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37
Efficient computation of a simplified medial axis
 Proc. of ACM Solid Modeling
, 2003
"... Applications of of the medial axis have been limited because of its instability and algebraic complexity. In this paper, we use a simplification of the medial axis, the θSMA, that is parameterized by a separation angle (θ) formed by the vectors connecting a point on the medial axis to the closest p ..."
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Cited by 57 (4 self)
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Applications of of the medial axis have been limited because of its instability and algebraic complexity. In this paper, we use a simplification of the medial axis, the θSMA, that is parameterized by a separation angle (θ) formed by the vectors connecting a point on the medial axis to the closest points on the boundary. We present a formal characterization of the degree of simplification of the θSMA as a function of θ, and we quantify the degree to which the simplified medial axis retains the features of the original polyhedron. We present a fast algorithm to compute an approximation of the θSMA. It is based on a spatial subdivision scheme, and uses fast computation of a distance field and its gradient using graphics hardware. The complexity of the algorithm varies based on the error threshold that is used, and is a linear function of the input size. We have applied this algorithm to approximate the SMA of models with tens or hundreds of thousands of triangles. Its running time varies from a few seconds, for a model consisting of hundreds of triangles, to minutes for highly complex models.
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 hybrid approach for computing visual hulls of complex objects
 In Computer Vision and Pattern Recognition
, 2003
"... This paper addresses the problem of computing visual hulls from image contours. We propose a new hybrid approach which overcomes the precisioncomplexity tradeoff inherent to voxel based approaches by taking advantage of surface based approaches. To this aim, we introduce a space discretization whi ..."
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Cited by 46 (8 self)
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This paper addresses the problem of computing visual hulls from image contours. We propose a new hybrid approach which overcomes the precisioncomplexity tradeoff inherent to voxel based approaches by taking advantage of surface based approaches. To this aim, we introduce a space discretization which does not rely on a regular grid, where most cells are ineffective, but rather on an irregular grid where sample points lie on the surface of the visual hull. Such a grid is composed of tetrahedral cells obtained by applying a Delaunay triangulation on the sample points. These cells are carved afterward according to image silhouette information. The proposed approach keeps the robustness of volumetric approaches while drastically improving their precision and reducing their time and space complexities. It thus allows modeling of objects with complex geometry, and it also makes real time feasible for precise models. Preliminary results with synthetic and real data are presented. 1.
Dense Point Sets Have Sparse Delaunay Triangulations
"... Delaunay triangulations and Voronoi diagrams are one of the most thoroughly studies objects in computational geometry, with numerous applications including nearestneighbor searching, clustering, finiteelement mesh generation, deformable surface modeling, and surface reconstruction. Many algorithms ..."
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Cited by 29 (2 self)
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Delaunay triangulations and Voronoi diagrams are one of the most thoroughly studies objects in computational geometry, with numerous applications including nearestneighbor searching, clustering, finiteelement mesh generation, deformable surface modeling, and surface reconstruction. Many algorithms in these application domains begin by constructing the Delaunay triangulation or Voronoi diagram of a set of points in R³. Since threedimensional Delaunay triangulations can have complexity Ω(n²) in the worst case, these algorithms have worstcase running time \Omega (n2). However, this behavior is almost never observed in practice except for highlycontrived inputs. For all practical purposes, threedimensional Delaunay triangulations appear to have linear complexity. This frustrating
Efficient Maximal PoissonDisk Sampling
, 2011
"... We solve the problem of generating a uniform Poissondisk sampling that is both maximal and unbiased over bounded nonconvex domains. To our knowledge this is the first provably correct algorithm with time and space dependent only on the number of points produced. Our method has two phases, both b ..."
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Cited by 26 (10 self)
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We solve the problem of generating a uniform Poissondisk sampling that is both maximal and unbiased over bounded nonconvex domains. To our knowledge this is the first provably correct algorithm with time and space dependent only on the number of points produced. Our method has two phases, both based on classical dartthrowing. The first phase uses a background grid of square cells to rapidly create an unbiased, nearmaximal covering of the domain. The second phase completes the maximal covering by calculating the connected components of the remaining uncovered voids, and by using their geometry to efficiently place unbiased samples that cover them. The second phase converges quickly, overcoming a common difficulty in dartthrowing methods. The deterministic memory is O(n) and the expected running time is O(n log n), where n is the output size, the number of points in the final sample. Our serial implementation verifies that the log n dependence is minor, and nearly O(n) performance for both time and memory is achieved in practice. We also present a parallel implementation on GPUs to demonstrate the parallelfriendly nature of our method, which achieves 2.4 × the performance of our serial version.
VietorisRips Complexes also Provide Topologically Correct Reconstructions of Sampled Shapes
, 2012
"... Given a point set that samples a shape, we formulate conditions under which the Rips complex of the point set at some scale reflects the homotopy type of the shape. For this, we associate with each compact set X of Rn two realvalued functions cX and hX defined on R+ which provide two measures of ho ..."
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Cited by 22 (8 self)
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Given a point set that samples a shape, we formulate conditions under which the Rips complex of the point set at some scale reflects the homotopy type of the shape. For this, we associate with each compact set X of Rn two realvalued functions cX and hX defined on R+ which provide two measures of how much the set X fails to be convex at a given scale. First, we show that, when P is a finite point set, an upper bound on cP (t) entails that the Rips complex of P at scale r collapses to the Čech complex of P at scale r for some suitable values of the parameters t and r. Second, we prove that, when P samples a compact set X, an upper bound on hX over some interval guarantees a topologically correct reconstruction of the shape X either with a Čech complex of P or with a Rips complex of P. Regarding the reconstruction with Čech complexes, our work compares well with previous approaches when X is a smooth set and surprisingly enough, even improves constants when X has a positive µreach. Most importantly, our work shows that Rips complexes can also be used to provide shape reconstructions having the correct homotopy type. This may be of some computational interest in high dimensions.
Complexity of Delaunay triangulation for points on lowerdimensional polyhedra
 PROC. 18TH ANNU. ACMSIAM SYMPOS. DISCRETE ALGO
, 2007
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On the combinatorial complexity of Euclidean Voronoi cells and convex hulls of ddimensional spheres
"... In this paper we show an equivalence relationship between additively weighted Voronoi cells in Rd, power diagrams in Rd and convex hulls of spheres in Rd. An immediate consequence of this equivalence relationship is a tight bound on the complexity of: (1) a single additively weighted Voronoi cell in ..."
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Cited by 20 (8 self)
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In this paper we show an equivalence relationship between additively weighted Voronoi cells in Rd, power diagrams in Rd and convex hulls of spheres in Rd. An immediate consequence of this equivalence relationship is a tight bound on the complexity of: (1) a single additively weighted Voronoi cell in dimension d; (2) the convex hull of a set of ddimensional spheres. In particular, given a set of n spheres in dimension d, we show that the worst case complexity of both a single additively weighted Voronoi cell and the convex hull of the set of spheres is d ⌈ Θ(n 2 ⌉). The equivalence between additively weighted Voronoi cells and convex hulls of spheres permits us to compute a single additively weighted Voronoi cell in did ⌈ mension d in worst case optimal time O(n log n+n 2 ⌉).
Delaunay Triangulations in O(sort(n)) Time and More
"... We present several results about Delaunay triangulations (DTs) and convex hulls in transdichotomous and hereditary settings: (i) the DT of a planar point set can be computed in expected time O(sort(n)) on a word RAM, where sort(n) is the time to sort n numbers. We assume that the word RAM supports ..."
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Cited by 12 (6 self)
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We present several results about Delaunay triangulations (DTs) and convex hulls in transdichotomous and hereditary settings: (i) the DT of a planar point set can be computed in expected time O(sort(n)) on a word RAM, where sort(n) is the time to sort n numbers. We assume that the word RAM supports the shuffleoperation in constant time; (ii) if we know the ordering of a planar point set in x and in ydirection, its DT can be found by a randomized algebraic computation tree of expected linear depth; (iii) given a universe U of points in the plane, we construct a data structure D for Delaunay queries: for any P ⊆ U, D can find the DT of P in time O(P  log log U); (iv) given a universe U of points in 3space in general convex position, there is a data structure D for convex hull queries: for any P ⊆ U, D can find the convex hull of P in time O(P (log log U) 2); (v) given a convex polytope in 3space with n vertices which are colored with χ> 2 colors, we can split it into the convex hulls of the individual color classes in time O(n(log log n) 2). The results (i)–(iii) generalize to higher dimensions. We need a wide range of techniques. Most prominently, we describe a reduction from DTs to nearestneighbor graphs that relies on a new variant of randomized incremental constructions using dependent sampling.
State of the Union (of Geometric Objects)
 CONTEMPORARY MATHEMATICS
"... Let C be a set of geometric objects in R d. The combinatorial complexity of the union of C is the total number of faces of all dimensions on its boundary. We survey the known upper bounds on the complexity of the union of n geometric objects satisfying various natural conditions. These bounds play ..."
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Cited by 11 (7 self)
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Let C be a set of geometric objects in R d. The combinatorial complexity of the union of C is the total number of faces of all dimensions on its boundary. We survey the known upper bounds on the complexity of the union of n geometric objects satisfying various natural conditions. These bounds play a central role in the analysis of many geometric algorithms, and the techniques used to attain these bounds are interesting in their own right.