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Total variation regularization for image denoising. I: Geometric theory. (English) Zbl 1185.49047

Let \(\Omega\) be an open subset of \(\mathbb{R}^n\), 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, No. 6, 1548–1562 (1992; Zbl 0859.93037)] 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 \(\mathcal{F}(\Omega)=\{f\in{\mathbf L}_1{(\Omega)}\cap{\mathbf L}_{\infty}{(\Omega)}:f\geq 0\}.\) 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 \(F(f)=\int_\Omega\gamma(f(x)-s(x))\,d\mathcal{L}^nx\) 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 \(\varepsilon>0\) and let \(F_\varepsilon(f)=\varepsilon\mathbf{TV}(f)+F(f)\) for \(f\in\mathcal{F}{\Omega}\); here \(\mathbf{TV}(f)\) is the total variation of \(f\). A minimizer of \(F_\varepsilon\) is called a total variation regularization of \(s\).
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 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 \(y\), \(\varepsilon\) 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.
[For Part II, see the author, SIAM J. Imaging Sci. 1, No. 4, 400–417, electronic only (2008; Zbl 1185.49048), III, SIAM J. Imaging Sci. 2, No. 2, 532-568, electronic only (2009; Zbl 1175.49038).]

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
58E30 Variational principles in infinite-dimensional spaces
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
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