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Graph-based shape indexing

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Abstract

Graphs have become growingly important in representing shapes in computer vision. Given a query graph, it is essential to retrieve similar database graphs efficiently from a large database. In this paper, we present a graph-based indexing technique which overcomes significant drawbacks of the previous work (Demirci et al. in Comput Vis Image Underst 110(3):312–325, 2008) using a recently developed theorem from the domain of matrix analysis. Our technique starts by representing the topological structure of a graph in a vector space. As done in the previous work, the topological structure of a graph is constructed using its Laplacian spectra. However, unlike the previous approach, which represents all sugraphs of a database graph in the vector space to account for local similarity, a database graph in the proposed framework is represented as a single vector. By performing a range search around the query, the proposed indexing technique returns a set with both partial and global similarity. Empirical evaluation of the algorithm on an extensive set of retrieval trials including a comparison with the previous approach in both 2D and 3D demonstrates the effectiveness, efficiency, and robustness of the overall approach.

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References

  1. Agarwal, P., Erickson, J.: Geometric range searching and its relatives. In: Chazelle, B., Goodman, R., Pollack, J.E. (eds.) Advances in Discrete and Computational Geometry. Contemporary Mathematics. American Mathematical Society Press, Providence, vol. 23 pp. 1–56 (1999)

  2. Almohamad H.A., Duffuaa S.O.: A linear programming approach for the weighted graph matching problem. IEEE Trans. Pattern Anal. Mach. Intell. 15(5), 522–525 (1993)

    Article  Google Scholar 

  3. Amir, A., Lipsky, O., Porat, E., Umanski, J.: Approximate matching in the L1 metric. In: Proceedings of the 16th Annual Symposium on Combinatorial Pattern Matching, pp. 91–103. Springer, Berlin (2005)

  4. Arya S., Mount D.: Approximate range searching. Comput. Geom. Theory Appl. 17(3–4), 135–152 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Attene M., Biasotti S., Spagnuolo M.: Shape understanding by contour-driven retiling. Vis Comput 19(2–3), 127–138 (2003)

    MATH  Google Scholar 

  6. Bhatia R.: Matrix Analysis. Springer, Berlin (1997)

    Book  Google Scholar 

  7. Biasotti, S.: Reeb graph representation of surfaces with boundary. In: SMI ’04: Proceedings of the Shape Modeling International 2004, pp. 371–374, Washington, DC, USA. IEEE Computer Society (2004)

  8. Biasottia S., Marini S., Spagnuoloa M., Falcidieno B.: Sub-part correspondence by structural descriptors of 3D shapes. Comput. Aided Design 38(9), 1002–1019 (2006)

    Article  Google Scholar 

  9. Blum H.: Biological shape and visual science (part I). J. Theoret. Biol. 38(2), 205–287 (1973)

    Article  Google Scholar 

  10. Bunke H.: Error correcting graph matching: on the influence of the underlying cost function. IEEE Trans. Pattern Anal. Mach. Intell. 21(9), 917–922 (1999)

    Article  Google Scholar 

  11. Cáceres J., Grima C.I., Márquez A., Moreno-González A.: Dilation-free graphs in the l1 metric. Network 49(2), 168–174 (2007)

    Article  MATH  Google Scholar 

  12. Chen, Q., Lim, A., Ong, K.: D(k)-index: an adaptive structural summary for graph-structured data. In: Proceedings of the 2003 ACM SIGMOD international conference on Management of data, pp. 134–144. ACM, New York (2003)

  13. Chuang J., Tsai C., Ko M.: Skeletonization of three-dimensional object using generalized potential field. IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1241–1251 (2000)

    Article  Google Scholar 

  14. Cole-McLaughlin, K., Edelsbrunner, H., Harer, J., Natarajan, V., Pascucci, V.: Loops in Reeb graphs of 2-manifolds. In: Discrete and Computational Geometry, pp. 231–244 (2004)

  15. Costa, M.S., Shapiro, L.G.: Relational indexing. In: SSPR ’96: Proceedings of the 6th International Workshop on Advances in Structural and Syntactical Pattern Recognition, pp. 130–139, London, UK, 1996. Springer, Berlin (1996)

  16. Demirci M.F., van Leuken R.H., Veltkamp R.C.: Indexing through laplacian spectra. Comput. Vis. Image Underst. 110(3), 312–325 (2008)

    Article  Google Scholar 

  17. Fortin, S.: The graph isomorphism problem. Technical report, The University of Alberta, Edmonton, Alberta, Canada (1996)

  18. Gabow, H., Bentley, J., Tarjan, R.: Scaling and related techniques for geometry problems. In: Proceedings of the Sixteenth Annual ACM Symposium on Theory of Computing, pp. 135–143. ACM, New York (1984)

  19. Garey M.R., Johnson D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York (1979)

    MATH  Google Scholar 

  20. Giblin P., Kimia B.: On the local form and transitions of symmetry sets, medial axes, and shocks. Int. J. Comput. Vis. 54(1–3), 143–156 (2003)

    Article  MATH  Google Scholar 

  21. Godsil, C.D., McKay, B.D.: Constructing cospectral graphs. In: Aequationes Mathematicae, vol. 25, pp. 257–268. Birkhãuser Basel (1982)

  22. Gohberg, J.C., Krein, M.G.: Introduction to the theory of linear non-selfadjoint operators. In: Translations of Mathematical Monographs, vol. 18. AMS, New York (1969)

  23. Gold S., Rangarajan A.: A graduated assignment algorithm for graph matching. IEEE Trans. Pattern Anal. Mach. Intell. 18, 377–388 (1996)

    Article  Google Scholar 

  24. Goldman, R., Widom, J.: Dataguides: Enabling query formulation and optimization in semistructured databases. In Proceedings of 23rd International Conference on Very Large Data Bases, August 25–29, 1997, pp. 436–445. Morgan Kaufmann, Menlo Park (1997)

  25. Grimson W.E.L., Lozano-Perez T., Huttenlocher D.P.: Object Recognition by Computer: The Role of Geometric Constraints. MIT Press, Cambridge (1990)

    Google Scholar 

  26. Haemers W.H., Spence E.: Enumeration of cospectral graphs. Eur. J. Comb. 25(2), 199–211 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  27. Hardy G.H., Littlewood J.E., Pólya G.: Inequalities. Cambridge University Press, Cambridge (1959)

    Google Scholar 

  28. Hilaga, M., Shinagawa, Y., Kohmura, T., Kunii, T.L.: Topology matching for fully automatic similarity estimation of 3D shapes. In: SIGGRAPH ’01: Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques, pp. 203–212. ACM Press, New York (2001)

  29. Horaud, R., Skordas, T.: Structural matching for stereo vision. In: Nineth International Conference on Pattern Recognition, pp. 439–445, Rome, Italy (1988)

  30. Horn R.A., Johnson C.R.: Topics in Matrix Analysis. Cambridge University Press, New York (1999)

    Google Scholar 

  31. Kittler, J., Christmas, W.J., Petrou, M.: Probabilistic relaxation for matching of symbolic structures. In: Workshop Structural and Syntactic Pattern Recognition, pp. 471–480, Bern, Switzerland (1992)

  32. Knyazev A.V., Argentati M.E.: Majorization for changes in angles between subspaces, ritz values, and graph laplacian spectra. SIAM J. Matrix Anal. Appl. 29(1), 15–32 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  33. Lee S.W., Kim J.H.: Attributed stroke graph matching for seal imprint verification. Pattern Recognit. Lett. 9, 137–145 (1989)

    Article  Google Scholar 

  34. Lueker, G.S.: A data structure for orthogonal range queries. In: Proceedings of 19th IEEE Symposium on Foundations of Computer Science, pp. 28–34 (1978)

  35. Marshall A.W., Olkin I.: Inequalities: Theory of Majorization and Its Applications. Mathematics in Science and Engineering, vol. 143. Academic Press, New York (1979)

    Google Scholar 

  36. Merris R.: Laplacian matrices of graphs: a survey. Linear Algebra Appl. 197, 143–176 (1994)

    Article  MathSciNet  Google Scholar 

  37. Messmer B.T., Bunke H.: A decision tree approach to graph and subgraph isomorphism detection. Pattern Recognit. 32(12), 1979–1998 (1999)

    Article  Google Scholar 

  38. Min J., Chung C., Shim K.: An adaptive path index for xml data using the query workload. Inf. Syst. 30(6), 467–487 (2005)

    Article  Google Scholar 

  39. Mohar, B.: The Laplacian spectrum of graphs. In: Sixth International Conference on the Theory and Applications of Graphs, pp. 871–898 (1988)

  40. Mohar B.: Laplace eigenvalues of graphs: a survey. Discrete Math 109(1–3), 171–183 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  41. Nievergelt, J., Widmayer, P.: Spatial data structures: Concepts and design choices. In: Algorithmic Foundations of Geographic Information Systems, this book originated from the CISM Advanced School on the Algorithmic Foundations of Geographic Information Systems, pp. 153–197, London, UK. Springer, Berlin (1997)

  42. Sengupta K., Boyer K.L.: Modelbase partitioning using property matrix spectra. Comput. Vis. Image Underst. 70(2), 177–196 (1998)

    Article  MATH  Google Scholar 

  43. Shapiro L.G., Haralick R.M.: Structural descriptions and inexact matching. IEEE Trans. Pattern Anal. Mach. Intell. 3(5), 504–519 (1981)

    Article  Google Scholar 

  44. Shapiro L.S., Brady J.M.: Feature-based correspondence: an eigenvector approach. Image Vision Computing 10(5), 283–288 (1992)

    Article  Google Scholar 

  45. Shasha, D., Wang, J., Giugno, R.: Algorithmics and applications of tree and graph searching. In: Proceedings of the 21st ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems, pp. 39–52. ACM, New York (2002)

  46. Shi J., Malik J.: Normalized cuts and image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 22(8), 888–905 (2000)

    Article  Google Scholar 

  47. Shmulevich, I., Yli-harja, O., Coyle, E.: Perceptual issues in music pattern recognition—complexity of rhythm and key finding. In: Computers and the Humanities, pp. 23–35 (2001)

  48. Shokoufandeh A., Macrini D., Dickinson S., Siddiqi K., Zucker S.W.: Indexing hierarchical structures using graph spectra. IEEE Trans. Pattern Anal. Mach. Intell. 27(7), 1125–1140 (2005)

    Article  Google Scholar 

  49. Siddiqi K., Shokoufandeh A., Dickinson S., Zucker S.: Shock graphs and shape matching. Int. J. Comput. Vis. 35(1), 13–32 (1999)

    Article  Google Scholar 

  50. Sossa, H., Horaud, R.: Model indexing: the graph-hashing approach. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 811–814, Champaign, IL, USA (1992)

  51. Sparaggis, P.D., Cassandras C.G., Towsley, D.: Very weak, p-weak majorization and their application. In: Proceedings of the 31st IEEE Conference on Decision and Control, Tucson, AZ, USA (1992)

  52. Turk M., Pentland A.P.: Eigenfaces for recognition. J. Cognit. Neurosci. 3(1), 71–86 (1991)

    Article  Google Scholar 

  53. Umeyama S.: An eigendecomposition approach to weighted graph matching problems. IEEE Trans. Pattern Anal. Mach. Intell. 10(5), 695–703 (1988)

    Article  MATH  Google Scholar 

  54. Valiente, G.: A general method for graph isomorphism. In: Proceedings of the 13th International Symposium on Fundamentals of Computation Theory, pp. 428–431, London, UK. Springer, Berlin (2001)

  55. Veltkamp, R., ter Haar, F.: Shrec 2007: 3D Shape Retrieval Contest. Technical Report UU-CS-2007-015, Department of Information and Computing Sciences, Utrecht University (2007)

  56. Wang, Y.K., Fan, KC., Horng, JT.: Genetic-based search for error-correcting graph isomorphism. IEEE Trans. Syst. Man Cybernet. 27 (1997)

  57. Willard D.E.: New data structures for orthogonal range queries. SIAM J. Comput. 14(1), 232–253 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  58. Williams M.L., Wilson R.C., Hancock E.R.: Deterministic search for relational graph matching. Pattern Recognit. 32(7), 1255–1271 (1999)

    Article  Google Scholar 

  59. Wilson R., Hancock E.: Structural matching by discrete relaxation. IEEE Trans. Pattern Anal. Mach. Intell. 19, 634–648 (1997)

    Article  Google Scholar 

  60. Wong E.K.: Model matching in robot vision by subgraph isomorphism. Pattern Recognit. 25(3), 287–303 (1992)

    Article  Google Scholar 

  61. Yan, X., Yu, P., Han, J.: Graph indexing: a frequent structure-based approach. In: Proceedings of the 2004 ACM SIGMOD International Conference on Management of data, pp. 335–346. ACM, New York (2004)

  62. Zhang, J., Siddiqi, K., Macrini, D., Shokoufandeh, A., Dickinson, S.J.: Retrieving articulated 3D models using medial surfaces and their graph spectra. In: International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition, pp. 285–300, St. Augustine, FL, USA (2005)

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  1. Department of Computer Engineering, TOBB University of Economics and Technology, Ankara, Turkey

    M. Fatih Demirci

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  1. M. Fatih Demirci
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Correspondence to M. Fatih Demirci.

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Demirci, M.F. Graph-based shape indexing. Machine Vision and Applications 23, 541–555 (2012). https://doi.org/10.1007/s00138-010-0290-z

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