Estimation of signals containing strongly homogeneous zones

We consider the generic problem of the recovery of signals comprising large strongly homogeneous zones, e.g., locally constant, locally linear, etc., from noisy data given at the output of a linear system. The solution is defined as the minimizer of an objective function combining a data-fidelity term and a regularization term. In our approach, prior knowledge about the presence of large strongly homogeneous zones in the estimate is incorporated directly in the estimator function, through the shape of the regularization term. The latter bears on the differences between consecutive samples. Our main result states that a necessary and sufficient condition that strongly homogeneous zones are recovered in the solution is that the regularization function is nonsmooth for zero-valued differences. In fact, the strongly homogeneous zones involved in such an estimate remain unchanged under small variations of the data. Equivalently, the data space is partitioned into volumes yielding solutions which are strongly homogeneous over the same zones. These theoretical results are illustrated on the deconvolution of a signal. They can be generalized to nonlinear observation systems as well as to the estimation of images