Image restoration by minimizing 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 regularisation term and α>0 is a parameter. Generally Ψ is a smooth function; only a few papers make an exception. Non-smooth data-fidelity terms are avoided in image processing. In spite of this, we consider both smooth and non-smooth data-fidelity terms. Our ambition is to catch essential features exhibited by the local minimizers of F in relation with the smoothness of Ψ. Cost-functions with non-smooth data-fidelity 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-fidelity terms