Structural graph matching using the EM algorithm and singular value decomposition

This paper describes an efficient algorithm for inexact graph matching. The method is purely structural, that is, it uses only the edge or connectivity structure of the graph and does not draw on node or edge attributes. We make two contributions: 1) commencing from a probability distribution for matching errors, we show how the problem of graph matching can be posed as maximum-likelihood estimation using the apparatus of the EM algorithm; and 2) we cast the recovery of correspondence matches between the graph nodes in a matrix framework. This allows one to efficiently recover correspondence matches using the singular value decomposition. We experiment with the method on both real-world and synthetic data. Here, we demonstrate that the method offers comparable performance to more computationally demanding method

Correspondence matching with modal clusters

The modal correspondence method of Shapiro and Brady aims to match point-sets by comparing the eigenvectors of a pairwise point proximity matrix. Although elegant by means of its matrix representation, the method is notoriously susceptible to differences in the relational structure of the point-sets under consideration. In this paper, we demonstrate how the method can be rendered robust to structural differences by adopting a hierarchical approach. To do this, we place the modal matching problem in a probabilistic setting in which the correspondences between pairwise clusters can be used to constrain the individual point correspondences. We demonstrate the utility of the method on a number of synthetic and real-world point-pattern matching problems

Graph edit distance from spectral seriation

This paper is concerned with computing graph edit distance. One of the criticisms that can be leveled at existing methods for computing graph edit distance is that they lack some of the formality and rigor of the computation of string edit distance. Hence, our aim is to convert graphs to string sequences so that string matching techniques can be used. To do this, we use a graph spectral seriation method to convert the adjacency matrix into a string or sequence order. We show how the serial ordering can be established using the leading eigenvector of the graph adjacency matrix. We pose the problem of graph-matching as a maximum a posteriori probability (MAP) alignment of the seriation sequences for pairs of graphs. This treatment leads to an expression in which the edit cost is the negative logarithm of the a posteriori sequence alignment probability. We compute the edit distance by finding the sequence of string edit operations which minimizes the cost of the path traversing the edit lattice. The edit costs are determined by the components of the leading eigenvectors of the adjacency matrix and by the edge densities of the graphs being matched. We demonstrate the utility of the edit distance on a number of graph clustering problems

Pattern vectors from algebraic graph theory

Graphstructures have proven computationally cumbersome for pattern analysis. The reason for this is that, before graphs can be converted to pattern vectors, correspondences must be established between the nodes of structures which are potentially of different size. To overcome this problem, in this paper, we turn to the spectral decomposition of the Laplacian matrix. We show how the elements of the spectral matrix for the Laplacian can be used to construct symmetric polynomials that are permutation invariants. The coefficients of these polynomials can be used as graph features which can be encoded in a vectorial manner. We extend this representation to graphs in which there are unary attributes on the nodes and binary attributes on the edges by using the spectral decomposition of a Hermitian property matrix that can be viewed as a complex analogue of the Laplacian. To embed the graphs in a pattern space, we explore whether the vectors of invariants can be embedded in a low- dimensional space using a number of alternative strategies, including principal components analysis ( PCA), multidimensional scaling ( MDS), and locality preserving projection ( LPP). Experimentally, we demonstrate that the embeddings result in well- defined graph clusters. Our experiments with the spectral representation involve both synthetic and real- world data. The experiments with synthetic data demonstrate that the distances between spectral feature vectors can be used to discriminate between graphs on the basis of their structure. The real- world experiments show that the method can be used to locate clusters of graphs

Graph matching with a dual-step EM algorithm

This paper describes a new approach to matching geometric structure in 2D point-sets. The novel feature is to unify the tasks of estimating transformation geometry and identifying point-correspondence matches. Unification is realized by constructing a mixture model over the bipartite graph representing the correspondence match and by affecting optimization using the EM algorithm. According to our EM framework, the probabilities of structural correspondence gate contributions to the expected likelihood function used to estimate maximum likelihood transformation parameters. These gating probabilities measure the consistency of the matched neighborhoods in the graphs. The recovery of transformational geometry and hard correspondence matches are interleaved and are realized by applying coupled update operations to the expected log-likelihood function. In this way, the two processes bootstrap one another. This provides a means of rejecting structural outliers. We evaluate the technique on two real-world problems. The first involves the matching of different perspective views of 3.5-inch floppy discs. The second example is furnished by the matching of a digital map against aerial images that are subject to severe barrel distortion due to a line-scan sampling process. We complement these experiments with a sensitivity study based on synthetic data

Robust Rotation Synchronization via Low-rank and Sparse Matrix Decomposition

This paper deals with the rotation synchronization problem, which arises in global registration of 3D point-sets and in structure from motion. The problem is formulated in an unprecedented way as a "low-rank and sparse" matrix decomposition that handles both outliers and missing data. A minimization strategy, dubbed R-GoDec, is also proposed and evaluated experimentally against state-of-the-art algorithms on simulated and real data. The results show that R-GoDec is the fastest among the robust algorithms.Comment: The material contained in this paper is part of a manuscript submitted to CVI