Statistical computations on grassmann and stiefel manifolds for image and video-based recognition

Pavan Turaga, Ashok Veeraraghavan, Anuj Srivastava, Rama Chellappa

Research output: Contribution to journalArticlepeer-review

275 Scopus citations


In this paper, we examine image and video-based recognition applications where the underlying models have a special structurethe 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.

Original languageEnglish (US)
Article number5740915
Pages (from-to)2273-2286
Number of pages14
JournalIEEE Transactions on Pattern Analysis and Machine Intelligence
Issue number11
StatePublished - 2011
Externally publishedYes


  • Grassmann
  • Image and video models
  • Stiefel
  • feature representation
  • manifolds
  • statistical models

ASJC Scopus subject areas

  • Software
  • Computer Vision and Pattern Recognition
  • Computational Theory and Mathematics
  • Artificial Intelligence
  • Applied Mathematics


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