In this work, we present a novel and practical approach to address one of the longstanding problems in computer vision: 2D and 3D affine invariant feature matching. Our Grassmannian Graph (GrassGraph) framework employs a two stage procedure that is capable of robustly recovering correspondences between two unorganized, affinely related feature (point) sets. The first stage maps the feature sets to an affine invariant Grassmannian representation, where the features are mapped into the same subspace. It turns out that coordinate representations extracted from the Grassmannian differ by an arbitrary orthonormal matrix. In the second stage, by approximating the Laplace-Beltrami operator (LBO) on these coordinates, this extra orthonormal factor is nullified, providing true affine-invariant coordinates which we then utilize to recover correspondences via simple nearest neighbor relations. The resulting GrassGraph algorithm is empirically shown to work well in non-ideal scenarios with noise, outliers, and occlusions. Our validation benchmarks use an unprecedented 440,000+ experimental trials performed on 2D and 3D datasets, with a variety of parameter settings and competing methods. State-of-the-art performance in the majority of these extensive evaluations confirm the utility of our method.
Algorithms that address non-rigid transformations inherently have an advantage over strictly affine methods, here we highlight some notable non-rigid methods that can be used to address the affine correspondence problem. @cite_28 form an equi-affine (volume preserving) invariant Laplacian for surfaces, with applications in shape matching. Their method, however, requires explicit metric tensor calculations on mesh surfaces which can lead to further complications of singular points on the surfaces. We avoid surfaces parameterization issues, offering an approach that works directly on point sets. Popular non-rigid matching algorithms include CPD @cite_17 , gmmreg @cite_10 and TPS-RPM @cite_26 . Graph matching methods have also been quite popular recently. Zhou and de la Torre @cite_31 @cite_35 presented the factorized graph matching (FGM) algorithm in which the affinity matrix is factorized as a Kronecker product of smaller matrices. Although the factorization of affinity matrices makes large graph matching problems more tractable, the method is still computationally expensive.
@cite_35 @cite_26 @cite_28 @cite_31 @cite_10 @cite_17
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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. This paper proposes an affine invariant matching algorithm for shape correspondence problems in arbitrary dimensions. Formulating shapes by configuration matrices of landmarks, and using the fact that subspaces (e.g. range spaces) of these matrices are invariant to affine transformations, the shape correspondence is modelled as a permutation relation between orthogonal projection matrices of the subspaces. Then the matching result is solved by an efficient factorization procedure for rank-deficient matrices. The algorithm is compact, fast, and independent of dimensions. Experimental results for 1D, 2D and 3D matchings of synthetic and real data are provided, which demonstrate potential applications of the algorithm to shape analysis, and to other related problems like wide baseline stereo matching and range data registration. We present a novel Object Recognition approach based on affine invariant regions. It actively counters the problems related to the limited repeatability of the region detectors, and the difficulty of matching, in the presence of large amounts of background clutter and particularly challenging viewing conditions. After producing an initial set of matches, the method gradually explores the surrounding image areas, recursively constructing more and more matching regions, increasingly farther from the initial ones. This process covers the object with matches, and simultaneously separates the correct matches from the wrong ones. Hence, recognition and segmentation are achieved at the same time. The approach includes a mechanism for capturing the relationships between multiple model views and exploiting these for integrating the contributions of the views at recognition time. This is based on an efficient algorithm for partitioning a set of region matches into groups lying on smooth surfaces. Integration is achieved by measuring the consistency of configurations of groups arising from different model views. Experimental results demonstrate the stronger power of the approach in dealing with extensive clutter, dominant occlusion, and large scale and viewpoint changes. Non-rigid deformations are explicitly taken into account, and the approximative contours of the object are produced. All presented techniques can extend any view-point invariant feature extractor. Graph matching (GM) is a fundamental problem in computer science, and it has been successfully applied to many problems in computer vision. Although widely used, existing GM algorithms cannot incorporate global consistence among nodes, which is a natural constraint in computer vision problems. This paper proposes deformable graph matching (DGM), an extension of GM for matching graphs subject to global rigid and non-rigid geometric constraints. The key idea of this work is a new factorization of the pair-wise affinity matrix. This factorization decouples the affinity matrix into the local structure of each graph and the pair-wise affinity edges. Besides the ability to incorporate global geometric transformations, this factorization offers three more benefits. First, there is no need to compute the costly (in space and time) pair-wise affinity matrix. Second, it provides a unified view of many GM methods and extends the standard iterative closest point algorithm. Third, it allows to use the path-following optimization algorithm that leads to improved optimization strategies and matching performance. Experimental results on synthetic and real databases illustrate how DGM outperforms state-of-the-art algorithms for GM. The code is available at http: humansensing.cs.cmu.edu fgm. In this paper we propose an inexact spectral matching algorithm that embeds large graphs on a low-dimensional isometric space spanned by a set of eigenvectors of the graph Laplacian. Given two sets of eigenvectors that correspond to the smallest non-null eigenvalues of the Laplacian matrices of two graphs, we project each graph onto its eigenenvectors. We estimate the histograms of these one-dimensional graph projections (eigenvector histograms) and we show that these histograms are well suited for selecting a subset of significant eigenvectors, for ordering them, for solving the sign-ambiguity of eigenvector computation, and for aligning two embeddings. This results in an inexact graph matching solution that can be improved using a rigid point registration algorithm. We apply the proposed methodology to match surfaces represented by meshes. 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. Feature-based methods for non-rigid registration frequently encounter the correspondence problem. Regardless of whether points, lines, curves or surface parameterizations are used, feature-based non-rigid matching requires us to automatically solve for correspondences between two sets of features. In addition, there could be many features in either set that have no counterparts in the other. This outlier rejection problem further complicates an already difficult correspondence problem. We formulate feature-based non-rigid registration as a non-rigid point matching problem. After a careful review of the problem and an in-depth examination of two types of methods previously designed for rigid robust point matching (RPM), we propose a new general framework for non-rigid point matching. We consider it a general framework because it does not depend on any particular form of spatial mapping. We have also developed an algorithm--the TPS-RPM algorithm--with the thin-plate spline (TPS) as the parameterization of the non-rigid spatial mapping and the softassign for the correspondence. The performance of the TPS-RPM algorithm is demonstrated and validated in a series of carefully designed synthetic experiments. In each of these experiments, an empirical comparison with the popular iterated closest point (ICP) algorithm is also provided. Finally, we apply the algorithm to the problem of non-rigid registration of cortical anatomical structures which is required in brain mapping. While these results are somewhat preliminary, they clearly demonstrate the applicability of our approach to real world tasks involving feature-based non-rigid registration. Analysis of deformable two-dimensional shapes is an important problem, encountered in numerous pattern recognition, computer vision and computer graphics applications. In this paper, we address three major problems in the analysis of non-rigid shapes: similarity, partial similarity, and correspondence. We present an axiomatic construction of similarity criteria for deformation-invariant shape comparison, based on intrinsic geometric properties of the shapes, and show that such criteria are related to the Gromov-Hausdorff distance. Next, we extend the problem of similarity computation to shapes which have similar parts but are dissimilar when considered as a whole, and present a construction of set-valued distances, based on the notion of Pareto optimality. Finally, we show that the correspondence between non-rigid shapes can be obtained as a byproduct of the non-rigid similarity problem. As a numerical framework, we use the generalized multidimensional scaling (GMDS) method, which is the numerical core of the three problems addressed in this paper. In this paper we propose a novel approach for detecting interest points invariant to scale and affine transformations. Our scale and affine invariant detectors are based on the following recent results: (1) Interest points extracted with the Harris detector can be adapted to affine transformations and give repeatable results (geometrically stable). (2) The characteristic scale of a local structure is indicated by a local extremum over scale of normalized derivatives (the Laplacian). (3) The affine shape of a point neighborhood is estimated based on the second moment matrix. Our scale invariant detector computes a multi-scale representation for the Harris interest point detector and then selects points at which a local measure (the Laplacian) is maximal over scales. This provides a set of distinctive points which are invariant to scale, rotation and translation as well as robust to illumination changes and limited changes of viewpoint. The characteristic scale determines a scale invariant region for each point. We extend the scale invariant detector to affine invariance by estimating the affine shape of a point neighborhood. An iterative algorithm modifies location, scale and neighborhood of each point and converges to affine invariant points. This method can deal with significant affine transformations including large scale changes. The characteristic scale and the affine shape of neighborhood determine an affine invariant region for each point. We present a comparative evaluation of different detectors and show that our approach provides better results than existing methods. The performance of our detector is also confirmed by excellent matching resultss the image is described by a set of scale affine invariant descriptors computed on the regions associated with our points. 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. In this paper we present a robust and efficient shape matching approach for Marker-less Motion Capture. Extracted features such as contour, gradient orientations and the turning function of the shape are embedded in a 1-D string. We formulate shape matching as a Linear Assignment Problem and propose to use Dynamic Time Warping on the string representation of shapes to discard unlikely correspondences and thereby to reduce ambiguities and spurious local minima. Furthermore, the proposed cost matrix pruning results in robustness to scaling, rotation and topological changes and allows to greatly reduce the computational cost. We show that our approach can track fast human motions where standard articulated Iterative Closest Point algorithms fail. Copyright © 2017, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. Motivated by the fact that in several cases a matching in a graph is stable if and only if it is produced by a greedy algorithm, we study the problem of computing a maximum weight greedy matching on weighted graphs, termed GREEDY-MATCHING. In wide contrast to the maximum weight matching problem, for which many efficient algorithms are known, we prove that GREEDYMATCHING is strongly NP-hard and APX-complete, and thus it does not admit a PTAS unless P=NP, even on graphs with maximum degree at most 3 and with at most three different integer edge weights. Furthermore we prove that GREEDYMATCHING is strongly NP-hard if the input graph is in addition bipartite. Moreover we consider three natural parameters of the problem, for which we establish a sharp threshold behavior between NP-hardness and computational tractability. On the positive side, we present a randomized approximation algorithm (RGMA) for GREEDYMATCHING on a special class of weighted graphs, called bush graphs. We highlight an unexpected connection between RGMA and the approximation of maximum cardinality matching in unweighted graphs via randomized greedy algorithms. We show that, if the approximation ratio of RGMA is θ, then for every ϵ > 0 the randomized MRG algorithm of ( 1995) gives a (ρ - ϵ)-approximation for the maximum cardinality matching. We conjecture that a tight bound for ρ is 2 3; we prove our conjecture true for four subclasses of bush graphs. Proving a tight bound for the approximation ratio of MRG on unweighted graphs (and thus also proving a tight value for ρ) is a long-standing open problem (Poloczek and Szegedy 2012). This unexpected relation of our RGMA algorithm with the MRG algorithm may provide new insights for solving this problem. The problem of graph matching (GM) in general is nondeterministic polynomial-complete and many approximate pairwise matching techniques have been proposed. For a general setting in real applications, it typically requires to find the consistent matching across a batch of graphs. Sequentially performing pairwise matching is prone to error propagation along the pairwise matching sequence, and the sequences generated in different pairwise matching orders can lead to contradictory solutions. Motivated by devising a robust and consistent multiple-GM model, we propose a unified alternating optimization framework for multi-GM. In addition, we define and use two metrics related to graphwise and pairwise consistencies. The former is used to find an appropriate reference graph, which induces a set of basis variables and launches the iteration procedure. The latter defines the order in which the considered graphs in the iterations are manipulated. We show two embodiments under the proposed framework that can cope with the nonfactorized and factorized affinity matrix, respectively. Our multi-GM model has two major characters: 1) the affinity information across multiple graphs are explored in each iteration by fixing part of the matching variables via a consistency-driven mechanism and 2) the framework is flexible to incorporate various existing pairwise GM solvers in an out-of-box fashion, and also can proceed with the output of other multi-GM methods. The experimental results on both synthetic data and real images empirically show that the proposed framework performs competitively with the state-of-the-art. The goal of our work is to develop an algorithm for automatic and robust detection of global intrinsic symmetries in 3D surface meshes. Our approach is based on two core observations. First, symmetry invariant point sets can be detected robustly using critical points of the Average Geodesic Distance (AGD) function. Second, intrinsic symmetries are self-isometries of surfaces and as such are contained in the low dimensional group of Mobius transformations. Based on these observations, we propose an algorithm that: 1) generates a set of symmetric points by detecting critical points of the AGD function, 2) enumerates small subsets of those feature points to generate candidate Mobius transformations,and 3) selects among those candidate Mobius transformations the one(s) that best map the surface onto itself. The main advantages of this algorithm stem from the stability of the AGD in predicting potential symmetric point features and the low dimensionality of the Mobius group for enumerating potential self-mappings. During experiments with a benchmark set of meshes augmented with human-specified symmetric correspondences, we find that the algorithm is able to find intrinsic symmetries for a wide variety of object types with moderate deviations from perfect symmetry. 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 cast the problem of point cloud matching as a shape matching problem by transforming each of the given point clouds into a shape representation called the Schrodinger distance transform (SDT) representation. This is achieved by solving a static Schrodinger equation instead of the corresponding static Hamilton-Jacobi equation in this setting. The SDT representation is an analytic expression and following the theoretical physics literature, can be normalized to have unit 2 norm -- making it a square-root density, which is identified with a point on a unit Hilbert sphere, whose intrinsic geometry is fully known. The Fisher-Rao metric, a natural metric for the space of densities leads to analytic expressions for the geodesic distance between points on this sphere. In this paper, we use the well known Riemannian framework never before used for point cloud matching, and present a novel matching algorithm. We pose point set matching under rigid and non-rigid transformations in this framework and solve for the transformations using standard nonlinear optimization techniques. Finally, to evaluate the performance of our algorithm -- dubbed SDTM -- we present several synthetic and real data examples along with extensive comparisons to state-of-the-art techniques. The experiments show that our algorithm outperforms state-of the-art point set registration algorithms on many quantitative metrics. We address the problem of finding reliable dense correspondences between a pair of images. This is a challenging task due to strong appearance differences between the corresponding scene elements and ambiguities generated by repetitive patterns. The contributions of this work are threefold. First, inspired by the classic idea of disambiguating feature matches using semi-local constraints, we develop an end-to-end trainable convolutional neural network architecture that identifies sets of spatially consistent matches by analyzing neighbourhood consensus patterns in the 4D space of all possible correspondences between a pair of images without the need for a global geometric model. Second, we demonstrate that the model can be trained effectively from weak supervision in the form of matching and non-matching image pairs without the need for costly manual annotation of point to point correspondences. Third, we show the proposed neighbourhood consensus network can be applied to a range of matching tasks including both category- and instance-level matching, obtaining the state-of-the-art results on the PF Pascal dataset and the InLoc indoor visual localization benchmark. We develop a novel technique for class-based matching of object parts across large changes in viewing conditions. Given a set of images of objects from a given class under different viewing conditions, the algorithm identifies corresponding regions depicting the same object part in different images. The technique is based on using the equivalence of corresponding features in different viewing conditions. This equivalence-based matching scheme is not restricted to planar components or affine transformations. As a result, it identifies corresponding parts more accurately and under more general conditions than previous methods. The scheme is general and works for a variety of natural object classes. We demonstrate that using the proposed methods, a dense set of accurate correspondences can be obtained. Experimental comparisons to several known techniques are presented. An application to the problem of invariant object recognition is shown, and additional applications to wide-baseline stereo are discussed. This paper offers the first variational approach to the problem of dense 3D reconstruction of non-rigid surfaces from a monocular video sequence. We formulate non-rigid structure from motion (nrsfm) as a global variational energy minimization problem to estimate dense low-rank smooth 3D shapes for every frame along with the camera motion matrices, given dense 2D correspondences. Unlike traditional factorization based approaches to nrsfm, which model the low-rank non-rigid shape using a fixed number of basis shapes and corresponding coefficients, we minimize the rank of the matrix of time-varying shapes directly via trace norm minimization. In conjunction with this low-rank constraint, we use an edge preserving total-variation regularization term to obtain spatially smooth shapes for every frame. Thanks to proximal splitting techniques the optimization problem can be decomposed into many point-wise sub-problems and simple linear systems which can be easily solved on GPU hardware. We show results on real sequences of different objects (face, torso, beating heart) where, despite challenges in tracking, illumination changes and occlusions, our method reconstructs highly deforming smooth surfaces densely and accurately directly from video, without the need for any prior models or shape templates. This paper presents a novel pattern recognition framework by capitalizing on dimensionality increasing techniques. In particular, the framework integrates Gabor image representation, a novel multiclass kernel Fisher analysis (KFA) method, and fractional power polynomial models for improving pattern recognition performance. Gabor image representation, which increases dimensionality by incorporating Gabor filters with different scales and orientations, is characterized by spatial frequency, spatial locality, and orientational selectivity for coping with image variabilities such as illumination variations. The KFA method first performs nonlinear mapping from the input space to a high-dimensional feature space, and then implements the multiclass Fisher discriminant analysis in the feature space. The significance of the nonlinear mapping is that it increases the discriminating power of the KFA method, which is linear in the feature space but nonlinear in the input space. The novelty of the KFA method comes from the fact that 1) it extends the two-class kernel Fisher methods by addressing multiclass pattern classification problems and 2) it improves upon the traditional generalized discriminant analysis (GDA) method by deriving a unique solution (compared to the GDA solution, which is not unique). The fractional power polynomial models further improve performance of the proposed pattern recognition framework. Experiments on face recognition using both the FERET database and the FRGC (face recognition grand challenge) databases show the feasibility of the proposed framework. In particular, experimental results using the FERET database show that the KFA method performs better than the GDA method and the fractional power polynomial models help both the KFA method and the GDA method improve their face recognition performance. Experimental results using the FRGC databases show that the proposed pattern recognition framework improves face recognition performance upon the BEE baseline algorithm and the LDA-based baseline algorithm by large margins.