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50
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.
Content based retrieval of VRML objects  an iterative and interactive approach
, 2001
"... Abstract. We examine the problem of searching a database of threedimensional objects (given in VRML) for objects similar to a given object. We introduce an algorithm which is both iterative andinteractive. Rather than base the search solely on geometric feature similarity, we propose letting the use ..."
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Cited by 92 (6 self)
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Abstract. We examine the problem of searching a database of threedimensional objects (given in VRML) for objects similar to a given object. We introduce an algorithm which is both iterative andinteractive. Rather than base the search solely on geometric feature similarity, we propose letting the user in uence future search results by marking some of the results of the current search as `relevant ' or `irrelevant', thus indicating personal preferences. A novel approach, based on SVM, is used for the adaptation of the distance measure consistently with these markings, which brings the `relevant ' objects closer and pushes the `irrelevant' objects farther. We show that in practice very few iterations are needed for the system to converge well on what the user \had in mind". 1
Shape Matching: Similarity Measures and Algorithms
, 2001
"... Shape matching is an important ingredient in shape retrieval, recognition and classification, alignment and registration, and approximation and simplification. This paper treats various aspects that are needed to solve shape matching problems: choosing the precise problem, selecting the properties o ..."
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Cited by 91 (1 self)
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Shape matching is an important ingredient in shape retrieval, recognition and classification, alignment and registration, and approximation and simplification. This paper treats various aspects that are needed to solve shape matching problems: choosing the precise problem, selecting the properties of the similarity measure that are needed for the problem, choosing the specific similarity measure, and constructing the algorithm to compute the similarity. The focus is on methods that lie close to the field of computational geometry.
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...
A survey of shape similarity assessment algorithms for product design and manufacturing applications
 Journal of Computing and Information Science in Engineering
, 2003
"... This document contains the draft version of the following paper: A. Cardone, S.K. Gupta, and M. Karnik. A survey of shape similarity assessment algorithms for product design and manufacturing applications. ASME Journal of ..."
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Cited by 45 (13 self)
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This document contains the draft version of the following paper: A. Cardone, S.K. Gupta, and M. Karnik. A survey of shape similarity assessment algorithms for product design and manufacturing applications. ASME Journal of
Geometric matching under noise: combinatorial bounds and algorithms
 ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS
, 1999
"... In geometric pattern matching, we are given two sets of points P and Q in d dimensions, and the problem is to determine the rigid transformation that brings P closest to Q, under some distance measure. More generally, each point can be modelled as a ball of small radius, and we may wish to nd a tran ..."
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Cited by 40 (9 self)
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In geometric pattern matching, we are given two sets of points P and Q in d dimensions, and the problem is to determine the rigid transformation that brings P closest to Q, under some distance measure. More generally, each point can be modelled as a ball of small radius, and we may wish to nd a transformation approximating the closest distance between P and Q. This problem has many applications in domains such as computer vision and computational chemistry In this paper we present improved algorithms for this problem, by allowing the running time of our algorithms to depend not only on n, (the number of points in the sets), but also on, the diameter of the point set. The dependence on also allows us to e ectively process point sets that occur in practice, where diameters tend to be small ([EVW94]). Our algorithms are also simple to implement, in contrast to much of the earlier work. To obtain the abovementioned results, we introduce a novel discretization technique to reduce geometric pattern matching to combinatorial pattern matching. In addition, we address various generalizations of the classical problem rst posed by Erdos: \Given a set of n points in the plane, how many pairs of points can be exactly a unit distance apart?". The combinatorial bounds we prove enable us to obtain improved results for geometric pattern matching and may have other applications.
Matching Shapes with a Reference Point
, 1994
"... For two given point sets, we present a very simple (almost trivial) algorithm to translate one set so that the Hausdorff distance between the two sets is not larger than a constant factor times the minimum Hausdorff distance which can be achieved in this way. The algorithm just matches the socal ..."
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Cited by 40 (4 self)
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For two given point sets, we present a very simple (almost trivial) algorithm to translate one set so that the Hausdorff distance between the two sets is not larger than a constant factor times the minimum Hausdorff distance which can be achieved in this way. The algorithm just matches the socalled Steiner points of the two sets. The focus of our paper is the general study of reference points (like the Steiner point) and their properties with respect to shape matching. For more general transformations than just translations, our method eliminates several degrees of freedom from the problem and thus yields good matchings with improved time bounds.
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.
Matching Convex Shapes with Respect to the Symmetric Difference
, 1998
"... This paper deals with questions from convex geometry related to shape matching. In particular, we consider the problem of moving one convex figure over another, minimizing the area of their symmetric difference. We show that if we just let the two centers of gravity coincide, the resulting symmetric ..."
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Cited by 36 (5 self)
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This paper deals with questions from convex geometry related to shape matching. In particular, we consider the problem of moving one convex figure over another, minimizing the area of their symmetric difference. We show that if we just let the two centers of gravity coincide, the resulting symmetric difference is within a factor of 11/3 of the optimum. This leads to efficient approximate matching algorithms for convex figures.
Reliable and Efficient Pattern Matching Using an Affine Invariant Metric
 International Journal of Computer Vision
, 1997
"... In the field of pattern matching, there is a clear tradeoff between effectiveness, accuracy and robustness on one hand and efficiency and simplicity on the other hand. For example, matching patterns more effectively by using a more general class of transformations usually results in a considera ..."
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Cited by 30 (1 self)
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In the field of pattern matching, there is a clear tradeoff between effectiveness, accuracy and robustness on one hand and efficiency and simplicity on the other hand. For example, matching patterns more effectively by using a more general class of transformations usually results in a considerable increase of computational complexity. In this paper, we introduce a general pattern matching approach which will be applied to a new measure called the absolute difference. This patternsimilarity measure is affine invariant, which stands out favourably in practical use. The problem of finding a transformation mapping to the minimal absolute difference, like many pattern matching problems, has a high computational complexity. Therefore, we base our algorithm on a hierarchical subdivision of transformation space. The method applies to any affine group of transformations, allowing optimisations for rigid motion. Our implementation of the method performs well in terms of reliabilit...