Partial Regularity for a Selective Smoothing Functional for Image Restoration in BV Space
Yunmei Chen, Murali Rao, Yoshihiro Tonegawa, Thomas Wunderli · SIAM Journal on Mathematical Analysis · 2005
In this paper we study the partial regularity of a functional on BV space proposed by Chambolle and Lions Numer. Math., 76 (1997), pp. 167--188] 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 $\epsilon$ and retain edges where gradients are above the threshold using the total variation. Here we prove that if the solution $u\in BV$ of the model minimization problem, defined on an open set $\Omega$, is such that the Lebesgue measure of the set where the gradient of u is below the threshold $\epsilon$ is positive, then there exists a nonempty open region E for which $u\in C^{1,\alpha}$ on E and $| abla u| < \epsilon$, and $| abla u|\ge \epsilon$ on $\Omega\backslash E$ almost everywhere. Thus we indeed have smoothing where $| abla u| < \epsilon$.