On the regularity of level sets of minimizers for denoising models using total variation regularization.

William K. Allard

Let s be a noisy grayscale image. For our purposes s : R^{2} → [0, 1], s is measurable and s vanishes off a bounded set. Various people have proposed using minimizers of

ϵTV(f) + β (∫ γ(f(x) − s(x)) dL^{2}x)

as denoising models for s; here ϵ > 0; β : [0, ∞) → [0, ∞) is nondecreasing and zero at zero; and γ : R → [0, ∞) is even, nondecreasing and zero at zero. For instance, Rudin, Osher and Fatemi take β(y) = y, y ∈ [0, ∞), and γ(y) = y^{2}, y ∈ R and Chan and Esedoglu take the same β but let γ(y) = |y|, y ∈ R.

In this work we we give detailed geometric information on the reduced boundaries of the sets {f ≥ y}, y ∈ R. We show these reduced boundaries are C^{1} curves (hypersurfaces if R^{2} above is replaced by R^{n}, 3 ≤ n ≤ 7) and, in many circumstances, give detailed information on their curvature. In addition, several examples of minimizers will be given for the models mentioned above.