Multidimensional directional steerable filters - Theory and application to 3D flow estimation

In this paper, a thorough theoretical analysis on the construction of multi-dimensional directional steerable filters is given. Steerable filters have been constructed for up to three dimensions. We extend the relevant theory to multiple dimensions and construct multi-dimensional steerable filters, as well as quadrature pairs of such filters. Formulating the multi-dimensional motion estimation problem in the spatiotemporal frequency domain, it is shown that motion manifests itself as energy concentration along “motion hyper-planes” in that domain. Subsequently, using the constructed multi-dimensional filters, we formulate the “hyper-donut” mechanism, i.e. a mechanism to efficiently “measure” the “motion energy” on a “motion hyper-plane”. On top of that, rigorous mathematical analysis on the use of the constructed filters in the dense flow estimation task is given. Based on the theoretical developments, a steerable filter-based algorithm is formulated, in its simplest possible form, for estimating 3D flow in sequences of volumetric or point-cloud data. Experimental results on simulated and real-world data verify the validity of our arguments and the effectiveness of the proposed method.