International Journal of Computer Vision

, Volume 109, Issue 3, pp 169–186 | Cite as

Graph Matching by Simplified Convex-Concave Relaxation Procedure

  • Zhi-Yong Liu
  • Hong Qiao
  • Xu Yang
  • Steven C. H. Hoi


The convex and concave relaxation procedure (CCRP) was recently proposed and exhibited state-of-the-art performance on the graph matching problem. However, CCRP involves explicitly both convex and concave relaxations which typically are difficult to find, and thus greatly limit its practical applications. In this paper we propose a simplified CCRP scheme, which can be proved to realize exactly CCRP, but with a much simpler formulation without needing the concave relaxation in an explicit way, thus significantly simplifying the process of developing CCRP algorithms. The simplified CCRP can be generally applied to any optimizations over the partial permutation matrix, as long as the convex relaxation can be found. Based on two convex relaxations, we obtain two graph matching algorithms defined on adjacency matrix and affinity matrix, respectively. Extensive experimental results witness the simplicity as well as state-of-the-art performance of the two simplified CCRP graph matching algorithms.


Graph matching Combinatorial optimization Deterministic annealing Graduated optimization Feature correspondence 



The authors thank Dr. Feng Zhou at Carnegie Mellon University for some helpful discussions on his factorized graph matching algorithm Zhou and De la Torre (2012). Many thanks also go to the anonymous reviewers and associate editor whose comments and suggestions greatly improved the manuscripts. This work was supported by the National Science Foundation of China (NSFC) (grants 61375005, 61033011, 61210009), and by Singapore MOE tier 1 research grant (RG33/11).


  1. Blake, A., & Zisserman, A. (1987). Visual Reconstruction. Cambridge, MA, USA: MIT Press.Google Scholar
  2. Boyd, S., & Vandenberghe, L. (2004). Convex Optimization. New York: Cambridge University Press.CrossRefzbMATHGoogle Scholar
  3. Cho, M., Alahari, K., Ponce, J., et al. (2013). Learning graphs to match. In: ICCV 2013-IEEE International Conference on Computer Vision.Google Scholar
  4. Cho, M., Lee, J., & Lee, K. M. (2010). Reweighted random walks for graph matching. In: Computer Vision-ECCV 2010. Berlin: Springer.Google Scholar
  5. Cho, M., Lee, K.M. (2012). Progressive graph matching: Making a move of graphs via probabilistic voting. In: Computer Vision and Pattern Recognition (CVPR), 2012 IEEE Conference on, pp. 398–405. IEEE.Google Scholar
  6. Conte, D., Foggia, P., Sansone, C., & Vento, M. (2004). Thirty years of graph matching in pattern recognition. International Journal of Pattern Recognition and Artificial Intelligence, 18(3), 265–298.CrossRefGoogle Scholar
  7. Cour, T., Srinivasan, P., & Shi, J. (2007). Balanced graph matching. Advances in Neural Information Processing Systems, 19, 313.Google Scholar
  8. Demirci, M. F., Shokoufandeh, A., Keselman, Y., Bretzner, L., & Dickinson, S. (2006). Object recognition as many-to-many feature matching. International Journal of Computer Vision, 69(2), 203–222.CrossRefGoogle Scholar
  9. Duchenne, O., Joulin, A., Ponce, J. (2011). A graph-matching kernel for object categorization. IEEE International Conference on Computer Vision pp. 1792–1799.Google Scholar
  10. Egozi, A., Keller, Y., & Guterman, H. (2013). A probabilistic approach to spectral graph matching. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35(1), 18–27.CrossRefGoogle Scholar
  11. Fischler, M. A., & Elschlager, R. A. (1973). The representation and matching of pictorial structures. IEEE Transactions on Computers, C–22(1), 67–92.CrossRefGoogle Scholar
  12. Frank, M., & Wolfe, P. (1956). An algorithm for quadratic programming. Naval Research Logistics Quarterly, 3(1–2), 95–110.CrossRefMathSciNetGoogle Scholar
  13. Geiger, D., & Yuille, A. (1991). A common framework for image segmentation. International Journal of Computer Vision, 6(3), 227–243.CrossRefGoogle Scholar
  14. Gold, S., & Rangarajan, A. (1996). A graduated assignment algorithm for graph matching. IEEE Transactions on Pattern Analysis and Machine Intelligence, 18(4), 377–388.CrossRefGoogle Scholar
  15. Kuhn, H. W. (1955). The hungarian method for the assignment problem. Naval Research Logistics Quarterly, 2(1–2), 83–97.CrossRefMathSciNetGoogle Scholar
  16. Leordeanu, M., & Hebert, M. (2005). A spectral technique for correspondence problems using pairwise constraints. Tenth IEEE International Conference on Computer Vision, 2, 1482–1489.CrossRefGoogle Scholar
  17. Leordeanu, M., Herbert, M., Sukthankar, R. (2009). An integer projected fixed point method for graph matching and map inference. Advances in Neural Information Processing Systems p. 1114C1122.Google Scholar
  18. Leordeanu, M., Sukthankar, R., & Hebert, M. (2012). Unsupervised learning for graph matching. International journal of computer vision, 96(1), 28–45.CrossRefzbMATHMathSciNetGoogle Scholar
  19. Liu, C. L., Yin, F., Wang, D. H., & Wang, Q. F. (2011). Casia online and offline chinese handwriting databases. In: Preceedings of the International Conference on Document Analysis and Recognition, 2011, 37–41.Google Scholar
  20. Liu, Z. Y., & Qiao, H. (2012). A convex-concave relaxation procedure based subgraph matching algorithm. Journal of Machine Learning Research: W&CP, 25, 237–252.Google Scholar
  21. Liu, Z. Y., Qiao, H., & Xu, L. (2012). An extended path following algorithm for graph matching problem. IEEE Transactions on Pattern Analysis and Machine Intelligence, 34(7), 1451–1456.CrossRefGoogle Scholar
  22. Maciel, J., & Costeira, J. P. (2003). A global solution to sparse correspondence problems. IEEE Transactions on Pattern Analysis and Machine Intelligence, 25(2), 187–199. Google Scholar
  23. Philbin, G., Sivic, J., & Zisserman, A. (2011). Geometric latent dirichlet allocation on a matching graph for large-scale image datasets. International Journal of Computer Vision, 95(2), 138–153.CrossRefzbMATHMathSciNetGoogle Scholar
  24. Ravikumar, P., Lakerty, J. (2006). Quadratic programming relaxations for metric labeling and markov random field map estimation. International Conference on Machine Learning.Google Scholar
  25. Rose, K. (1998). Deterministic annealing for clustering, compression, classification, regression, and related optimization problems. Proceedings of the IEEE, 86(11), 2210–2239.CrossRefGoogle Scholar
  26. Suh, Y., Cho, M., & Lee, K. M. (2012). Graph matching via sequential monte carlo. In: Computer Vision-ECCV 2012. Berlin: Springer.Google Scholar
  27. Tian, Y., Yan, J., Zhang, H., Zhang, Y., Yang, X., & Zha, H. (2012). On the convergence of graph matching: graduated assignment revisited. In: Computer Vision-ECCV 2012. Berlin: Springer.Google Scholar
  28. Torresani, L., Kolmogorov, V., Rother, C. (2008). Feature correspondence via graph matching: Models and global optimization. In D. Forsyth, P. Torr, A. Ziseerman (eds.), ECCV 2008, Part II, LNCS 5303, (pp. 596–609).Google Scholar
  29. Umeyama, S. (1988). An eigendecomposition approach to weighted graph matching problems. IEEE Transactions on Pattern Analysis and Machine Intelligence, 10(5), 695–703.CrossRefzbMATHGoogle Scholar
  30. Zaslavskiy, M., Bach, F., & Vert, J. P. (2009). A path following algorithm for the graph matching problem. IEEE Transactions on Pattern Analysis and Machine Intelligence, 31(12), 2227–2242.CrossRefGoogle Scholar
  31. Zhou, F., De la Torre, F. (2012). Factorized graph matching. In: IEEE International Conference on Computer Vision and Pattern Recognition, pp. 127–134.Google Scholar
  32. Zhou, F., De la Torre, F. (2013). Deformable graph matching. In: Computer Vision and Pattern Recognition (CVPR), 2013 IEEE Conference on, pp. 2922–2929. IEEE.Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Zhi-Yong Liu
    • 1
  • Hong Qiao
    • 1
  • Xu Yang
    • 1
  • Steven C. H. Hoi
    • 2
  1. 1.The State Key Laboratory of Management and Control for Complex SystemsInstitute of Automation, Chinese Academy of SciencesBeijingChina
  2. 2.School of Computer EngineeringNanyang Technological UniversitySingaporeSingapore

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