Total Variation Regularization for Image Denoising, III. Examples

Let F(R-2) be the family of bounded nonnegative Lebesgue measurable functions on R-2. Suppose s is an element of F(R-2) and gamma : R -> [0,infinity) is zero at zero, positive away from zero, and convex. For f is an element of F(Omega) let F(f) = integral(Omega)gamma(f(x)-s(x)) dL(2)x; here L-2 is Lebesgue measure on R-2. In the denoising literature F would be called a fidelity in that it measures how much f differs from s which could be a noisy grayscale image. Suppose 0 < epsilon < infinity and let m(epsilon)(loc)(F) be the set of those f is an element of F(R-2) such that TV(f) < infinity and epsilon TV(f) + F(f) <= epsilon TV(g) + F(g) for g is an element of k(f); here TV(f) is the total variation of f and k(f) is the set of g is an element of F(R-2) such that g = f off some compact subset of R-2. A member of m(epsilon)(loc)(F) is called a total variation regularization of s (with smoothing parameter epsilon). Rudin, Osher, and Fatemi in [Phys. D, 60 (1992), pp. 259-268] and Chan and Esedoglu in [SIAM J. Appl. Math., 65 (2005), pp. 1817-1837] have studied total variation regularizations of s where gamma(y) = y(2)/2 and gamma(y) = y, y is an element of R, respectively. It turns out that the character of a member of m(epsilon)(loc)(F) changes quite a bit as gamma changes. In [SIAM J. Imaging Sci., 1 (2008), pp. 400-417] the family m(epsilon)(loc)(F) was described in complete detail when s is the indicator function of a compact convex subset of R-2. Our main purpose in this paper is to describe, in complete detail, m(epsilon)(loc)(F) when s is the indicator function of either S = ([0, 1] x[0,-1]) boolean OR ([-1, 0] x [0, 1]) or S = {x is an element of R-2 : vertical bar x-c(+)vertical bar <= 1} boolean OR {x is an element of R-2 : vertical bar x-c(-)vertical bar <= 1}, where, for some l is an element of [1,infinity), c(+/-) = (+/- l, 0). We believe these examples reveal a great deal about the nature of total variation regularizations. For example, it is said that total variation denoising preserves edges; while this is certainly true in many cases and in comparison with other denoising methods, the examples given in sections 8 and 9 provide evidence to the contrary. In addition, one can test computational schemes for total variation regularization against these examples. We will also establish what we believe to be a number of interesting properties of m(epsilon)(loc)(F).