The Community for Technology Leaders
RSS Icon
Issue No.05 - May (2009 vol.31)
pp: 769-782
Matching and analysis of patterns or shapes in the digital plane are of utmost importance in various problems of computer vision and pattern recognition. A digital point set is such a pattern that corresponds to an object in the digital plane. Although there exist several data structures that can be employed for Approximate Point Set Pattern Matching (APSPM) in the real domain, they require substantial modification to support algorithms in the digital domain. To bridge this gap, a novel data structure called "angular tree” is proposed, targeting an efficient and error-controllable circular range query in the digital plane. The farthest pair of points may be used as the starting correspondence between the pattern set and the background set. Several classical discrete structures and methodologies of computational geometry, as well as some topological features of circles/discs in digital geometry, have been used in tandem, for successful realization of the proposed APSPM algorithm in the digital plane. The APSPM algorithm based on the angular tree has been implemented and tested on various point sets and the reported results demonstrate the efficiency and versatility of the new data structure for supporting APSPM algorithms.
Approximate matching, circular range query, digital geometry, point set pattern matching, polygonal range query.
Partha Bhowmick, Bhargab Bhattacharya, "Approximate Matching of Digital Point Sets Using a Novel Angular Tree", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.31, no. 5, pp. 769-782, May 2009, doi:10.1109/TPAMI.2007.70812
[1] P.K. Agarwal and J. Erickson, “Geometric Range Searching and Its Relatives,” Advances in Discrete and Computational Geometry, B.Chazelle, J. Goodman, and R. Pollack, eds., pp. 1-56, Am. Math. Soc., 1998.
[2] H. Alt and L.J. Guibas, “Discrete Geometric Shapes: Matching, Interpolation, and Approximation—A Survey,” Report B 96-11, Freie Universität, 1996.
[3] H. Alt, K. Mehlhorn, H. Wagener, and E. Welzel, “Congruence, Similarity and Symmetries of Geometric Objects,” Discrete and Computational Geometry, vol. 3, pp. 237-256, 1988.
[4] E. Andres, “Discrete Circles, Rings and Spheres,” Computers and Graphics, vol. 18, no. 5, pp. 695-706, 1994.
[5] E. Andres and M. Jacob, “The Discrete Analytical Hyperspheres,” IEEE Trans. Visualization and Computer Graphics, vol. 3, no. 1, pp.75-86, Mar. 1997.
[6] E.M. Arkin, K. Kedmen, J.S.B. Mitchell, J. Sprinzak, and Werman, “Matching Points into Pair-Wise Disjoint Noise Regions: Combinatorials Bounds and Algorithms,” ORSA J. Computing, vol. 4, no. 4, pp. 375-386, 1992.
[7] M.D. Atkinson, “An Optional Algorithm for Geometric Congruence,” J. Algorithms, vol. 8, pp. 159-172, 1998.
[8] N.I. Badler, “Disk Generators for a Raster Display Device,” Computer Graphics and Image Processing, vol. 6, pp. 589-593, 1977.
[9] H.S. Baird, “Model-Based Image Matching Using Location,” MIT Press, 1985.
[10] J.L. Bentley, “Multidimensional Binary Search Trees Used for Associative Searching,” Comm. ACM, vol. 18, pp. 509-517, 1975.
[11] J.L. Bentley, “Multidimensional Binary Search Trees Used in Database Applications,” IEEE Trans. Software Eng., vol. 4, no. 5, pp. 333-340, 1979.
[12] M.D. Berg, M.V. Kreveld, M. Overmars, and O. Schwarzkopf, Computational Geometry Algorithms and Applications. Springer, 2000.
[13] P. Bhowmick and B.B. Bhattacharya, “Approximate Fingerprint Matching Using Kd-Tree,” Proc. 17th Int'l Conf. Pattern Recognition, vol. 1, pp. 544-547, 2004.
[14] P. Bhowmick and B.B. Bhattacharya, “Approximation of Digital Circles by Regular Polygons,” Proc. Int'l Conf. Advances in Pattern Recognition, pp. 257-267, 2005.
[15] P. Bhowmick, A. Bishnu, B.B. Bhattacharya, M.K. Kundu, C.A. Murthy, and T. Acharya, “Determination of Minutiae Scores for Fingerprint Image Applications,” Int'l J. Image and Graphics, vol. 5, pp. 1-35, 2005.
[16] A. Bishnu, S. Das, S.C. Nandy, and B.B. Bhattacharya, “An Improved Algorithm for Point Set Pattern Matching Under Rigid Motion,” Proc. Fifth Italian Conf. Algorithms and Complexity, pp. 36-45, 2003.
[17] J.F. Blinn, “How Many Ways Can You Draw a Circle,” IEEE Computer Graphics and Applications, vol. 7, no. 8, pp. 39-44, Aug. 1987.
[18] J.E. Bresenham, “A Linear Algorithm for Incremental Digital Display of Circular Arcs,” Comm. ACM, vol. 20, no. 2, pp. 100-106, 1977.
[19] T.H. Cormen, C.E. Leiserson, and R.L. Rivest, Introduction to Algorithms. Prentice Hall of India, 2000.
[20] P.J. de Rezende and D.T. Lee, “Point Set Pattern Matching in d-Dimensions,” Algorithmica, vol. 13, pp. 387-404, 1995.
[21] M. Doros, “Algorithms for Generation of Discrete Circles, Rings, and Disks,” Computer Graphics and Image Processing, vol. 10, pp.366-371, 1979.
[22] A. Erafat and A. Itai, “Improvements on Bottleneck Matching and Related Problems Using Geometry,” Proc. 12th Ann. ACM Symp. Computational Geometry, pp. 301-310, 1996.
[23] P.W. Finn, L.E. Kavraki, J. Latombe, R. Motwani, C.R. Shelton, S. Venkatasubramanian, and A. Yao, “RAPID: Randomized Pharmacophore Identification for Drug Design,” Proc. Symp. Computational Geometry, pp. 324-333, 1997.
[24] J.D. Foley, A. van Dam, S.K. Feiner, and J.F. Hughes, Computer Graphics—Principles and Practice. Addison-Wesley, 1993.
[25] M. Gleicher, “Animation from Observation: Motion Capture and Motion Editing,” Computer Graphics, vol. 33, no. 4, pp. 51-55, 1999.
[26] M.T. Goodrich, J.B. Mitchel, and M.W. Orletsky, “Practical Methods for Approximate Geometric Pattern Matching Under Rigid Motion,” Proc. 10th Ann. ACM Symp. Computational Geometry, pp. 103-113, 1994.
[27] M. Hagedoorn and R.C. Veltkamp, “Reliable and Efficient Pattern Matching Using an Affine Invariant Metric,” Int'l J. Computer Vision, vol. 31, nos. 2/3, pp. 203-225, 1999.
[28] R.M. Haralick, “A Measure for Circularity of Digital Figures,” IEEE Trans. Systems, Man, and Cybernetics, vol. 4, pp. 394-396, 1974.
[29] P.J. Heffernan and S. Schirra, “Approximate Decision Algorithms for Point Set Congruence,” Computational Geometry: Theory and Applications, vol. 4, no. 3, pp. 137-156, 1994.
[30] F. Hoffmann, K. Kriegel, and C. Wenk, “An Applied Point Pattern Matching Problem: Comparing 2D Patterns of Protein Spots,” Discrete Applied Math., vol. 93, pp. 75-88, 1999.
[31] L. Holm and C. Sander, “Mapping the Protein Universe,” Science, vol. 273, no. 5275, pp. 595-602, Aug. 1996.
[32] P.I. Hosur and K.-K. Ma, “A Novel Scheme for Progressive Polygon Approximation of Shape Contours,” Proc. IEEE Third Workshop Multimedia Signal Processing, pp. 309-314, 1999.
[33] S.Y. Hsu, L.R. Chow, and C.H. Liu, “A New Approach for the Generation of Circles,” Computer Graphics Forum 12, vol. 2, pp. 105-109, 1993.
[34] D.P. Huttenlocher, K. Kedem, and J.M. Kleinberg, “On Dynamic Voronoi Diagrams and the Minimum Hausdorff Distance for Point Sets under Euclidean Motion in the Plane,” Proc. Eighth Ann. ACM Symp. Computational Geometry, pp. 110-120, 1992.
[35] D.P. Huttenlocher and W.T. Rucklidge, “A Multi-Resolution Technique for Comparing Images Using the Hausdorff Distance,” Proc. Int'l Conf. Computer Vision and Pattern Recognition, pp. 705-706, 1993.
[36] P. Indyk, R. Motwani, and S. Venkatasubramanian, “Geometric Matching under Noise: Combinatorial Bounds and Algorithm,” Proc. 10th Ann. SIAM-ACM Symp. Discrete Algorithms, 1999.
[37] A. Jobbágy, E. Furnée, B. Romhányi, L. Gyöngy, and G. Soós, “Resolution and Accuracy of Passive Marker-Based Motion Analysis,” Automatika, vol. 40, pp. 25-29, 1999.
[38] R. Klette and A. Rosenfeld, Digital Geometry: Geometric Methods for Digital Picture Analysis. Morgan Kaufmann, 2004.
[39] B. Li and H. Holstein, “Using k-d Trees for Robust 3D Point Pattern Matching,” Proc. Fourth Int'l Conf. 3-D Digital Imaging and Modeling, 2003.
[40] B. Likar and F. Pernus, “Automatic Extraction of Corresponding Points for the Registration of Medical Images,” Medical Physics, vol. 26, no. 8, pp. 1678-1686, 1999.
[41] J. Maintz and M. Viergever, “A Survey of Medical Image Registration,” IEEE Eng. in Medicine and Biology Magazine, vol. 2, no. 1, pp. 1-36, 1998.
[42] D. Maltoni, D. Maio, A.K. Jain, and S. Prabhakar, Handbook of Fingerprint Recognition. Springer, 2003.
[43] M.D. Mcllroy, “Best Approximate Circles on Integer Grids,” ACM Trans. Graphics, vol. 2, no. 4, pp. 237-263, 1983.
[44] D. Mount, N. Netanyahu, and J. Lemoigne, “Improved Algorithms for Robust Point Pattern Matching and Applications to Image Registration,” Proc. 14th Ann. ACM Symp. Computational Geometry, pp. 155-164, 1998.
[45] J. Panek and J. Vohradsky, “Point Pattern Matching in the Analysis of Two-Dimensional Gel Electropherograms,” Electrophoresis, vol. 20, pp. 3483-3491, 1999.
[46] S.K. Parui, S. Sarkar, and B.B. Chaudhuri, “Computing the Shape of a Point Set in Digital Images,” Pattern Recognition Letters, vol. 14, no. 2, pp. 89-94, 1993.
[47] M.L.V. Pitteway, “Algorithm for Drawing Ellipses or Hyperbolae with a Digital Plotter,” The Computer J., vol. 10, no. 3, pp. 282-289, 1967.
[48] B. Soifman, D. Beymer, P. McLauchlan, and J. Malik, A Real-Time Computer Vision System for Vehicle Tracking and Traffic Surveillance,, 2004.
[49] I. Szekely, “Crossing Numbers and Hard Erdös Problems in Discrete Geometry,” Combinatorics, Probability, and Computing, vol. 6, pp. 36-47, 1997.
[50] S.M. Thomas and Y.T. Chan, “A Simple Approach for the Estimation of Circular Arc Center and Its Radius,” Computer Vision, Graphics, and Image Processing, vol. 45, no. 3, pp. 362-370, 1989.
[51] G. Tian, D. Gledhill, and D. Taylor, “Comprehensive Interest Points Based Imaging Mosaic,” Pattern Recognition Letters, vol. 24, pp. 1171-1179, 2003.
[52] M. Worring and A.W.M. Smeulders, “Digitized Circular Arcs: Characterization and Parameter Estimation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 17, no. 6, pp. 587-598, June 1995.
[53] C. Yao and J.G. Rokne, “Hybrid Scan-Conversion of Circles,” IEEE Trans. Visualization and Computer Graphics, vol. 1, no. 4, pp. 311-318, Dec. 1995.
[54] P.C. Yuen and G.C. Feng, “A Novel Method for Parameter Estimation of Digital Arc,” Pattern Recognition Letters, vol. 17, no. 9, pp. 929-938, 1996.
5 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool