Supervised learning for parameterized Koopmans–Beckmann’s graph matching☆
Section snippets
Instruction
Graph matching involves establishing correspondences between the nodes of two graphs, usually with the consideration of graph structures. It is a fundamental problem closely related to many famous problems, including, for instance, MAP estimation of Markov random field [25] and quadratic assignment problem [12]. Moreover, it plays a central role in many practical applications also, such as visual object tracking [31], human activity recognition [33], 2D or 3D reconstruction [23], and image
Literature review and preliminaries
Graph matching is in general an NP-hard problem such that many computationally efficient methods have been proposed to seek inexact solutions. For example, the graduated assignment method [13], spectral methods [9], [18], and reweighted random walks [6] are some typical methods for LGM; the path following method [35] for KBGM; the GNCCP based methods [20], [32] for both LGM and KBGM. The key point of these approximate methods is to relax the discrete graph matching problem to be a continuous
Problem formulation
We define KBGMw on a set of adjacency matrices as follows, where and are respectively denote the th adjacency matrices of graphs and measures the weight of the th KBGM.
KBGMw can be seen as a generalization of KBGM, which allows considering different kinds of graph adjacency matrices in a single unifying model. Mathematically it can achieve performance better than, or at least equivalent to
Experimental results and analysis
We evaluate the performance of the proposed method and several peer methods both on synthetic and real-world image data sets. The peer methods involved in our experiments are as follows.
- SM [18]: The most classic spectral method.
- SMAC [27]: A spectral method with affine constraints.
- PGM [9]: A spectral method with a probabilistic matching scheme.
- IPFP [19]: The integer projected fixed point method, which solves graph matching problem iteratively by optimizing its first-order Taylor
Conclusion
In this paper, we propose a supervised learning method for the parameterized Koopmans–Beckmann’s graph matching problem (KBGMw). Firstly, we show that KBGMw can be taken as a special case of the parameterized Lawler’s graph matching (LGMw), subject to certain conditions. Then, based on the structured SVM, we propose a supervised learning method to automatically estimate its parameters. We evaluate the proposed method on two synthetic data sets with rotation and noise transformations, and two
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work was supported in part by the National Key Research and Development Plan of China (Grants 2016YFC0300801 and 2017YFB1300202), and in part by the National Natural Science Foundation of China (NSFC) (Grants 61503383, 61633009, and U1613213).
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Editor: Andrea F. Abate