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87
Discrete Geometric Shapes: Matching, Interpolation, and Approximation: A Survey
 Handbook of Computational Geometry
, 1996
"... In this survey we consider geometric techniques which have been used to measure the similarity or distance between shapes, as well as to approximate shapes, or interpolate between shapes. Shape is a modality which plays a key role in many disciplines, ranging from computer vision to molecular biolog ..."
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Cited by 126 (10 self)
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In this survey we consider geometric techniques which have been used to measure the similarity or distance between shapes, as well as to approximate shapes, or interpolate between shapes. Shape is a modality which plays a key role in many disciplines, ranging from computer vision to molecular biology. We focus on algorithmic techniques based on computational geometry that have been developed for shape matching, simplification, and morphing. 1 Introduction The matching and analysis of geometric patterns and shapes is of importance in various application areas, in particular in computer vision and pattern recognition, but also in other disciplines concerned with the form of objects such as cartography, molecular biology, and computer animation. The general situation is that we are given two objects A, B and want to know how much they resemble each other. Usually one of the objects may undergo certain transformations like translations, rotations or scalings in order to be matched with th...
Efficient algorithms for geometric optimization
 ACM Comput. Surv
, 1998
"... We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, pruneandsearch techniques for linear progra ..."
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Cited by 94 (12 self)
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We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, pruneandsearch techniques for linear programming and related problems, and LPtype problems and their efficient solution. We then describe a variety of applications of these and other techniques to numerous problems in geometric optimization, including facility location, proximity problems, statistical estimators and metrology, placement and intersection of polygons and polyhedra, and ray shooting and other querytype problems.
Applications of parametric searching in geometric optimization
 J. Algorithms
, 1994
"... z Sivan Toledo x ..."
Geometric Pattern Matching under Euclidean Motion
, 1993
"... Given two planar sets A and B, we examine the problem of determining the smallest " such that there is a Euclidean motion (rotation and translation) of A that brings each member of A within distance " of some member of B. We establish upper bounds on the combinatorial complexity of this subproblem i ..."
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Cited by 73 (2 self)
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Given two planar sets A and B, we examine the problem of determining the smallest " such that there is a Euclidean motion (rotation and translation) of A that brings each member of A within distance " of some member of B. We establish upper bounds on the combinatorial complexity of this subproblem in modelbased computer vision, when the sets A and B contain points, line segments, or (filledin) polygons. We also show how to use our methods to substantially improve on existing algorithms for finding the minimum Hausdorff distance under Euclidean motion. 1 Author's address: Department of Computer Science, Cornell University, Ithaca, NY 14853. This work was supported by the Advanced Research Projects Agency of the Department of Defense under ONR Contract N0001492J1989, and by ONR Contract N0001492J1839, NSF Contract IRI9006137, and AFOSR Contract AFOSR910328. 2 Author's address: Department of Computer Science, Johns Hopkins University, Baltimore, MD 21218. This work was suppo...
OutputSensitive Results on Convex Hulls, Extreme Points, and Related Problems
, 1996
"... . We use known data structures for rayshooting and linearprogramming queries to derive new outputsensitive results on convex hulls, extreme points, and related problems. We show that the f face convex hull of an npoint set P in a fixed dimension d # 2 can be constructed in O(n log f + (nf) ..."
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Cited by 65 (13 self)
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. We use known data structures for rayshooting and linearprogramming queries to derive new outputsensitive results on convex hulls, extreme points, and related problems. We show that the f face convex hull of an npoint set P in a fixed dimension d # 2 can be constructed in O(n log f + (nf) 11/(#d/2#+1) log O(1) n) time; this is optimal if f = O(n 1/#d/2# / log K n) for some sufficiently large constant K . We also show that the h extreme points of P can be computed in O(n log O(1) h + (nh) 11/(#d/2#+1) log O(1) n) time. These results are then applied to produce an algorithm that computes the vertices of all the convex layers of P in O(n 2# ) time for any constant #<2/(#d/2# 2 + 1). Finally, we obtain improved time bounds for other problems including levels in arrangements and linear programming with few violated constraints. In all of our algorithms the input is assumed to be in general position. 1. Introduction Let P be a set of n points in ddimen...
Approximate Nearest Neighbor Queries Revisited
, 1998
"... This paper proposes new methods to answer approximate nearest neighbor queries on a set of n points in ddimensional Euclidean space. For any fixed constant d, a data structure with O(" (1\Gammad)=2 n log n) preprocessing time and O(" (1\Gammad)=2 log n) query time achieves approximation factor ..."
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Cited by 57 (3 self)
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This paper proposes new methods to answer approximate nearest neighbor queries on a set of n points in ddimensional Euclidean space. For any fixed constant d, a data structure with O(" (1\Gammad)=2 n log n) preprocessing time and O(" (1\Gammad)=2 log n) query time achieves approximation factor 1 + " for any given 0 ! " ! 1; a variant reduces the "dependence by a factor of " \Gamma1=2 . For any arbitrary d, a data structure with O(d 2 n log n) preprocessing time and O(d 2 log n) query time achieves approximation factor O(d 3=2 ). Applications to various proximity problems are discussed. 1 Introduction Let P be a set of n point sites in ddimensional space IR d . In the wellknown post office problem, we want to preprocess P into a data structure so that a site closest to a given query point q (called the nearest neighbor of q) can be found efficiently. Distances are measured under the Euclidean metric. The post office problem has many applications within computational...
Iterated Nearest Neighbors and Finding Minimal Polytopes
, 1994
"... Weintroduce a new method for finding several types of optimal kpoint sets, minimizing perimeter, diameter, circumradius, and related measures, by testing sets of the O(k) nearest neighbors to each point. We argue that this is better in a number of ways than previous algorithms, whichwere based o ..."
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Cited by 56 (6 self)
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Weintroduce a new method for finding several types of optimal kpoint sets, minimizing perimeter, diameter, circumradius, and related measures, by testing sets of the O(k) nearest neighbors to each point. We argue that this is better in a number of ways than previous algorithms, whichwere based on high order Voronoi diagrams. Our technique allows us for the first time to efficiently maintain minimal sets as new points are inserted, to generalize our algorithms to higher dimensions, to find minimal convex kvertex polygons and polytopes, and to improvemany previous results. Weachievemany of our results via a new algorithm for finding rectilinear nearest neighbors in the plane in time O(n log n+kn). We also demonstrate a related technique for finding minimum area kpoint sets in the plane, based on testing sets of nearest vertical neighbors to each line segment determined by a pair of points. A generalization of this technique also allows us to find minimum volume and boundary measure sets in arbitrary dimensions.
Geometric Applications of a Randomized Optimization Technique
 Discrete Comput. Geom
, 1999
"... We propose a simple, general, randomized technique to reduce certain geometric optimization problems to their corresponding decision problems. These reductions increase the expected time complexity by only a constant factor and eliminate extra logarithmic factors in previous, often more complicated, ..."
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Cited by 53 (6 self)
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We propose a simple, general, randomized technique to reduce certain geometric optimization problems to their corresponding decision problems. These reductions increase the expected time complexity by only a constant factor and eliminate extra logarithmic factors in previous, often more complicated, deterministic approaches (such as parametric searching). Faster algorithms are thus obtained for a variety of problems in computational geometry: finding minimal kpoint subsets, matching point sets under translation, computing rectilinear pcenters and discrete 1centers, and solving linear programs with k violations. 1 Introduction Consider the classic randomized algorithm for finding the minimum of r numbers minfA[1]; : : : ; A[r]g: Algorithm randmin 1. randomly pick a permutation hi 1 ; : : : ; i r i of h1; : : : ; ri 2. t /1 3. for k = 1; : : : ; r do 4. if A[i k ] ! t then 5. t / A[i k ] 6. return t By a wellknown fact [27, 44], the expected number of times that step 5 is execut...
Faster Construction of Planar Twocenters
, 1997
"... Improving on a recent breakthrough of Sharir, we find two minimumradius circular disks covering a planar point set, in randomized expected time O(n log 2 n). 1 Introduction The kcenter problem for a point set S is to find k points (called centers, usually not required to be a subset of S) such ..."
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Cited by 47 (0 self)
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Improving on a recent breakthrough of Sharir, we find two minimumradius circular disks covering a planar point set, in randomized expected time O(n log 2 n). 1 Introduction The kcenter problem for a point set S is to find k points (called centers, usually not required to be a subset of S) such that the maximum distance from any point in S to the nearest center is minimized. A case of particular interest is the planar twocenter problem [4], which can be viewed less abstractly as one of covering a set of points in the plane by two congruent circular disks, in such a way as to minimize the radius r # of the disks. For a long time the best algorithms for this problem had time bounds of the form O(n 2 log c n) [1, 5, 12, 11]. In a recent breakthrough, Sharir [16] greatly improved all of these algorithms, giving a twocenter algorithm with running time O(n log c n). The basic idea is to search for different types of partition depending on the relative positions of the two disk...
An Optimal Randomized Algorithm for Maximum Tukey Depth
, 2004
"... We present the first optimal algorithm to compute the maximum Tukey depth (also known as location or halfspace depth) for a nondegenerate point set in the plane. The algorithm is randomized and requires O(n log n) expected time for n data points. In a higher fixed dimension d 3, the expected tim ..."
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Cited by 46 (4 self)
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We present the first optimal algorithm to compute the maximum Tukey depth (also known as location or halfspace depth) for a nondegenerate point set in the plane. The algorithm is randomized and requires O(n log n) expected time for n data points. In a higher fixed dimension d 3, the expected time bound is O(n ), which is probably optimal as well. The result is obtained using an interesting variant of the author's randomized optimization technique, capable of solving "implicit" linearprogrammingtype problems; some other applications of this technique are briefly mentioned.