Suppose s is a noisy grayscale image which we wish to denoise; mathematically, s will be a bounded real valued function on R^2. One way to do this is as follows. Let \gamma: R --> [0,\infty) be zero at zero, positive away from zero and convex. Let \epsilon >0 be a ``smoothing'' parameter which we will tune appropriately and let
F_\epsilon(f) = \epsilon TV(f) + \int \gamma(f(x)-s(x)) dx
for f: R^2 --> [0,\infty); here TV(f) is the total variation of f. Minimizers of F_\epsilon are used to denoise s.
In this talk we will describe some notable geometric properties of minimizers and will provide some interesting examples. It turns out that these results rest on the theory of area minimization which has been in the literature for over 35 years.
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