Local continuity and thresholding using truncated quadratic regularization

We consider the estimation of a signal, containing homogeneous zones and sharp edges, from noisy data given at the output of a linear system. The solution is defined as the global minimizer of an objective function combining a quadratic data-fidelity term and a regularization term. The latter term is a sum over the differences between adjacent samples, weighted using a truncated quadratic potential function. Such estimators are formulated either in a statistical or in a variational framework and they are frequently used in practice. The resultant objective function is highly nonconvex and nonsmooth. We show that its global minimizers determine implicit functions which are locally continuous with respect to the data. Furthermore, we find that the magnitude of the differences between adjacent samples of a global minimizer are either smaller than a first threshold, or larger than a second threshold, which is strictly larger than the first one. No difference at a global minimizer can thus be placed between these thresholds, independently of the data. This result explains how edges are detected in an estimate involving a truncated quadratic regularization