An Interdisciplinary Workshop on Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy (IMRT)

# seminar: Markovian a priori models and Variational Bayesian algorithms for 2D and 3D computed tomography

speaker: Ali Mohammad-Djafari (CNRS)

abstract: In a non supervised Bayesian estimation approach for inverse problem of computed tomography, one tries to estimate jointly the unknown image pixels or voxels $f$ and the hyperparameters $\theta$ through the joint posterior law $p(f,\theta g)$ where $g$ represents the observed data. The expression of this joint law depends on the likelihood and the a priori model. Sample or compound markovian models has been used very often in many imaging applications. Here, we propose a mixture of Gaussians with hidden Potts-Markov region labels prior model which gives us the possibility to account for the fact that, very often, we may know that the unknown object under the test is composed of a finite number of homogeneous compact regions. However, with this kind of a priori the expression of the joint a posteriori becomes very complex and its exploration through sampling and computation of the point estimators such as MAP and posterior means need either optimization of non convex criteria or integration of non Gaussian and multi variate probability laws. In any of these cases, we need to do approximations. We had explored before the possibilities of Laplace approximation and sampling by MCMC. In this paper, we explore the possibility of approximating this joint law by a separable one in $\fb$ and in $\thetab$. This gives the possibility of developing iterative algorithms with more reasonable computational cost, in particular, if the approximating laws are choosed in the exponential conjugate families. The main objective of this paper is to give details of different algorithms we obtain with different choices of these families. The proposed algorithms are used for 2D and 3D CT. We show a few simulation results to illustrate the performances of these algorithms.

timetable:
Tue 16 Oct, 14:00 - 15:00, Aula Dini
documents: