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 (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. 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. 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. 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. 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. 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. We propose a novel framework to build descriptors of local intensity that are invariant to general deformations. In this framework, an image is embedded as a 2D surface in 3D space, with intensity weighted relative to distance in x-y. We show that as this weight increases, geodesic distances on the embedded surface are less affected by image deformations. In the limit, distances are deformation invariant. We use geodesic sampling to get neighborhood samples for interest points, and then use a geodesic-intensity histogram (GIH) as a deformation invariant local descriptor. In addition to its invariance, the new descriptor automatically finds its support region. This means it can safely gather information from a large neighborhood to improve discriminability. Furthermore, we propose a matching method for this descriptor that is invariant to affine lighting changes. We have tested this new descriptor on interest point matching for two data sets, one with synthetic deformation and lighting change, and another with real non-affine deformations. Our method shows promising matching results compared to several other approaches 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. 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. Image texture analysis has received a lot of attention in the past years. Researchers have developed many texture signatures based on texture measurements, for the purpose of uniquely characterizing the texture. Existing texture signatures, in general, are not invariant to 3D transforms such as view-point changes and non-rigid deformations of the texture surface, which is a serious limitation for many applications. In this paper, we introduce a new texture signature, called the multifractal spectrum (MFS). It provides an efficient framework combining global spatial invariance and local robust measurements. The MFS is invariant under the bi-Lipschitz map, which includes view-point changes and non-rigid deformations of the texture surface, as well as local affine illumination changes. Experiments demonstrate that the MFS captures the essential structure of textures with quite low dimension. 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. 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. Feature matching is a challenging problem at the heart of numerous computer graphics and computer vision applications. We present the SuperMatching algorithm for finding correspondences between two sets of features. It does so by considering triples or higher order tuples of points, going beyond the pointwise and pairwise approaches typically used. SuperMatching is formulated using a supersymmetric tensor representing an affinity metric that takes into account feature similarity and geometric constraints between features: Feature matching is cast as a higher order graph matching problem. SuperMatching takes advantage of supersymmetry to devise an efficient sampling strategy to estimate the affinity tensor, as well as to store the estimated tensor compactly. Matching is performed by an efficient higher order power iteration approach that takes advantage of this compact representation. Experiments on both synthetic and real data show that SuperMatching provides more accurate feature matching than other state-of-the-art approaches for a wide range of 2D and 3D features, with competitive computational cost. In this paper, the problem of non-rigid shape recognition is studied from the perspective of metric geometry. In particular, we explore the applicability of diffusion distances within the Gromov-Hausdorff framework. While the traditionally used geodesic distance exploits the shortest path between points on the surface, the diffusion distance averages all paths connecting the points. The diffusion distance constitutes an intrinsic metric which is robust, in particular, to topological changes. Such changes in the form of shortcuts, holes, and missing data may be a result of natural non-rigid deformations as well as acquisition and representation noise due to inaccurate surface construction. The presentation of the proposed framework is complemented with examples demonstrating that in addition to the relatively low complexity involved in the computation of the diffusion distances between surface points, its recognition and matching performances favorably compare to the classical geodesic distances in the presence of topological changes between the non-rigid shapes. We have presented a new problem -- the wide multiple baseline stereo (WxBS) -- which considers matching of images that simultaneously differ in more than one image acquisition factor such as viewpoint, illumination, sensor type or where object appearance changes significantly, e.g. over time. A new dataset with the ground truth for evaluation of matching algorithms has been introduced and will be made public. We have extensively tested a large set of popular and recent detectors and descriptors and show than the combination of RootSIFT and HalfRootSIFT as descriptors with MSER and Hessian-Affine detectors works best for many different nuisance factors. We show that simple adaptive thresholding improves Hessian-Affine, DoG, MSER (and possibly other) detectors and allows to use them on infrared and low contrast images. A novel matching algorithm for addressing the WxBS problem has been introduced. We have shown experimentally that the WxBS-M matcher dominantes the state-of-the-art methods both on both the new and existing datasets. We present an approach to find dense point-to-point correspondences between two deformed surfaces corresponding to different postures of the same non-rigid object in a fully automatic way. The approach requires no prior knowledge about the shapes being registered or the initial alignment of the shapes. We consider surfaces that are represented by possibly incomplete triangular meshes. We model the deformations of an object as isometries. To solve the correspondence problem, our approach maps the intrinsic geometries of the surfaces into a low-dimensional Euclidean space via multi-dimensional scaling. This results in posture-invariant shapes that can be registered using rigid correspondence algorithms. 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. We present a new method for 3D shape modeling that achieves intuitive and robust deformations by emulating physically plausible surface behavior inspired by thin shells and plates. The surface mesh is embedded in a layer of volumetric prisms, which are coupled through non-linear, elastic forces. To deform the mesh, prisms are rigidly transformed to satisfy user constraints while minimizing the elastic energy. The rigidity of the prisms prevents degenerations even under extreme deformations, making the method numerically stable. For the underlying geometric optimization we employ both local and global shape matching techniques. Our modeling framework allows for the specification of various geometrically intuitive parameters that provide control over the physical surface behavior. While computationally more involved than previous methods, our approach significantly improves robustness and simplifies user interaction for large, complex deformations. The fundamental matrix is a basic tool in the analysis of scenes taken with two uncalibrated cameras, and the eight-point algorithm is a frequently cited method for computing the fundamental matrix from a set of eight or more point matches. It has the advantage of simplicity of implementation. The prevailing view is, however, that it is extremely susceptible to noise and hence virtually useless for most purposes. This paper challenges that view, by showing that by preceding the algorithm with a very simple normalization (translation and scaling) of the coordinates of the matched points, results are obtained comparable with the best iterative algorithms. This improved performance is justified by theory and verified by extensive experiments on real images. Localizing facial landmarks is a fundamental step in facial image analysis. However, the problem is still challenging due to the large variability in pose and appearance, and the existence of occlusions in real-world face images. In this paper, we present exemplar-based graph matching (EGM), a robust framework for facial landmark localization. Compared to conventional algorithms, EGM has three advantages: (1) an affine-invariant shape constraint is learned online from similar exemplars to better adapt to the test face, (2) the optimal landmark configuration can be directly obtained by solving a graph matching problem with the learned shape constraint, (3) the graph matching problem can be optimized efficiently by linear programming. To our best knowledge, this is the first attempt to apply a graph matching technique for facial landmark localization. Experiments on several challenging datasets demonstrate the advantages of EGM over state-of-the-art methods.