Abstract
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.
Similar content being viewed by others
Notes
The convex relaxation (7) is derived by adding some dummy nodes into the smaller graph to obtain an equal-sized matching problem, such that \(\mathbf {X}\) is constrained as a permutation instead of a partial permutation matrix. Such an expansion is appropriate in case \(\mathbf {K}\) is constructed following (4), because it is straightforward to check that adding dummy nodes will not change the problem. However, if we define the partial matching based on objective function (1) by similarly adding some dummy nodes such that the convex relaxation (5) still holds, it can be shown that it in general changes the objective function.
For convenience sake and without loss of generality, we consider minimization problem here, since the maximization problem such as (2) can be transferred to be a minimization one by setting \(\mathbf {K}\leftarrow -\mathbf {K}\).
Actually, if \(\zeta \) is further increased from \(\eta \) to be \(1\), the resulted \(\mathbf {P}\in \Pi \) will retain since it remains to be a local minimum of the concave function \(F_{\zeta }(\mathbf {P})\).
Codes of SM and RRWM are available at http://cv.snu.ac.kr/research/~RRWM/.
All of the codes of the ten algorithms are available at http://www.escience.cn/people/zyliu/SCCRP.html.
References
Blake, A., & Zisserman, A. (1987). Visual Reconstruction. Cambridge, MA, USA: MIT Press.
Boyd, S., & Vandenberghe, L. (2004). Convex Optimization. New York: Cambridge University Press.
Cho, M., Alahari, K., Ponce, J., et al. (2013). Learning graphs to match. In: ICCV 2013-IEEE International Conference on Computer Vision.
Cho, M., Lee, J., & Lee, K. M. (2010). Reweighted random walks for graph matching. In: Computer Vision-ECCV 2010. Berlin: Springer.
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.
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.
Cour, T., Srinivasan, P., & Shi, J. (2007). Balanced graph matching. Advances in Neural Information Processing Systems, 19, 313.
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.
Duchenne, O., Joulin, A., Ponce, J. (2011). A graph-matching kernel for object categorization. IEEE International Conference on Computer Vision pp. 1792–1799.
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.
Fischler, M. A., & Elschlager, R. A. (1973). The representation and matching of pictorial structures. IEEE Transactions on Computers, C–22(1), 67–92.
Frank, M., & Wolfe, P. (1956). An algorithm for quadratic programming. Naval Research Logistics Quarterly, 3(1–2), 95–110.
Geiger, D., & Yuille, A. (1991). A common framework for image segmentation. International Journal of Computer Vision, 6(3), 227–243.
Gold, S., & Rangarajan, A. (1996). A graduated assignment algorithm for graph matching. IEEE Transactions on Pattern Analysis and Machine Intelligence, 18(4), 377–388.
Kuhn, H. W. (1955). The hungarian method for the assignment problem. Naval Research Logistics Quarterly, 2(1–2), 83–97.
Leordeanu, M., & Hebert, M. (2005). A spectral technique for correspondence problems using pairwise constraints. Tenth IEEE International Conference on Computer Vision, 2, 1482–1489.
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.
Leordeanu, M., Sukthankar, R., & Hebert, M. (2012). Unsupervised learning for graph matching. International journal of computer vision, 96(1), 28–45.
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.
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.
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.
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.
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.
Ravikumar, P., Lakerty, J. (2006). Quadratic programming relaxations for metric labeling and markov random field map estimation. International Conference on Machine Learning.
Rose, K. (1998). Deterministic annealing for clustering, compression, classification, regression, and related optimization problems. Proceedings of the IEEE, 86(11), 2210–2239.
Suh, Y., Cho, M., & Lee, K. M. (2012). Graph matching via sequential monte carlo. In: Computer Vision-ECCV 2012. Berlin: Springer.
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.
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).
Umeyama, S. (1988). An eigendecomposition approach to weighted graph matching problems. IEEE Transactions on Pattern Analysis and Machine Intelligence, 10(5), 695–703.
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.
Zhou, F., De la Torre, F. (2012). Factorized graph matching. In: IEEE International Conference on Computer Vision and Pattern Recognition, pp. 127–134.
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.
Acknowledgments
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).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by M. Hebert.
Appendix: Proof of Corollary 1
Appendix: Proof of Corollary 1
To prove Corollary 1, we prove \(-\mathbf {x}^{\top } \mathbf {K} \mathbf {x}=\parallel \mathbf {A}_{D}-\mathbf {X}\mathbf {A}_{M} \mathbf {X}^{\top }\parallel _{F}^2\), where \(\mathbf {K}\) is given by (3), and \(\mathbf {x}=\mathrm {vec}(\mathbf {X})\). Writing the partial permutation matrix \(\mathbf {X}\in \mathbb {R}^{N\times M}\) as
where \(\mathbf {e}_{\pi (i)}\) denotes a column vector of length \(M\) with 1 at the position \(\pi (i)\) and 0 every other position, it can be then shown that
On the other hand, the term \(\mathbf {X}\mathbf {A}_{M}\mathbf {X}^{\top }\) can be equivalently written as \(\{{\mathbf {A}_{M}}_{\pi (i)\pi (j)}\}^{N\times N}\), and consequently,
Thus, \(-\mathbf {x}^{\top }\mathbf {K}\mathbf {x}=\parallel \mathbf {A}_{D}-\mathbf {X}\mathbf {A}_{M}\mathbf {X}^{\top } \parallel _{F}^2\), and the proof is accomplished.\(\square \)
Rights and permissions
About this article
Cite this article
Liu, ZY., Qiao, H., Yang, X. et al. Graph Matching by Simplified Convex-Concave Relaxation Procedure. Int J Comput Vis 109, 169–186 (2014). https://doi.org/10.1007/s11263-014-0707-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11263-014-0707-7