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Voronoi Diagrams
 Handbook of Computational Geometry
"... Voronoi diagrams can also be thought of as lower envelopes, in the sense mentioned at the beginning of this subsection. Namely, for each point x not situated on a bisecting curve, the relation p x q defines a total ordering on S. If we construct a set of surfaces H p , p S,in3space such t ..."
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Cited by 143 (19 self)
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Voronoi diagrams can also be thought of as lower envelopes, in the sense mentioned at the beginning of this subsection. Namely, for each point x not situated on a bisecting curve, the relation p x q defines a total ordering on S. If we construct a set of surfaces H p , p S,in3space such that H p is below H q i# p x q holds, then the projection of their lower envelope equals the abstract Voronoi diagram.
MAPRM: A probabilistic roadmap planner with sampling on the medial axis of the free space
 In Proc. IEEE Int. Conf. Robot. Autom. (ICRA
, 1999
"... Probabilistic roadmap planning methods have been shown to perform well in a number of practical situations, but their performance degrades when paths are required to pass through narrow passages in the free space. We propose a new method of sampling the configuration space in which randomly generate ..."
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Cited by 142 (32 self)
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Probabilistic roadmap planning methods have been shown to perform well in a number of practical situations, but their performance degrades when paths are required to pass through narrow passages in the free space. We propose a new method of sampling the configuration space in which randomly generated configurations, free or not, are retracted onto the medial axis of the free space. We give algorithms that perform this retraction while avoiding explicit computation of the medial axis, and we show that sampling and retracting in this manner increases the number of nodes found in small volume corridors in a way that is independent of the volume of the corridor and depends only on the characteristics of the obstacles bounding it. Theoretical and experimental results are given to show that this improves performance on problems requiring traversal of narrow passages. 1
A Simple Algorithm for Nearest Neighbor Search in High Dimensions
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 1997
"... Abstract—The problem of finding the closest point in highdimensional spaces is common in pattern recognition. Unfortunately, the complexity of most existing search algorithms, such as kd tree and Rtree, grows exponentially with dimension, making them impractical for dimensionality above 15. In ne ..."
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Cited by 126 (1 self)
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Abstract—The problem of finding the closest point in highdimensional spaces is common in pattern recognition. Unfortunately, the complexity of most existing search algorithms, such as kd tree and Rtree, grows exponentially with dimension, making them impractical for dimensionality above 15. In nearly all applications, the closest point is of interest only if it lies within a userspecified distance e. We present a simple and practical algorithm to efficiently search for the nearest neighbor within Euclidean distance e. The use of projection search combined with a novel data structure dramatically improves performance in high dimensions. A complexity analysis is presented which helps to automatically determine e in structured problems. A comprehensive set of benchmarks clearly shows the superiority of the proposed algorithm for a variety of structured and unstructured search problems. Object recognition is demonstrated as an example application. The simplicity of the algorithm makes it possible to construct an inexpensive hardware search engine which can be 100 times faster than its software equivalent. A C++ implementation of our algorithm is available upon request to search@cs.columbia.edu/CAVE/.
Panoramic Mosaics by Manifold Projection
, 1997
"... As the field of view of a picture is much smaller than our own visual field of view, it is common to paste together several pictures to create a panoramic mosaic having a larger field of view. Images with a wider field of view can be generated by using fisheye lens, or panoramic mosaics can be crea ..."
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Cited by 121 (6 self)
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As the field of view of a picture is much smaller than our own visual field of view, it is common to paste together several pictures to create a panoramic mosaic having a larger field of view. Images with a wider field of view can be generated by using fisheye lens, or panoramic mosaics can be created by special devices which rotate around the camera's optical center (Quicktime VR, Surround Video), or by aligning, and pasting, frames in a video sequence to a single reference frame. Existing mosaicing methods have strong limitations on imaging conditions, and distortions are common. Manifold projection enables the creation of panoramic mosaics from video sequences under more general conditions, and in particular the unrestricted motion of a handheld camera. The panoramic mosaic is a projection of the scene into a virtual manifold whose structure depends on the camera's motion. This manifold is more general than the customary projections onto a single image plane or onto a cylinder.
Dynamic clustering for acoustic target tracking in wireless sensor networks
, 2003
"... In the paper, we devise and evaluate a fully decentralized, lightweight, dynamic clustering algorithm for target tracking. Instead of assuming the same role for all the sensors, we envision a hierarchical sensor network that is composed of (a) a static backbone of sparsely placed highcapability se ..."
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Cited by 92 (1 self)
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In the paper, we devise and evaluate a fully decentralized, lightweight, dynamic clustering algorithm for target tracking. Instead of assuming the same role for all the sensors, we envision a hierarchical sensor network that is composed of (a) a static backbone of sparsely placed highcapability sensors which will assume the role of a cluster head (CH) upon triggered by certain signal events; and (b) moderately to densely populated lowend sensors whose function is to provide sensor information to CHs upon request. A cluster is formed and a CH becomes active, when the acoustic signal strength detected by the CH exceeds a predetermined threshold. The active CH then broadcasts an information solicitation packet, asking sensors in its vicinity to join the cluster and provide their sensing information. We address and devise solution approaches (with the use of Voronoi diagram) to realize dynamic clustering: (I1) how CHs cooperate with one another to ensure that only one CH (preferably the CH that is closest to the target) is active with high probability; (I2) when the active CH solicits for sensor information, instead of having all the sensors in its vicinity reply, only a sufficient number of sensors respond with nonredundant, essential information to determine the target location; and and (I3) both the packets that sensors send to their CHs and packets that CHs report to subscribers do not incur significant collision. Through both probabilistic analysis and ns2 simulation, we show with the use of Voronoi diagram, the CH that is usually closest to the target is (implicitly) selected as the leader and and that the proposed dynamic clustering algorithm effectively eliminates contention among sensors and renders more accurate estimates of target locations as a result of better quality data collected and less collision incurred.
A Replacement for Voronoi Diagrams of Near Linear Size
 In Proc. 42nd Annu. IEEE Sympos. Found. Comput. Sci
, 2001
"... For a set P of n points in R^d, we define a new type of space decomposition. The new diagram provides an εapproximation to the distance function associated with the Voronoi diagram of P, while being of near linear size, for d ≥ 2. This contrasts with the standard Voronoi diagram that has ..."
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Cited by 87 (6 self)
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For a set P of n points in R^d, we define a new type of space decomposition. The new diagram provides an εapproximation to the distance function associated with the Voronoi diagram of P, while being of near linear size, for d ≥ 2. This contrasts with the standard Voronoi diagram that has complexity Ω(n^⌈d/2⌉) in the worst case.
Sensor Based Motion Planning: The Hierarchical Generalized Voronoi Graph
, 1996
"... The hierarchical generalized Voronoi graph (HGVG) is a roadmap that can serve as a basis for sensor based robot motion planning. A key feature of the HGVG is its incremental construction procedure that uses only line of sight distance information. This work describes basic properties of the HGVG and ..."
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Cited by 79 (9 self)
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The hierarchical generalized Voronoi graph (HGVG) is a roadmap that can serve as a basis for sensor based robot motion planning. A key feature of the HGVG is its incremental construction procedure that uses only line of sight distance information. This work describes basic properties of the HGVG and the procedure for its incremental construction using local range sensors. Simulations and experiments verify this approach. 1 Introduction Sensor based motion planning incorporates sensor information, reflecting the current state of the environment, into a robot's planning process, as opposed to classical planning, which assumes full knowledge of the world's geometry prior to planning. Sensor based planning is important for realistic deployment of robots because: (1) the robot often has no a priori knowledge of the world; (2) the robot may have only a coarse knowledge of the world because of limited computer memory; (3) the world model is bound to contain inaccuracies which can be overcom...
Locationbased Spatial Queries
 In SIGMOD
, 2003
"... In this paper we propose an approach that enables mobile clients to determine the validity of previous queries based on their current locations. In order to make this possible, the server returns in addition to the query result, a validity region around the client's location within which the result ..."
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Cited by 78 (11 self)
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In this paper we propose an approach that enables mobile clients to determine the validity of previous queries based on their current locations. In order to make this possible, the server returns in addition to the query result, a validity region around the client's location within which the result remains the same. We focus on two of the most common spatial query types, namely nearest neighbor and window queries, define the validity region in each case and propose the corresponding query processing algorithms. In addition, we provide analytical models for estimating the expected size of the validity region. Our techniques can significantly reduce the number of queries issued to the server, while introducing minimal computational and network overhead compared to traditional spatial queries.
Arrangements and Their Applications
 Handbook of Computational Geometry
, 1998
"... The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arr ..."
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Cited by 78 (22 self)
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The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A. was supported by Army Research Office MURI grant DAAH049610013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR9122103 and CCR9311127, by a MaxPlanck Research Award, and by grants from the U.S.Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...