Statistical Computations on Grassmann and Stiefel Manifolds for Image and Video-Based Recognition

TitleStatistical Computations on Grassmann and Stiefel Manifolds for Image and Video-Based Recognition
Publication TypeJournal Articles
Year of Publication2011
AuthorsTuraga P, Veeraraghavan A, Srivastava A, Chellappa R
JournalIEEE Transactions on Pattern Analysis and Machine Intelligence
Pagination2273 - 2286
Date Published2011/11//
ISBN Number0162-8828
Keywordsactivity based video clustering, activity recognition, computational geometry, Computational modeling, Data models, face recognition, feature representation, finite dimensional linear subspaces, geometric properties, Geometry, Grassmann Manifolds, Grassmann., HUMANS, Image and video models, image recognition, linear dynamic models, linear subspace structure, Manifolds, maximum likelihood classification, maximum likelihood estimation, Object recognition, Riemannian geometry, Riemannian metrics, SHAPE, statistical computations, statistical models, Stiefel, Stiefel Manifolds, unsupervised clustering, video based face recognition, video based recognition, video signal processing

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.