The authors propose a new family of cost functions for signal and image recovery: they are composed of \(\ell_1\) data fitting terms combined with concave regularization. They exhibit when and how to employ such cost functions. Their theoretical results show that the minimizers of these cost functions are such that each one of their entries is involved either in an exact data fitting component or in a null component of the regularization part. This is a strong and particular property that can be useful for various image recovery problems. The minimization of such cost functions presents a computational challenge. Also they propose a fast minimization algorithm to solve this numerical problem. The experimental results show the effectiveness of the proposed algorithm. All illustrations and numerical experiments give a flavor of the possibilities offered by the minimizers of this new family of cost functions in solving specialized image processing tasks.