Journal of Mathematical Imaging and Vision
Integro-Differential Equations Based on $(BV, L^1)$ Image Decomposition
SIAM Journal on Imaging Sciences
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Let $\mathcal{F}(\mathbb{R}^{2})=\{f\in\mathbf{L}_{\infty}(\mathbb{R}^{2})\cap\mathbf{L}_{1}(\mathbb{R}^{2}):f\geq 0\}$. Suppose $s\in\mathcal{F}(\mathbb{R}^{2})$ and $\gamma:\mathbb{R}\rightarrow[0,\infty)$. Suppose $\gamma$ is zero at zero, positive away from zero, and convex. For $f\in\mathcal{F}(\mathbb{R}^{2})$ let $F(f)=\int_{\mathbb{R}^2}\gamma(f(x)-s(x))\,d\mathcal{L}^{2}x$; $\mathcal{L}^{2}$ here is Lebesgue measure on $\mathbb{R}^2$. In the denoising literature $F$ would be called a fidelity in that it measures how much $f$ differs from $s$, which could be a noisy grayscale image. Suppose $0total variation regularization of $s$ (with smoothing parameter $\epsilon$). Rudin, Osher, and Fatemi in [Phys. D, 60 (1992), pp. 259-268] and Chan and Esedog¯lu in [SIAM J. Appl. Math., 65 (2005), pp. 1817-1837] have studied total variation regularizations of $F$ where $\gamma(y)=y^2$ and $\gamma(y)=y$, $y\in\mathbb{R}$, respectively. Our purpose in this paper is to determine $\mathbf{m}^{\text{{\it loc\/}}}_{\epsilon}(F)$ when $s$ is the indicator function of a compact convex subset of $\mathbb{R}^{2}$. It will turn out that if $f\in\mathbf{m}^{\text{{\it loc\/}}}_{\epsilon}(F)$, then, for $0y\}$ is essentially empty or is essentially the union of the family of closed balls of a certain radius depending in a simple way on $\gamma$, $\epsilon$, and $y$. While taking $s=1_S$, $S$ compact and convex, is certainly not representative of the functions $s$ which occur in image denoising, we hope this result sheds some light on the nature of total variation regularizations. In addition, one can test computational schemes for total variation regularization against these examples. Examples where $S$ is not convex will appear in a later paper.