Convergent iterative algorithms for joint reconstruction of activity and attenuation from time-of-flight PET data

Joint reconstruction of activity and attenuation maps from emission data only, while being a long-standing problem in emission tomography, has been gaining recent interests because of its application to MR-based attenuation correction in PET/MR scanners where CT images are not available. Furthermore, recent studies showed that TOF (time-of-flight) information can substantially reduce, or completely remove in theory, crosstalk artifacts, which had been one of the hurdles preventing joint reconstruction techniques from being used clinically. Nonetheless, estimating both activity and attenuation from TOF emission data is a computationally challenging nonconvex optimization problem with high-dimensional data size. Therefore, as a tool for investigating and optimizing the joint estimation techniques for clinical use, we need numerical algorithms that are derived in a principled way and guaranteed to converge to a solution. Here, in a PL (penalized-likelihood) framework, we present a block alternating MM (minorization maximization) algorithm, which is provably globally convergent although the PL objective function is nonconcave. By using linear parameterization of attenuation maps, the algorithm applies to a variety of scenarios, depending on the type of and the degree of prior information, in a unified way. In addition, we provide a proof of the uniqueness of solutions to joint estimation problems for a continuous-space TOF PET system whose TOF kernels do not need to be Gaussian.