Tuesday, April 8, 1997
We are concerned with the class of estimators defined as the minimizer of a coupled criterion, joining a quadratic data-fidelity term and a regularization term. The latter term is added to favour the recovery of features known a priori, and it sums the contributions of the differences between neighbouring pixels, weighted using general potential functions (PFs). Such estimators arise in both the regularization framework and in Bayesian MAP estimation, the regularization term being a Markovian energy. Quite a variety of PFs, convex or bounded, non-smooth or discontinuous at some points, are used in practice which correspond to different prior models. Here we explore the mutual relation between the shape of the PF and the main characteristics of the estimate such as recovery of homogeneous regions, stability, edge recovery, bias over edges, and resolution.
First we give necessary conditions for local minima of a piecewise smooth
multimodal criterion. Then, we show that the global minimum is regularized
while strict local minima evolve continuously with the data. Estimators
incorporating non-convex PFs are piecewise continuous, so they are locally
stable. Furthermore, a solution presenting strictly homogeneous regions
can be obtained if and only if the PF is non-smooth at zero. These
strictly homogeneous regions remain unchanged under small variations of
the data. The presence of local concavities in the PF non-convex PFs
affect edge detection performance. Moreover, concave
non-differentiability implies a classification of the differences w.r.t. a
fixed threshold, thus refining the resolution. In particular, we show that
edges may be recovered without bias only if the PF is constant beyond a
Mila Nikolova received the Ph.D. degree in Signal Processing from the Universite de Paris-Sud in 1995. Currently, she is teaching and research assistant at the Universite Paris V. Her research interests are in inverse problems and image reconstruction.