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Probabilistic Matching of Planar Regions
, 2009
"... We analyze a probabilistic algorithm for matching shapes modeled by planar regions under translations and rigid motions (rotation and translation). Given shapes A and B, the algorithm computes a transformation t such that with high probability the area of overlap of t(A) and B is close to maximal. I ..."
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Cited by 4 (3 self)
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We analyze a probabilistic algorithm for matching shapes modeled by planar regions under translations and rigid motions (rotation and translation). Given shapes A and B, the algorithm computes a transformation t such that with high probability the area of overlap of t(A) and B is close to maximal. In the case of polygons, we give a time bound that does not depend significantly on the number of vertices.
A probabilistic approach to optimal estimation  Part I: Problem formulation adn methodology
 In Proceedings IEEE Conference on Decision and Control
, 2012
"... Abstract — In this paper, we develop randomized and deterministic algorithms for computing the probabilistic radius of information associated to an identification problem, and the corresponding optimal probabilistic estimate. To compute this estimate, in the companion paper [11] the concept of optim ..."
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Abstract — In this paper, we develop randomized and deterministic algorithms for computing the probabilistic radius of information associated to an identification problem, and the corresponding optimal probabilistic estimate. To compute this estimate, in the companion paper [11] the concept of optimal violation function is introduced. Moreover, for the case of uniform distributions, it is shown how its computation is related to the solution of a (quasi) concave optimization problem, based on to the maximization of the volume of a specially constructed polytope. In this second paper, we move a step further and develop specific algorithms for addressing this problem. In particular, since the problem is NPhard, we propose both randomized relaxations (based on a probabilistic volume oracle and stochastic optimization algorithms), and deterministic ones (based on semidefinite programming). Finally, we present a numerical example illustrating the performance of the proposed algorithms.
Overlap of Convex Polytopes under Rigid Motion
, 2012
"... We present an algorithm to compute an approximate overlap of two convex polytopes P1 and P2 in R³ under rigid motion. Given any ε ∈ (0, 1/2], our algorithm runs in O(ε −3 n log 3.5 n) time with probability 1 − n −O(1) and returns a (1 − ε)approximate maximum overlap, provided that the maximum overl ..."
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We present an algorithm to compute an approximate overlap of two convex polytopes P1 and P2 in R³ under rigid motion. Given any ε ∈ (0, 1/2], our algorithm runs in O(ε −3 n log 3.5 n) time with probability 1 − n −O(1) and returns a (1 − ε)approximate maximum overlap, provided that the maximum overlap is at least λ · max{P1, P2} for some given constant λ ∈ (0, 1].
Scandinavian Thins on Top of Cake: New and improved algorithms for stacking and packing
"... We show how to compute the smallest rectangle that can enclose any polygon, from a given set of polygons, in nearly linear time; we also present a PTAS for the problem, as well as a lineartime algorithm for the case when the polygons are rectangles themselves. We prove that finding a smallest conve ..."
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We show how to compute the smallest rectangle that can enclose any polygon, from a given set of polygons, in nearly linear time; we also present a PTAS for the problem, as well as a lineartime algorithm for the case when the polygons are rectangles themselves. We prove that finding a smallest convex polygon that encloses any of the given polygons is NPhard, and give a PTAS for minimizing the perimeter of the convex enclosure. We also give efficient algorithms to find the smallest rectangle simultaneously enclosing a given pair of convex polygons.
Approximating the Maximum Overlap of Polygons under Translation
, 2014
"... Let P and Q be two simple polygons in the plane of total complexity n, each of which can be decomposed into at most k convex parts. We present an (1 − ε)approximation algorithm, for finding the translation of Q, which maximizes its area of overlap with P. Our algorithm runs in O(cn) time, where c i ..."
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Let P and Q be two simple polygons in the plane of total complexity n, each of which can be decomposed into at most k convex parts. We present an (1 − ε)approximation algorithm, for finding the translation of Q, which maximizes its area of overlap with P. Our algorithm runs in O(cn) time, where c is a constant that depends only on k and ε. This suggest that for polygons that are “close” to being convex, the problem can be solved (approximately), in near linear time.