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12
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 422 (9 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.
A Simple Algorithm for Homeomorphic Surface Reconstruction
 International Journal of Computational Geometry and Applications
, 2000
"... The problem of computing a piecewise linear approximation to a surface from a set of sample points is important in solid modeling, computer graphics and computer vision. A recent algorithm [1] using the Voronoi diagram of the sample points gave a guarantee on the distance of the output surface from ..."
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Cited by 238 (29 self)
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The problem of computing a piecewise linear approximation to a surface from a set of sample points is important in solid modeling, computer graphics and computer vision. A recent algorithm [1] using the Voronoi diagram of the sample points gave a guarantee on the distance of the output surface from the original sampled surface assuming the sample was `suciently dense'. We give a similar algorithm, simplifying the computation and the proof of the geometric guarantee. In addition, we guarantee that our output surface is homeomorphic to the original surface; to our knowledge this is the rst such topological guarantee for this problem. 1 Introduction A number of applications in CAD, computer graphics, computer vision and mathematical modeling involve the computation of a piecewise lin Dept. of Computer Science, U. of Texas, Austin TX 78712. email: amenta@cs.utexas.edu, supported by NSF grant CCR9731977 y Dept. of Computer Science, U. of Texas, Austin, TX 78712. email: sunghe...
A Simple Provable Algorithm for Curve Reconstruction
, 1998
"... We present an algorithm that provably reconstructs a curve in the framework introduced by Amenta, Bern and Eppstein. The highlights of the algorithm are: (i) it is simple, (ii) it requires a sampling density better than previously known, (iii) it can be adapted for curve reconstruction in higher dim ..."
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Cited by 54 (12 self)
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We present an algorithm that provably reconstructs a curve in the framework introduced by Amenta, Bern and Eppstein. The highlights of the algorithm are: (i) it is simple, (ii) it requires a sampling density better than previously known, (iii) it can be adapted for curve reconstruction in higher dimensions straightforwardly.
Curve Reconstruction: Connecting Dots with Good Reason
 IN PROC. 15TH ANNU. ACM SYMPOS. COMPUT. GEOM
, 1999
"... Curve reconstruction algorithms are supposed to reconstruct curves from point samples. Recent papers present algorithms that come with a guarantee: Given a suciently dense sample of a closed smooth curve, the algorithms construct the correct polygonal reconstruction. Nothing is claimed about the ..."
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Cited by 52 (8 self)
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Curve reconstruction algorithms are supposed to reconstruct curves from point samples. Recent papers present algorithms that come with a guarantee: Given a suciently dense sample of a closed smooth curve, the algorithms construct the correct polygonal reconstruction. Nothing is claimed about the output of the algorithms, if the input is not a dense sample of a closed smooth curve, e.g., a sample of a curve with endpoints. We present an algorithm that comes with a guarantee for any set P of input points. The algorithm constructs a polygonal reconstruction G and a smooth curve that justies G as the reconstruction from P.
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 35 (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.
Shape reconstruction with Delaunay complex
 PROCEEDINGS OF LATIN’98: THEORETICAL INFORMATICS, LECTURE NOTES IN COMPUTER SCIENCE
, 1998
"... The reconstruction of a shape or surface from a finite set of points is a practically significant and theoretically challenging problem. This paper presents a unified view of algorithmic solutions proposed in the computer science literature that are based on the Delaunay complex of the points. ..."
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Cited by 33 (0 self)
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The reconstruction of a shape or surface from a finite set of points is a practically significant and theoretically challenging problem. This paper presents a unified view of algorithmic solutions proposed in the computer science literature that are based on the Delaunay complex of the points.
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 27 (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.
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 22 (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.
Surface Reconstruction by Wrapping Finite Sets in Space
"... Given a finite point set in R³, the surface reconstruction problem asks for a surface that passes through many but not necessarily all points. We describe an unambiguous definition of such a surface in geometric and topological terms, and sketch a fast algorithm for constructing it. Our solution ove ..."
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Cited by 21 (4 self)
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Given a finite point set in R³, the surface reconstruction problem asks for a surface that passes through many but not necessarily all points. We describe an unambiguous definition of such a surface in geometric and topological terms, and sketch a fast algorithm for constructing it. Our solution overcomes past limitations to special point distributions and heuristic design decisions.
Reconstructing a Collection of Curves with Corners and Endpoints
 Proc. 12th Annu. ACMSIAM Sympos. Discrete Alg
, 2001
"... We present an algorithm which provably reconstructs a collection of curves with corners and endpoints from a sample set that satisfies a certain sampling condition. The algorithm outputs a polygonal reconstruction that contains the edges in the correct reconstruction of the curves and such that any ..."
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Cited by 13 (4 self)
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We present an algorithm which provably reconstructs a collection of curves with corners and endpoints from a sample set that satisfies a certain sampling condition. The algorithm outputs a polygonal reconstruction that contains the edges in the correct reconstruction of the curves and such that any additional edge between sample points is justified. Furthermore, we show that for any such collection of curves, there exists a sample set such that a slightly modified version of our algorithm outputs exactly the correct reconstruction. The algorithm also performs quite well in practice. 1