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162
Primitives for the manipulation of general subdivisions and the computations of Voronoi diagrams
 ACM Tmns. Graph
, 1985
"... The following problem is discussed: Given n points in the plane (the sites) and an arbitrary query point 4, find the site that is closest to q. This problem can be solved by constructing the Voronoi diagram of the given sites and then locating the query point in one of its regions. Two algorithms ar ..."
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Cited by 491 (10 self)
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The following problem is discussed: Given n points in the plane (the sites) and an arbitrary query point 4, find the site that is closest to q. This problem can be solved by constructing the Voronoi diagram of the given sites and then locating the query point in one of its regions. Two algorithms are given, one that constructs the Voronoi diagram in O(n log n) time, and another that inserts a new site in O(n) time. Both are based on the use of the Voronoi dual, or Delaunay triangulation, and are simple enough to be of practical value. The simplicity of both algorithms can be attributed to the separation of the geometrical and topological aspects of the problem and to the use of two simple but powerful primitives, a geometric predicate and an operator for manipulating the topology of the diagram. The topology is represented by a new data structure for generalized diagrams, that is, embeddings of graphs in twodimensional manifolds. This structure represents simultaneously an embedding, its dual, and its mirror image. Furthermore, just two operators are sufficient for building and modifying arbitrary diagrams.
An optimal algorithm for intersecting line segments in the plane
 J. ACM
, 1992
"... Abstract. Themain contribution ofthiswork is an O(nlogr ~ +k)timeal gorithmfo rcomputingall k intersections among n line segments in the plane, This time complexity IS easdy shown to be optimal. Within thesame asymptotic cost, ouralgorithm canalso construct thesubdiwslon of theplancdefmed by the se ..."
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Cited by 170 (2 self)
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Abstract. Themain contribution ofthiswork is an O(nlogr ~ +k)timeal gorithmfo rcomputingall k intersections among n line segments in the plane, This time complexity IS easdy shown to be optimal. Within thesame asymptotic cost, ouralgorithm canalso construct thesubdiwslon of theplancdefmed by the segments and compute which segment (if any) lies right above (or below) each intersection and each endpoint. The algorithm has been implemented and performs very well. The storage requirement is on the order of n + k in the worst case, but it is considerably lower in practice. To analyze the complexity of the algorithm, an amortization argument based on a new combinatorial theorem on line arrangements is used.
Exact polyhedral visual hulls
 In British Machine Vision Conference
, 2003
"... We propose an exact method for efficiently and robustly computing the visual hull of an object from image contours. Unlike most existing approaches, ours computes an exact description of the visual hull polyhedron associated to polygonal image contours. Furthermore, the proposed approach is fast and ..."
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Cited by 66 (12 self)
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We propose an exact method for efficiently and robustly computing the visual hull of an object from image contours. Unlike most existing approaches, ours computes an exact description of the visual hull polyhedron associated to polygonal image contours. Furthermore, the proposed approach is fast and allows realtime recovery of both manifold and watertight visual hull polyhedra. The process involves three main steps. First, a coarse geometrical approximation of the visual hull is computed by retrieving its viewing edges, an unconnected subset of the wanted mesh. Then, local orientation and connectivity rules are used to walk along the relevant viewing cone intersection boundaries, so as to iteratively generate the missing surface points and connections. A final connection walkthrough allows us to identify the planar contours for each face of the polyhedron. Implementation details and results with synthetic and real data are presented. 1
Planar Separators and Parallel Polygon Triangulation
"... We show how to construct an O ( p n)separator decomposition of a planar graph G in O(n) time. Such a decomposition defines a binary tree where each node corresponds to a subgraph of G and stores an O ( p n)separator of that subgraph. We also show how to construct an O(n)way decomposition tree in ..."
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Cited by 51 (8 self)
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We show how to construct an O ( p n)separator decomposition of a planar graph G in O(n) time. Such a decomposition defines a binary tree where each node corresponds to a subgraph of G and stores an O ( p n)separator of that subgraph. We also show how to construct an O(n)way decomposition tree in parallel in O(log n) time so that each node corresponds to a subgraph of G and stores an O(n 1=2+)separator of that subgraph. We demonstrate the utility of such a separator decomposition by showing how it can be used in the design of a parallel algorithm for triangulating a simple polygon deterministically in O(log n) time using O(n = log n) processors on a CRCW PRAM.
Using Generic Programming for Designing a Data Structure for Polyhedral Surfaces
 Comput. Geom. Theory Appl
, 1999
"... Appeared in Computational Geometry  Theory and Applications 13, 1999, 6590. Software design solutions are presented for combinatorial data structures, such as polyhedral surfaces and planar maps, tailored for program libraries in computational geometry. Design issues considered are flexibility, ..."
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Cited by 49 (5 self)
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Appeared in Computational Geometry  Theory and Applications 13, 1999, 6590. Software design solutions are presented for combinatorial data structures, such as polyhedral surfaces and planar maps, tailored for program libraries in computational geometry. Design issues considered are flexibility, time and space efficiency, and easeofuse. We focus on topological aspects of polyhedral surfaces and evaluate edgebased representations with respect to our design goals. A design for polyhedral surfaces in a halfedge data structure is developed following the generic programming paradigm known from the Standard Template Library STL for C++. Connections are shown to planar maps and facebased structures. Key words: Library design; Generic programming; Combinatorial data structure; Polyhedral surface; Halfedge data structure 1 Introduction Combinatorial structures, such as planar maps, are fundamental in computational geometry. In order to be useful in practice, a solid library for compu...
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 41 (7 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.
Designing a Data Structure for Polyhedral Surfaces
 In Proc. 14th Annu. ACM Sympos. Comput. Geom
, 1998
"... Design solutions for a program library are presented for combinatorial data structures in computational geometry, such as planar maps and polyhedral surfaces. Design issues considered are genericity, flexibility, time and space efficiency, and easeofuse. We focus on topological aspects of polyhedr ..."
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Cited by 31 (2 self)
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Design solutions for a program library are presented for combinatorial data structures in computational geometry, such as planar maps and polyhedral surfaces. Design issues considered are genericity, flexibility, time and space efficiency, and easeofuse. We focus on topological aspects of polyhedral surfaces. Edgebased representations for polyhedrons are evaluated with respect to the design goals. A design for polyhedral surfaces in a halfedge data structure is developed following the generic programming paradigm known from the Standard Template Library STL for C++. Connections are shown to planar maps and facebased structures managing holes in facets. 1 Introduction Combinatorial structures, such as planar maps, are fundamental in computational geometry. In order to use computational geometry in practice, a solid library must provide generic and flexible solutions as one of its fundamental cornerstones. Other design criteria are time and space efficiency. Easeofuse is necessar...
Simplification and Compression of 3D Meshes
 In Proceedings of the European Summer School on Principles of Multiresolution in Geometric Modelling (PRIMUS
, 1998
"... We survey recent developments in compact representations of 3D mesh data. This includes: Methods to reduce the complexity of meshes by simplification, thereby reducing the number of vertices and faces in the mesh; Methods to resample the geometry in order to optimize the vertex distribution; Methods ..."
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Cited by 31 (5 self)
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We survey recent developments in compact representations of 3D mesh data. This includes: Methods to reduce the complexity of meshes by simplification, thereby reducing the number of vertices and faces in the mesh; Methods to resample the geometry in order to optimize the vertex distribution; Methods to compactly represent the connectivity data (the graph structure defined by the edges) of the mesh; Methods to compactly represent the geometry data (the vertex coordinates) of a mesh.
FeatureBased Cellular Texturing for Architectural Models
 In Proceedings of ACM SIGGRAPH 2001, ACM
, 2001
"... Cellular patterns are all around us, in masonry, tiling, shingles, and many other materials. Such patterns, especially in architectural settings, are influenced by geometric features of the underlying shape. Bricks turn corners, stones frame windows and doorways, and patterns on disconnected portion ..."
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Cited by 30 (2 self)
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Cellular patterns are all around us, in masonry, tiling, shingles, and many other materials. Such patterns, especially in architectural settings, are influenced by geometric features of the underlying shape. Bricks turn corners, stones frame windows and doorways, and patterns on disconnected portions of a building align to achieve a particular aesthetic goal. We present a strategy for featurebased cellular texturing, where the resulting texture is derived from both patterns of cells and the geometry to which they are applied. As part of this strategy, we perform texturing operations on features in a welldefined order that simplifies the interdependence between cells of adjacent patterns. Occupancy maps are used to indicate which regions of a feature are already occupied by cells of its neighbors, and which regions remain to be textured. We also introduce the notion of a pattern generator  the cellular texturing analogy of a shader used in local illumination  and show how several can be used together to build complex textures. We present results obtained with an implementation of this strategy and discuss details of some example pattern generators.