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Geometric SpeedUp Techniques for Finding Shortest Paths in Large Sparse Graphs
, 2003
"... In this paper, we consider Dijkstra's algorithm for the single source single target shortest paths problem in large sparse graphs. The goal is to reduce the response time for online queries by using precomputed information. For the result of the preprocessing, we admit at most linear space. We as ..."
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Cited by 53 (14 self)
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In this paper, we consider Dijkstra's algorithm for the single source single target shortest paths problem in large sparse graphs. The goal is to reduce the response time for online queries by using precomputed information. For the result of the preprocessing, we admit at most linear space. We assume that a layout of the graph is given. From this layout, in the preprocessing, we determine for each edge a geometric object containing all nodes that can be reached on a shortest path starting with that edge. Based on these geometric objects, the search space for online computation can be reduced significantly. We present an extensive experimental study comparing the impact of different types of objects. The test data we use are traffic networks, the typical field of application for this scenario.
The Sensor Selection Problem for Bounded Uncertainty Sensing Models
 IEEE Tran. Automation Science and Engineering
, 2005
"... We address the problem of selecting sensors so as to minimize the error in estimating the position of a target. We consider a generic sensor model where the measurements can be interpreted as polygonal, convex subsets of the plane. This model applies to a large class of sensors including cameras. We ..."
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Cited by 49 (2 self)
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We address the problem of selecting sensors so as to minimize the error in estimating the position of a target. We consider a generic sensor model where the measurements can be interpreted as polygonal, convex subsets of the plane. This model applies to a large class of sensors including cameras. We present an approximation algorithm which guarantees that the resulting error in estimation is within a factor 2 of the least possible error. In establishing this result, we formally prove that a constant number of sensors suffice for a good estimate  an observation made by many researchers. In the second part of the paper, we study the scenario where the target's position is given by an uncertainty region and present algorithms for both probabilistic and online versions of this problem.
The virtual mesh: A geometric abstraction for efficiently computing radiosity
 ACM Transactions on Graphics
"... In this paper, we introduce a generalpurpose method for computing radiosity on scenes made of parametric surfaces with arbitrary trimming curves. By contrast with past approaches that require a tessellation of the input surfaces (be it made up of triangles or patches with simple trimming curves) or ..."
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Cited by 13 (6 self)
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In this paper, we introduce a generalpurpose method for computing radiosity on scenes made of parametric surfaces with arbitrary trimming curves. By contrast with past approaches that require a tessellation of the input surfaces (be it made up of triangles or patches with simple trimming curves) or some form of geometric approximation, our method takes fully advantage of the rich and compact mathematical representation of objects. At its core lies the virtual mesh, an abstraction of the input geometry that allows complex shapes to be illuminated as if they were simple primitives. The virtual mesh is a collection of normalized square domains to which the input surfaces are mapped while preserving their energy properties. Radiosity values are then computed on these supports before being lifted back to the original surfaces. To demonstrate the power of our method, we describe a highorder wavelet radiosity implementation that uses the virtual mesh. Examples of objects and environments, designed for interactive applications or virtual reality, are presented. They prove that, by exactly integrating curved surfaces in the resolution process, the virtual mesh allows complex scenes to be rendered more quickly, more accurately and much more naturally than with previously known methods.
Wavelet radiosity on arbitrary planar surfaces
 In Rendering Techniques 2000: 11th Eurographics Workshop on Rendering, Eurographics, 161–172. ISBN
, 2000
"... Abstract. Wavelet radiosity is, by its nature, restricted to parallelograms or triangles. This paper presents an innovative technique enabling wavelet radiosity computations on planar surfaces of arbitrary shape, including concave contours or contours with holes. This technique replaces the need for ..."
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Cited by 9 (3 self)
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Abstract. Wavelet radiosity is, by its nature, restricted to parallelograms or triangles. This paper presents an innovative technique enabling wavelet radiosity computations on planar surfaces of arbitrary shape, including concave contours or contours with holes. This technique replaces the need for triangulating such complicated shapes, greatly reducing the complexity of the wavelet radiosity algorithm and the computation time. It also gives a better approximation of the radiosity function, resulting in better visual results. Our technique works by separating the radiosity function from the surface geometry, extending the radiosity function defined on the original shape onto a simpler domain – a parallelogram – better behaved for hierarchical refinement and wavelet computations. 1
ArticulationInvariant Representation of Nonplanar Shapes
"... Abstract. Given a set of points corresponding to a 2D projection of a nonplanar shape, we would like to obtain a representation invariant to articulations (under no selfocclusions). It is a challenging problem since we need to account for the changes in 2D shape due to 3D articulations, viewpoint ..."
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Cited by 7 (1 self)
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Abstract. Given a set of points corresponding to a 2D projection of a nonplanar shape, we would like to obtain a representation invariant to articulations (under no selfocclusions). It is a challenging problem since we need to account for the changes in 2D shape due to 3D articulations, viewpoint variations, as well as the varying effects of imaging process on different regions of the shape due to its nonplanarity. By modeling an articulating shape as a combination of approximate convex parts connected by nonconvex junctions, we propose to preserve distances between a pair of points by (i) estimating the parts of the shape through approximate convex decomposition, by introducing a robust measure of convexity and (ii) performing partwise affine normalization by assuming a weak perspective camera model, and then relating the points using the inner distance which is insensitive to planar articulations. We demonstrate the effectiveness of our representation on a dataset with nonplanar articulations, and on standard shape retrieval datasets like MPEG7.
Geometric Shortest Path Containers
, 2004
"... In this paper, we consider Dijkstra's algorithm for the single source single target shortest path problem in large sparse graphs. The goal is to reduce the response time for online queries by using precomputed information. Due to the size of the graph, preprocessing space requirements can be onl ..."
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Cited by 2 (1 self)
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In this paper, we consider Dijkstra's algorithm for the single source single target shortest path problem in large sparse graphs. The goal is to reduce the response time for online queries by using precomputed information. Due to the size of the graph, preprocessing space requirements can be only linear in the number of nodes. We assume that a layout of the graph is given. In the preprocessing, we determine from this layout a geometric object for each edge containing all nodes that can be reached by a shortest path starting with that edge.
Laboratoire de l'Informatique du Paralllisme
, 1998
"... This paper discusses some algorithmic issues when computing with a heterogeneous network of workstations (the typical poor man's parallel computer). Dealing with processors of dioeerent speeds requires to use more involved strategies than blockcyclic data distributions ..."
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This paper discusses some algorithmic issues when computing with a heterogeneous network of workstations (the typical poor man's parallel computer). Dealing with processors of dioeerent speeds requires to use more involved strategies than blockcyclic data distributions
Minimal enclosing parallelepiped in 3D
"... We investigate the problem of nding a minimal volume parallelepiped enclosing a given set of n threedimensional points. We give two mathematical properties of these parallelepipeds, from which we derive two algorithms of theoretical complexity O(n ). Experiments show that in practice our quickes ..."
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We investigate the problem of nding a minimal volume parallelepiped enclosing a given set of n threedimensional points. We give two mathematical properties of these parallelepipeds, from which we derive two algorithms of theoretical complexity O(n ). Experiments show that in practice our quickest algorithm runs in O(n ) (at least for n 10 ). We also present our application in structural biology.
www.elsevier.com/locate/comgeo Minimal enclosing parallelepiped in 3D
, 2003
"... We investigate the problem of finding a minimal volume parallelepiped enclosing a given set of n threedimensional points. We give two mathematical properties of these parallelepipeds, from which we derive two algorithms of theoretical complexity O(n 6). Experiments show that in practice our quickest ..."
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We investigate the problem of finding a minimal volume parallelepiped enclosing a given set of n threedimensional points. We give two mathematical properties of these parallelepipeds, from which we derive two algorithms of theoretical complexity O(n 6). Experiments show that in practice our quickest algorithm runs in O(n 2) (at least for n � 10 5). We also present our application in structural biology. © 2004 Elsevier B.V. All rights reserved.
SUBMITTED TO IEEE TASE – REGULAR PAPER 1 Sensor Selection in Arbitrary Dimensions
"... Abstract — We address the sensor selection problem which arises in tracking and localization applications. In sensor selection, the goal is to select a small number of sensors whose measurements provide a good estimate of a target’s state (such as location). We focus on the bounded uncertainty sensi ..."
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Abstract — We address the sensor selection problem which arises in tracking and localization applications. In sensor selection, the goal is to select a small number of sensors whose measurements provide a good estimate of a target’s state (such as location). We focus on the bounded uncertainty sensing model where the target is a point in the d dimensional Euclidean space. Each sensor measurement corresponds to a convex, polyhedral subset of the space. The measurements are merged by intersecting corresponding sets. We show that, on the plane, four sensors are sufficient (and sometimes necessary) to obtain an estimate whose area is at most twice the area of the best possible estimate (obtained by intersecting all measurements). We also extend this result to arbitrary dimensions and show that a constant number of sensors suffice for a constant factor approximation in arbitrary dimensions. Both constants depend on the dimensionality of the space but are independent of the total number of sensors in the network. Note to Practitioners In many applications, sensing and communication constraints may render using all available sensors infeasible. In such scenarios, selecting a small number of sensors – whose collaborative performance in estimating the state of a target is comparable to the best possible achievable error – becomes important. This paper focuses on sensors whose measurements can be specified as an intersection of halfspaces (e.g. cameras, whose measurements correspond to cones). It is proven that a “small ” set of good sensors can be selected from an arbitrary set of measurements in any dimension d. Of practical importance are the two cases: d = 2 (where four sensors suffice for a good estimate) and d = 3 (eight sensors are enough). Index Terms — Sensor networks: camera networks and sensor