A Functional Representation for Graph Matching

Abstract

Graph matching is an important and persistent problem in computer vision and pattern recognition for finding node-to-node correspondence between graph-structured data. However, as widely used, graph matching that incorporates pairwise constraints can be formulated as a quadratic assignment problem (QAP), which is NP-complete and results in intrinsic computational difficulties. In this paper, we present a functional representation for graph matching (FRGM) that aims to provide more geometric insights on the problem and reduce the space and time complexities of corresponding algorithms. To achieve these goals, we represent a graph endowed with edge attributes by a linear function space equipped with a functional such as inner product or metric, that has an explicit geometric meaning. Consequently, the correspondence between graphs can be represented as a linear representation map of that functional. Specifically, we reformulate the linear functional representation map as a new parameterization for Euclidean graph matching, which is associative with geometric parameters for graphs under rigid or nonrigid deformations. This allows us to estimate the correspondence and geometric deformations simultaneously. The use of the representation of edge attributes rather than the affinity matrix enables us to reduce the space complexity by two orders of magnitudes. Furthermore, we propose an efficient optimization strategy with low time complexity to optimize the objective function. The experimental results on both synthetic and real-world datasets demonstrate that the proposed FRGM can achieve state-of-the-art performance.

Pairwise Point Cloud Registration using Graph Matching and Rotation-invariant Features

“…Then, an objective function is formulated to find the optimal correspondence, which maximizes the total similarity between the corresponding graphs formed by keypoints. The optimization problem is solved using a fast approximation of the Frank-Wolfe method [17] and transformation parameters are estimated in closed form with given correspondence. Finally, the correspondence and the transformation parameters are updated and getting closer to the final optimal solution in the iterative process.…”

“…2) Problem formulation: Although the original GM methods can provide good solutions for graphs embedded in explicit or implicit feature spaces, the geometric nature of the real data is not fully considered when we apply this type of method to point cloud registration. Inspired by FRGM [17], we improve the graphical model by building connections between correspondence with geometric transformation parameters.…”

Zero-Assignment Constraint for Graph Matching with Outliers

“…The max-pooling-based method [8] was proposed to avoid the adverse effect of false matches of outliers. A domain adaptation-based outlierremoval strategy proposed in [34] aimed to remove outliers as a pre-processing step. However, they directly rely on empirical criterions of outliers and can not deal with complicated situations.…”

“…In many real applications of computer vision and pattern recognition, the feature sets of interest represented as graphs are usually cluttered with numerous outliers [3,42,38,30], which often reduce the accuracy of GM. Although recent works on GM [7,11,21,22,34,44] can achieve satisfactory results for simple graphs that consist of only inliers or a few outliers, they still lack of ability to tolerate numerous outliers arising in complicated graphs. Empirically, the inliers in one graph are nodes that have highly-similar corresponding nodes in the other graph, while the outliers do not.…”

Joint Symmetry Detection and Shape Matching for Non-Rigid Point Cloud

“…Shape correspondence is a fundamental problem in computer vision, computer graphics and related fields [44], since it facilitates many applications such as texture or deformation transfer and statistical shape analysis [1] to name a few. Although shape correspondence has been studied from many viewpoints, we focus here on a functional mapbased approaches [28] as this framework is quite general, scalable and thus, has been extended to various other applications such as pose estimation [26], matrix completion [42] and graph matching [46].…”