Image restoration by minimizing objective functions with nonsmooth data-£delity terms

We present a theoretical study of the recovery of images x from noisy data y by minimizing a regularized cost-function F(x, y) = Ψ(x, y) + αΦ(x), where Ψ is a data-£delity term, Φ is a smooth regularization term and α > 0 is a parameter. Generally, Ψ is a smooth function; only a few papers make an exception. Non-smooth data-£delity terms are avoided in image processing. In spite of this, we consider both smooth and non-smooth data-£delity terms. Our ambition is to catch essential features exhibited by the local minimizers of F in relation with the smoothness of Ψ. We focus on Ψ(x, y) = ∑ i ψ(a T i x−yi) where a T i are linear operators and ψ is C-smooth on IR\{0}. We show that if ψ(0) < ψ(0), typical data y lead to local minimizers x̂ of F(., y) which £t exactly part of the data entries: there is a possibly large set ĥ such that ai x̂= yi for every i∈ ĥ. In contrast, if ψ is smooth on IR, for almost every y, the local minimizers of F(., y) do not £t any entry of y. Cost-functions with non-smooth data-£delity exhibit a strong mathematical property which can be used in various ways. We then construct a cost-function allowing aberrant data to be detected and selectively smoothed. The obtained results advocate the use of non-smooth data-£delity terms.