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54
A Replacement for Voronoi Diagrams of Near Linear Size
 In Proc. 42nd Annu. IEEE Sympos. Found. Comput. Sci
, 2001
"... For a set P of n points in R^d, we define a new type of space decomposition. The new diagram provides an εapproximation to the distance function associated with the Voronoi diagram of P, while being of near linear size, for d ≥ 2. This contrasts with the standard Voronoi diagram ..."
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Cited by 97 (8 self)
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For a set P of n points in R^d, we define a new type of space decomposition. The new diagram provides an &epsilon;approximation to the distance function associated with the Voronoi diagram of P, while being of near linear size, for d &ge; 2. This contrasts with the standard Voronoi diagram that has complexity &Omega;(n^&lceil;d/2&rceil;) in the worst case.
Complexity of the Delaunay triangulation of points on surfaces: the smooth case
 In Annual Symposium on Computational Geometry
, 2003
"... It is well known that the complexity of the Delaunay triangulation of N points in 3, i.e. the number of its faces, can be (N2). The case of points distributed on a surface is of great practical importance in reverse engineering since most surface reconstruction algorithms rst construct the Delaun ..."
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Cited by 54 (15 self)
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It is well known that the complexity of the Delaunay triangulation of N points in 3, i.e. the number of its faces, can be (N2). The case of points distributed on a surface is of great practical importance in reverse engineering since most surface reconstruction algorithms rst construct the Delaunay triangulation of a set of points measured on a surface. In this paper, we bound the complexity of the Delaunay triangulation of points distributed on generic smooth surfaces of 3. Under a mild uniform sampling condition, we show that the complexity of the 3D Delaunay triangulation of the points is O(N log N). Categories and Subject Descriptors F.2.2 [Theory of Computation]: Analysis of Algorithms and Problem ComplexityGeometrical problems and com
A geometric convection approach of 3D reconstruction
 EUROGRAPHICS SYMPOSIUM ON GEOMETRY PROCESSING (2003)
, 2003
"... This paper introduces a fast and efficient algorithm for surface reconstruction. As many algorithms of this kind, it produces a piecewise linear approximation of a surface S from a finite, sufficiently dense, subset of its points. Originally, the starting point of this work does not come from the co ..."
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Cited by 46 (6 self)
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This paper introduces a fast and efficient algorithm for surface reconstruction. As many algorithms of this kind, it produces a piecewise linear approximation of a surface S from a finite, sufficiently dense, subset of its points. Originally, the starting point of this work does not come from the computational geometry field. It is inspired by an existing numerical scheme of surface convection developed by Zhao, Osher and Fedkiw. We have translated this scheme to make it depend on the geometry of the input data set only, and not on the precision of some grid around the surface. Our algorithm deforms a closed oriented pseudosurface embedded in the 3D Delaunay triangulation of the sampled points, and the reconstructed surface consists of a set of oriented facets located in this 3D Delaunay triangulation. This paper provides an appropriate data structure to represent a pseudosurface, together with operations that manage deformations and topological changes. The algorithm can handle surfaces with boundaries, surfaces of high genus and, unlike most of the other existing schemes, it does not involve a global heuristic. Its complexity is that of the 3D Delaunay triangulation of the points. We present some results of the method, which turns out to be efficient even on noisy input data.
On ConflictFree Coloring of Points and Simple Regions in the Plane
"... In this paper, we study coloring problems related to frequency assignment problems in cellular networks. In abstract setting, the problems are of the following two types: CFcoloring of regions: Given a finite family of n regions of some fixed type (such as discs, pseudodiscs, axisparallel rec ..."
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Cited by 46 (6 self)
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In this paper, we study coloring problems related to frequency assignment problems in cellular networks. In abstract setting, the problems are of the following two types: CFcoloring of regions: Given a finite family of n regions of some fixed type (such as discs, pseudodiscs, axisparallel rectangles, etc.), what is the minimum integer k, such that one can assign a color to each region of using a total of at most k colors, such that the resulting coloring has the following property: For each point p b#S b there is at least one region b # S that contains p in its interior, whose color is unique among all regions in that contain p in their interior (in this case we say that p is being `served' by that color). We refer to such a coloring as a conflictfree coloring of (CFcoloring in short).
The Delaunay hierarchy
 Internat. J. Found. Comput. Sci
"... We propose a new data structure to compute the Delaunay triangulation of a set of points in the plane. It combines good worst case complexity, fast behavior on real data, small memory occupation and the possibility of fully dynamic insertions and deletions. The location structure is organized into s ..."
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Cited by 43 (5 self)
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We propose a new data structure to compute the Delaunay triangulation of a set of points in the plane. It combines good worst case complexity, fast behavior on real data, small memory occupation and the possibility of fully dynamic insertions and deletions. The location structure is organized into several levels. The lowest level just consists of the triangulation, then each level contains the triangulation of a small sample of the level below. Point location is done by walking in a triangulation to determine the nearest neighbor of the query at that level, then the walk restarts from that neighbor at the level below. Using a small subset (3%) to sample a level allows a small memory occupation; the walk and the use of the nearest neighbor to change levels quickly locate the query.
Provably Good Surface Sampling and Approximation
, 2003
"... We present an algorithm for meshing surfaces that is a simple adaptation of a greedy "farthest point" technique proposed by Chew. Given a surface S, it progressively adds points on S and updates the 3dimensional Delaunay triangulation of the points. The method is very simple and works in ..."
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Cited by 42 (1 self)
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We present an algorithm for meshing surfaces that is a simple adaptation of a greedy "farthest point" technique proposed by Chew. Given a surface S, it progressively adds points on S and updates the 3dimensional Delaunay triangulation of the points. The method is very simple and works in 3dspace without requiring to parameterize the surface. Taking advantage of recent results on the restricted Delaunay triangulation, we prove that the algorithm can generate good samples on S as well as triangulated surfaces that approximate S. More precisely, we show that the restricted Delaunay triangulation Del # S of the points has the same topology type as S, that the Hausdorff distance between Del # S and S can be made arbitrarily small, and that we can bound the aspect ratio of the facets of Del # S . The algorithm has been implemented and we report on experimental results that provide evidence that it is very effective in practice. We present results on implicit surfaces, on CSG models and on polyhedra. Although most of our theoretical results are given for smooth closed surfaces, the method is quite robust in handling smooth surfaces with boundaries, and even nonsmooth surfaces.
Sparse Voronoi Refinement
 IN PROCEEDINGS OF THE 15TH INTERNATIONAL MESHING ROUNDTABLE
, 2006
"... ... a conformal Delaunay mesh in arbitrary dimension with guaranteed mesh size and quality. Our algorithm runs in outputsensitive time O(nlog(L/s) + m), with constants depending only on dimension and on prescribed element shape quality bounds. For a large class of inputs, including integer coordina ..."
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Cited by 42 (26 self)
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... a conformal Delaunay mesh in arbitrary dimension with guaranteed mesh size and quality. Our algorithm runs in outputsensitive time O(nlog(L/s) + m), with constants depending only on dimension and on prescribed element shape quality bounds. For a large class of inputs, including integer coordinates, this matches the optimal time bound of Θ(n log n + m). Our new technique uses interleaving: we maintain a sparse mesh as we mix the recovery of input features with the addition of Steiner vertices for quality improvement.
SmoothSurface Reconstruction in Near Linear Time
, 2001
"... A surface reconstruction algorithm takes as input a set of sample points from an unknown closed and smooth surface in 3d space, and produces a piecewise linear approximation of the surface that contains the sample points. Variants of this problem have received considerable attention in computer ..."
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Cited by 41 (5 self)
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A surface reconstruction algorithm takes as input a set of sample points from an unknown closed and smooth surface in 3d space, and produces a piecewise linear approximation of the surface that contains the sample points. Variants of this problem have received considerable attention in computer vision and computer graphics and more recently in computational geometry. In the latter area, three different algorithms (Amenta and Bern `98, and refined in Amenta, Choi, Dey and Leekha `00; Amenta, Choi and Kolluri `00; Boissonnat and Cazals `00) have been proposed. These algorithms have a correctness guarantee: if the sample is sufficiently dense then the output is a good approximation to the original surface. They have unfortunately a worstcase running time that is quadratic in the size of the input. This is so because they are based on the construction of 3d Voronoi diagrams or Delaunay tetrahedrizations, which can have quadratic size. Even worse, according to recent work (Erickson `01), there are surfaces for which this is the case even when the sample set is "locally uniform" on the surface. In this paper, we describe a new algorithm that also has a correctness guarantee but whose worstcase running time is almost linear. In fact, O(n log n) where n is the input size. As in some of the previous algorithms, the piecewise linear approximation produced by the new algorithm is a subset of the 3d Delaunay tetrahedrization; however, this is obtained by computing only the relevant parts of the 3d Delaunay structure. The algorithm first estimates for each sample point the surface normal and a parameter that is then used to "decimate" the set of samples. The resulting subset of sample points is locally uniform and so a reconstruction based on it can be compu...
Provably good sampling and meshing of surfaces
 Graphical Models
, 2005
"... The notion of εsample, introduced by Amenta and Bern, has proven to be a key concept in the theory of sampled surfaces. Of particular interest is the fact that, if E is an εsample of a C2continuous surface S for a sufficiently small ε, then the Delaunay triangulation of E restricted to S is a goo ..."
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Cited by 37 (9 self)
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The notion of εsample, introduced by Amenta and Bern, has proven to be a key concept in the theory of sampled surfaces. Of particular interest is the fact that, if E is an εsample of a C2continuous surface S for a sufficiently small ε, then the Delaunay triangulation of E restricted to S is a good approximation of S, both in a topological and in a geometric sense. Hence, if one can construct an εsample, one also gets a good approximation of the surface. Moreover, correct reconstruction is ensured by various algorithms. In this paper, we introduce the notion of loose εsample. We show that the set of loose εsamples contains and is asymptotically identical to the set of εsamples. The main advantage of loose εsamples over εsamples is that they are easier to check and to construct. We also present a simple algorithm that constructs provably good surface samples and meshes. Given a C2continuous surface S without boundary, the algorithm generates a sparse εsample E and at the same time a triangulated surface DelS(E). The triangulated surface has the same topological type as S, is close to S for the Hausdorff distance and can provide good approximations of normals, areas and curvatures. A notable feature of the algorithm is that the surface needs only to be known through an oracle that, given a line segment, detects whether the segment intersects the surface and, in the affirmative, returns the intersection points. This makes the algorithm useful in a wide variety of contexts and for a large class of surfaces. Keywords: Surface mesh generation, εsampling, surface approximation, restricted Delaunay triangulation, mesh refinement
A linear bound on the complexity of the Delaunay triangulations of points on polyhedral surfaces
 Proc. 7th Annu. ACM Sympos. Solid Modeling Appl
"... Delaunay triangulations and Voronoi diagrams have found numerous applications in surface modeling, surface mesh generation, deformable surface modeling and surface reconstruction. Many algorithms in these applications begin by constructing the threedimensional Delaunay triangulation of a finite set ..."
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Cited by 37 (10 self)
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Delaunay triangulations and Voronoi diagrams have found numerous applications in surface modeling, surface mesh generation, deformable surface modeling and surface reconstruction. Many algorithms in these applications begin by constructing the threedimensional Delaunay triangulation of a finite set of points scattered over a surface. Their runningtime therefore depends on the complexity of the Delaunay triangulation of such point sets. Although the complexity of the Delaunay triangulation of points in may be quadratic in the worstcase, we show in this paper that it is only linear when the points are distributed on a fixed number of wellsampled facets of (e.g. the facets of a polyhedron). Our bound is deterministic and the constants are explicitly given. Categories and Subject Descriptors I.3.5 [Computing Methodologies]: Computational Geometry and