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190
The Quickhull algorithm for convex hulls
 ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE
, 1996
"... The convex hull of a set of points is the smallest convex set that contains the points. This article presents a practical convex hull algorithm that combines the twodimensional Quickhull Algorithm with the generaldimension BeneathBeyond Algorithm. It is similar to the randomized, incremental algo ..."
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Cited by 501 (0 self)
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The convex hull of a set of points is the smallest convex set that contains the points. This article presents a practical convex hull algorithm that combines the twodimensional Quickhull Algorithm with the generaldimension BeneathBeyond Algorithm. It is similar to the randomized, incremental algorithms for convex hull and Delaunay triangulation. We provide empirical evidence that the algorithm runs faster when the input contains nonextreme points and that it uses less memory. Computational geometry algorithms have traditionally assumed that input sets are well behaved. When an algorithm is implemented with floatingpoint arithmetic, this assumption can lead to serious errors. We briefly describe a solution to this problem when computing the convex hull in two, three, or four dimensions. The output is a set of “thick ” facets that contain all possible exact convex hulls of the input. A variation is effective in five or more dimensions.
Topological Simultaneous Localization and Mapping (SLAM): Toward Exact Localization Without Explicit Localization
 IEEE Transactions on Robotics and Automation
, 2001
"... One of the critical components of mapping an unknown environment is the robot's ability to locate itself on a partially explored map. This becomes challenging when the robot experiences positioning error, does not have an external positioning device, nor the luxury of engineered landmarks place ..."
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Cited by 193 (10 self)
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One of the critical components of mapping an unknown environment is the robot's ability to locate itself on a partially explored map. This becomes challenging when the robot experiences positioning error, does not have an external positioning device, nor the luxury of engineered landmarks placed in its free space. This paper presents a new method for simultaneous localization and mapping that exploits the topology of the robot's free space to localize the robot on a partially constructed map. The topology of the environment is encoded in a topological map; the particular topological map used in this paper is the generalized Voronoi graph (GVG), which also encodes some metric information about the robot's environment, as well. In this paper, we present the lowlevel control laws that generate the GVG edges and nodes, thereby allowing for exploration of an unknown space. With these prescribed control laws, the GVG (or other topological map) can be viewed as an arbitrator for a hybrid control system that determines when to invoke a particular lowlevel controller from a set of controllers all working toward the highlevel capability of mobile robot exploration. The main contribution, however, is using the graph structure of the GVG, via a graph matching process, to localize the robot. Experimental results verify the described work. Index TermsExploration, localization, mapping, mobile robots, motion planning, tologoical maps, Voronoi diagrams. I.
Mesh Generation And Optimal Triangulation
, 1992
"... We survey the computational geometry relevant to finite element mesh generation. We especially focus on optimal triangulations of geometric domains in two and threedimensions. An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some cri ..."
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Cited by 188 (7 self)
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We survey the computational geometry relevant to finite element mesh generation. We especially focus on optimal triangulations of geometric domains in two and threedimensions. An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some criterion that measures the size, shape, or number of triangles. We discuss algorithms both for the optimization of triangulations on a fixed set of vertices and for the placement of new vertices (Steiner points). We briefly survey the heuristic algorithms used in some practical mesh generators.
Movementassisted sensor deployment
, 2006
"... Adequate coverage is very important for sensor networks to fulfill the issued sensing tasks. In many working environments, it is necessary to make use of mobile sensors, which can move to the correct places to provide the required coverage. In this paper, we study the problem of placing mobile senso ..."
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Cited by 167 (8 self)
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Adequate coverage is very important for sensor networks to fulfill the issued sensing tasks. In many working environments, it is necessary to make use of mobile sensors, which can move to the correct places to provide the required coverage. In this paper, we study the problem of placing mobile sensors to get high coverage. Based on Voronoi diagrams, we design two sets of distributed protocols for controlling the movement of sensors, one favoring communication and one favoring movement. In each set of protocols, we use Voronoi diagrams to detect coverage holes and use one of three algorithms to calculate the target locations of sensors if holes exist. Simulation results show the effectiveness of our protocols and give insight on choosing protocols and calculation algorithms under different application requirements and working conditions.
Incremental Topological Flipping Works for Regular Triangulations
 ALGORITHMICA
, 1996
"... A set of n weighted points in general position in Rd defines a unique regular triangulation. This paper proves that if the points are added one by one, then flipping in a topological order will succeed in constructing this triangulation. If, in addition, the points are added in a random sequence an ..."
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Cited by 161 (7 self)
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A set of n weighted points in general position in Rd defines a unique regular triangulation. This paper proves that if the points are added one by one, then flipping in a topological order will succeed in constructing this triangulation. If, in addition, the points are added in a random sequence and the history of the flips is used for locating the next point, then the algorithm takes expected time at most O(n log n+n ⌈d/2 ⌉). Under the assumption that the points and weights are independently and identically distributed, the expected running time is between proportional to and a factor log n more than the expected size of the regular triangulation. The expectation is over choosing the points and over independent coinflips performed by the algorithm.
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 134 (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/.
SelfConfiguring Localization Systems: Design and Experimental Evaluation
 ACM TRANSACTIONS ON EMBEDDED COMPUTING SYSTEMS
, 2003
"... ..."
Finding the Medial Axis of a Simple Polygon in Linear Time
 Discrete Comput. Geom
, 1995
"... We give a lineartime algorithm for computing the medial axis of a simple polygon P , This answers a longstanding open question  previously, the best deterministic algorithm ran in O(n log n) time. We decompose P into pseudonormal histograms, then influence histograms and xy monotone histograms. ..."
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Cited by 73 (4 self)
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We give a lineartime algorithm for computing the medial axis of a simple polygon P , This answers a longstanding open question  previously, the best deterministic algorithm ran in O(n log n) time. We decompose P into pseudonormal histograms, then influence histograms and xy monotone histograms. We can compute the medial axes for xy monotone histograms and merge to obtain the medial axis for P .
A linear time algorithm for computing exact Euclidean distance transforms of binary images in arbitrary dimensions
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2003
"... Abstract—A sequential algorithm is presented for computing the exact Euclidean distance transform (DT) of a kdimensional binary image in time linear in the total number of voxelsN. The algorithm, which is based on dimensionality reduction and partial Voronoi diagram construction, can be used for co ..."
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Cited by 62 (3 self)
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Abstract—A sequential algorithm is presented for computing the exact Euclidean distance transform (DT) of a kdimensional binary image in time linear in the total number of voxelsN. The algorithm, which is based on dimensionality reduction and partial Voronoi diagram construction, can be used for computing the DT for a wide class of distance functions, including the Lp and chamfer metrics. At each dimension level, the DT is computed by constructing the intersection of the Voronoi diagram whose sites are the feature voxels with each row of the image. This construction is performed efficiently by using the DT in the next lower dimension. The correctness and linear time complexity are demonstrated analytically and verified experimentally. The algorithm may be of practical value since it is relatively simple and easy to implement and it is relatively fast (not only does it run in O N time but the time constant is small). A simple modification of the algorithm computes the weighted Euclidean DT, which is useful for images with anisotropic voxel dimensions. A parallel version of the algorithm runs in O
N=p time with p processors. Index Terms—Euclidean distance transform, closest feature transform, Voronoi diagram. æ
The Farthest Point Strategy for Progressive Image Sampling
 IEEE TRANS. ON IMAGE PROCESSING
, 1997
"... A new method of farthest point strategy (FPS) for progressive image acquisition—an acquisition process that enables an approximation of the whole image at each sampling stage—is presented. Its main advantage is in retaining its uniformity with the increased density, providing efficient means for s ..."
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Cited by 58 (1 self)
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A new method of farthest point strategy (FPS) for progressive image acquisition—an acquisition process that enables an approximation of the whole image at each sampling stage—is presented. Its main advantage is in retaining its uniformity with the increased density, providing efficient means for sparse image sampling and display. In contrast to previously presented stochastic approaches, the FPS guarantees the uniformity in a deterministic minmax sense. Within this uniformity criterion, the sampling points are irregularly spaced, exhibiting antialiasing properties comparable to those characteristic of the best available method (Poisson disk). A straightforward modification of the FPS yields an imagedependent adaptive sampling scheme. An efficient O(N log N) algorithm for both versions is introduced, and several applications of the FPS are discussed.