Summary: 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 \(F(f)=\int_{\mathbb{R}^2}\gamma(f(x)-s(x))\,d\mathcal{L}^2x\); \(\mathcal L^2\) 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 \(0<\varepsilon<\infty\), and let \(\mathbf{m}^{\text{loc}}_{\varepsilon}(F)\) be the set of those \(f\in\mathcal{F}(\mathbb{R}^2)\) such that \(\mathbf{TV}(f)<\infty\) and \(\varepsilon\mathbf{TV}(f)+F(f)\leq\varepsilon\mathbf{TV}(g)+F(g)\) for \(g\in\mathbf{k}(f)\); here \(\mathbf{TV}(f)\) is the total variation of \(f\), and \(\mathbf{k}(f)\) is the set of \(g\in\mathcal{F}(\mathbb{R}^2)\) such that \(g=f\) off some compact subset of \(\mathbb{R}^2\). A member of \(\mathbf{m}^{\text{loc}}_{\varepsilon}(F)\) is called a total variation regularization of \(s\) (with smoothing parameter \(\varepsilon\)). L. I. Rudin, S. Osher and E. Fatemi in [Physica D 60, No. 1–4, 259–268 (1992; Zbl 0780.49028)] and T. F. Chan and S. Esedoglu in [SIAM J. Appl. Math. 65, No. 5, 1817–1837 (2005; Zbl 1096.94004)] 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 \(\mathbf{m}^{\text{loc}}_{\varepsilon}(F)\) when \(s\) is the indicator function of a compact convex subset of \(\mathbb{R}^2\). It will turn out that if \(f\in\mathbf{m}^{\text{loc}}_{\varepsilon}(F)\), then, for \(0<y<1\), the set \(\{f>y\}\) is essentially empty or is essentially the union of the family of closed balls of a certain radius depending in a simple way on \(\gamma, \varepsilon\), 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 [cf. Part III, SIAM J. Imaging Sci. 2, No. 2, 532–568, electronic only (2009; Zbl 1175.49038)].
For Part I, see the author, SIAM J. Math. Anal. 39, No. 4, 1150–1190 (2007; Zbl 1185.49047).