Integro-Differential Equations Based on $(BV, L^1)$ Image Decomposition
SIAM Journal on Imaging Sciences
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Let $\mathcal{F}(\mathbb{R}^{2})$ be the family of bounded nonnegative Lebesgue measurable functions on $\mathbb{R}^{2}$. Suppose $s\in\mathcal{F}(\mathbb{R}^{2})$ and $\gamma:\mathbb{R}\rightarrow[0,\infty)$ is zero at zero, positive away from zero, and convex. For $f\in\mathcal{F}(\Omega)$ let $F(f)=\int_\Omega\gamma(f(x)-s(x))\,d\mathcal{L}^{2}x$; here $\mathcal{L}^{2}$ 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 $s$ where $\gamma(y)=y^2/2$ and $\gamma(y)=y$, $y\in\mathbb{R}$, respectively. It turns out that the character of a member of ${\bf m}^{loc}_{\epsilon}(F)$ changes quite a bit as $\gamma$ changes. In [SIAM J. Imaging Sci., 1 (2008), pp. 400-417] the family ${\bf m}^{loc}_{\epsilon}(F)$ was described in complete detail when $s$ is the indicator function of a compact convex subset of $\mathbb{R}^{2}$. Our main purpose in this paper is to describe, in complete detail, ${\bf m}^{loc}_{\epsilon}(F)$ when $s$ is the indicator function of either $S=([0,1]\times[0,-1])\cup([-1,0]\times[0,1])$ or $S=\{x\in\mathbb{R}^{2}:|x-{\bf c}_+|\leq 1\}\cup\{x\in\mathbb{R}^{2}:|x-{\bf c}_-|\leq 1\}$, where, for some $l\in[1,\infty)$, ${\bf c}_\pm=(\pm l,0)$. We believe these examples reveal a great deal about the nature of total variation regularizations. For example, it is said that total variation denoising preserves edges; while this is certainly true in many cases and in comparison with other denoising methods, the examples given in sections 2squares and 2circles provide evidence to the contrary. In addition, one can test computational schemes for total variation regularization against these examples. We will also establish what we believe to be a number of interesting properties of ${\bf m}^{loc}_{\epsilon}(F)$.