Summary: SIAM J. MATH. ANAL. c XXXX Society for Industrial and Applied Mathematics
Vol. 0, No. 0, pp. 000­000
TOTAL VARIATION REGULARIZATION FOR IMAGE DENOISING,
I. GEOMETRIC THEORY
WILLIAM K. ALLARD
Abstract. Let 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. 1548­1562] 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() = {f L1()L() : f 0}.
Suppose s F() and : R [0, ) is locally Lipschitzian, positive on R {0}, and zero at zero.
Let F(f) = (f(x) - s(x)) dLnx for f F(); 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 > 0 and let F (f) = TV(f) + F(f) for f F(); here TV(f) is the total variation of f. A
minimizer of F is called a total variation regularization of s. Rudin, Osher, and Fatemi and Chan
and Esedo¯glu have studied total variation regularizations where (y) = y2 and (y) = |y|, y R,
respectively. As these and other examples show, the geometry of a total variation regularization
is quite sensitive to changes in . 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

Source: Allard, William K. - Department of Mathematics, Duke University

Collections: Mathematics