Total Variation Regularization for Image Denoising, I. Geometric Theory

Abstract

Let $\Omega$ be an open subset of ${\bf R}^n$, where $2\leq n\leq 7$; we assume $n\geq 2$ because the case $n=1$ has been treated elsewhere (see [S. S. Alliney, IEEE Trans. Signal Process., 40 (1992), pp. 1548–1562] and is quite different from the case $n>1$; we assume $n\leq 7$ because we will make use of the regularity theory for area minimizing hypersurfaces. Let $\mathcal{F}(\Omega)=\{f\in{\bf L}_{1}{\Omega}\cap{\bf L}_{\infty}{\Omega} :f\geq 0\}.$ Suppose $s\in\mathcal{F}(\Omega)$ and $\gamma:\mathbb{R}\rightarrow[0,\infty)$ is locally Lipschitzian, positive on $\mathbb{R}\sim\{0\}$, and zero at zero. Let $F(f)=\int_\Omega\gamma(f(x)-s(x))\,d\mathcal{L}^{n}x$ for $f\in\mathcal{F}(\Omega)$; here $\mathcal{L}^{n}$ is Lebesgue measure on $\mathbb{R}^{n}$. Note that $F(f)=0$ if and only if $f(x)=s(x)$ for $\mathcal{L}^{n}$ almost all $x\in\mathbb{R}^{n}$. In the denoising literature F would be called a fidelity in that it measures deviation from s, which could be a noisy grayscale image. Let $\epsilon>0$ and let $F_\epsilon(f)=\epsilon{bf TV}(f)+F(f)$ for $f\in\mycal{F}{\Omega}$; here ${\bf TV}(f)$ is the total variation of f. A minimizer of $F_\epsilon$ is called a total variation regularization of s. Rudin, Osher, and Fatemi and Chan and Esedolu have studied total variation regularizations where $\gamma(y)=y^2$ and $\gamma(y)=|y|$, $y\in\mathbb{R}$, respectively. As these and other examples show, the geometry of a total variation regularization is quite sensitive to changes in $\gamma$. Let f be a total variation regularization of s. The first main result of this paper is that the reduced boundaries of the sets $\{f>y\}$, $0<y<\infty$, are embedded $C^{1,\mu}$ hypersurfaces for any $\mu\in(0,1)$ where $n>2$ and any $\mu\in(0,1]$ where $n=2$; moreover, the generalized mean curvature of the sets $\{f\geq y\}$ will be bounded in terms of y, $\epsilon$ and the magnitude of $|s|$ near the point in question. In fact, this result holds for a more general class of fidelities than those described above. A second result gives precise curvature information about the reduced boundary of $\{f>y\}$ in regions where s is smooth, provided F is convex. This curvature information will allow us to construct a number of interesting examples of total variation regularizations in this and in a subsequent paper. In addition, a number of other theorems about regularizations are proved.