# 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);

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## Abstract

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 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 non-empty 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 $|\nabla u|<\ \epsilon$.

Item Type: Preprint 10 copies bounded variation, selective smoothing, image processing, image restoration, noise removal, partial regularity 49-xx CALCULUS OF VARIATIONS AND OPTIMAL CONTROL; OPTIMIZATION35-xx PARTIAL DIFFERENTIAL EQUATIONS 860