Feature detection and tracking in optical flow on non-flat manifolds

Optical flow is a classical tool to estimate velocity vector fields from objects on a manifold. The Helmholtz-Hodge Decomposition (HHD) of vector fields has been used to isolate rotational and divergential features in vector fields, such as vortices in vector fields. However, the existing HHD techniques operate on flat, 2D domains, which is non-adequate to many potential applications. In this paper, we extend the Helmholtz-Hodge decomposition to vector fields defined over any arbitrary surface manifolds. This is achieved using a Riemannian variational formalism. We illustrate the proposed methodology with the decomposition of optical flow vector fields defined on a variety of surface objects using synthetic and experimental data. ∗Mailing Address: Department of Neurology, Massachusetts General Hospital, Harvard Medical School, Boston, MA 02129, Phone: 617-643-5634, Fax 617-948-5966 Preprint submitted to Pattern recognition Letters April 25, 2011