Improved higher degree total variation (HDTV) regularization

The main focus of this paper is to further improve the performance of the recently introduced higher degree total variation (HDTV) penalties, which are L₁-L_p; p ≥ 1 norms of directional image derivatives. We generalize this class as the L₁-L_p norms of image responses to rotated versions of an arbitrary derivative operator. We show that several penalties proposed by other researchers are special cases of the generalized isotropic penalties (p = 2), when the derivative operator is chosen appropriately; our experiments show that the anisotropic (p = 1) versions of these penalties provide improved reconstructions. In addition, we optimize the derivative operator for improved orientation selectivity, thus further improving the ability of the resulting penalties to provide high quality image reconstructions. We also focus on the efficient discretization of HDTV penalties, which are specified in the continuous domain. Specifically, we approximate the derivative operators as the sum of partial derivatives of an almost isotropic B-spline window. Our numerical experiments confirm the benefit of the improved discretization and the optimization of the operator.