Nonparametric posterior learning for emission tomography with multimodal data

In this work we continue studies of the uncertainty quantification problem in emission tomographies such as PET or SPECT. In particular, we consider a scenario when additional multimodal data (e.g., anatomical MRI images) are available. To solve the aforementioned problem we adapt the recently proposed nonparametric posterior learning technique to the context of Poisson-type data in emission tomography. Using this approach we derive sampling algorithms which are trivially parallelizable, scalable and very easy to implement. In addition, we prove conditional consistency and tightness for the distribution of produced samples in the small noise limit (i.e., when the acquisition time tends to infinity) and derive new geometrical and necessary condition on how MRI images must be used. This condition arises naturally in the context of misspecified generalized Poisson models. We also contrast our approach with bayesian MCMC sampling based a data augmentation scheme which is very popular in the context of EM-type algorithms for PET or SPECT. We show theoretically and also numerically that such data augmentation significantly increases mixing times for the Markov chain. In view of this, our algorithms seem to give a reasonable trade-off between design complexity, scalability, numerical load and asessement for the uncertainty quantification.