An Alternating Direction Method for Optical Flow Estimation with lp Regularization

We consider the optical flow estimation problem with lp sub-quadratic regularization, where 0 ≤ p ≤ 1. As in other image analysis tasks based on functional minimization, sub-quadratic regularization is expected to admit discontinuities and avoid oversmoothing of the estimated optical flow field. The problem is mathematically challenging, since the regularization term is non-differentiable. It is harder than the l1 case, that can be addressed via Moreau proximal mapping with a closed form solution. In this paper, we propose a novel approach, based on variable splitting and the Alternating Direction Method of Multipliers (ADMM). We exemplify that our method can outperform optical flow with l1 regularization, but this is not the essence of this paper. The contribution is in demonstrating that state of the art optimization methods can be harnessed to solve a mathematically-challenging class of important image processing problems, and to highlight crucial numerical aspects that are often obscured in the image processing literature.