Total variation regularization for image denoising, I. Geometric theory

Let Omega be an open subset of R-n, 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(Omega) = {f is an element of L-1(Omega) boolean AND L-infinity(Omega) : f >= 0}. Suppose s is an element of F(Omega) and gamma : R -> [0,infinity) is locally Lipschitzian, positive on R similar to {0}, and zero at zero. Let F(f) = integral(Omega) gamma(f(x)-s(x)) dL(n)x for f is an element of F(Omega); here L-n is Lebesgue measure on R-n. Note that F(f) = 0 if and only if f(x) = s(x) for L-n almost all x is an element of 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 epsilon > 0 and let F-epsilon(f) = epsilon TV(f) + F(f) for f is an element of F(Omega); here TV(f) is the total variation of f. A minimizer of F-epsilon is called a total variation regularization of s. Rudin, Osher, and Fatemi and Chan and Esedoglu have studied total variation regularizations where gamma(y) = y(2) and gamma(y) = |y|, y is an element of 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 < infinity, are embedded C-1,C-mu hypersurfaces for any mu is an element of (0, 1) where n > 2 and any mu is an element of (0, 1] where n = 2; moreover, the generalized mean curvature of the sets {f >= y} will be bounded in terms of y, epsilon 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.