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

Optical flow is a classical approach to estimating the velocity vector fields associated to illuminated objects traveling onto manifolds. The extraction of rotational (vortices) or curl-free (sources or sinks) features of interest from these vector fields can be obtained from their Helmholtz–Hodge decomposition (HHD). However, the applications of existing HHD techniques are limited to flat, 2D domains. Here we demonstrate the extension of the HHD to vector fields defined over arbitrary surface manifolds. We propose a Riemannian variational formalism, and illustrate the proposed methodology with synthetic and empirical examples of optical-flow vector field decompositions obtained on a variety of surface objects.

► Feature detection of critical points in optical flow on non-flat surfaces. ► Helmholtz Hodge decomposition (HHD) in Riemannian formulism. ► Vector fields on non-flat surfaces. ► Dimensionality reduction in optical flow, by defining its salient feature in few equivalent feature sets. ► Application of optical flow/HHD in structural and functional brain imaging.