Abstract
This paper presents a simple algorithm for the partial point set pattern matching in 2D. Given a set P of n points, called sample set, and a query set Q of k points (n ≥ k), the problem is to find a matching of Q with a subset of P under rigid motion. In other words, whether each point in Q is matched with corresponding point in P under translation and/or rotation. The proposed algorithm requires O(n 2) space and O(n 2logn) preprocessing time, and the worst case query time complexity is O(kαlogn), where α is the maximum number of equidistant pairs of points. For a set of n points, α may be O(n 4/3) in the worst case. Experimental results on random point sets and fingerprint databases show that it needs much less time in actual practice. The algorithm is then extended for checking the existence of a matching among two sets of line segments under rigid motion in O(knlogn) time, and locating a query polygon among a set of sample polygons in O(kn) time under rigid motion.
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References
P. K. Agarwal and M. Sharir, Efficient Algorithm for Geometric Optimization, ACM Computing Surveys, vol. 30, No. 4, pp. 412–458, 1998.
P. K. Agarwal, B. Aronov, M. Sharir and S. Suri, Selecting Distances in the Plane, Algorithmica, vol. 9, pp. 495–514, 1993.
T. Akutsu, H. Tamaki, and T. Tokuyama, Distribution of Distances and Triangles in a Plane Set and Algorithms for Computing the Largest Common Point Sets, Discrete Computational Geometry, vol. 20, pp. 307–331, 1998.
H. Alt and L. J. Guibas, Discrete Geometric Shapes: Matching, Interpolation, and Approximation — A Survey, Technical Report B 96-11, Freie Universität Berlin, 1996.
H. Alt, B. Behrends, and J. Blömer, Approximate Matching of Polygonal Shapes, in Proc. ACM Symposium on Computational Geometry, pp. 186–193, 1991.
H. Alt, K. Mehlhorn, H. Wagener, and E. Welzl, Congruence, Similarity and Symmetries of Geometric Objects, Discrete Computational Geometry, vol. 3, pp. 237–256, 1988.
M. D. Atkinson, An Optimal Algorithm for Geometrical Congruence, J. Algorithms, vol. 8, pp. 159–172, 1987.
A. Bishnu, P. Bhowmick, J. Dey, B. B. Bhattacharya, M. K. Kundu, C. A. Murthy, and T. Acharya, Combinatorial Classification of Pixels for Ridge Extraction in a Gray-scale Fingerprint Image, accepted in: The 3rd Indian Conference on Computer Vision, Graphics and Image Processing, Ahmedabad, India, Dec. 2002.
J. Beck and J. Spencer, Unit Distances, J. Combinatorial Theory A, vol. 37, pp. 231–238, 1984.
G. T. Candela, P. J. Grother, C. I. Watson, R. A. Wilkinson, and C. L. Wilson, PCAzSYS — A Pattern-Level Classification Automation System for Fingerprints, NISTIR 5647, National Institute of Standards and Technology, August 1995.
A. Farina, Zs. M. Kovacs-Vajna, and A. Leone, Fingerprint Minutiae Extraction from Skeletonized Binary Images, Pattern Recognition, vol. 32, pp. 877–889, 1999.
D. Forsyth, J. L. Mundy, A. Zisserman, C. Coelho, A. Heller, and C. Rothwell, Invariant Descriptors for 3-D Object Recognition and Pose, IEEE Trans. PAMI, vol. 13, pp. 971–991, 1991.
M. Gavrilov, P. Indyk, R. Motwani, and S. Venkatasubramanian, Geometric Pattern Matching: A Performance Study, in Proc. ACM Symposium on Computational Geometry, pp. 79–85, 1999.
J. E. Goodman and R. Pollack, Multidimensional Sorting, SIAM Journal on Computing, vol. 12, pp. 484–507, 1983.
M. T. Goodrich, J. S. B. Mitchell, and M. W. Orletsky, Approximate Geometric Pattern Matching under Rigid Motions, IEEE Trans. PAMI, vol. 21, No. 4, pp. 371–379, April, 1999.
R. M. Haralick, C. N. Lee, X. Zhuang, V. G. Vaidya, and M. B. Kim, Pose Estimation from Corresponding Point Data, IEEE Trans. SMC, vol. 19, pp. 1426–1446, 1989.
S. Jozsa and E. Szemeredi, The Number of Unit Distances on the Plane, in: Infinite and Finite Sets Coll. Math. Soc. J. Bolyai, vol. 10, pp. 939–950, 1973.
D. E. Knuth, The Art of Computer Programming, Vol. 3: Sorting and Searching, Addison-Wesley Publishing Company, Inc., 1973.
D. E. Knuth, Jr. J. H. Morris, and V. R. Pratt, Fast Pattern Matching in Strings, SIAM J. on Computing, vol. 6, no. 2, pp. 240–267, 1977.
D. T. Lee and Y. T. Ching, The Power of Geometric Duality Revisited, Inform. Process. Lett., vol 21, pp. 117–122, 1985.
J. Matousek, On Enclosing k Points by a Circle, Information Processing Letters, vol. 53, pp. 217–221, 1995.
N. Megiddo, Applying Parallel Computation Algorithms in the Design of Serial Algorithms, J. Assoc. Comput. Mach, vol 30, pp. 852–865, 1983.
P. J. Rezende and D. T. Lee, Point Set Pattern Matching in d-dimensions, Algorithmica, vol. 13, pp. 387–404, 1995.
L. Szekely, Crossing numbers and Hard Erdös Problems in Discrete Geometry, Combinatorics, Probability and Computing, vol. 6, pp. 353–358, 1997.
J. Spencer, E. Szemeredi, and W. T. Trotter, Unit Distances in the Euclidean Plane, in: Graph Theory and Combinatorics (B. Bollobas ed.) Academic Press, London, pp. 293–308, 1984.
C. I. Watson, Mated Fingerprint Card Pairs 2, Technical Report Special Database 14, MFCP2, National Institute of Standards and Technology, September 1993.
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Bishnu, A., Das, S., Nandy, S.C., Bhattacharya, B.B. (2003). An Improved Algorithm for Point Set Pattern Matching under Rigid Motion. In: Petreschi, R., Persiano, G., Silvestri, R. (eds) Algorithms and Complexity. CIAC 2003. Lecture Notes in Computer Science, vol 2653. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44849-7_11
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