Steerable Filters in Motion Estimation

This paper deals with steerable filters in context of motion estimation. Whereas the classical brightness constancy constraint equation (BCCE) connects the optical flow with the first order directional derivative filter of the observed signal, we abstract to the most general class of filters nullifying the signal when applied in the direction of motion. We developed a method to adjust linear combinations of such directional filters to the characteristics of the signal and noise. Steerability, one of the essential property of this class of filters is not fully covered by recent steerable approaches. Thus, we extended recent steerable approaches to arbitrary Lie groups, like the important case of the rotation group SO(3) in three dimensions. In contrast to recent theories which either do not cover the case of non-Abelian groups [16] or do not deliver the minimum number of basis functions [11] our approach uses the full power of Lie group theory to generate the minimum number of basis functions also for non-Abelian compact Lie groups . It is a direct extension of the approach of [16] in which the basis functions are generated from the eigenfunctions of the generators of the Lie group. In our approach a Casimir operator is used to generate the basis functions also for non-Abelian compact Lie groups. For non-compact groups it turned out that at least any polynomial can be steered by any Lie group transformation.