TY - JOUR T1 - Statistical Computations on Grassmann and Stiefel Manifolds for Image and Video-Based Recognition JF - IEEE Transactions on Pattern Analysis and Machine Intelligence Y1 - 2011 A1 - Turaga,P. A1 - Veeraraghavan,A. A1 - Srivastava, A. A1 - Chellapa, Rama KW - activity based video clustering KW - activity recognition KW - computational geometry KW - Computational modeling KW - Data models KW - face recognition KW - feature representation KW - finite dimensional linear subspaces KW - geometric properties KW - Geometry KW - Grassmann Manifolds KW - Grassmann. KW - HUMANS KW - Image and video models KW - image recognition KW - linear dynamic models KW - linear subspace structure KW - Manifolds KW - maximum likelihood classification KW - maximum likelihood estimation KW - Object recognition KW - Riemannian geometry KW - Riemannian metrics KW - SHAPE KW - statistical computations KW - statistical models KW - Stiefel KW - Stiefel Manifolds KW - unsupervised clustering KW - video based face recognition KW - video based recognition KW - video signal processing AB - In this paper, we examine image and video-based recognition applications where the underlying models have a special structure-the linear subspace structure. We discuss how commonly used parametric models for videos and image sets can be described using the unified framework of Grassmann and Stiefel manifolds. We first show that the parameters of linear dynamic models are finite-dimensional linear subspaces of appropriate dimensions. Unordered image sets as samples from a finite-dimensional linear subspace naturally fall under this framework. We show that an inference over subspaces can be naturally cast as an inference problem on the Grassmann manifold. To perform recognition using subspace-based models, we need tools from the Riemannian geometry of the Grassmann manifold. This involves a study of the geometric properties of the space, appropriate definitions of Riemannian metrics, and definition of geodesics. Further, we derive statistical modeling of inter and intraclass variations that respect the geometry of the space. We apply techniques such as intrinsic and extrinsic statistics to enable maximum-likelihood classification. We also provide algorithms for unsupervised clustering derived from the geometry of the manifold. Finally, we demonstrate the improved performance of these methods in a wide variety of vision applications such as activity recognition, video-based face recognition, object recognition from image sets, and activity-based video clustering. VL - 33 SN - 0162-8828 CP - 11 M3 - 10.1109/TPAMI.2011.52 ER -