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16
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 359 (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.
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 ..."
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Cited by 344 (11 self)
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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...
The Power Crust, Unions of Balls, and the Medial Axis Transform
 Computational Geometry: Theory and Applications
, 2000
"... The medial axis transform (or MAT) is a representation of an object as an infinite union of balls. We consider approximating the MAT of a threedimensional object, and its complement, with a finite union of balls. Using this approximate MAT we define a new piecewiselinear approximation to the objec ..."
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Cited by 172 (5 self)
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The medial axis transform (or MAT) is a representation of an object as an infinite union of balls. We consider approximating the MAT of a threedimensional object, and its complement, with a finite union of balls. Using this approximate MAT we define a new piecewiselinear approximation to the object surface, which we call the power crust. We assume that we are given as input a suficiently dense sample of points from the object surface. We select a subset of the Voronoi balls of the sample, the polar balls, as the union of balls representation. We bound the geometric error of the union, and of the corresponding power crust, and show that both representations are topologically correct as well. Thus, our results provide a new algorithm for surface reconstruction from sample points. By construction, the power crust is always the boundary of a solid, so we avoid the holefilling or manifold extraction steps used in previous algorithms. The union of balls representation and the power crust have corresponding piecewiselinear dual representations, which in some sense approximate the medial axis. We show a geometric relationship between these duals and the medial axis by proving that, as the sampling density goes to infinity, the set of poles, the centers of the polar balls, converge to the medial axis.
Reconstructing Curves with Sharp Corners
 Comput. Geom. Theory & Appl
, 2000
"... In this paper we present a new algorithm for curve reconstruction that has multiple applications in image processing, geographic information systems, pattern recognition and mathematical modeling. The algorithm can deal with nonsmooth curves with multiple components that cannot be handled by existin ..."
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Cited by 30 (1 self)
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In this paper we present a new algorithm for curve reconstruction that has multiple applications in image processing, geographic information systems, pattern recognition and mathematical modeling. The algorithm can deal with nonsmooth curves with multiple components that cannot be handled by existing algorithms. Experiments with several input data reveals the effectiveness of the algorithm in contrast with the other competitive algorithms for the problem. An attractive feature of the algorithm is that it is extendible to three dimensions for surface reconstructions.
Curve and Surface Reconstruction
, 2004
"... The problem of reconstructing a shape from its sample appears in many scientific and engineering applications. Because of the variety in shapes and applications, many algorithms have been proposed over the last two decades, some of which exploit applicationspecific information and some of which are ..."
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Cited by 17 (0 self)
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The problem of reconstructing a shape from its sample appears in many scientific and engineering applications. Because of the variety in shapes and applications, many algorithms have been proposed over the last two decades, some of which exploit applicationspecific information and some of which are more general. We will concentrate on techniques that apply to the general setting and have proved to provide some guarantees on the quality of reconstruction.
OnePass Delaunay Filtering for Homeomorphic 3D Surface Reconstruction
, 1999
"... We give a simple algorithm for surface reconstruction from a set of point samples in R , using only one threedimensional Voronoi diagram computation. We also give a fairly simple proof that the reconstruction is topologically correct when the input is a sufficiently dense sample from a smoot ..."
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Cited by 8 (0 self)
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We give a simple algorithm for surface reconstruction from a set of point samples in R , using only one threedimensional Voronoi diagram computation. We also give a fairly simple proof that the reconstruction is topologically correct when the input is a sufficiently dense sample from a smooth surface.
Combinatorial Curve Reconstruction in Hilbert Spaces: A New Sampling Theory and an Old Result Revisited
 Computational Geometry: Theory and Applications
, 2002
"... The goals of this paper are twofold. The first is to present a new sampling theory for curves, based on a new notion of local feature size. The properties of this new feature size are investigated, and are compared with the standard feature size definitions. The second goal is to revisit an existing ..."
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Cited by 5 (0 self)
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The goals of this paper are twofold. The first is to present a new sampling theory for curves, based on a new notion of local feature size. The properties of this new feature size are investigated, and are compared with the standard feature size definitions. The second goal is to revisit an existing algorithm for combinatorial curve reconstruction in spaces of arbitrary dimension, the Nearest Neighbour Crust of Dey and Kumar [8], and to prove its validity under the new sampling conditions. Because the new sampling theory can imply less dense sampling, the new proof is, in some cases, stronger than that presented in [8]. Also of interest are the techniques used to prove the theorem, as they are unlike those used used in the curve reconstruction literature to date.
Computational Geometry Column 38
, 2000
"... Recent results on curve reconstruction are described. Reconstruction of a curve from sample points ("connectthedots") is an important problem studied now for twenty years. Early efforts, primarily by researchers in computer vision, pattern recognition, and computational morphology, reli ..."
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Cited by 4 (0 self)
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Recent results on curve reconstruction are described. Reconstruction of a curve from sample points ("connectthedots") is an important problem studied now for twenty years. Early efforts, primarily by researchers in computer vision, pattern recognition, and computational morphology, relied on ad hoc heuristics (e.g., my own [OBW87]). The heuristics were placed on a firmer footing with ffshapes [EKS83] and fiskeletons [KR85] and other structures, whose underlying proximity graphs were later shown to support accurate reconstruction from uniformly sampled curves [FMG95, Att98, BB97]. User selection of the ff or fi parameter is still necessary. A breakthrough was achieved by Amenta, Bern, and Eppstein [ABE98], who designed two algorithms that guarantee correct reconstruction of smooth closed curves even with (sufficiently dense) nonuniform samples, and which lift the burden of selecting a parameter. One of their algorithms computes what they call the crust , a subgraph of the complete...
Analysis of Curve Reconstruction by Meshless Parameterization
 Numerical Algorithms
, 2003
"... Abstract: This paper proposes and analyzes a method called meshless parameterization for reconstructing curves from unordered point samples. The method solves a linear system of equations based on convex combinations so as to map the sampled points into corresponding parameter values, whose natural ..."
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Cited by 1 (0 self)
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Abstract: This paper proposes and analyzes a method called meshless parameterization for reconstructing curves from unordered point samples. The method solves a linear system of equations based on convex combinations so as to map the sampled points into corresponding parameter values, whose natural ordering provides the ordering of the points. Using the theory of Mmatrices, we derive natural conditions on the point sample which guarantee the correct ordering. A sufficient condition is that the underlying curve be tangentcontinuous and free of selfintersections and that the sample is dense enough. AMS subject classification: 65D05, 65D10. Key words: parameterization, curve reconstruction, monotonicity, Mmatrix, surface reconstruction, triangulation. 1.