Boundary Preserving Variational Image Differentiation

Abstract

The purpose of this study is to investigate image differentiation by a boundary preserving variational method. The method minimizes a functional composed of an anti-differentiation data discrepancy term and an $L^1$ regularization term. For each partial derivative of the image, the anti-differentiation term biases the minimizer toward a function which integrates to the image up to an additive constant, while the regularization term biases it toward a function smooth everywhere except across image edges. A discretization of the functional Euler-Lagrange equations gives a large scale system of nonlinear equations that, however, is sparse, and “almost” linear, which directs to a resolution by successive linear approximations. The method is investigated in two important computer vision problems, namely optical flow and scene flow estimation, where image differentiation is used and ordinarily done by local averaging of finite image differences. We present several experiments, which show that motion fields are more accurate when computed using image derivatives evaluated by regularized variational differentiation than with conventional averaging of finite differences.