This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Recognition of Shapes by Editing Their Shock Graphs
May 2004 (vol. 26 no. 5)
pp. 550-571

Abstract—This paper presents a novel framework for the recognition of objects based on their silhouettes. The main idea is to measure the distance between two shapes as the minimum extent of deformation necessary for one shape to match the other. Since the space of deformations is very high-dimensional, three steps are taken to make the search practical: 1) define an equivalence class for shapes based on shock-graph topology, 2) define an equivalence class for deformation paths based on shock-graph transitions, and 3) avoid complexity-increasing deformation paths by moving toward shock-graph degeneracy. Despite these steps, which tremendously reduce the search requirement, there still remain numerous deformation paths to consider. To that end, we employ an edit-distance algorithm for shock graphs that finds the optimal deformation path in polynomial time. The proposed approach gives intuitive correspondences for a variety of shapes and is robust in the presence of a wide range of visual transformations. The recognition rates on two distinct databases of 99 and 216 shapes each indicate highly successful within category matches (100 percent in top three matches), which render the framework potentially usable in a range of shape-based recognition applications.

[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, 1986.
[2] R. Basri, L. Costa, D. Geiger, and D. Jacobs, Determining the Similarity of Deformable Shapes Vision Research, vol. 38, pp. 2365-2385, 1998.
[3] S. Belongie, J. Puzhicha, and J. Malik, Shape Matching and Object Recognition Using Shape Contexts IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 24, no. 4, pp. 509-522, Apr. 2002.
[4] S. Belongie and J. Malik, Matching with Shape Contexts Proc. IEEE Workshop Content-based Access of Image and Video Libraries, 2000.
[5] H. Blum, Biological Shape and Visual Science J. Theoretical Biology, vol. 38, pp. 205-287, 1973.
[6] J. Bruce, P. Giblin, and C. Gibson, Symmetry Sets Proc. Royal Soc. Edinburgh, vol. 101A, pp. 163-186, 1985.
[7] H. Bunke, On a Relation between Graph Edit Distance and Maximum Common Subgraph Pattern Recognition Letters, vol. 18, no. 8, pp. 689-694, Aug. 1997.
[8] H. Bunke and K. Shearer, A Graph Distance Metric Based on the Maximal Common Subgraph Pattern Recognition Letters, vol. 19, nos. 3-4, pp. 255-259, 1998.
[9] H. Chui and A. Rangarajan, “A New Algorithm for Non-Rigid Point Matching,” Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 44-51, June 2000.
[10] I. Cohen, N. Ayache, and P. Sulger, Tracking Points on Deformable Objects Using Curvature Information Proc. European Conf. Computer Vision, pp. 458-466, 1992.
[11] C.M. Cyr and B.B. Kimia, 3D Object Recognition Using Shape Similarity-Based Aspect Graph Proc. Int'l Conf. Computer Vision, pp. 254-261, 2001.
[12] W. Dewaard, An Optimized Minimal Edit Distance for Hand-Written Word Recognition Pattern Recognition Letters, vol. 16, no. 10, pp. 1091-1096, 1995.
[13] 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.
[14] P.J. Giblin and B.B. Kimia, On the Intrinsic Reconstruction of Shape from Its Symmetries Proc. IEEE Computer Soc. Conf. Computer Vision and Pattern Recognition, pp. 79-84, June 1999. IEEE Trans. Pattern Analysis and Machine Intelligence, to appear.
[15] P.J. Giblin and B.B. Kimia, On the Local Form and Transitions of Symmetry Sets, and Medial Axes, and Shocks in 2D Proc. Int'l Conf. Computer Vision, pp. 385-391, 1999.
[16] P.J. Giblin and B.B. Kimia, Transitions of the 3D Medial Axis under a One-Parameter Family of Deformations Proc. European Conf. Computer Vision, pp. 718-724, 2002.
[17] S. Gold and A. Rangarajan, “A Graduated Assignment Algorithm for Graph Matching,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 18, no. 4, pp. 377-388, Apr. 1996.
[18] D.G. Kendall, D. Barden, T.K. Carne, and H. Le, Shape and Shape Theory. Chichester, UK: John Wiley and Sons, Inc., 1999.
[19] B.B. Kimia, Conservation Laws and a Theory of Shape PhD dissertation, McGill Center for Intelligent Machines, McGill Univ., Montreal, Canada, 1990.
[20] B.B. Kimia, J. Chan, D. Bertrand, S. Coe, Z. Roadhouse, and H. Tek, A Shock-Based Approach for Indexing of Image Databases Using Shape SPIE: Multimedia Storage and Archiving Systems II, vol. 3229, pp. 288-302, 1997.
[21] P. Klein, Computing the Edit Distance between Unrooted Ordered Trees Proc. European Symp. Algorithms, pp. 91-102, 1998.
[22] P. Klein, T. Sebastian, and B. Kimia, Shape Matching Using Edit-Distance: An Implementation ACM-SIAM Symp. Discrete Algorithms, pp. 781-790, 2001.
[23] P. Klein, S. Tirthapura, D. Sharvit, and B. Kimia, A Tree-Edit Distance Algorithm for Comparing Simple, Closed Shapes Proc. ACM-SIAM Symp. Discrete Algorithms, pp. 696-704, 2000.
[24] 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.
[25] T. Liu and D. Geiger, Approximate Tree Matching and Shape Similarity Proc. Int'l Conf. Computer Vision, pp. 456-462, 1999.
[26] 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.
[27] R. Myers, R. Wilson, and E. Hancock, Bayesian Graph Edit Distance IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 22, no. 6, pp. 628-635, June 2000.
[28] R.L. Ogniewicz, Discrete Voronoi Skeletons. Hartung-Gorre, 1993.
[29] P.J. Giblin and B.B. Kimia, On the Intrinsic Reconstruction of Shape from Its Symmetries IEEE Trans. Pattern Analysis and Machine Intelligence, 2003.
[30] M. Pelillo, Matching Free Trees, Maximal Cliques, and Monotone Game Dynamics IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 24, no. 11, pp. 1535-1541, Nov. 2002.
[31] M. Pelillo, K. Siddiqi, and S.W. Zucker, “Matching Hierarchical Structures Using Association Graphs,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 21, no. 11, pp. 1105-1120, 1999.
[32] M. Pelillo, K. Siddiqi, and S.W. Zucker, Many-to-Many Matching of Attributed Trees Using Association Graphs and Game Dynamics Proc. Int'l Workshop Visual Form, pp. 583-593, 2001.
[33] T.B. Sebastian, P.N. Klein, and B.B. Kimia, On Aligning Curves IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 25, no. 1, pp. 116-125, Jan. 2003.
[34] T.B. Sebastian, J.J. Crisco, P.N. Klein, and B.B. Kimia, Construction of 2D Curve Atlases Proc. IEEE Workshop Math. Methods in Biomedical Image Analysis, pp. 70-77, 2000.
[35] T.B. Sebastian and B.B. Kimia, Metric-Based Shape Retrieval in Large Databases Proc. Int'l Conf. Pattern Recognition, vol. 3, pp. 291-296, 2002.
[36] T.B. Sebastian, P.N. Klein, and B.B. Kimia, Alignment-Based Recognition of Shape Outlines Proc. Int'l Workshop Visual Form, pp. 606-618, 2001.
[37] T.B. Sebastian, P.N. Klein, and B.B. Kimia, Recognition of Shapes by Editing Shock Graphs Proc. Eighth Int'l Conf. Computer Vision, pp. 755-762, July 2001.
[38] T.B. Sebastian, P.N. Klein, and B.B. Kimia, Shock-Based Indexing into Large Shape Databases Proc. European Conf. Computer Vision, pp. 731-746, 2002.
[39] D. Sharvit, J. Chan, H. Tek, and B.B. Kimia, Symmetry-Based Indexing of Image Databases J. Visual Comm. and Image Representation, vol. 9, no. 4, pp. 366-380, 1998.
[40] K. Siddiqi, B. Kimia, A. Tannenbaum, and S. Zucker, Shocks, Shapes, and Wiggles Image and Vision Computing, vol. 17, nos. 5-6, pp. 365-373, 1999.
[41] K. Siddiqi and B.B. Kimia, A Shock Grammar for Recognition Proc. Conf. Computer Vision and Pattern Recognition, pp. 507-513, 1996.
[42] K. Siddiqi, A. Shokoufandeh, S. Dickinson, and S. Zucker, Shock Graphs and Shape Matching Int'l J. Computer Vision, vol. 35, no. 1, pp. 13-32, 1999.
[43] H. Tagare, Shape-Based Nonrigid Correspondence with Application to Heart Motion Analysis IEEE Trans. Medical Imaging, vol. 18, no. 7, pp. 570-579, 1999.
[44] K.-C. Tai, The Tree-to-Tree Correction Problem J. Assoc. Computing Machinery, vol. 26, pp. 422-433, 1979.
[45] H. Tek, The Role of Symmetry Maps in Representating Objects in Images PhD dissertation, Division of Eng., Brown Univ., Providence, R.I., July 1999.
[46] H. Tek and B.B. Kimia, Symmetry Maps of Free-Form Curve Segments via Wave Propagation Proc. Int'l Conf. Computer Vision, pp. 362-369, 1999.
[47] A. Torsello and E.R. Hancock, Computing Approximate Tree Edit Distance Using Relaxation Labeling Pattern Recognition Letters, pp. 1089-1097, 2003.
[48] A. Torsello and E.R. Hancock, A Skeletal Measure of 2D Shape Similarity Proc. Int'l Workshop Visual Form, pp. 260-271, 2001.
[49] R. Wagner and M. Fischer, The String-to-String Correction Problem J. Assoc. Computing Machinery, vol. 21, pp. 168-173, 1974.
[50] W. Wallis, P. Shoubridge, M. Kraetz, and D. Ray, Graph Distances Using Graph Union Pattern Recognition Letters, vol. 22, no. 6-7, pp. 701-704, 2001.
[51] K. Zhang and D. Sasha, Simple Fast Algorithms for the Editing Distance between Trees and Related Problems SIAM J. Computing, vol. 18, pp. 1245-1262, 1989.
[52] S.C. Zhu and A.L. Yuille, FORMS: A Flexible Object Recognition and Modeling System Int'l J. Computer Vision, vol. 20, no. 3, pp. 187-212, 1996.
[53] P.J. Giblin and B.B. Kimia, On the Local Form and Transitions of Symmetry Sets, Medial Axes, and Shocks Int'l J. Computer Vision, vol. 54, no. 1-3, pp. 143-157, 2003.
[54] H. Tek and B.B. Kimia, Symmetry Maps of Free-Form Curve Segments Via Wave Propagation Int'l J. Computer Vision, vol. 54, no. 1-3, pp. 35-81, 2003.

Index Terms:
Shape deformation, shock graphs, graph matching, edit distance, shape matching, object recognition, dynamic programming.
Citation:
Thomas B. Sebastian, Philip N. Klein, Benjamin B. Kimia, "Recognition of Shapes by Editing Their Shock Graphs," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 26, no. 5, pp. 550-571, May 2004, doi:10.1109/TPAMI.2004.1273924
Usage of this product signifies your acceptance of the Terms of Use.