| The maximum likelihood (ML) approach using the expectation maximization (EM) algorithm has been useful in reconstruction for emission tomography. However, due to the ill-posed nature of the problem, the ML-EM suffers from instability. In contrast, Bayesian approaches overcome this instability by introducing prior information, often in the form of a spatial smoothness regularizer. More elaborate forms of smoothness constraints may be used to extend the role of the prior beyond that of a stabilizer in order to capture actual spatial information about the object. Previously proposed forms of such prior distributions were based on the assumption of a nearly piecewise constant source distribution. Here, we propose an extension to a piecewise linear model--the weak plate (WP)--which is more expressive than the piecewise constant model. The WP prior not only preserves edges but also allows for piecewise ramplike regions in the reconstruction. For our application in SPECT, such ramplike regions are observed in "ground-truth" source distributions in the form of primate autoradiographs. To incorporate the WP prior in a maximum a posteriori (MAP) approach, we model the prior as a Gibbs distribution and use a generalized EM (GEM) formulation for the optimization. We compare quantitative performance of the ML-EM algorithm, a GEM algorithm with the weak membrane (WM) prior favoring piecewise constant regions, and a GEM algorithm with our WP prior. Pointwise and regional bias and variance of ensemble image reconstructions are used as indications of image quality. Our results show that the WP exhibits improved variance relative to the WM and ML-EM techniques at approximately no cost in bias error. We also propose a method for estimating hyper-parameters for prior models. Our method uses an ML technique to fit hyperparameters to a training set that is itself representative of "ground-truth" objects to be reconstructed. Numerical experiments demonstrate that parameters derived from training sets are fairly close to parameter values optimal in the sense of various measures of image quality. |
