As two fundamental problems, graph cuts and graph matching have been investigated over decades, resulting in vast literature in these two topics respectively. However the way of jointly applying and solving graph cuts and matching receives few attention. In this paper, we first formalize the problem of simultaneously cutting a graph into two partitions i.e. graph cuts and establishing their correspondence i.e. graph matching. Then we develop an optimization algorithm by updating matching and cutting alternatively, provided with theoretical analysis. The efficacy of our algorithm is verified on both synthetic dataset and real-world images containing similar regions or structures.
Graph matching is mostly considered in the two-graph setting. Since the problem can in general be reformulated as a quadratic assignment task which is NP-hard, different approximate solvers @cite_2 @cite_25 @cite_32 are devised with empirical success. A line of work @cite_22 @cite_12 @cite_31 @cite_20 @cite_24 moves from the two-graph setting to a collection of graphs. While all these work assume each graph is known and focus on finding the node correspondences among graphs. There are also end-to-end approaches @cite_0 @cite_3 on joint node detection (in their respective candidate sets) and matching in an iterative fashion. However, none of these work involves graph partition as they still assume the two graphs are separate in advance. Another orthogonal area is graph structure learning @cite_30 @cite_26 whereby the edge weights on each separate graph is learned to improve the matching accuracy. Readers are referred to @cite_16 for a comprehensive survey.
@cite_30 @cite_26 @cite_22 @cite_32 @cite_3 @cite_16 @cite_24 @cite_0 @cite_2 @cite_31 @cite_20 @cite_25 @cite_12
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As a fundamental problem in pattern recognition, graph matching has applications in a variety of fields, from computer vision to computational biology. In graph matching, patterns are modeled as graphs and pattern recognition amounts to finding a correspondence between the nodes of different graphs. Many formulations of this problem can be cast in general as a quadratic assignment problem, where a linear term in the objective function encodes node compatibility and a quadratic term encodes edge compatibility. The main research focus in this theme is about designing efficient algorithms for approximately solving the quadratic assignment problem, since it is NP-hard. In this paper we turn our attention to a different question: how to estimate compatibility functions such that the solution of the resulting graph matching problem best matches the expected solution that a human would manually provide. We present a method for learning graph matching: the training examples are pairs of graphs and the 'labels' are matches between them. Our experimental results reveal that learning can substantially improve the performance of standard graph matching algorithms. In particular, we find that simple linear assignment with such a learning scheme outperforms Graduated Assignment with bistochastic normalisation, a state-of-the-art quadratic assignment relaxation algorithm. Many tasks in computer vision are formulated as graph matching problems. Despite the NP-hard nature of the problem, fast and accurate approximations have led to significant progress in a wide range of applications. Learning graph models from observed data, however, still remains a challenging issue. This paper presents an effective scheme to parameterize a graph model, and learn its structural attributes for visual object matching. For this, we propose a graph representation with histogram-based attributes, and optimize them to increase the matching accuracy. Experimental evaluations on synthetic and real image datasets demonstrate the effectiveness of our approach, and show significant improvement in matching accuracy over graphs with pre-defined structures. The problem of matching not just two, but m different sets of objects to each other arises in many contexts, including finding the correspondence between feature points across multiple images in computer vision. At present it is usually solved by matching the sets pairwise, in series. In contrast, we propose a new method, Permutation Synchronization, which finds all the matchings jointly, in one shot, via a relaxation to eigenvector decomposition. The resulting algorithm is both computationally efficient, and, as we demonstrate with theoretical arguments as well as experimental results, much more stable to noise than previous methods. Graph matching is an essential problem in computer vision and machine learning. In this paper, we introduce a random walk view on the problem and propose a robust graph matching algorithm against outliers and deformation. Matching between two graphs is formulated as node selection on an association graph whose nodes represent candidate correspondences between the two graphs. The solution is obtained by simulating random walks with reweighting jumps enforcing the matching constraints on the association graph. Our algorithm achieves noise-robust graph matching by iteratively updating and exploiting the confidences of candidate correspondences. In a practical sense, our work is of particular importance since the real-world matching problem is made difficult by the presence of noise and outliers. Extensive and comparative experiments demonstrate that it outperforms the state-of-the-art graph matching algorithms especially in the presence of outliers and deformation. An error occurs in graph-based keypoint matching when keypoints in two different images are matched by an algorithm but do not correspond to the same physical point. Most previous methods acquire keypoints in a black-box manner, and focus on developing better algorithms to match the provided points. However to study the complete performance of a matching system one has to study errors through the whole matching pipeline, from keypoint detection, candidate selection to graph optimisation. We show that in the full pipeline there are six different types of errors that cause mismatches. We then present a matching framework designed to reduce these errors. We achieve this by adapting keypoint detectors to better suit the needs of graph-based matching, and achieve better graph constraints by exploiting more information from their keypoints. Our framework is applicable in general images and can handle clutter and motion discontinuities. We also propose a method to identify many mismatches a posteriori based on Left-Right Consistency inspired by stereo matching due to the asymmetric way we detect keypoints and define the graph. Graph matching, which refers to a class of computational problems of finding an optimal correspondence between the vertices of graphs to minimize (maximize) their node and edge disagreements (affinities), is a fundamental problem in computer science and relates to many areas such as combinatorics, pattern recognition, multimedia and computer vision. Compared with the exact graph (sub)isomorphism often considered in a theoretical setting, inexact weighted graph matching receives more attentions due to its flexibility and practical utility. A short review of the recent research activity concerning (inexact) weighted graph matching is presented, detailing the methodologies, formulations, and algorithms. It highlights the methods under several key bullets, e.g. how many graphs are involved, how the affinity is modeled, how the problem order is explored, and how the matching procedure is conducted etc. Moreover, the research activity at the forefront of graph matching applications especially in computer vision, multimedia and machine learning is reported. The aim is to provide a systematic and compact framework regarding the recent development and the current state-of-the-arts in graph matching. 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. Graph matching is widely used in a variety of scientific fields, including computer vision, due to its powerful performance, robustness, and generality. Its computational complexity, however, limits the permissible size of input graphs in practice. Therefore, in real-world applications, the initial construction of graphs to match becomes a critical factor for the matching performance, and often leads to unsatisfactory results. In this paper, to resolve the issue, we propose a novel progressive framework which combines probabilistic progression of graphs with matching of graphs. The algorithm efficiently re-estimates in a Bayesian manner the most plausible target graphs based on the current matching result, and guarantees to boost the matching objective at the subsequent graph matching. Experimental evaluation demonstrates that our approach effectively handles the limits of conventional graph matching and achieves significant improvement in challenging image matching problems. A graduated assignment algorithm for graph matching is presented which is fast and accurate even in the presence of high noise. By combining graduated nonconvexity, two-way (assignment) constraints, and sparsity, large improvements in accuracy and speed are achieved. Its low order computational complexity [O(lm), where l and m are the number of links in the two graphs] and robustness in the presence of noise offer advantages over traditional combinatorial approaches. The algorithm, not restricted to any special class of graph, is applied to subgraph isomorphism, weighted graph matching, and attributed relational graph matching. To illustrate the performance of the algorithm, attributed relational graphs derived from objects are matched. Then, results from twenty-five thousand experiments conducted on 100 mode random graphs of varying types (graphs with only zero-one links, weighted graphs, and graphs with node attributes and multiple link types) are reported. No comparable results have been reported by any other graph matching algorithm before in the research literature. Twenty-five hundred control experiments are conducted using a relaxation labeling algorithm and large improvements in accuracy are demonstrated. 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. The problem of graph matching in general is NP-hard and approaches have been proposed for its sub optimal solution, most focusing on finding the one-to-one node mapping between two graphs. A more general and challenging problem arises when one aims to find consistent mappings across a number of graphs more than two. Conventional graph pair matching methods often result in mapping inconsistency since the mapping between two graphs can either be determined by pair mapping or by an additional anchor graph. To address this issue, a novel formulation is derived which is maximized via alternating optimization. Our method enjoys several advantages: 1) the mappings are jointly optimized rather than sequentially performed by applying pair matching, allowing the global affinity information across graphs can be propagated and explored, 2) the number of concerned variables to optimize is in linear with the number of graphs, being superior to local pair matching resulting in O(n2) variables, 3) the mapping consistency constraints are analytically satisfied during optimization, and 4) off-the-shelf graph pair matching solvers can be reused under the proposed framework in an out-of-the-box' fashion. Competitive results on both the synthesized data and the real data are reported, by varying the level of deformation, outliers and edge densities.