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Comparing Images Using the Hausdorff Distance
 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE
, 1993
"... The Hausdorff distance measures the extent to which each point of a `model' set lies near some point of an `image' set and vice versa. Thus this distance can be used to determine the degree of resemblance between two objects that are superimposed on one another. In this paper we provide ef ..."
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Cited by 659 (10 self)
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The Hausdorff distance measures the extent to which each point of a `model' set lies near some point of an `image' set and vice versa. Thus this distance can be used to determine the degree of resemblance between two objects that are superimposed on one another. In this paper we provide efficient algorithms for computing the Hausdorff distance between all possible relative positions of a binary image and a model. We focus primarily on the case in which the model is only allowed to translate with respect to the image. Then we consider how to extend the techniques to rigid motion (translation and rotation). The Hausdorff distance computation differs from many other shape comparison methods in that no correspondence between the model and the image is derived. The method is quite tolerant of small position errors as occur with edge detectors and other feature extraction methods. Moreover, we show how the method extends naturally to the problem of comparing a portion of a model against an i...
Applications of parametric searching in geometric optimization
 J. Algorithms
, 1994
"... z Sivan Toledo x ..."
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Distance Measures for Point Sets and Their Computation
 Acta Informatica
, 1997
"... We consider the problem of measuring the similarity or distance between two finite sets of points in a metric space, and computing the measure. This problem has applications in, e.g., computational geometry, philosophy of science, updating or changing theories, and machine learning. We review some o ..."
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Cited by 71 (2 self)
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We consider the problem of measuring the similarity or distance between two finite sets of points in a metric space, and computing the measure. This problem has applications in, e.g., computational geometry, philosophy of science, updating or changing theories, and machine learning. We review some of the distance functions proposed in the literature, among them the minimum distance link measure, the surjection measure, and the fair surjection measure, and supply polynomial time algorithms for the computation of these measures. Furthermore, we introduce the minimum link measure, a new distance function which is more appealing than the other distance functions mentioned. We also present a polynomial time algorithm for computing this new measure. We further address the issue of defining a metric on point sets. We present the metric infimum method that constructs a metric from any distance functions on point sets. In particular, the metric infimum of the minimum link measure is a quite int...
On Dynamic Voronoi Diagrams and the Minimum Hausdorff Distance for Point Sets Under Euclidean Motion in the Plane
, 1992
"... We show that the dynamic Voronoi diagram of k sets of points in the plane, where each set consists of n points moving rigidly, has complexity O(n 2 k 2 s (k)) for some fixed s, where s (n) is the maximum length of a (n; s) DavenportSchinzel sequence. This improves the result of Aonuma et. al ..."
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Cited by 51 (3 self)
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We show that the dynamic Voronoi diagram of k sets of points in the plane, where each set consists of n points moving rigidly, has complexity O(n 2 k 2 s (k)) for some fixed s, where s (n) is the maximum length of a (n; s) DavenportSchinzel sequence. This improves the result of Aonuma et. al., who show an upper bound of O(n 3 k 4 log k) for the complexity of such Voronoi diagrams. We then apply this result to the problem of finding the minimum Hausdorff distance between two point sets in the plane under Euclidean motion. We show that this distance can be computed in time O((m + n) 6 log(mn)), where the two sets contain m and n points respectively. This work was supported in part by NSF grant IRI9057928 and matching funds from General Electric and Kodak, and in part by AFOSR under contract AFOSR910328. The second author was also supported by the Eshkol grant 0460190 from The Israeli Ministry of Science and Technology. 1. Introduction Determining the degree to ...
Geometry helps in bottleneck matching and related problems
 Algorithmica
, 2001
"... This paper is accepted for publication in ALGORITHMICA Abstract Let A and B be two sets of n objects in Rd, and let Match be a (onetoone)matching between A and B. Let min(Match), max(Match), and \Sigma (Match) denote thelength of the shortest edge, the length of the longest edge, and the sum of th ..."
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Cited by 36 (5 self)
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This paper is accepted for publication in ALGORITHMICA Abstract Let A and B be two sets of n objects in Rd, and let Match be a (onetoone)matching between A and B. Let min(Match), max(Match), and \Sigma (Match) denote thelength of the shortest edge, the length of the longest edge, and the sum of the lengths of the edges of Match respectively. Bottleneck matchinga matching that minimizesmax( Match)is suggested as a convenient way for measuring the resemblance between A and B. Several algorithms for computing, as well as approximating, this resemblanceare proposed. The running time of all the algorithms involving planar objects is roughly O(n1.5). For instance, if the objects are points in the plane, the running time of the exactalgorithm is O(n1.5 log n). A semidynamic datastructure for answering containmentproblems for a set of congruent disks in the plane is developed. This data structure may be of independent interest.Next, the problem of finding a translation of B that maximizes the resemblance to A under the bottleneck matching criterion is considered. When A and B are pointsetsin the plane, an O(n5 log n) time algorithm for determining whether for some translatedcopy the resemblance gets below a given ae is presented, thus improving the previousresult of Alt, Mehlhorn, Wagener and Welzl by a factor of almost n. This result is usedto compute the smallest such ae in time O(n5 log2 n), and an efficient approximationscheme for this problem is also given. The uniform matching problem (also called the balanced assignment problem, or thefair matching problem) is to find Match*U, a matching that minimizes max(Match)min ( Match). A minimum deviation matching Match*D is a matching that minimizes(1 /n)\Sigma (Match) min(Match). Algorithms for computing Match*U and Match*D inroughly O(n10/3) time are presented. These algorithms are more efficient than theprevious
Pattern Matching for Spatial Point Sets
 PROC. 39TH ANNU. IEEE SYMPOS. FOUND. COMPUT. SCI
, 1998
"... Two sets of points in ddimensional space are given: a data set D consisting of N points, and a pattern set or probe P consisting of k points. We address the problem of determining whether there is a transformation, among a specified group of transformations of the space, carrying P into or near (me ..."
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Cited by 36 (0 self)
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Two sets of points in ddimensional space are given: a data set D consisting of N points, and a pattern set or probe P consisting of k points. We address the problem of determining whether there is a transformation, among a specified group of transformations of the space, carrying P into or near (meaning at a small directed Hausdorff distance of) D. The groups we consider are translations and rigid motions. Runtimes of approximately O(n log n) and O(n d log n) respectively are obtained (letting n = maxfN; kg and omitting the effects of several secondary parameters). For translations, a runtime of approximately O(n(ak + 1) log² n) is obtained for the case that a constant fraction a ! 1 of the points of the probe is allowed to fail to match.
Mobile robot selflocalization without explicit landmarks
 Algorithmica
, 2000
"... Abstract. Localization is the process of determining the robot’s location within its environment. More precisely, it is a procedure which takes as input a geometric map, a current estimate of the robot’s pose, and sensor readings, and produces as output an improved estimate of the robot’s current po ..."
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Cited by 15 (2 self)
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Abstract. Localization is the process of determining the robot’s location within its environment. More precisely, it is a procedure which takes as input a geometric map, a current estimate of the robot’s pose, and sensor readings, and produces as output an improved estimate of the robot’s current pose (position and orientation). We describe a combinatorially precise algorithm which performs mobile robot localization using a geometric model of the world and a pointandshoot ranging device. We also describe a rasterized version of this algorithm which we have implemented on a real mobile robot equipped with a laser rangefinder we designed. Both versions of the algorithm allow for uncertainty in the data returned by the range sensor. We also present experimental results for the rasterized algorithm, obtained using our mobile robots at Cornell. Key Words. Navigation, Mobile robots, Rasterized algorithms, Localization. 1. Introduction. Localization
Algorithmic techniques for geometric optimization
 In Computer Science Today: Recent Trends and Developments, Lecture Notes in Computer Science
, 1995
"... ..."
Noncrossing Matchings of Points with Geometric Objects
, 2011
"... Given an ordered set of points and an ordered set of geometric objects in the plane, we are interested in finding a noncrossing matching between pointobject pairs. In this paper, we address the algorithmic problem of determining whether a noncrossing matching exists between a given pointobject p ..."
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Cited by 3 (0 self)
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Given an ordered set of points and an ordered set of geometric objects in the plane, we are interested in finding a noncrossing matching between pointobject pairs. In this paper, we address the algorithmic problem of determining whether a noncrossing matching exists between a given pointobject pair. We show that when the objects we match the points to are finite point sets, the problem is NPcomplete in general, and polynomial when the ob