Hypergraph matching has recently become a popular approach for solving correspondence problems in computer vision as it allows the use of higher-order geometric information. Hypergraph matching can be formulated as a third-order optimization problem subject to assignment constraints which turns out to be NP-hard. In recent work, we have proposed an algorithm for hypergraph matching which first lifts the third-order problem to a fourth-order problem and then solves the fourth-order problem via optimization of the corresponding multilinear form. This leads to a tensor block coordinate ascent scheme which has the guarantee of providing monotonic ascent in the original matching score function and leads to state-of-the-art performance both in terms of achieved matching score and accuracy. In this paper we show that the lifting step to a fourth-order problem can be avoided yielding a third-order scheme with the same guarantees and performance but being two times faster. Moreover, we introduce a homotopy type method which further improves the performance.
The graph resp. hypergraph matching problem is known to be NP-hard except for special cases where polynomial-time algorithms exist for planar graphs @cite_33 . In order to make the problem computationally tractable, a myriad of approximate algorithms have been proposed over the last three decades aiming at an acceptable trade-off between the complexity of the algorithm and matching accuracy. They can be categorized from different perspectives and we refer to @cite_26 @cite_10 for an extensive review. In this section, we review only those approximate algorithms that use Lawler's formulations for both graph and hypergraph matching as they are closely related to our work. In particular, the graph matching problem can be formulated as a quadratic assignment problem (QAP) @math while the hypergraph matching problem is formulated as a higher-order assignment problem (HAP) @math In these formulations, @math and @math refer to the affinity matrix and affinity tensor respectively, and @math is some matching constraint set depending on specific applications. Also, depending on whether the problem is graph matching or hypergraph matching, the corresponding algorithms will be called second-order methods or higher-order methods.
@cite_26 @cite_10 @cite_33
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Graph matching plays a central role in solving correspondence problems in computer vision. Graph matching problems that incorporate pair-wise constraints can be cast as a quadratic assignment problem (QAP). Unfortunately, QAP is NP-hard and many algorithms have been proposed to solve different relaxations. This paper presents factorized graph matching (FGM), a novel framework for interpreting and optimizing graph matching problems. In this work we show that the affinity matrix can be factorized as a Kronecker product of smaller matrices. There are three main benefits of using this factorization in graph matching: (1) There is no need to compute the costly (in space and time) pair-wise affinity matrix; (2) The factorization provides a taxonomy for graph matching and reveals the connection among several methods; (3) Using the factorization we derive a new approximation of the original problem that improves state-of-the-art algorithms in graph matching. Experimental results in synthetic and real databases illustrate the benefits of FGM. The code is available at http: humansensing.cs.cmu.edu fgm. Graph matching has a wide spectrum of computer vision applications such as finding feature point correspondences across images. The problem of graph matching is generally NP-hard, so most existing work pursues suboptimal solutions between two graphs. This paper investigates a more general problem of matching N attributed graphs to each other, i.e. labeling their common node correspondences such that a certain compatibility affinity objective is optimized. This multi-graph matching problem involves two key ingredients affecting the overall accuracy: a) the pairwise affinity matching score between two local graphs, and b) global matching consistency that measures the uniqueness and consistency of the pairwise matching results by different sequential matching orders. Previous work typically either enforces the matching consistency constraints in the beginning of iterative optimization, which may propagate matching error both over iterations and across different graph pairs; or separates score optimizing and consistency synchronization in two steps. This paper is motivated by the observation that affinity score and consistency are mutually affected and shall be tackled jointly to capture their correlation behavior. As such, we propose a novel multi-graph matching algorithm to incorporate the two aspects by iteratively approximating the global-optimal affinity score, meanwhile gradually infusing the consistency as a regularizer, which improves the performance of the initial solutions obtained by existing pairwise graph matching solvers. The proposed algorithm with a theoretically proven convergence shows notable efficacy on both synthetic and public image datasets. In this paper, we propose a general framework for graph matching which is suitable for different problems of pattern recognition. The pattern representation we assume is at the same time highly structured, like for classic syntactic and structural approaches, and of subsymbolic nature with real-valued features, like for connectionist and statistic approaches. We show that random walk based models, inspired by Google's PageRank, give rise to a spectral theory that nicely enhances the graph topological features at node level. As a straightforward consequence, we derive a polynomial algorithm for the classic graph isomorphism problem, under the restriction of dealing with Markovian spectrally distinguishable graphs (MSD), a class of graphs that does not seem to be easily reducible to others proposed in the literature. The experimental results that we found on different test-beds of the TC-15 graph database show that the defined MSD class "almost always" covers the database, and that the proposed algorithm is significantly more efficient than top scoring VF algorithm on the same data. Most interestingly, the proposed approach is very well-suited for dealing with partial and approximate graph matching problems, derived for instance from image retrieval tasks. We consider the objects of the COIL-100 visual collection and provide a graph-based representation, whose node's labels contain appropriate visual features. We show that the adoption of classic bipartite graph matching algorithms offers a straightforward generalization of the algorithm given for graph isomorphism and, finally, we report very promising experimental results on the COIL-100 visual collection. Due to its wide range of applications, matching between two graphs has been extensively studied and remains an active topic. By contrast, it is still under-exploited on how to jointly match multiple graphs, partly due to its intrinsic combinatorial intractability. In this work, we address this challenging problem in a principled way under the rank-1 tensor approximation framework. In particular, we formulate multi-graph matching as a combinational optimization problem with two main ingredients: unary matching over graph vertices and structure matching over graph edges, both of which across multiple graphs. Then we propose an efficient power iteration solution for the resulting NP-hard optimization problem. The proposed algorithm has several advantages: 1) the intrinsic matching consistency across multiple graphs based on the high-order tensor optimization, 2) the free employment of powerful high-order node affinity, 3) the flexible integration between various types of node affinities and edge hyper-edge affinities. Experiments on diverse and challenging datasets validate the effectiveness of the proposed approach in comparison with state-of the-arts.