Total variation regularization for image denoising, 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.) = {f ∈ L1 (Ω)∩lL∞(Ω) : f ≥ 0} Suppose s ∈ ℱ(Ω) and γ : ℝ → [0, ∞) is locally Lipschitzian, positive on ℝ ∼ {0}, and zero at zero. Let F(f) = ∫Ω γ(f(x) - s(x)) dℒn x for f ∈ 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 x ∈ ℝn. 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) = εTV(f) + F(f) for f ∈ ℱ(Ω); here TV(f) is the total variation of f. A minimizer of Fε is called a total variation regularization of s. Rudin, Osher, and Fatemi and Chan and Esedoglu have studied total variation regularizations where γ(y) = y 2 and γ(y) = |y|, y ∈ ℝ, 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 {f > y}, 0 < y < ∞, are embedded C1, μ hypersurfaces for any μ ∈ (0,1) where n > 2 and any μ ∈ (0,1] where n = 2; moreover, the generalized mean curvature of the sets {f ≥ 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 {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. © 2007 Society for Industrial and Applied Mathematics.

Duke Scholars

Author William K. Allard Mathematics