text: 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.
  @cite_35: 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.
  @cite_26: 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.
  @cite_28: 
  @cite_31: 
  @cite_10: In this paper, we present a unified framework for the rigid and nonrigid point set registration problem in the presence of significant amounts of noise and outliers. The key idea of this registration framework is to represent the input point sets using Gaussian mixture models. Then, the problem of point set registration is reformulated as the problem of aligning two Gaussian mixtures such that a statistical discrepancy measure between the two corresponding mixtures is minimized. We show that the popular iterative closest point (ICP) method and several existing point set registration methods in the field are closely related and can be reinterpreted meaningfully in our general framework. Our instantiation of this general framework is based on the the L2 distance between two Gaussian mixtures, which has the closed-form expression and in turn leads to a computationally efficient registration algorithm. The resulting registration algorithm exhibits inherent statistical robustness, has an intuitive interpretation, and is simple to implement. We also provide theoretical and experimental comparisons with other robust methods for point set registration.
  @cite_17: We introduce Coherent Point Drift (CPD), a novel probabilistic method for non-rigid registration of point sets. The registration is treated as a Maximum Likelihood (ML) estimation problem with motion coherence constraint over the velocity field such that one point set moves coherently to align with the second set. We formulate the motion coherence constraint and derive a solution of regularized ML estimation through the variational approach, which leads to an elegant kernel form. We also derive the EM algorithm for the penalized ML optimization with deterministic annealing. The CPD method simultaneously finds both the non-rigid transformation and the correspondence between two point sets without making any prior assumption of the transformation model except that of motion coherence. This method can estimate complex non-linear non-rigid transformations, and is shown to be accurate on 2D and 3D examples and robust in the presence of outliers and missing points.
summary: 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.