Posterior Expectation of the Total Variation Model

The total variation image (or signal) denoising model is a variational approach that can be interpreted, in a Bayesian framework, as a search for the maximum point of the posterior density (maximum a posteriori estimator). This maximization aspect is partly responsible for a restoration bias called the “staircasing effect,” that is, the outbreak of quasi-constant regions separated by sharp edges in the intensity map. In this paper we study a variant of this model that considers the expectation of the posterior distribution instead of its maximum point. Apart from the least square error optimality, this variant seems to better account for the global properties of the posterior distribution. We present theoretical and numerical results that demonstrate in particular that images denoised with this model do not suffer from the staircasing effect.