Smoothing of outliers in image restoration by minimizing regularized objective functions with nonsmooth data-fidelity 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-fidelity term, Φ is a smooth regularization term and a>0 is a parameter. Generally, Ψ is a smooth function; only a few papers make an exception. Nonsmooth data-fidelity terms are avoided in image processing. In spite of this, 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)=Σ_iψ(a _i^Tx-y_i) where a_i^T are linear operators and ψ is C^m-smooth on R{0}. We show that if ψ´(0^-)<ψ´(0⁺), typical data y lead to local minimizers xˆ of F(., y) which fit exactly part of the data entries: there is a possibly large set hˆ such that a_i^Txˆ=y_i for every i∈hˆ. In contrast, this effect does not occur if F is smooth. We thus have a strong mathematical property which can be used in various ways. Based on it, we construct a cost-function allowing aberrant data to be detected and selectively smoothed either from images or from noisy data, while preserving efficiently all non-aberrant pixels. This is illustrated by numerical examples. The obtained results advocate the use of non-smooth data-fidelity terms in image processing