A variational approach to maximum

*a posteriori* estimation for image denoising.

*(English)* Zbl 1001.68630
Figueiredo, Mário (ed.) et al., Energy minimization methods in computer vision and pattern recognition. 3rd international workshop, EMMCVPR 2001, Sophia Antipolis, France, September 3-5, 2001. Proceedings. Berlin: Springer. Lect. Notes Comput. Sci. 2134, 19-33 (2001).

Summary: Using first principles, we establish in this paper a connection between the maximum *a posteriori* (MAP) estimator and the variational formulation of optimizing a given functional subject to some noise constraints. A MAP estimator which uses a Markov or a maximum entropy random field model for a prior distribution can be viewed as a minimizer of a variational problem. Using notions from robust statistics, a variational filter called *Huber gradient descent flow* is proposed. It yields the solution to a Huber type functional subject to some noise constraints, and the resulting filter behaves like a total variation anisotropic diffusion for large gradient magnitudes and like an isotropic diffusion for small gradient magnitudes. Using some of the gained insight, we are also able to propose an information-theoretic gradient descent flow whose functional turns out to be a compromise between a neg-entropy variational integral and a total variation. Illustrating examples demonstrate a much improved performance of the proposed filters in the presence of Gaussian and heavy tailed noise.

##### MSC:

68U99 | Computing methodologies |

68T10 | Pattern recognition, speech recognition |

68T45 | Machine vision and scene understanding |