Partial Regularity for a Selective Smoothing Functional for Image Restoration in BV Space

In this paper we study the partial regularity of a functional on BV space proposed by Chambolle and Lions [3] for the purposes of image restoration. The functional is designed to smooth corrupted images using isotropic diffusion via the Laplacian where the gradients of the image are below a certain threshold and retain edges where gradients are above the threshold using the total variation. Here we prove that if the solution u ∈ BV of the model minimization problem, defined on an open set Ω, is such that the Lebesgue measure of the set where the gradient of u is below the threshold is positive, then there exists a non-empty open region E for which u ∈ C1,α on E and |∇u| < , and |∇u| ≥ on Ω\E a.e. Thus we indeed have smoothing where |∇u| < .