The University of Montana
Department of Mathematical Sciences

Technical report #1/2009

Hierarchical regularization for edge-preserving reconstruction of PET images

Johnathan M. Bardsley , Daniela Calvetti, and Erkki Somersalo


Abstract

The data in PET emission and transmission tomography and in low dose X-ray tomography, consists of counts of photons originating from random events. The need to model the data as a Poisson process poses a challenge for traditional integral geometry-based reconstruction algorithms. Although qualitative a priori information of the target may be available, it may be difficult to encode it as a regularization functional in a minimization algorithm. This is the case, for example, when the target is known to consist of well defined structures, but how many, and their location, form and size are not specified. Following the Bayesian paradigm, we model the data and the target as random variables, and we account for the qualitative nature of the a priori information by introducing a hierarchical model in which the a priori variance is unknown and therefore part of the estimation problem. We present a numerically effective algorithm for estimating both the target and its prior variance. Computed examples with simulated and real data demonstrate that the algorithm gives good quality reconstruction for both emission and transmission PET problems at very low computational cost.

Keywords: edge-preserving regularization, positron emission tomography, Bayesian statistical methods, nonnegatively constrained optimization

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