Narrow directional steerable filters in motion estimation

We extend the mathematical analysis of previous work [M.T. Andersson, Controllable multi-dimensional filters and models in low-level computer vision, Ph.D. Thesis, Department of Electrical Engineering, Linkonping University, Sweden, 1992] and we give rigorous, general, mathematical formulas for the construction of 3-D steerable directional cosine filters of arbitrary higher order. Furthermore, we present the mathematical analysis for the construction of arbitrary narrow, steerable directional quadrature pairs. Incorporating the “Donut Mechanism” of Simoncelli [E.P. Simoncelli, Distributed representation and analysis of visual motion, Ph.D. Thesis, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, 1993] and extending it for quadrature pairs, we present a unified theory and a simple algorithm for using the constructed filters to estimate the motion in image sequences. Based on simple theoretical analysis, we explain the advantages of using higher order filters. Experimental results on synthetic, realistic, and natural sequences verify the effectiveness of the main algorithm and our arguments.