Recognition of Humans and their Activities using Statistical analysis on Stiefel and Grassmann Manifolds

Many applications in computer vision involve learning and recognition of patterns from exemplars which lie on certainmanifolds. Given a database of examples and a query, the following two questions are usually addressed – a) what is the
‘closest’ example to the query in the database ? b) what is the ‘most probable’ class to which the query belongs ? The answer
to the first question involves study of the geometric properties of the manifold, which then leads to appropriate definitions
of distance metrics on the manifold (geodesics etc). The answer to the second question involves statistical modeling of inter-
and intra-class variations on the manifold. In this paper, we concern ourselves with two related manifolds that often appear in
several vision applications – the Stiefel Manifold and the Grassmann Manifold. We describe statistical modeling and inference
tools on these manifolds which result in significant improvements in performance over traditional distance-based classifiers.
We illustrate applications to video-based face recognition and activity recognition.