Partial regularity for a selective smoothing functional for image restoration in BV space
Preprint Series # 693 Chen, Yunmei and Rao, Murali and Tonegawa, Yoshihiro and Wunderli, T Partial regularity for a selective smoothing functional for image restoration in BV space. (2005); AbstractIn 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 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 ther exists a nonempty open region $E$ for which $u \in C^{1,\alpha}$ on $E$ and $\nabla u<\epsilon$, and $\nabla u \geq \epsilon $ on $\Omega\setminus E $ a.e. Thus we indeed have smoothing where $\nabla u<\ \epsilon$.
