Statistical methods on special manifolds for image and video understanding

In this chapter we describe some recent advances in image and video understanding applications based on statistical inference methods over nonlinear manifolds. These tools are developed by adapting techniques from multivariate statistics to the differential geometries of the manifold of interest. These manifolds include: (1) transformation groups, such as rotation, translation, and scale, (2) their quotient spaces, and (3) shape manifolds of planar curves. We discuss applications of statistical methods on the aforementioned manifolds for face recognition, shape analysis, activity recognition, age estimation, and gesture recognition. In each case we motivate the need for manifold-based methods and discuss techniques from differential geometry that are needed to design appropriate statistical inference algorithms for these applications. These discussions contain some illustrative examples and references to the literature for additional reading.