Total Variation Regularization for Image Denoising, I. Geometric Theory

Let Ω be an open subset of Rⁿ, where 2≤ n≤ 7; we assume n≥ 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≤ 7 because we will make use of the regularity theory for area minimizing hypersurfaces. Let F(Ω)=∈ L₁Ω∩ L∞Ω :f≥ 0\. Suppose s(Ω) and γ:R→[0,∞) is locally Lipschitzian, positive on R~\0\, and zero at zero. Let F(f)=∫_Ωγ(f(x)-s(x))\,dLⁿx for f(Ω); here Lⁿ is Lebesgue measure on Rⁿ. Note that $F(f)=0$ if and only if $f(x)=s(x)$ for Lⁿ almost all xⁿ. 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 ε>0 and let F_ε(f)=εbf TV(f)+F(f) for f∈FΩ; here TV(f) is the total variation of f. A minimizer of F_ε is called a total variation regularization of s. Rudin, Osher, and Fatemi and Chan and Esedolu have studied total variation regularizations where γ(y)=y² and γ(y)=|y|, y, respectively. As these and other examples show, the geometry of a total variation regularization is quite sensitive to changes in γ. Let f be a total variation regularization of s. The first main result of this paper is that the reduced boundaries of the sets >y\, $0 2$ and any μ∈(0,1] where $n=2$; moreover, the generalized mean curvature of the sets ≥ y\ will be bounded in terms of y, ε 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 >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.