 
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 EsedoŻglu 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
