Many computer vision algorithms employ subspace models to represent data. The Low-rank representation (LRR) has been successfully applied in subspace clustering for which data are clustered according to their subspace structures. The possibility of extending LRR on Grassmann manifold is explored in this paper. Rather than directly embedding Grassmann manifold into a symmetric matrix space, an extrinsic view is taken by building the self-representation of LRR over the tangent space of each Grassmannian point. A new algorithm for solving the proposed Grassmannian LRR model is designed and implemented. Several clustering experiments are conducted on handwritten digits dataset, dynamic texture video clips and YouTube celebrity face video data. The experimental results show our method outperforms a number of existing methods.
. The Stiefel Manifold Representation @cite_2 : Denote by @math the set of all the @math dimension bases, called Stiefel manifold. We identify a point @math on Grassmann manifold @math as an equivalent class under orthogonal transform of Stiefel manifold. Generally we have [ G (p,d) ST (p,d) O (p). ]
@cite_2
2108036604 177021162 2092050166 567868760
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. Motivated by image perturbation and the geometry of manifolds, we present a novel method combining these two elements. First, we form a tangent space from a set of perturbed images and observe that the tangent space admits a vector space structure. Second, we embed the approximated tangent spaces on a Grassmann manifold and employ a chordal distance as the means for comparing subspaces. The matching process is accelerated using a coarse to fine strategy. Experiments on the FERET database suggest that the proposed method yields excellent results using both holistic and local features. Specifically, on the FERET Dup2 data set, our proposed method achieves 83.8 rank 1 recognition: to our knowledge the currently the best result among all non-trained methods. Evidence is also presented that peak recognition performance is achieved using roughly 100 distinct perturbed images. 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. We investigate representations of mapping class groups of surfaces that arise from the untwisted Drinfeld double of a finite group G, focusing on surfaces without marked points or with one marked point. We obtain concrete descriptions of such representations in terms of finite group data. This allows us to establish various properties of these representations. In particular we show that they have finite images, and that for surfaces of genus at least 3 their restriction to the Torelli group is non-trivial iff G is non-abelian.