Regularization parameter selection for Bayesian reconstruction of attenuation maps

Previously we developed algorithms to obtain transmission reconstructions from truncated projections and from emission data without transmission measurements. The optimal basis set derived from "knowledge set" was used to create an approximate attenuation map, and the expansion coefficients were estimated using optimization algorithms. Since a truncated expansion does not represent an image precisely, and the projections of the basis vectors are not orthogonal, the estimated coefficients may be unstable in the presence of systematic errors. In addition to this, a nonlinear problem of reconstruction of an attenuation map from emission data has a nonunique solution. A constraint based on distribution of the expansion coefficients is considered in this paper to regularize the estimation problems. The parameter selection methods based on different assumptions are applied to find the optimum regularization parameter. The selected regularization parameter obtained from a projection data set has been shown to provide satisfactory reconstruction results.