K. Fredriksson and E. Ukkonen. Combinatorial methods for approximate pattern matching under rotations and translations in 3D arrays. Submitted, 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

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K. Fredriksson and E. Ukkonen. Combinatorial methods for approximate pattern matching under rotations and translations in 3D arrays. Submitted, 2000.

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K. Fredriksson and E. Ukkonen. Combinatorial methods for approximate pattern matching under rotations and translations in 3D arrays. In SPIRE2000.

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K. Fredriksson and E. Ukkonen. Combinatorial methods for approximate pattern matching under rotations and translations in 3D arrays. In Proc. 7th International Symposium on String Processing and Information Retrieval (SPIRE 2000.

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K. Fredriksson and E. Ukkonen. Combinatorial methods for approximate pattern matching under rotations and translations in 3d arrays. In Proc. 7th String Processing and Information Retrieval (SPIRE'2000), pages 96-104. IEEE CS Press, 2000.

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K. Fredriksson and E. Ukkonen. Combinatorial methods for approximate pattern matching under rotations and translations in 3d arrays. In Proc. 7th String Processing and Information Retrieval (SPIRE'2000.

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K. Fredriksson and E. Ukkonen. Combinatorial methods for approximate pattern matching under rotations and translations in 3D arrays. 1999.

....partially supported by the Academy of Finland and Fundaci on Andes. Dept. of Computer Science, University of Helsinki. Work supported by the Academy of Finland. and vice versa, it is not clear what should match with what. Among the different matching models considered in previous work [7, 8, 9], we stick to the simplest one in this paper: 1) the geometric center of the pattern has to align with the center of a text cell; 2) the text cells involved in the match are those whose geometric centers are covered by the pattern; 3) each text cell involved in a match should match the value of ....

....(3) each text cell involved in a match should match the value of the pattern cell that covers its center. Under this exact matching model, an online algorithm is presented in [7] for searching a pattern allowing rotations in O(n ) average time. The model (a 3D version) was extended in [9] such that there may be a limited number (k) of mismatches between the pattern and its occurrence. Under this mismatches model a O(k average time (2D version) of the algorithm can be obtained to find the occurrences. This works for any 0 k m . For small k, an O(k 1=2 ) average time ....

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K. Fredriksson and E. Ukkonen. Combinatorial methods for approximate pattern matching under rotations and translations in 3d arrays. In Proceedings of the 7th International Symposium on String Processing and Information Retrieval (SPIRE'2000.

....the pattern and text as regular grids, then de ning the notion of matching becomes nontrivial when we rotate the pattern: since every pattern cell intersects several text cells and vice versa, it is not clear what should match what. Among the di erent matching models considered in previous work [10 12], we stick to the simplest one in this paper: 1) the geometric center of the pattern has to align with the center of a text cell; 2) the text cells involved in the match are those whose geometric centers are covered by the pattern; 3) each text cell involved in a match should match the value of ....

....(3) each text cell involved in a match should match the value of the pattern cell that covers its center. Under this exact matching model, an online algorithm is presented in [10] to search for a pattern allowing rotations in O(n ) average time. The model (a 3D version) was extended in [12] such that there may be a limited number k of mismatches between the pattern and its occurrence. Under this mismatches model an O(k ) average time algorithm was obtained, as well as an O(k ) average time algorithm for computing the lower bound of the distance; here we will develop a 2D ....

[Article contains additional citation context not shown here]

K. Fredriksson and E. Ukkonen. Combinatorial methods for approximate pattern matching under rotations and translations in 3d arrays. In Proc. 7th String Processing and Information Retrieval (SPIRE'2000), pages 96-104. IEEE CS Press, 2000.

....2. The threshold problem is solved in sublinear O(n 2 ) expected time, for 1. The above expected time results hold only for low enough values of . All the results have their counterparts in three (and more) dimensions. Similar results hold for some other distance functions as well [14, 16, 17, 11, 12]. The article is organized as follows. Sec. 2 gives the basic de nitions and formulation of the problem and gives some analyses of the complexity of the problem. In Sec. 3 a simple (but optimal) algorithm for evaluating the distance between the pattern and the image in each position and ....

....of P . All the algorithms to be presented have also versions that work in three dimensional inputs. The de nitions above have their obvious counterparts in 3D. In 3D, we speak of volume V = V [1: n; 1: n; 1: n] and the pattern P = P [1: m; 1: m; 1: m] that are 3D arrays of voxels. Theorem 3 ([17]) Assuming the center to center translation, there are O(jP j Note that although there are O(m ) matching functions, the number of di erent rotations is only ) because some of the rotations for the matching functions are the same. Theorem 4 ( 17] Without the center to center ....

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K. Fredriksson and E. Ukkonen. Combinatorial methods for approximate pattern matching under rotations and translations in 3d arrays. In Proceedings of the 7th International Symposium on String Processing and Information Retrieval (SPIRE'2000.

....can be e.g. the number of mismatching pixels between the image and the pattern. Hamming distance) In [3] it was shown how one can filter a 2D image in time O(jIj 1=2 ) to find the image locations that may have distance at most , for any orientation of the pattern. In 3D the algorithm of [4] works in time O(jV j 2=3 ) where jV j is the number of voxels in the 3D volume. The image volume locations that pass the filter, are then inspected more carefully to find the orientation that the pattern should have the verification phase. For this, one may use the filtering algorithm again. ....

....other methods, but this could be improved, see Sec. 2.2. The SAD matching code is much more complex than the code for the other algorithms, also in terms of constant factors. For the verification phase of the filtering algorithms, an almost brute force method was applied (that is, the method of [4], without the incrementalization) using the same rotational sampling as was used in the FFT based method (sampling step 1=w radians, where w is the width of the template) For the 3D experiments the test data was a reconstruction of sus1 mutant of the PRD1 bacteriophage [9] The size of the ....

K. Fredriksson and E. Ukkonen, "Combinatorial methods for approximate pattern matching under rotations and translations in 3d arrays," in SPIRE'

....there are O( 7=2 ) sistrings to search at depth . The search time becomes O(log oe n) 9=2 . By indexing all the rotations and center displacements we get O(log oe n) time again, but at a space cost of O(n 2 (log oe n) 7=2 ) It is also possible to extend the method to 3 dimensions [3]. With center to center assumption we have O(m 1 1) rotations. This means O( 11=3 ) sistrings at depth . Therefore, at O(n 3 ) space the total search time becomes O(log oe n) 14=3 , and if we index all the rotations up to H = x log oe n with x 3 we will have a space requirement of O(n ....

K. Fredriksson and E. Ukkonen. Combinatorial methods for approximate pattern matching under rotations and translations in 3D arrays. year?

....) sistrings to search at depth . The search time for the Exact model becomes O(log oe n) 9=2 . By indexing all the rotations and center displacements we get O(log oe n) time again, but at a space cost of O(n 2 (log oe n) 7=2 ) It is also possible to extend the methods to three dimensions [6]. With the center to center assumption we have O(m 11 ) rotations. This means O( 11=3 ) sistrings at depth . Therefore, at O(n 3 ) space the total search time becomes O( log oe n) 14=3 ) for exact searching. If we index all the rotations up to H = x log oe n with x 3 we will have a ....

K. Fredriksson and E. Ukkonen. Combinatorial methods for approximate pattern matching under rotations and translations in 3D arrays. Submitted, 2000.

....make, and 2r = O(k 1=2 ) p jP j = m. SAD and SSD are sum of absolute differences of pixel values, and sum of squared differences, respectively. Delta was first defined in [7] see Sec. 4.2. 2. Problem definition For simplicity, we give the algorithms in the two dimensionsional case. See [9] for a 3D version of the basic Hamming distance filter. All the other filters generalize also, in a straight forward way. Let I = I[1: n; 1: n] and P = P [1: m; 1: m] be two dimensional arrays of point samples, such that m n. Each sample has a color in a finite ordered alphabet Sigma. The ....

K. Fredriksson and E. Ukkonen. Combinatorial methods for approximate pattern matching under rotations and translations in 3D arrays. In SPIRE2000, 2000. (These proceedings) .

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K. Fredriksson and E. Ukkonen. Combinatorial methods for approximate pattern matching under rotations and translations in 3d arrays. In Proc. 7th Symposium on String Processing and Information Retrieval (SPIRE'2000.

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