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On Aligning Curves
January 2003 (vol. 25 no. 1)
pp. 116-125

Abstract—We present a novel approach to finding a correspondence (alignment) between two curves. The correspondence is based on a notion of an alignment curve which treats both curves symmetrically. We then define a similarity metric based on the alignment curve using two intrinsic properties of the curve, namely, length and curvature. The optimal correspondence is found by an efficient dynamic-programming method both for aligning pairs of curve segments and pairs of closed curves, and is effective in the presence of a variety of transformations of the curve. Finally, the correspondence is shown in application to handwritten character recognition, prototype formation, and object recognition, and is potentially useful in other applications such as registration and tracking.

[1] N. Ayache and O. Faugeras, “HYPER: A New Approach for the Recognition and Positioning of Two-Dimensional Objects,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 8, no. 1, pp. 44-54, Jan. 1986.
[2] Y. Gdalyahu and D. Weinshall, “Flexible Syntactic Matching of Curves and its Application to Automatic Hierarchical Classification of Silhouettes,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 21, no. 12, pp. 1312-1328, Dec. 1999.
[3] E. Milios and E.G.M. Petrakis, “Shape Retrieval Based on Dynamic Programming,” IEEE Trans. Image Processing, vol. 9, no. 1, pp. 141-146, 2000.
[4] C.C. Tappert, “Cursive Script Recognition by Elastic Matching,” IBM J. Research Development, vol. 26, no. 6, pp. 765-771, 1982.
[5] S.D. Connell and A.K. Jain, Learning Prototypes for On-Line Handwriting Digits Proc. 12th Int'l Conf. Pattern Recognition, pp. 182-184, 1994.
[6] B. Wirtz, “Average Prototypes for Stroke-Based Signature Verification,” Proc. Int'l Conf. Document Analysis and Recognition, pp. 268-272, 1997.
[7] I. Cohen, N. Ayache, and P. Sulger, “Tracking Points on Deformable Objects Using Curvature Information,” Proc. European Conf. Computer Vision, pp. 458-466, 1992.
[8] H. Tagare, Shape-Based Nonrigid Correspondence with Application to Heart Motion Analysis IEEE Trans. Medical Imaging, vol. 18, no. 7, pp. 570-579, 1999.
[9] J.T. Schwartz and M. Sharir, "Identification of Partially Obscured Objects in Two and Three Dimensions by Matching Noisy Characteristic Curves," Int'l J. Robotics Research, Vol. 6, No. 2, Summer 1987, pp. 29-44.
[10] S. Umeyama, "Parameterized Point Pattern Matching and its Application to Recognition of Object Families," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 15, no. 2, pp. 136-144, 1993.
[11] R. Basri, L. Costa, D. Geiger, and D. Jacobs, “Determining the Similarity of Deformable Shapes,” Vision Research, vol. 38, pp. 2365-2385, 1998.
[12] L. Younes, “Computable Elastic Distance between Shapes,” SIAM J. Applied Math., vol. 58, pp. 565-586, 1998.
[13] M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: Active Contour Models,” Int'l J. Computer Vision, vol. 1, no. 4, pp. 321-331, 1988.
[14] D.W. Jacobs, D. Weinshall, and Y. Gdalyahu, Classification with Nonmetric Distances: Image Retrieval and Class Representation IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 22, no. 6, pp. 583-600, June 2000.
[15] A. Tversky, “Features of Similarity,” Psychological Rev., vol. 84, no. 4, pp. 327-352, 1977.
[16] T.B. Sebastian, P.N. Klein, and B.B. Kimia, “Shock-Based Indexing into Large Shape Databases,” Proc. Seventh European Conf. Computer Vision, May 2002.
[17] S. Brin, “Near Neighbour Search in Large Metric Spaces,” Proc. 21st Int'l Conf. Very Large Data Bases, pp. 574-584, Sept. 1995.
[18] J. Uhlmann, “Satisfying General Proximity/Similarity Queries with Metric Trees,” Information Processing Letters, vol. 40, pp. 175-179, 1991.
[19] P. Yianilos, “Data Structures and Algorithms for Nearest Neighbor Search in General Metric Spaces,” Proc. Third Ann. ACM-SIAM Symp. Discrete Algorithms, pp. 311-321, 1993.
[20] A. Marzal and E. Vidal, "Computation of Normalized Edit Distance and Applications," IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. 15, pp. 926-932, 1993.
[21] K.S. Arun, T.S. Huang, and S.D. Blostein, "Least Squares Fitting of Two 3-(D) Point Sets," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 9, pp. 698-700, 1987.
[22] H. Liuand and M. Srinath, “Partial Shape Classification Using Contour Matching in Distance Transformation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 12, pp. 1,072-1,079, 1990.
[23] W.H. Tsai and S.S. Yu, “Attributed String Matching with Merging for Shape Recognition,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 7, no. 4, pp. 453-462, 1985.
[24] N. Ueda and S. Suzuki, "Learning Visual Models from Shape Contours Using Multiscale Convex/Concave Structure Matching," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 15, no. 4, pp. 337-352, Apr. 1995.
[25] H.D. Tagare, “Deformable 2-D Template Matching Using Orthogonal Curves,” IEEE Trans. Medical Imaging, vol. 16, no. 1, pp. 108-117, 1997.
[26] H. Sakoe and S. Chiba, "Dynamic Programming Optimization for Spoken Word Recognition," IEEE Trans. ASSP, vol. 26, pp. 623-625, 1980.
[27] R.E. Bellman and S.E. Dreyfus, Applied Dynamic Programming. Princeton Univ. Press, 1962.
[28] T. Sebastian, P. Klein, and B. Kimia, “Curve Matching Using Alignment Curve,” Technical Report LEMS 184, LEMS, Brown Univ., 2000.
[29] R.A. Wagner and M.J. Fischer, "The String-to-String Correction Problem," J. ACM, vol. 21, no. 1, pp. 168-78, 1974.
[30] M. Maes, “On a Cyclic String-to-String Correction Problem,” Information Processing Letters, vol. 35, pp. 73-78, June 1990.
[31] S. Belongie and J. Malik, “Matching with Shape Contexts,” IEEE Workshop Content-Based Access of Image and Video Libraries, 2000.
[32] P.J. Besl and N.D. McKay, "A Method for Registration of 3D Shapes," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 14, no. 2, pp. 239-256, Feb. 1992.
[33] A. Rangagajan, H. Chui, and F.L. Bookstein, “The Softassign Procrustes Matching Algorithm,” Proc. 15th Conf Information Processing in Medical Imaging, pp. 29-42, 1997.
[34] L.J. Latecki, R. Lakämper, and U. Eckhardt, “Shape Descriptors for Non-Rigid Shapes with a Single Closed Contour,” Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 424-429, 2000.
[35] F. Mokhtarian, S. Abbasi, and J. Kittler, “Efficient and Robust Shape Indexing through Curvature Scale Space,” Int'l Workshop Image Databases and Multimedia Search, pp. 35-42, 1996.
[36] S. Belongie, J. Malik, and J. Puzicha, “Matching Shapes,” Proc. Eighth Int'l. Conf. Computer Vision, pp. 454-461, July 2001.
[37] T.P. Wallace and P.A. Wintz, “An Efficient Three-Dimensional Aircraft Recognition Algorithm Using Fourier Descriptors,” Computer Graphics and Image Processing, vol. 13, no. 1, pp. 99-126, 1980.
[38] L. Gupta and M.D. Srinath, “Contour Sequence Moments for the Classification of Closed Planar Shapes,” Pattern Recognition, vol. 20, pp. 267-272, 1987.
[39] M.K. Hu, “Pattern Recognition by Moment Invariants,” Proc. IRE Trans. Information Theory, vol. 8, pp. 179-187, 1962.

Index Terms:
Curve alignment, recognition, dynamic programming, prototypes, correspondence.
Citation:
Thomas B. Sebastian, Philip N. Klein, Benjamin B. Kimia, "On Aligning Curves," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 25, no. 1, pp. 116-125, Jan. 2003, doi:10.1109/TPAMI.2003.1159951
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