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_34: 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.
  @cite_22: Many applications in computer vision and pattern recognition involve drawing inferences on certain manifold-valued parameters. In order to develop accurate inference algorithms on these manifolds we need to a) understand the geometric structure of these manifolds b) derive appropriate distance measures and c) develop probability distribution functions (pdf) and estimation techniques that are consistent with the geometric structure of these manifolds. In this paper, we consider two related manifolds - the Stiefel manifold and the Grassmann manifold, which arise naturally in several vision applications such as spatio-temporal modeling, affine invariant shape analysis, image matching and learning theory. We show how accurate statistical characterization that reflects the geometry of these manifolds allows us to design efficient algorithms that compare favorably to the state of the art in these very different applications. In particular, we describe appropriate distance measures and parametric and non-parametric density estimators on these manifolds. These methods are then used to learn class conditional densities for applications such as activity recognition, video based face recognition and shape classification.
  @cite_12: 
  @cite_2: Abstract Scene geometry, imaging laws, as well as computational mechanisms generate mathematical constraints on both raw data and computed features. In many cases, these constraints place the patterns on geometrically well-defined spaces, described as manifolds. In such cases, we argue that exploiting the geometry of these manifolds is important to our understanding of the objects and semantics in the imagery. Statistics and algorithms accounting for manifolds also yield improved performance in many vision applications. We justify these arguments by presenting our recent research efforts based on manifold theory for addressing a variety of pattern recognition and computer vision problems, including hashing on manifolds for efficient search, statistical modeling on Grassmann Stiefel manifolds for activity recognition, discriminative learning for group motion recognition, stochastic optimization for spatio-temporal alignment, and shape matching. We also discuss the manifolds of re-parameterizations and elastic shapes, as well as applications of manifolds to face recognition and unsupervised adaptation of classification model from one domain to another.
summary: 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 .