input, the ratio of the largest to smallest pairwise distance. We reduce this dependence to O(logn) by using a sequence of ǫnets to determine input insertion order into a incremental Voronoi diagram. We generate a hierarchy of wellspaced meshes and use these to show that the complexity of the Voronoi diagram stays linear in the number of points throughout the construction.