  In this work, we investigate hybrid PET reconstruction algorithms based on
coupling a model-based variational reconstruction and the application of a
separately learnt Deep Neural Network operator (DNN) in an ADMM Plug and Play
framework. Following recent results in optimization, fixed point convergence of
the scheme can be achieved by enforcing an additional constraint on network
parameters during learning. We propose such an ADMM algorithm and show in a
realistic [18F]-FDG synthetic brain exam that the proposed scheme indeed lead
experimentally to convergence to a meaningful fixed point. When the proposed
constraint is not enforced during learning of the DNN, the proposed ADMM
algorithm was observed experimentally not to converge.
