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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 560 (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.
Fast Computation of Generalized Voronoi Diagrams Using Graphics Hardware
, 1999
"... We present a new approach for computing generalized 2D and 3D Voronoi diagrams using interpolationbased polygon rasterization hardware. We compute a discrete Voronoi diagram by rendering a three dimensional distance mesh for each Voronoi site. The polygonal mesh is a boundederror approximation of ..."
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Cited by 195 (26 self)
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We present a new approach for computing generalized 2D and 3D Voronoi diagrams using interpolationbased polygon rasterization hardware. We compute a discrete Voronoi diagram by rendering a three dimensional distance mesh for each Voronoi site. The polygonal mesh is a boundederror approximation of a (possibly) nonlinear function of the distance between a site and a 2D planar grid of sample points. For each sample point, we compute the closest site and the distance to that site using polygon scanconversion and the Zbuffer depth comparison. We construct distance meshes for points, line segments, polygons, polyhedra, curves, and curved surfaces in 2D and 3D. We generalize to weighted and farthestsite Voronoi diagrams, and present efficient techniques for computing the Voronoi boundaries, Voronoi neighbors, and the Delaunay triangulation of points. We also show how to adaptively refine the solution through a simple windowing operation. The algorithm has been implemented on SGI workstations and PCs using OpenGL, and applied to complex datasets. We demonstrate the application of our algorithm to fast motion planning in static and dynamic environments, selection in complex userinterfaces, and creation of dynamic mosaic effects.
Closest Point Search in Lattices
 IEEE TRANS. INFORM. THEORY
, 2000
"... In this semitutorial paper, a comprehensive survey of closestpoint search methods for lattices without a regular structure is presented. The existing search strategies are described in a unified framework, and differences between them are elucidated. An efficient closestpoint search algorithm, ba ..."
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Cited by 194 (1 self)
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In this semitutorial paper, a comprehensive survey of closestpoint search methods for lattices without a regular structure is presented. The existing search strategies are described in a unified framework, and differences between them are elucidated. An efficient closestpoint search algorithm, based on the SchnorrEuchner variation of the Pohst method, is implemented. Given an arbitrary point x 2 R m and a generator matrix for a lattice , the algorithm computes the point of that is closest to x. The algorithm is shown to be substantially faster than other known methods, by means of a theoretical comparison with the Kannan algorithm and an experimental comparison with the Pohst algorithm and its variants, such as the recent ViterboBoutros decoder. The improvement increases with the dimension of the lattice. Modifications of the algorithm are developed to solve a number of related search problems for lattices, such as finding a shortest vector, determining the kissing number, compu...
Sliver Exudation
 ANNUAL SYMPOSIUM ON COMPUTATIONAL GEOMETRY
, 1999
"... A sliver is a tetrahedron whose four vertices lie close to a plane and whose orthogonal projection to that plane is a convex quadrilateral with no short edge. Slivers are notoriously common in 3dimensional Delaunay triangulations even for wellspaced point sets. We show that if the Delaunay triangu ..."
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Cited by 81 (11 self)
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A sliver is a tetrahedron whose four vertices lie close to a plane and whose orthogonal projection to that plane is a convex quadrilateral with no short edge. Slivers are notoriously common in 3dimensional Delaunay triangulations even for wellspaced point sets. We show that if the Delaunay triangulation has the ratio property introduced in [15] then there is an assignment of weights so the weighted Delaunay triangulation contains no slivers. We also give an algorithm to compute such a weight assignment.
On the Definition and the Construction of Pockets in Macromolecules
, 1995
"... The shape of a protein is important for its functions. This includes the location and size of identifiable regions in its complement space. We formally define pockets as regions in the complement with limited accessibility from the outside. Pockets can be efficiently constructed by an algorithm base ..."
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Cited by 78 (23 self)
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The shape of a protein is important for its functions. This includes the location and size of identifiable regions in its complement space. We formally define pockets as regions in the complement with limited accessibility from the outside. Pockets can be efficiently constructed by an algorithm based on alpha complexes. The algorithm is implemented and applied to proteins with known threedimensional conformations.
Linear Time Euclidean Distance Transform Algorithms
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 1995
"... Two linear time (and hence asymptotically optimal) algorithms for computing the Euclidean distance transform of a twodimensional binary image are presented. The algorithms are based on the construction and regular sampling of the Voronoi diagram whose sites consist of the unit (feature) pixels in t ..."
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Cited by 62 (0 self)
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Two linear time (and hence asymptotically optimal) algorithms for computing the Euclidean distance transform of a twodimensional binary image are presented. The algorithms are based on the construction and regular sampling of the Voronoi diagram whose sites consist of the unit (feature) pixels in the image. The first algorithm, which is of primarily theoretical interest, constructs the complete Voronoi diagram. The second, more practical, algorithm constructs the Voronoi diagram where it intersects the horizontal lines passing through the image pixel centres. Extensions to higher dimensional images and to other distance functions are also discussed. 1 Introduction A twodimensional binary image is a function, I, from the elements of an n by m array, referred to as pixels, to f0; 1g. Pixels of unit (respectively, zero) value are referred to as feature (respectively, background) pixels of the image. We associate the pixel in row r and column c with the Cartesian point (c; r). Thus, an...
Aspects of Unstructured Grids and FiniteVolume Solvers for the Euler and NavierStokes Equations (Part 4)
, 1995
"... this report, the model was tested on various subsonic and transonic flow problems: flat plates, airfoils, wakes, etc. The model consists of a single advectiondiffusion equation with source term for a field variable which is the product of turbulence Reynolds number and kinematic viscosity, e RT . ..."
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Cited by 56 (0 self)
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this report, the model was tested on various subsonic and transonic flow problems: flat plates, airfoils, wakes, etc. The model consists of a single advectiondiffusion equation with source term for a field variable which is the product of turbulence Reynolds number and kinematic viscosity, e RT . This variable is proportional to the eddy viscosity except very near a solid wall. The model equation is of the form: D( e RT ) Dt =(c ffl 2 f 2 (y + ) \Gamma c ffl 1 ) q e RT P +( + t oe R )r 2 ( e RT ) \Gamma 1 oe ffl (r t ) \Delta r( e RT ): (6:3:3) In this equation P is the production of turbulent kinetic energy and is related to the mean flow velocity rateofstrain and the kinematic eddy viscosity t . Equation (6.3.3) depends on distance to solid walls in two ways. First, the damping function f 2 appearing in equation (6.3.3) depends directly on distance to the wall (in wall units). Secondly, t depends on e R t and damping functions which require distance to the wall
The structure of multineuron firing patterns in primate retina
 Petrusca D, Sher A, Litke AM & Chichilnisky EJ
, 2006
"... Synchronized firing among neurons has been proposed to constitute an elementary aspect of the neural code in sensory and motor systems. However, it remains unclear how synchronized firing affects the largescale patterns of activity and redundancy of visual signals in a complete population of neuron ..."
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Cited by 54 (7 self)
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Synchronized firing among neurons has been proposed to constitute an elementary aspect of the neural code in sensory and motor systems. However, it remains unclear how synchronized firing affects the largescale patterns of activity and redundancy of visual signals in a complete population of neurons. We recorded simultaneously from hundreds of retinal ganglion cells in primate retina, and examined synchronized firing in completely sampled populations of �50–100 ONparasol cells, which form a major projection to the magnocellular layers of the lateral geniculate nucleus. Synchronized firing in pairs of cells was a subset of a much larger pattern of activity that exhibited local, isotropic spatial properties. However, a simple model based solely on interactions between adjacent cells reproduced 99 % of the spatial structure and scale of synchronized firing. No more than 20 % of the variability in firing of an individual cell was predictable from the activity of its neighbors. These results held both for spontaneous firing and in the presence of independent visual modulation of the firing of each cell. In sum, largescale synchronized firing in the entire population of ONparasol cells appears to reflect simple neighbor interactions, rather than a unique visual signal or a highly redundant coding scheme.