Nonlocal Structure Tensor Functionals for Image Regularization

We present a nonlocal regularization framework that we apply to inverse imaging problems. As opposed to existing nonlocal regularization methods that rely on the graph gradient as the regularization operator, we introduce a family of nonlocal energy functionals that involves the standard image gradient. Our motivation for designing these functionals is to exploit at the same time two important properties inherent in natural images, namely the local structural image regularity and the nonlocal image self-similarity. To this end, our regularizers employ as their regularization operator a novel nonlocal version of the structure tensor. This operator performs a nonlocal weighted average of the image gradients computed at every image location and, thus, is able to provide a robust measure of image variation. Furthermore, we show a connection of the proposed regularizers to the total variation semi-norm and prove convexity. The convexity property allows us to employ powerful tools from convex optimization to design an efficient minimization algorithm. Our algorithm is based on a splitting variable strategy, which leads to an augmented Lagrangian formulation. To solve the corresponding optimization problem, we employ the alternating-direction methods of multipliers. Finally, we present extensive experiments on several inverse imaging problems, where we compare our regularizers with other competing local and nonlocal regularization approaches. Our results are shown to be systematically superior, both quantitatively and visually.