A property of the minimum vectors of a regularizing functional defined by means of the absolute norm

We consider a regularizing functional defined by means of the l ₁ norm, where the regularization is obtained using first differences; as is well-known, such a functional can be put in relation with recursive median filters of appropriate window length. We show that at least one of the minima is reached at a vector, whose components have values over the same discrete set of the given signal. This suggests a simple method to refine the approximate solution to the regularization problem, which can be obtained with recursive median filters of increasing order. We also report an example of application, where the refinement method is employed for a signal detection problem