Minimizers of Cost-Functions Involving Nonsmooth Data-Fidelity Terms. Application to the Processing of Outliers

We present a theoretical study of the recovery of an unknown vector x ∈ IR (a signal, an image) from noisy data y ∈ IR by minimizing with respect to x 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 α > 0 is a parameter. Typically, Ψ(x, y) = ‖Ax − y‖ where A is a linear operator. The data-fidelity terms Ψ involved in regularized costfunctions are generally smooth functions; only a few papers make an exception and they consider restricted situations. 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 regularized cost-functions in relation with the smoothness of the data-fidelity term. In order to fix the context of our study, we consider Ψ(x, y) = ∑ i ψ(aTi x− yi), where a T i are the rows of A and ψ is C on IR\{0}. We show that if ψ′(0−) < ψ(0), typical data y give rise to local minimizers x̂ of F(., y) which fit exactly a certain number of the data entries: there is a possibly large set ĥ of indexes such that aTi 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 fit any entry of y. Thus, the possibility that a local minimizer fits some data entries is due to the non-smoothness of the data-fidelity term. This is a strong mathematical property which is useful in practice. By way of application, we construct a cost-function allowing aberrant data (outliers) to be detected and to be selectively smoothed. Our numerical experiments advocate the use of non-smooth data-fidelity terms in regularized cost-functions for special purposes in image and signal processing.