A concave prior penalizing relative differences for maximum-a-posteriori reconstruction in emission tomography

A well-known problem with maximum likelihood reconstruction in emission tomography is the excessive noise propagation. To prevent this, the objective function is often extended with a Gibbs prior favoring smooth solutions. We hypothesize that the following three requirements should produce a useful and conservative Gibbs prior for emission tomography: 1) the prior function should be concave to ensure that the posterior has a unique maximum; 2) the prior should penalize relative differences rather than absolute differences; 3) the prior should be tolerant for "large" differences between neighboring pixels. The second requirement should avoid tuning problems caused by the large dynamic range of activity values in the reconstructed image. A simple function has been derived that meets these three requirements. Our initial evaluations indicate that the prior behaves as intended.