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
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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.