A partial regularity result for an anisotropic smoothing functional for image restoration in BV space

Here we examine the partial regularity of minimizers of a functional used for image restoration in BV space. This functional is a combination of a regularized p-Laplacian for the part of the image with small gradient and a total variation functional for the part with large gradient. This model was originally introduced in Chambolle and Lions using the Laplacian. Due to the singular nature of the p-Laplacian we study a regularized p-Laplacian. We show that where the gradient is small, the regularized p-Laplacian smooths the image u, in the sense that u∈C1,α for some 0<α<1. This functional thus anisotropically smooths the image where the gradient is small and preserves edges via total variation where the gradient is large.