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18
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 ..."
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Cited by 355 (8 self)
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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.
Fast surface reconstruction using the level set method
 In VLSM ’01: Proceedings of the IEEE Workshop on Variational and Level Set Methods
, 2001
"... In this paper we describe new formulations and develop fast algorithms for implicit surface reconstruction based on variational and partial differential equation (PDE) methods. In particular we use the level set method and fast sweeping and tagging methods to reconstruct surfaces from scattered data ..."
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Cited by 117 (11 self)
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In this paper we describe new formulations and develop fast algorithms for implicit surface reconstruction based on variational and partial differential equation (PDE) methods. In particular we use the level set method and fast sweeping and tagging methods to reconstruct surfaces from scattered data set. The data set might consist of points, curves and/or surface patches. A weighted minimal surfacelike model is constructed and its variational level set formulation is implemented with optimal efficiency. The reconstructed surface is smoother than piecewise linear and has a natural scaling in the regularization that allows varying flexibility according to the local sampling density. As is usual with the level set method we can handle complicated topology and deformations, as well as noisy or highly nonuniform data sets easily. The method is based on a simple rectangular grid, although adaptive and triangular grids are also possible. Some consequences, such as hole filling capability, are demonstrated, as well as the viability and convergence of our new fast tagging algorithm.
Nice Point Sets Can Have Nasty Delaunay Triangulations
 In Proc. 17th Annu. ACM Sympos. Comput. Geom
, 2001
"... We consider the complexity of Delaunay triangulations of sets of points in IR 3 under certain practical geometric constraints. The spread of a set of points is the ratio between the longest and shortest pairwise distances. We show that in the worst case, the Delaunay triangulation of u points in ..."
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Cited by 49 (5 self)
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We consider the complexity of Delaunay triangulations of sets of points in IR 3 under certain practical geometric constraints. The spread of a set of points is the ratio between the longest and shortest pairwise distances. We show that in the worst case, the Delaunay triangulation of u points in IR 3 with spread A has complexity il(min{A 3 , uA, u2}) and O (min{A 4, u2}). For the case A = D(v/), our lower bound construction consists of a gridlike sample of a right circular cylinder with constant height and radius. We also construct a family of smooth connected surfaces such that the Delaunay triangulation of any good point sample has nearquadratic complexity.
Approximating the Medial Axis from the Voronoi Diagram with a Convergence Guarantee
 Algorithmica
, 2004
"... The medial axis of a surface in 3D is the closure of all points that have two or more closest points on the surface. It is an essential geometric structure in a number of applications involving 3D geometric shapes. Since exact computation of the medial axis is difficult in general, efforts continue ..."
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Cited by 34 (7 self)
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The medial axis of a surface in 3D is the closure of all points that have two or more closest points on the surface. It is an essential geometric structure in a number of applications involving 3D geometric shapes. Since exact computation of the medial axis is difficult in general, efforts continue to improve their approximations. Voronoi diagrams turn out to be useful for this approximation. Although it is known that Voronoi vertices for a sample of points from a curve in 2D approximate its medial axis, similar result does not hold in 3D. Recently, it has been discovered that only a subset of Voronoi vertices converge to the medial axis as sample density approaches infinity. However, most applications need a nondiscrete approximation as opposed to a discrete one. To date no known algorithm can compute this approximation straight from the Voronoi diagram with a guarantee of convergence. We present such an algorithm and its convergence analysis in this paper. One salient feature of the algorithm is that it is scale and density independent. Experimental results corroborate our theoretical claims.
Shape Dimension and Approximation from Samples
, 2003
"... There are many scientific and engineering applications where an automatic detection of shape dimension from sample data is necessary. Topological dimensions of shapes constitute an important global feature of them. We present a Voronoi based dimension detection algorithm that assigns a dimension to ..."
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Cited by 31 (6 self)
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There are many scientific and engineering applications where an automatic detection of shape dimension from sample data is necessary. Topological dimensions of shapes constitute an important global feature of them. We present a Voronoi based dimension detection algorithm that assigns a dimension to a sample point which is the topological dimension of the manifold it belongs to. Based on this dimension detection, we also present an algorithm to approximate shapes of arbitrary dimension from their samples. Our empirical results with data sets in three dimensions support our theory.
New Techniques for Topologically Correct Surface Reconstruction
, 2000
"... We present a new approach to surface reconstruction based on the Delaunay complex. First we give a simple and fast algorithm that picks locally a surface at each vertex. For that, we introduce the concept of lambdaintervals. It turns out that for smooth regions of the surface this method works very ..."
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Cited by 30 (4 self)
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We present a new approach to surface reconstruction based on the Delaunay complex. First we give a simple and fast algorithm that picks locally a surface at each vertex. For that, we introduce the concept of lambdaintervals. It turns out that for smooth regions of the surface this method works very well and at difficult parts of the surface yields an output wellsuited for postprocessing. As a postprocessing step we propose a topological clean up and a new technique based on linear programming in order to establish a topologically correct surface. These techniques should be useful also for many other reconstruction schemes.
Regular and NonRegular Point Sets: Properties and Reconstruction
 IN "COMPUTATIONAL GEOMETRY  THEORY AND APPLICATION"
"... In this paper, we address the problem of curve and surface reconstruction from sets of points. We introduce regular interpolants, which are polygonal approximations of curves and surfaces satisfying a new regularity condition. This new condition, which is an extension of the popular notion ofsampli ..."
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Cited by 20 (0 self)
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In this paper, we address the problem of curve and surface reconstruction from sets of points. We introduce regular interpolants, which are polygonal approximations of curves and surfaces satisfying a new regularity condition. This new condition, which is an extension of the popular notion ofsampling to the practical case of discrete shapes, seems much more realistic than previously proposed conditions based on properties of the underlying continuous shapes. Indeed, contrary to previous sampling criteria, our regularity condition can be checked on the basis of the samples alone and can be turned into a provably correct curve and surface reconstruction algorithm. Our reconstruction methods can also be applied to nonregular and unorganized point sets, revealing a larger part of the inner structure of such point sets than past approaches. Several realsize reconstruction examples validate the new method.
Visualization, Analysis and Shape Reconstruction of Unorganized Data Sets
"... In this chapter mathematical models and efficient algorithms are developed for the visualization, analysis and shape reconstruction for an arbitrary data set that can include unorganized points or continuous manifolds of any codimension, such as pieces of curves and surface patches. The distance fun ..."
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Cited by 9 (0 self)
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In this chapter mathematical models and efficient algorithms are developed for the visualization, analysis and shape reconstruction for an arbitrary data set that can include unorganized points or continuous manifolds of any codimension, such as pieces of curves and surface patches. The distance function to the data set and its contours are used for fast visualization and analysis of the data set. A minimal surface and a convection model are used for shape reconstruction from the data set. All formulations and numerical algorithms are based on implicit representations on simple rectangular grids which extend to any number of dimensions and which also can easily be combined with the level set method for dynamic shape deformation and other manipulations.
Visual hull construction using adaptive sampling
 In IEEE Workshop on Applications of Computer Vision
, 2005
"... Volumetric visual hulls have become very popular in many computer vision applications including human body pose estimation and virtualized reality. In these applications, the visual hull is used to approximate the 3D geometry of an object. Existing volumetric visual hull construction techniques, how ..."
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Cited by 9 (0 self)
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Volumetric visual hulls have become very popular in many computer vision applications including human body pose estimation and virtualized reality. In these applications, the visual hull is used to approximate the 3D geometry of an object. Existing volumetric visual hull construction techniques, however, produce a 3color volume data that merely serves as a bounding volume. In other words it lacks an accurate surface representation. Polygonization can produce satisfactory results only at high resolutions. In this study we extend the binary visual hull to an implicit surface in order to capture the geometry of the visual hull itself. In particular, we introduce an octreebased visual hull specific adaptive sampling algorithm to obtain a volumetric representation that provides accuracy proportional to the level of detail. Moreover, we propose a method to process the resulting octree to extract a crackfree polygonal visual hull surface. Experimental results illustrate the performance of the algorithm. 1.
On the reflexivity of point sets
 Discrete and Computational Geometry: The GoodmanPollack Festschrift, 139–156
, 2003
"... Abstract. We introduce a new measure for planar point sets S. Intuitively, it describes the combinatorial distance from a convex set: The reflexivity ρ(S) ofS is given by the smallest number of reflex vertices in a simple polygonalization of S. We prove various combinatorial bounds and provide effic ..."
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Cited by 8 (1 self)
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Abstract. We introduce a new measure for planar point sets S. Intuitively, it describes the combinatorial distance from a convex set: The reflexivity ρ(S) ofS is given by the smallest number of reflex vertices in a simple polygonalization of S. We prove various combinatorial bounds and provide efficient algorithms to compute reflexivity, both exactly (in special cases) and approximately (in general). Our study naturally takes us into the examination of some closely related quantities, such as the convex cover number κ1(S) of a planar point set, which is the smallest number of convex chains that cover S, and the convex partition number κ2(S), which is given by the smallest number of disjoint convex chains that cover S. We prove that it is NPcomplete to determine the convex cover or the convex partition number, and we give logarithmicapproximation algorithms for determining each. 1