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 $x\in{\mathbb R}^p$ (such as a signal or an image) from noisy data $y\in{\mathbb R}^q$ by minimizing with respect to $x$ a regularized cost-function ${\cal F}(x,y)=\Psi(x,y)+\alpha\Phi(x)$, where $\Psi$ is a data-fidelity term, $\Phi$ is a smooth regularization term, and $\alpha>0$ is a parameter. Typically, $\Psi(x,y)=\|Ax-y \|^2$, where A is a linear operator. The data-fidelity terms $\Psi$ involved in regularized cost-functions are generally smooth functions; only a few papers make an exception to this and they consider restricted situations. Nonsmooth data-fidelity terms are avoided in image processing. In spite of this, we consider both smooth and nonsmooth data-fidelity terms. Our goal is to capture essential features exhibited by the local minimizers of regularized cost-functions in relation to the smoothness of the data-fidelity term. In order to fix the context of our study, we consider $\Psi(x,y)=\sum_i\psi(a_i^Tx-y_i)$, where $a_i^T$ are the rows of $A$ and $\psi$ is ${\cal C}^m$ on ${\mathbb R}\setminus \{0\}$. We show that if $\psi\1(0^-)<\psi\1(0^+)$, then typical data y give rise to local minimizers $\hat x$ of ${\cal F}(.,y)$ which fit exactly a certain number of the data entries: there is a possibly large set $\hat h$ of indexes such that $a_i^T\hat x=y_i$ for every $i\in\hat h$. In contrast, if $\psi$ is smooth on ${\mathbb R}$, for almost every y, the local minimizers of ${\cal F}(.,y)$ do not fit any entry of y. Thus, the possibility that a local minimizer fits some data entries is due to the nonsmoothness 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 nonsmooth data-fidelity terms in regularized cost-functions for special purposes in image and signal processing.