Minimization of cost-functions with a non-smooth data-fidelity term. A new approach to the processing of impulsive noise

We consider signal and image restoration using convex cost-functions composed of a non-smooth data-fidelity term and a smooth regularization term. First, we provide a convergent method to minimize such cost-functions. Then we propose an efficient method to remove impulsive noise by minimizing cost-functions composed of an l1 data-fidelity term and an edge-preserving regularization term. Their minimizers have the property to fit exactly uncorrupted (regular) data samples and to smooth aberrant data entries (outliers). This method furnishes a new approach to the processing of data corrupted with impulsive noise. A crucial advantage over alternative filtering methods is that such cost-functions can convey adequate priors about the sought signals and images—such as the presence of edges. We present numerical experiments where images and signals are efficiently restored from highly corrupted data. Although cost-functions with non-smooth data-fidelity terms are not popular, our results advocate their use for special tasks in signal and image processing.