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.
Begelfor and Werman @cite_12 popularized the use of Grassmannian subspaces as an affine invariant. Their work focused on developing clustering algorithms on the Grassmann manifold, but they did not utilize this invariance to solve the correspondence problem. A method that did try to address this correspondence problem through subspace invariance was @cite_34 . The robustness of their method was never evaluated, and, moreover, their approach of using QR factorizations of rank-deficient orthogonal projection matrices is quite different from our proposed two-stage approach. Finally, have shown the effectiveness of Grassmannian representations for object recognition @cite_22 and their use has expanded to other vision domains @cite_2 .
@cite_34 @cite_22 @cite_12 @cite_2
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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 give simple formulas for the canonical metric, gradient, Lie derivative, Riemannian connection, parallel translation, geodesics and distance on the Grassmann manifold of p-planes in R n . In these formulas, p-planes are represented as the column space of n  p matrices. The Newton method on abstract Riemannian manifolds proposed by Smith is made explicit on the Grassmann manifold. Two applications - computing an invariant subspace of a matrix and the mean of subspaces - are worked out. A convenient way of dealing with image sets is to represent them as points on Grassmannian manifolds. While several recent studies explored the applicability of discriminant analysis on such manifolds, the conventional formalism of discriminant analysis suffers from not considering the local structure of the data. We propose a discriminant analysis approach on Grassmannian manifolds, based on a graph-embedding framework. We show that by introducing within-class and between-class similarity graphs to characterise intra-class compactness and inter-class separability, the geometrical structure of data can be exploited. Experiments on several image datasets (PIE, BANCA, MoBo, ETH-80) show that the proposed algorithm obtains considerable improvements in discrimination accuracy, in comparison to three recent methods: Grassmann Discriminant Analysis (GDA), Kernel GDA, and the kernel version of Affine Hull Image Set Distance. We further propose a Grassmannian kernel, based on canonical correlation between subspaces, which can increase discrimination accuracy when used in combination with previous Grassmannian kernels. This paper aims to address a class of nuclear norm regularized least square (NNLS) problems. By exploiting the underlying low-rank matrix manifold structure, the problem with nuclear norm regularization is cast to a Riemannian optimization problem over matrix manifolds. Compared with existing NNLS algorithms involving singular value decomposition (SVD) of large-scale matrices, our method achieves significant reduction in computational complexity. Moreover, the uniqueness of matrix factorization can be guaranteed by our Grassmannian manifold method. In our solution, we first introduce the bilateral factorization into the original NNLS problem and convert it into a Grassmannian optimization problem by using a linearized technique. Then the conjugate gradient procedure on the Grassmannian manifold is developed for our method with a guarantee of local convergence. Finally, our method can be extended to address the graph regularized problem. Experimental results verified both the efficiency and effectiveness of our method.