H. Alt, B. Behrends, and J. Blomer. Approximate matching of polygonal shapes. Ann. Math. Artif. Intell., 13:251-266, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is the Euclidean distance. If we allow P to translate only, then we want to compute min v H(P v;Q) The problem has been solved by Agarwal et al. 19] using parametric searching, in O( mn) mn) time, which is significantly faster than the previously best known algorithm by Alt et al. [33]. If P and Q are finite sets of points, a more efficient solution, not based on parametric searching, is proposed by Huttenlocher et al. 163] Their solution, however, does not apply to the case of polygons. If we measure distance by the L1 metric, faster algorithms, based on parametric ....

H. Alt, B. Behrends, and J. Blomer, Approximate matching of polygonal shapes, Ann. Math. Artif. Intell., 13 (1995), 251--266.

....measure yield an optimal matching as well. Consider, for example, the algorithm for computing the Arkin metric [2] for polygons, which provides only the optimal rotation angle and no translation vector, in contrast to algorithms for computing the Minimum Hausdoroe distance (under rigid motions) [1], which provide the distance measure as well as the corresponding matching. The distance d(S; V ) should have the following properties: 1. It should be invariant under translations of the scan or the skeleton in their local coordinate systems. 2. The distance should be continuous. That is, ....

....That is, at least the triangle inequality d(S; V 2 ) d(S; V 1 ) D(V 2 ) should be satised. 4.3 Suitable distances for d(S; V It is hard to nd distance functions that meet al..l of the requirements from above. For example, the Minimum Hausdoroe distance (under rigid motions) [1] fullls the rst and the third property by denition and is also continuous, but it is very expensive to compute. That is, the costs are in O( ms) m s) log(m s) if m (s, respectively) denotes the complexity of the scan (skeleton, respectively) On the other hand, the computation of the ....

H. Alt, B. Behrends, and J. Bl#mer. Approximate Matching of Polygonal Shapes. In Proceedings of the 7th Annual ACM Symposium on Computational Geometry, pages 186193, 1991.

....determine how much these shapes resemble each other. More precisely, one wants to find a rigid motion of one shape that maximizes the resemblance with the other shape. There are several ways to measure resemblance. For example, for point sets or polygonal chains one can use the Hausdorff distance [2, 1, 9, 15, 16]; for polygonal chains one can also use the Frchet distance [3] The resemblance of two convex polygons can also be measured by looking at the Hausdorff or rchet distance between their boundaries. For an application in computer vision, however, it seems more appropriate to look at the area of ....

H. Alt, B. Behrends, and J. B15mer. Approximate matching of polygonal shapes. In Proc. 7th Annu. ACM Sympos. Cornput. Geom., pages 186-193, 1991.

....to determine how much these shapes resemble each other. More precisely, one wants to nd a rigid motion of one shape that maximizes the resemblance with the other shape. There are several ways to measure resemblance. For example, for point sets or polygonal chains one can use the Hausdoroe distance [2, 1, 9, 15, 16]; for polygonal chains one can also use the Fr#chet distance [3] The resemblance of two convex polygons can also be measured by looking at the Hausdoroe or Fr#chet distance between their boundaries. For an application in computer vision, however, it seems more appropriate to look at the area of ....

H. Alt, B. Behrends, and J. Bl#mer. Approximate matching of polygonal shapes. In Proc. 7th Annu. ACM Sympos. Comput. Geom., pages 186 193, 1991. 9

.... end, we first translate P such that the centroids of P and Q align (say at point s) Then, we find a rotation of P around s which minimizes its distance to Q; we perform it by reducing it to O(log Delta) instances of pattern matching under translations, as in [CS98, IMV99] By the argument of [ABB91] this approach results in a constant factor approximation algorithm; this can be further reduced to 1 ffl by exploring O(1=ffl 2 ) points s [ABB91] In this way we obtain an algorithm for approximate congruence with essentially the same performance guarantees as the algorithm for translations ....

.... to Q; we perform it by reducing it to O(log Delta) instances of pattern matching under translations, as in [CS98, IMV99] By the argument of [ABB91] this approach results in a constant factor approximation algorithm; this can be further reduced to 1 ffl by exploring O(1=ffl 2 ) points s [ABB91]. In this way we obtain an algorithm for approximate congruence with essentially the same performance guarantees as the algorithm for translations and with running time larger by a factor of 1=ffl 2 . 3 Near Linear Matching Schemes This section (and the next one) are devoted to improving the ....

H. Alt, B. Behrends, and J. Blomer. Approximate matching of polygonal shapes. In Proc. 7th Annu. ACM Sympos. Comput. Geom., pages 186--193, 1991.

....algorithm of [19] Finding the minimum Hausdor# distance under translation between two polygonal regions in the plane under the Euclidean metric. This is a hard instance of a general pattern matching problem. It was left untreated in [26] and solved by a brute force ine#cient method in [8]. We solve it in time O( mn) 2 log 3 (mn) where m and n are the number of edges of the given polygons. This is about 3 orders of magnitude faster than the algorithm of [8] Solving the 1 segment center problem given a set of n points in the plane, find a placement for a given line ....

....of a general pattern matching problem. It was left untreated in [26] and solved by a brute force ine#cient method in [8] We solve it in time O( mn) 2 log 3 (mn) where m and n are the number of edges of the given polygons. This is about 3 orders of magnitude faster than the algorithm of [8]. Solving the 1 segment center problem given a set of n points in the plane, find a placement for a given line segment (under translation and rotation) which minimizes the largest distance from the segment to the given points. We present an algorithm for this problem whose time complexity ....

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H. Alt, B. Behrends and J. Blomer, Approximate matching of polygonal shapes, Proc. 7th ACM Symp. on Computational Geometry, 1991, 186--193.

....the Euclidean distance. If we allow P to translate only, then we want to compute min v H(P v;Q) The problem has been solved by Agarwal et al. 18] using parametric searching, in O( mn) 2 log 3 (mn) time, which is significantly faster than the previously best known algorithm by Alt et al. [30]. If P and Q are finite sets of points, a more efficient solution, not based on parametric searching, is proposed by Huttenlocher et al. 140] Their solution, however, does not apply to the case of polygons. If we measure distance by the L1 metric, faster algorithms, based on parametric ....

H. Alt, B. Behrends, and J. Blomer, Approximate matching of polygonal shapes, Ann. Math. Artif. Intell., 13 (1995), 251--266.

....[14] we derive algorithms that are guaranteed to come close to the optimal value, In particular, each of our methods gives a rigid motion T such that h(T (P ) B) for some small constant 1. We note that a similar use of approximation algorithms is taken by Alt et al. [3] for the problem of polygon matching. Our results are summarized in Table 1. We justify the implementability of our methods through an empirical study of the running time and the quality of the match of our methods when run on various input instances. We compare the performance with that of a ....

H. Alt, B. Behrends, and J. Blomer. Approximate matching of polygonal shapes. Ann. Math. Artif. Intell., 13:251-266, 1995.

.... distance on which they are based may have high complexity in a small space: for a fixed ffl, we can make DG (g) have Omega Gamma n l ) complexity in an arbitrarily small region of transformation space (i.e. this does not depend on just shrinking ffl) This is motivated by the observations in [2] and [6] that for some groups G, if DG (g) ffl, then g must lie in a small region in transformation space, and thus, if the undirected Hausdorff distance could have only small complexity in a small area, we might be able to obtain efficient algorithms, as was done in [6] The lower bounds here ....

H. Alt, B. Behrends, and J. Blomer. Approximate matching of polygonal shapes. In Proc. Seventh ACM Symposium on Computational Geometry, 1991.

....each consisting of n points. Hence, D(A;B) can be computed in time O(n 3 log n) See also [76, 77] for some related results. The algorithm of [78] can be extended to compute the minimum Hausdorff distance D(A;B) for sets A; B of nonintersecting segments under the L 1 or L1 metric. Alt et al. [14] presented an O(n 7 log n) time algorithm for computing D(A;B) for sets of nonintersecting segments under the L 2 metric, which has been improved by Agarwal et al. 9] If we allow both translations and rotations, the problem of computing a placement that minimizes the Hausdorff distance becomes ....

H. Alt, B. Behrends, and J. Blomer, Approximate matching of polygonal shapes, Proc. 7th Annu. ACM Sympos. Comput. Geom., 1991, pp. 186--193.

....shape distances between the original and final shapes. Since two dimensional shapes are specified by the planar curves forming their boundaries, it is natural to seek a formal measure of how similar two given curves are to each other. There is a rich literature, encompassing the Hausdorff metric [2], the Frechet metric [3] similarity measures that are invariant under similitude transformations [20] 4] Fourier descriptors [26] 33] tree matching [35] etc. However, none of these metrics satisfies even simple desiderata for our PRFP context. An ideal shape metric should distinguish between ....

H. Alt, B. Behrends and J. Blomer, "Approximate Matching of Polygonal Shapes", Proc. ACM Symp. on Computational Geometry, 1991, pp. 186-193.

....problems [Mit94] Solutions to some closely related problems often provide large intersection area in practice. One possibility is to align the centroids of the polygons. Another is to maximize the Hausdorff distance between the polygons. An inexpensive approximation to this is given in [ABB91]. Their result states that if you superimpose the lower left corner of the bounding boxes of two polygons, the Hausdorff distance between them is (1 p 2)ffi, where ffi is the minimum Hausdorff distance possible for the two polygons. Of course, there is nothing special about the lower left ....

H. Alt, B. Behrends, and J. Blomer. Approximate matching of polygonal shapes. In Proceedings of the 7th Annual ACM Symposium on Computational Geometry, pages 186--193, 1991.

....if the width is fixed, our separation algorithm can fail to find a feasible placement. We can not expect to do better because the general problem is NP complete. 5.3. 2 Shape Matching Criteria Recent studies in computational geometry offer some general techniques on matching polygon shapes [ABB91] AG92] However, these techniques do not take advantage of the properties of the polygons in a specific application area possess and thus runs slowly. One the other hand, domain specific knowledge sometimes can provide much more direct and effective heuristics for matching and substituting ....

H. Alt, B. Behrends, and J. Blomer. Approximate matching of polygonal shapes. In Proc. 7th Annu. ACM Sympos. Comput. Geom., pages 186--193, 1991.

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H. Alt, B. Behrends, and J. Blomer. Approximate matching of polygonal shapes. Ann. Math. Artif. Intell., 13:251-266, 1995.

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H. Alt, B. Behrends, and J. Blomer. Approximate matching of polygonal shapes. Ann. Math. Artif. Intell., 13:251-266, 1995.

....as well. The directed Hausdor distance is interesting on its own because (P; Q) is a measure of similarity between P and some part of Q. First results on computing the Hausdor distance between two convex polygons in R were obtained in [7] and for two nite sets of points or line segments in [5]. Other previous research is concerned with matching shapes under certain allowable motions minimizing the Hausdor Free University of Berlin, Takustr. 9, 14195 Berlin, Germany; fnameg inf.fu berlin.de Part of this research was funded by the Deutsche Forschungsgemeinschaft (DFG) under grant ....

....P . The maximum over all distances of these pairs gives the directed Hausdor distance (Q; P ) P; Q) can be determined analogously and altogether we have an algorithm of runtime O( m n) log(m n) A generalization of this approach to sets of line segments in two dimensions can be found in [5]. For the sake of completeness we will present the main ideas of this algorithm here. We assume that the line segments are pairwise not properly intersecting which means that any pair of segments either does not intersect or has an intersection point that does not lie in the interior of both ....

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H. Alt, B. Behrends, and J. Blomer. Approximate matching of polygonal shapes. Ann. Math. Artif. Intell., 13:251-266, 1995.

....Two curves with small Hausdor distance having a large Fr echet distance. Figure 2: Two curves with small weak Fr echet distance having a large Fr echet distance. For given polygonal curves P; Q with n and m vertices, respectively, one can compute H (P; Q) in O (m n) log(m n) time, see [2], and F (P; Q) as well as F (P; Q) in O(mn log(m n) time, see [4] The following result from [3] see also [8] shows that for certain classes of curves the three distance measures are closely related, so we can do better than mn) when we want to compute the Fr echet distance. Theorem ....

....log(m n) O( m n) log(m n)2 (m n) bd (PL ) Q(m) Q Figure 12: The cell C in A. randomized time ( 7] see also [13] Theorem 6.15) where 4 (n) O(n2 (n) is the maximum length of a Davenport Schinzel sequence of order 4 over an n element alphabet. With an algorithm of Alt et al. [2] we can compute = H (P; Q) in O( m n) log(m n) time. Combining this with Theorems 2 and 3, we can nd a ( 1) approximation to F (P; Q) together with a reparametrization app that witnesses this fact, within the time bounds as stated in Theorem 3. Corollary 2. For any pair of ....

H. Alt, B. Behrends, and J. Blomer. Approximate matching of polygonal shapes. Ann. Math. Artif. Intell., 13:251-266, 1995.

....Berlin, Germany. falt,knauer,wenkg inf.fu berlin.de 166 backwards. Then the Fr echet distance of the curves is the minimal length of a leash that is necessary. For given polygonal curves P; Q with m and n vertices, respectively, one can compute H (P; Q) in O (m n) log(m n) time, see [1], and F (P; Q) in O(mn log(m n) time, see [3] Here, a polygonal curve is a curve P : 0; n] R 2 with n 2 N, such that for all i 2 f0; 1; n 1g each P i : P j [i;i 1] is ane, i.e. P (i ) 1 )P (i) P (i 1) for all 2 [0; 1] The following result from [2] see also [4] ....

H. Alt, B. Behrends, and J. Blomer. Approximate matching of polygonal shapes. Ann. Math. Artif. Intell., 13:251-266, 1995.

....ecient. To remedy this situation, we present approximation algorithms which do not necessarily compute the optimal transformation, but one that yields a Fr echet distance which di ers from the optimum value by a constant factor only. To this end, we generalize the notion of a reference point, c.f. [2] and [1] to the Fr echet metric and observe that all reference points for the Hausdor distance are also reference points for the Fr echet distance. We rst need the concept of a reference point that was introduced in [1] A reference point of a gure is a characteristic point with the property ....

H. Alt, B. Behrends, and J. Blomer. Approximate matching of polygonal shapes. Ann. Math. Artif. Intell., 13:251-266, 1995.

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H. Alt, B. Behrends and J. Blomer, Approximate matching of polygonal shapes, Ann. Math. Artif. Intell. 13 (1995), 251-266.

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Helmut Alt, Bernd Behrends, and Johannes Blomer. Approximate matching of polygonal shapes, Proceedings of the 7th Annual Symposium on Computational Geometry, (1991), pages 186-93.

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H. Alt, B. Behrends and J. Blomer. Approximate matching of polygonal shapes, Proc. 7th Annu. ACM Sympos. Comput. Geom, 1991, 186--193

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H. Alt, B. Behrends and J. Blomer, "Approximate Matching of Polygonal Shapes", Proc. ACM Symp. on Computational Geometry, 1991, pp. 186-193.

No context found.

Helmut Alt, Bernd Behrends, and Johannes Blomer. Approximate matching of polygonal shapes. Ann. Math. Artif. Intell., 13:251-266, 1995.

No context found.

Alt, H., B. Behrends, and J. Blomer. 1991. "Approximate Matching of Polygonal Shapes," Proc. Seventh ACM Symposium on Computational Geometry, pp. 186-193.

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