Tid: 21 april 2008 kl 15.15-16.00

Plats : Seminarierummet 3733, Institutionen för matematik, KTH, Lindstedts väg 25, plan 7. Karta!

Föredragshållare: Prof. Ali Mohammad-Djafari, Paris

Titel: Inverse Problems: From regularization theory to Bayesian inference

Sammanfattning: In this talk, after a synthetic analysis of the main deterministic methods (analytical inversion, parametric methods and regularization theory) used in inverse problems, the focus is given to the Bayesian inference approach. Then, in a first step, the link between Maximum A Posteriori (MAP) and the regularization criteria is described and we will see how different prior modeling result to different regularization criteria. In particular we consider the cases of separable Gaussian and Non-Gaussian, Gauss-Markov and more general Markovian prior models.

Then, the advantages of the Bayesian approach to deterministic methods are highlighted through the possibilities of accounting more precisely for uncertainties of the data and model parameters, hyper parameter estimation, marginalization of nuisance parameters and the possibilities of the exploration of the space of the possible solutions by the Markov Chain Monte Carlo (MCMC) methods. One last advantage is the possibility of accounting for more specific prior knowledge through the Markovian or mixture models with hidden Markovian variables of contours and region labels, which do not have equivalent in deterministic methods.

Finally, I introduce a class of Gauss-Markov-Potts prior models which we have developed and used effectively in many imaging applications which will give, I hope, the audience to come to the second talk which is focused more specifically on this model and on its application in Computed Tomography.

References:

A. Mohammad-Djafari, From Deterministic to Probabilistic Approaches to Solve Inverse Problems, In Bayesian Inference for Inverse Problems, SPIE 98, San Diego, CA, USA, pp: 2-11,1998

Ali Mohammad-Djafari. Bayesian inference for inverse problems in signal and image processing and applications. International Journal of Imaging Systems and Technology, 16:209-214, 2007.

Ali Mohammad-Djafari, Jean-François Giovannelli, Guy Demoment and Jérôme Idier. Regularization, maximum entropy and probabilistic methods in mass spectrometry data processing problems. Int. Journal of Mass Spectrometry, 215(1-3):175-193, April 2002.

Olivier Féron, Bernard Duchêne, and Ali Mohammad-Djafari. Microwave imaging of inhomogeneous objects made of a finite number of dielectric and conductive materials from experimental data. Inverse Problems, 21(6):95-115, Dec 2005.

Ali Mohammad-Djafari and Lionel Robillard. Hierarchical markovian models for 3D computed tomography in non destructive testing applications. In EUSIPCO 2006. EUSIPCO 2006, September 4-8, Florence, Italy, September 2006.

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