Quantitative effects of using thin-plate priors in Bayesian SPECT reconstruction

Maximum a posteriori approaches in the context of a Bayesian framework have played an important role in SPECT reconstruction. The major advantages of these approaches include not only the capability of modeling the character of the data in a natural way but also the allowance of the incorporation of a priori information. Here, we show that a simple modification of the conventional smoothing prior, such as the membrane prior, to one less sensitive to variations in first spatial derivatives - the thin plate (TP) prior - yields improved reconstructions in the sense of low bias at little change in variance. Although the nonquadratic priors, such as the weak membrane and the weak plate, can exhibit good performance, they suffer difficulties in optimization and hyperparameter estimation. On the other hand, the thin plate, which is a quadratic prior, leads to easier optimization and hyperparameter estimation. In this work, we evaluate and compare quantitative performance of MM, TP, and FBP (filtered backprojection) algorithms in an ensemble sense to validate advantages of the thin plate model. We also observe and characterize the behavior of the associated hyperparameters of the prior distributions in a systematic way. To incorporate our new prior in a MAP approach, we model the prior as a Gibbs distribution and embed the optimization within a generalized expectation-maximization algorithm. For optimization for the corresponding M-step objective function, we use a version of iterated conditional modes. We show that the use of second-derivatives yields "robustness" in both bias and variance by demonstrating that TP leads to very low bias error over a large range of smoothing parameter, while keeping a reasonable variance.