Simultaneous estimation of attenuation and activity images using optimization transfer

This paper addresses the application of optimization transfer to simultaneous statistical estimation of attenuation and activity images in tomographic image reconstruction. Although the technique we propose has wider applicability, we focus on the problem of reconstructing from data acquired via a post-injection transmission scan protocol. In this protocol, emission scan data is supplemented with transmission scan data that is acquired after the patient has received the injection of radio-tracer. The negative loglikelihood function for this data is a complicated function of the activity and attenuation images, leading to an objective function for the model that is difficult to minimize for the purpose of estimation. Previous work on this problem showed that when either the attenuation or activity image was held fixed, a paraboloidal surrogate could be found for the negative loglikelihood as a function of the remaining variables. This led to an algorithm in which the model's objective function is alternately minimized as a function of the attenuation and activity, using the optimization transfer technique. In the work we present here, however, we develop bivariate surrogates for the loglikelihood, i.e., functions that serve as surrogates with respect to both the attenuation and activity variables. Hence, simultaneous minimization in all variables can be carried out, potentially leading to convergence in fewer surrogate minimizations.