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Geometric Range Searching and Its Relatives
 CONTEMPORARY MATHEMATICS
"... ... process a set S of points in so that the points of S lying inside a query R region can be reported or counted quickly. Wesurvey the known techniques and data structures for range searching and describe their application to other related searching problems. ..."
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Cited by 256 (40 self)
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... process a set S of points in so that the points of S lying inside a query R region can be reported or counted quickly. Wesurvey the known techniques and data structures for range searching and describe their application to other related searching problems.
SelfCustomized BSP Trees for Collision Detection
, 2000
"... The ability to perform efficient collision detection is essential in virtual reality environments and their applications, such as walkthroughs. In this paper we reexplore a classical structure used for collision detection  the binary space partitioning tree. Unlike the common approach, which a ..."
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Cited by 14 (1 self)
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The ability to perform efficient collision detection is essential in virtual reality environments and their applications, such as walkthroughs. In this paper we reexplore a classical structure used for collision detection  the binary space partitioning tree. Unlike the common approach, which attributes equal likelihood to each possible query, we assume events that happened in the past are more likely to happen again in the future. This leads us to the definition of selfcustomized data structures. We report encouraging results obtained while experimenting with this concept in the context of selfcustomized bsp trees. Keywords: Collision detection, binary space partitioning, selfcustomization. 1 Introduction Virtual reality refers to the use of computer graphics to simulate physical worlds or to generate synthetic ones, where a user is to feel immersed in the environment to the extent that the user feels as if "objects" seen are really there. For example, "objects" should m...
When Crossings Count  Approximating the Minimum Spanning Tree
 In Proc. 16th Annu. ACM Sympos. Comput. Geom
, 2002
"... We present an (1+")approximation algorithm for computing the minimumspanning tree of points in a planar arrangement of lines, where the metric is the number of crossings between the spanning tree and the lines. The expected running time of the algorithm is near linear. We also show how to embed ..."
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Cited by 13 (5 self)
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We present an (1+")approximation algorithm for computing the minimumspanning tree of points in a planar arrangement of lines, where the metric is the number of crossings between the spanning tree and the lines. The expected running time of the algorithm is near linear. We also show how to embed such a crossing metric of hyperplanes in ddimensions, in subquadratic time, into highdimensions so that the distances are preserved. As a result, we can deploy a large collection of subquadratic approximations algorithms [IM98, GIV01] for problems involving points with the crossing metric as a distance function. Applications include MST, matching, clustering, nearestneighbor, and furthestneighbor.
Approximate Minimum Weight Steiner Triangulation in Three Dimensions
 In Proceedings of the Tenth Annual ACMSIAM Symposium on Discrete Algorithms
, 1999
"... Difficulty of minimum weight triangulation of a point set in R 2 is well known. In this paper we study the minimum weight triangulation problem for polyhedra and general obstacle set in three dimensions. The weight of a triangulation in three dimensions is assumed to be the total surface area of a ..."
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Cited by 2 (0 self)
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Difficulty of minimum weight triangulation of a point set in R 2 is well known. In this paper we study the minimum weight triangulation problem for polyhedra and general obstacle set in three dimensions. The weight of a triangulation in three dimensions is assumed to be the total surface area of all triangles involved. It is shown that a polyhedron P of size n can be triangulated with O(n 2 log n) tetrahedra in time O(n 2 log 3 n) approximating the minimum weight triangulation of P within a constant factor. No such prior result is known. The same bounds also hold for a 3D point set triangulation allowing Steiner points. We consider another setting called general obstacle set, where the convex hull of a set of n triangles is triangulated conforming to the input triangles. In this case we show that our method produces a triangulation of size O(n 3 log n) in time O(n 3 log 3 n) approximating the weight of the minimum weight triangulation within a constant factor. This is a ...
Quadtree Decomposition, Steiner Triangulation, and Ray shooting
 in Proc. 9th International Symposium on Algorithms and Computation
, 1998
"... . We present a new quadtreebased decomposition of a polygon possibly with holes. For a polygon of n vertices, a truncated decomposition can be computed in O(n log n) time which yields a Steiner triangulation of the interior of the polygon that has O(n log n) size and approximates the minimum we ..."
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Cited by 1 (1 self)
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. We present a new quadtreebased decomposition of a polygon possibly with holes. For a polygon of n vertices, a truncated decomposition can be computed in O(n log n) time which yields a Steiner triangulation of the interior of the polygon that has O(n log n) size and approximates the minimum weight Steiner triangulation (MWST) to within a constant factor. An approximate MWST is good for ray shooting in the average case as defined by Aronov and Fortune. The untruncated decomposition also yields an approximate MWST. Moreover, we show that this triangulation supports querysensitive ray shooting as defined by Mitchell, Mount, and Suri. Hence, there exists a Steiner triangulation that is simultaneously good for ray shooting in the querysensitive sense and in the average case. 1 Introduction Triangulation is a popular research topic because many problems call for a decomposition of a scene into simple elements that facilitate processing. In the plane, the focus has been on op...
FiniteResolution Hidden Surface Removal
"... We propose a hybrid imagespace/objectspace solution to the classical hidden surface removal problem: Given n disjoint triangles in IR 3 and p sample points (\pixels") in the xyplane, determine the rst triangle directly behind each pixel. Our algorithm constructs the sampled visibility map of th ..."
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We propose a hybrid imagespace/objectspace solution to the classical hidden surface removal problem: Given n disjoint triangles in IR 3 and p sample points (\pixels") in the xyplane, determine the rst triangle directly behind each pixel. Our algorithm constructs the sampled visibility map of the triangles with respect to the pixels, which is the subset of the trapezoids in a trapezoidal decomposition of the analytic visibility map that contain at least one pixel. The sampled visibility map adapts to local changes in image complexity, and its complexity is bounded both by the number of pixels and by the complexity of the analytic visibility map. Our algorithm runs in time O(n 1+" + n 2=3+" t 2=3 + p), where t is the output size. This is nearly optimal in the worst case and compares favorably with the best outputsensitive algorithms for both ray casting and analytic hidden surface removal. In the special case where the pixels form a regular grid, a sweepline variant of our a...
Sensor, motion and temporal planning
, 2006
"... We describe in this dissertation, planning strategies which enhance the accuracy with which visual surveillance can be conducted and which expand the capabilities of visual surveillance systems. Several classes of planning strategies are considered: sensor planning, motion planning and temporal plan ..."
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We describe in this dissertation, planning strategies which enhance the accuracy with which visual surveillance can be conducted and which expand the capabilities of visual surveillance systems. Several classes of planning strategies are considered: sensor planning, motion planning and temporal planning. Sensor planning is the study of the control of cameras to optimize information gathering for performing vision algorithms. The study of camera control spans camera placement strategies, active camera (specifically, PanTiltZoom or PTZ cameras) control, and, in some cases, camera selection from a collection of static cameras. Camera placement strategies have been employed previously for enhancing vision algorithms such as 3D reconstruction, area coverage in surveillance, occlusion and visibility analysis, etc. We will introduce a twocamera placement strategy that is utilized by a background subtraction algorithm, allowing it to achieve video rate performance and invariance to several illumination artifacts, such as lighting changes and shadows. While camera placement strategies can improve the performance of vision algorithms significantly, their utilities are limited in situations where it is more costeffective
Quadtree, Ray Shooting and Approximate Minimum Weight Steiner Triangulation
, 2001
"... We present a quadtreebased decomposition of the interior of a polygon with holes. The complete decomposition yields a constant factor approximation of the minimum weight Steiner triangulation (MWST) of the polygon. We show that this approximate MWST supports ray shooting queries in the queryse ..."
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We present a quadtreebased decomposition of the interior of a polygon with holes. The complete decomposition yields a constant factor approximation of the minimum weight Steiner triangulation (MWST) of the polygon. We show that this approximate MWST supports ray shooting queries in the querysensitive sense as defined by Mitchell, Mount and Suri. A proper truncation of our quadtreebased decomposition yields another constant factor approximation of the MWST. For a polygon with n vertices, the complexity of this approximate MWST is O(n log n) and it can be constructed in O(n log n) time. The running time is optimal in the algebraic decision tree model.