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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. ..."
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Cited by 753 (5 self)
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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.
Multiresolution Analysis of Arbitrary Meshes
, 1995
"... In computer graphics and geometric modeling, shapes are often represented by triangular meshes. With the advent of laser scanning systems, meshes of extreme complexity are rapidly becoming commonplace. Such meshes are notoriously expensive to store, transmit, render, and are awkward to edit. Multire ..."
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Cited by 605 (16 self)
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In computer graphics and geometric modeling, shapes are often represented by triangular meshes. With the advent of laser scanning systems, meshes of extreme complexity are rapidly becoming commonplace. Such meshes are notoriously expensive to store, transmit, render, and are awkward to edit. Multiresolution analysis offers a simple, unified, and theoretically sound approach to dealing with these problems. Lounsbery et al. have recently developed a technique for creating multiresolution representations for a restricted class of meshes with subdivision connectivity. Unfortunately, meshes encountered in practice typically do not meet this requirement. In this paper we present a method for overcoming the subdivision connectivity restriction, meaning that completely arbitrary meshes can now be converted to multiresolution form. The method is based on the approximation of an arbitrary initial mesh M by a mesh M that has subdivision connectivity and is guaranteed to be within a specified tolerance. The key
Computing geodesic distances on triangular meshes
 In Proc. of WSCG’2002
, 2002
"... We present an approximation method to compute geodesic distances on triangulated domains in the three dimensional space. Our particular approach is based on the Fast Marching Method for solving the Eikonal equation on triangular meshes. As such, the algorithm is a wavefront propagation method, a rem ..."
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Cited by 23 (2 self)
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We present an approximation method to compute geodesic distances on triangulated domains in the three dimensional space. Our particular approach is based on the Fast Marching Method for solving the Eikonal equation on triangular meshes. As such, the algorithm is a wavefront propagation method, a reminiscent of the Dijkstra algorithm which runs in O(n log n) steps.
Approximating Shortest Paths on a Nonconvex Polyhedron
 In Proc. 38th Annu. IEEE Sympos. Found. Comput. Sci
, 1997
"... We present an approximation algorithm that, given a simple, possibly nonconvex polyhedron P with n vertices in R 3 , and two points s and t on its surface @P , constructs a path on @P between s and t whose length is at most 7(1 + ")ae, where ae is the length of the shortest path between s a ..."
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Cited by 22 (4 self)
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We present an approximation algorithm that, given a simple, possibly nonconvex polyhedron P with n vertices in R 3 , and two points s and t on its surface @P , constructs a path on @P between s and t whose length is at most 7(1 + ")ae, where ae is the length of the shortest path between s and t on @P , and " ? 0 is an arbitararily small positive constant. The algorithm runs in O(n 5=3 log 5=3 n) time. We also present a slightly faster algorithm that runs in O(n 8=5 log 8=5 n) time and returns a path whose length is at most 15(1 + ")ae. Work on this paper has been supported by National Science Foundation Grant CCR9301259, by an Army Research Office MURI grant DAAH049610013, by a Sloan fellowship, by an NYI award, by matching funds from Xerox Corporation, and by a grant from the U.S.Israeli Binational Science Foundation. y Department of Computer Science, Box 90129, Duke University, krv@cs.duke.edu z Department of Computer Science, Box 90129, Duke University, pa...
Construction of IsoContours, Bisectors, and Voronoi Diagrams on Triangulated Surfaces
"... Abstract—In the research of computer vision and machine perception, 3D objects are usually represented by 2manifold triangular meshes M. In this paper, we present practical and efficient algorithms to construct isocontours, bisectors, and Voronoi diagrams of point sites on M, based on an exact geo ..."
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Cited by 15 (6 self)
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Abstract—In the research of computer vision and machine perception, 3D objects are usually represented by 2manifold triangular meshes M. In this paper, we present practical and efficient algorithms to construct isocontours, bisectors, and Voronoi diagrams of point sites on M, based on an exact geodesic metric. Compared to euclidean metric spaces, the Voronoi diagrams on M exhibit many special properties that fail all of the existing euclidean Voronoi algorithms. To provide practical algorithms for constructing geodesicmetricbased Voronoi diagrams on M, this paper studies the analytic structure of isocontours, bisectors, and Voronoi diagrams on M. After a necessary preprocessing of model M, practical algorithms are proposed for quickly obtaining full information about isocontours, bisectors, and Voronoi diagrams on M. The complexity of the construction algorithms is also analyzed. Finally, three interesting applications—surface sampling and reconstruction, 3D skeleton extraction, and point pattern analysis—are presented that show the potential power of the proposed algorithms in pattern analysis. Index Terms—Shape, geometric transformations, triangular meshes, exact geodesic metrics, point patterns. Ç 1
Surface Reconstruction Using Adaptive Clustering
 Geometric Modeling, Supplement to the Journal Computing
, 2001
"... We present an automatic method for the generation of surface triangulations from sets of scattered points. Given a set of scattered points in threedimensional space, without connectivity information, our method reconstructs a triangulated surface model in a twostep procedure. First, we apply an ad ..."
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Cited by 14 (4 self)
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We present an automatic method for the generation of surface triangulations from sets of scattered points. Given a set of scattered points in threedimensional space, without connectivity information, our method reconstructs a triangulated surface model in a twostep procedure. First, we apply an adaptive clustering technique to the given set of points, identifying point subsets in regions that are nearly planar. The output of this clustering step is a set of twomanifold "tiles" that locally approximate the underlying, unknown surface. Second, we construct a surface triangulation by triangulating the data within the individual tiles and the gaps between the tiles. This algorithm can generate multiresolution representations by applying the triangulation step to various resolution levels resulting from the hierarchical clustering step. We compute deviation measures for each cluster, and thus we can produce reconstructions with prescribed error bounds.
Facility location on terrains
 PROC. 9TH INTERNATIONAL SYMPOSIUM OF ALGORITHMS AND COMPUTATION, VOLUME 1533 OF LECTURE NOTES COMPUT. SCI
, 1998
"... Given a terrain defined as a piecewiselinear function with n triangles, and m point sites on it, we would like to identify the location on the terrain that minimizes the maximum distance to the sites. The distance is measured as the length of the Euclidean shortest path along the terrain. To simpli ..."
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Cited by 10 (1 self)
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Given a terrain defined as a piecewiselinear function with n triangles, and m point sites on it, we would like to identify the location on the terrain that minimizes the maximum distance to the sites. The distance is measured as the length of the Euclidean shortest path along the terrain. To simplify the problem somewhat, we extend the terrain to (the surface of) a polyhedron. To compute the optimum placement, we compute the furthestsite Voronoi diagram of the sites on the polyhedron. The diagram has maximum combinatorial complexity Θ(mn²), and the algorithm runs in O(mn² log² mlogn) time.
Texturing Surfaces Using ReactionDiffusion
, 1994
"... This dissenation introduces a new method of creating computer graphics textures that is based on simulaling a biological model of pa11em formation known as reactiondiffusion. Applied mathematicians and biologistli have shown how simple reactiondiffusion systems can create pan ems of spots or strip ..."
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Cited by 4 (0 self)
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This dissenation introduces a new method of creating computer graphics textures that is based on simulaling a biological model of pa11em formation known as reactiondiffusion. Applied mathematicians and biologistli have shown how simple reactiondiffusion systems can create pan ems of spots or stripes. Here we demonstr.lte that the range of patterns created by reactiondtffusion can be greatly expanded by cascading two or more reactiondiffusion systems. Cascaded systems can produce such complex paucms as the clusters of spots found on leopards and the mixture of spots and stripes found on ccnain squirrels. This disscnalion also presents a method for simulating reactiondiffusion systems directly on the surface of any given polygonal model. This IS done by creating a mesh for simulation that is specifically fit to a p<tnicular model. Such.a mesh is created by first randomly distribuLing points ovcnhc surface of the model and causing these pomts to repel one another so that they are evenly spaced over the surface. TI1en a mesh cell is created around each point by finding the Vorono • region for each pomt within a local planar approximation 10 the surface. Reactiondiffusion systems can then be simulated on this mesh of cells. The chemical
HigherOrder Voronoi Diagrams on Triangulated Surfaces
, 2009
"... We study the complexity of higherorder Voronoi diagrams on triangulated surfaces under the geodesic distance, when the sites may be polygonal domains of constant complexity. More precisely, we show that on a surface defined by n triangles the sum of the combinatorial complexities of the orderj Vor ..."
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Cited by 2 (1 self)
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We study the complexity of higherorder Voronoi diagrams on triangulated surfaces under the geodesic distance, when the sites may be polygonal domains of constant complexity. More precisely, we show that on a surface defined by n triangles the sum of the combinatorial complexities of the orderj Voronoi diagrams of m sites, for j = 1,...,k, is O(k 2 n 2 + k 2 m + knm), which is asymptotically tight in the worst case. 1
Medial Residues of Piecewise Linear Manifolds
"... Skeleton structures of objects are used in a wide variety of applications such as shape analysis and path planning. One of the most widely used skeletons is the medial axis, which is a thin structure centered within and homotopy equivalent to the object. However, on piecewise linear surfaces, which ..."
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Cited by 1 (0 self)
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Skeleton structures of objects are used in a wide variety of applications such as shape analysis and path planning. One of the most widely used skeletons is the medial axis, which is a thin structure centered within and homotopy equivalent to the object. However, on piecewise linear surfaces, which are one of the most common outputs from surface reconstruction algorithms, natural generalizations of typical medial axis definitions may fail to have these desirable properties. In this paper, we propose a new extension of the medial axis, called the medial residue, and prove that it is a finite curve network homotopy equivalent to the original surface when the input is a piecewise linear surface with boundary. We also develop an efficient algorithm to compute the medial residue on a triangulated mesh, building on previously known work to compute geodesic distances. 1