Unifying probabilistic and variational estimation

A maximum a posteriori (MAP) estimator using a Markov or a maximum entropy random field model for a prior distribution may be viewed as a minimizer of a variational problem.Using notions from robust statistics, a variational filter referred to as a Huber gradient descent flow is proposed. It is a result of optimizing a Huber functional subject to some noise constraints and takes a hybrid form of a total variation diffusion for large gradient magnitudes and of a linear diffusion for small gradient magnitudes. Using the gained insight, and as a further extension, we propose an information-theoretic gradient descent flow which is a result of minimizing a functional that is a hybrid between a negentropy variational integral and a total variation. Illustrating examples demonstrate a much improved performance of the approach in the presence of Gaussian and heavy tailed noise. In this article, we present a variational approach to MAP estimation with a more qualitative and tutorial emphasis. The key idea behind this approach is to use geometric insight in helping construct regularizing functionals and avoiding a subjective choice of a prior in MAP estimation. Using tools from robust statistics and information theory, we show that we can extend this strategy and develop two gradient descent flows for image denoising with a demonstrated performance.