Convex Generalizations of Total Variation Based on the Structure Tensor with Applications to Inverse Problems

Abstract

We introduce a generic convex energy functional that is suitable for both grayscale and vector-valued images. Our functional is based on the eigenvalues of the structure tensor, therefore it penalizes image variation at every point by taking into account the information from its neighborhood. It generalizes several existing variational penalties, such as the Total Variation and vectorial extensions of it. By introducing the concept of patch-based Jacobian operator, we derive an equivalent formulation of the proposed regularizer that is based on the Schatten norm of this operator. Using this new formulation, we prove convexity and develop a dual definition for the proposed energy, which gives rise to an efficient and parallelizable minimization algorithm. Moreover, we establish a connection between the minimization of the proposed convex regularizer and a generic type of nonlinear anisotropic diffusion that is driven by a spatially regularized and adaptive diffusion tensor. Finally, we perform extensive experiments with image denoising and deblurring for grayscale and color images. The results show the effectiveness of the proposed approach as well as its improved performance compared to Total Variation and existing vectorial extensions of it.

S.L. and A.R. contributed equally and have joint first authorship. S.L. and M.U. were supported (in part) by the Hasler Foundation and the Indo-Swiss Joint Research Program. A.R. was supported by the ERC Starting Grant 204871-HUMANIS. P.M. was partially supported by the Greek research grant COGNIMUSE under the ARISTEIA action.