Hypermodels in the Bayesian Imaging Framework

Start: 02/15/2008 - 1:30pm
End  : 02/15/2008 - 2:30pm

Applied Math Seminar

Prof. Daniela Calvetti (Case Western Reserve University)


We consider the problem of restoring an image from a noisy blurred
copy, with the additional qualitative information that the
image contains sharp discontinuities of unknown size and location.
The flexibility of the Bayesian imaging framework is
particularly convenient in the presence of such qualitative, rather
than quantitative, information. By using a non-stationary Markov
model with the variance of the innovation process also unknown, it
is possible to take advantage of the qualitative prior information,
and Bayesian techniques can be applied to estimate simultaneously
the unknown and the prior variance. Here we present a unified
approach to Bayesian signal and imaging, and show that with rather
standard choices of hyperpriors we obtain some classical
regularization methods as special cases. The application of Bayesian
hyperprior models to imaging applications requires a careful
organization of the computations to overcome the challenges coming
from the high dimensionality. We explain how the computation of
MAP estimates within the proposed Bayesian framework can be made
very efficiency by a judicious use of Krylov iterative methods and
priorconditioners. The Bayesian approach, unlike deterministic
methods which produce a single solution image, provides a
very natural way to assess the reliability of single image estimates
by a Markov Chain Monte Carlo (MCMC) based analysis of the
posterior. Computed examples illustrate the different features and
the computational properties of the Bayesian hypermodel approach to

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