Results 1  10
of
136
Voronoi diagrams  a survey of a fundamental geometric data structure
 ACM COMPUTING SURVEYS
, 1991
"... This paper presents a survey of the Voronoi diagram, one of the most fundamental data structures in computational geometry. It demonstrates the importance and usefulness of the Voronoi diagram in a wide variety of fields inside and outside computer science and surveys the history of its development. ..."
Abstract

Cited by 560 (5 self)
 Add to MetaCart
This paper presents a survey of the Voronoi diagram, one of the most fundamental data structures in computational geometry. It demonstrates the importance and usefulness of the Voronoi diagram in a wide variety of fields inside and outside computer science and surveys the history of its development. The paper puts particular emphasis on the unified exposition of its mathematical and algorithmic properties. Finally, the paper provides the first comprehensive bibliography on Voronoi diagrams and related structures.
Applications of Random Sampling in Computational Geometry, II
 Discrete Comput. Geom
, 1995
"... We use random sampling for several new geometric algorithms. The algorithms are "Las Vegas," and their expected bounds are with respect to the random behavior of the algorithms. These algorithms follow from new general results giving sharp bounds for the use of random subsets in geometric algorithms ..."
Abstract

Cited by 396 (12 self)
 Add to MetaCart
We use random sampling for several new geometric algorithms. The algorithms are "Las Vegas," and their expected bounds are with respect to the random behavior of the algorithms. These algorithms follow from new general results giving sharp bounds for the use of random subsets in geometric algorithms. These bounds show that random subsets can be used optimally for divideandconquer, and also give bounds for a simple, general technique for building geometric structures incrementally. One new algorithm reports all the intersecting pairs of a set of line segments in the plane, and requires O(A + n log n) expected time, where A is the number of intersecting pairs reported. The algorithm requires O(n) space in the worst case. Another algorithm computes the convex hull of n points in E d in O(n log n) expected time for d = 3, and O(n bd=2c ) expected time for d ? 3. The algorithm also gives fast expected times for random input points. Another algorithm computes the diameter of a set of n...
A New VoronoiBased Surface Reconstruction Algorithm
, 2002
"... We describe our experience with a new algorithm for the reconstruction of surfaces from unorganized sample points in R³. The algorithm is the first for this problem with provable guarantees. Given a “good sample” from a smooth surface, the output is guaranteed to be topologically correct and converg ..."
Abstract

Cited by 355 (8 self)
 Add to MetaCart
We describe our experience with a new algorithm for the reconstruction of surfaces from unorganized sample points in R³. The algorithm is the first for this problem with provable guarantees. Given a “good sample” from a smooth surface, the output is guaranteed to be topologically correct and convergent to the original surface as the sampling density increases. The definition of a good sample is itself interesting: the required sampling density varies locally, rigorously capturing the intuitive notion that featureless areas can be reconstructed from fewer samples. The output mesh interpolates, rather than approximates, the input points. Our algorithm is based on the threedimensional Voronoi diagram. Given a good program for this fundamental subroutine, the algorithm is quite easy to implement.
Surface Reconstruction by Voronoi Filtering
 Discrete and Computational Geometry
, 1998
"... We give a simple combinatorial algorithm that computes a piecewiselinear approximation of a smooth surface from a finite set of sample points. The algorithm uses Voronoi vertices to remove triangles from the Delaunay triangulation. We prove the algorithm correct by showing that for densely sampled ..."
Abstract

Cited by 341 (11 self)
 Add to MetaCart
We give a simple combinatorial algorithm that computes a piecewiselinear approximation of a smooth surface from a finite set of sample points. The algorithm uses Voronoi vertices to remove triangles from the Delaunay triangulation. We prove the algorithm correct by showing that for densely sampled surfaces, where density depends on "local feature size", the output is topologically valid and convergent (both pointwise and in surface normals) to the original surface. We describe an implementation of the algorithm and show example outputs. 1 Introduction The problem of reconstructing a surface from scattered sample points arises in many applications such as computer graphics, medical imaging, and cartography. In this paper we consider the specific reconstruction problem in which the input is a set of sample points S drawn from a smooth twodimensional manifold F embedded in three dimensions, and the desired output is a triangular mesh with vertex set equal to S that faithfully represen...
Smooth Surface Reconstruction via Natural Neighbour Interpolation of Distance Functions
, 2000
"... We present an algorithm to reconstruct smooth surfaces of arbitrary topology from unorganised sample points and normals. The method uses natural neighbour interpolation, works in any dimension and allows to deal with non uniform samples. The reconstructed surface is a smooth manifold passing through ..."
Abstract

Cited by 119 (4 self)
 Add to MetaCart
We present an algorithm to reconstruct smooth surfaces of arbitrary topology from unorganised sample points and normals. The method uses natural neighbour interpolation, works in any dimension and allows to deal with non uniform samples. The reconstructed surface is a smooth manifold passing through all the sample points. This surface is implicitly represented as the zeroset of some pseudodistance function. It can be meshed so as to satisfy a userdefined error bound. Experimental results are presented for surfaces in R³.
An Incremental Algorithm for Betti Numbers of Simplicial Complexes
, 1993
"... A general and direct method for computing the betti numbers of the homology groups of a finite simplicial complex is given. For subcomplexes of a triangulation of S³ this method has implementations that run in time 0(’na(n)) and O(n), where n is the number of simplices in the triangulation. If app!i ..."
Abstract

Cited by 94 (16 self)
 Add to MetaCart
A general and direct method for computing the betti numbers of the homology groups of a finite simplicial complex is given. For subcomplexes of a triangulation of S³ this method has implementations that run in time 0(’na(n)) and O(n), where n is the number of simplices in the triangulation. If app!ied to the family of ashapes of a finite point set in R³ ittakes time O(ncz(n)) to compute the betti numbers of all crshapes.
rRegular Shape Reconstruction from Unorganized Points
, 1997
"... In this paper, the problem of reconstructing a surface, given a set of scattered data points is addressed. First, a precise formulation of the reconstruction problem is proposed. The solution is mathematically defined as a particular mesh of the surface called the normalized mesh. This solution has ..."
Abstract

Cited by 76 (2 self)
 Add to MetaCart
In this paper, the problem of reconstructing a surface, given a set of scattered data points is addressed. First, a precise formulation of the reconstruction problem is proposed. The solution is mathematically defined as a particular mesh of the surface called the normalized mesh. This solution has the property to be included inside the Delaunay graph. A criterion to select boundary faces inside the Delaunay graph is proposed. This criterion is proven to provide the exact solution in 2D for points sampling a rregular shapes with a sampling path ffl ! 0:38r. In 3D, this results cannot be extended and the criterion cannot retrieve every faces. Some heuristics are then proposed in order to complete the surface. 1 Introduction This paper addresses the problem of meshing a surface only known by an unorganized set of points. Such a problem may occur in many domains including pattern recognition, computer vision, and graphics. Meshing the boundary of an object is useful to study its geometr...
The Crust and the BetaSkeleton: Combinatorial Curve Reconstruction
 Graphical Models and Image Processing
, 1998
"... We construct a graph on a planar point set, which captures its shape in the following sense: if a smooth curve is sampled densely enough, the graph on the samples is a polygonalization of the curve, with no extraneous edges. The required sampling density varies with the Local Feature Size on the cur ..."
Abstract

Cited by 49 (0 self)
 Add to MetaCart
We construct a graph on a planar point set, which captures its shape in the following sense: if a smooth curve is sampled densely enough, the graph on the samples is a polygonalization of the curve, with no extraneous edges. The required sampling density varies with the Local Feature Size on the curve, so that areas of less detail can be sampled less densely. We give two different graphs that, in this sense, reconstruct smooth curves: a simple new construction which we call the crust, and the fiskeleton, using a specific value of fi. 1 Introduction There are many situations in which a set of sample points lying on or near a surface is used to reconstruct a polygonal approximation to the surface. In the plane, this problem becomes a sort of unlabeled version of connectthedots: we are given a set of points and asked to connect them into the most likely polygonal curve. We show that under fairly generous and welldefined sampling conditions either of two proximitybased graphs defined ...