Towards a unified view of estimation: variational vs. statistical

A connection between the maximum a posteriori (MAP) estimation and the variational formulation based on the minimization of a given variational integral subject to some noise constraints is established in this paper. A MAP estimator which uses a Markov or a maximum entropy random field model for the prior distribution can be viewed as a minimizer of a variational problem. Inspired by the maximum entropy principle, a nonlinear variational filter called improved entropic gradient descent flow is proposed. It minimizes a hybrid functional between the neg-entropy variational integral and the total variation subject to some noise constraints. Simulation results showing a much improved performance of the proposed filter in the presence of Gaussian and Laplacian noise are analyzed and illustrated