Noise propagation from attenuation measurements into PET emission reconstructions

Approximations are presented for calculation of the covariance, mean estimator gradient, impulse response, and bias of three types of penalized “maximum likelihood” estimators when measured attenuation data is used. The three estimators examined are: (1) the conventional penalized maximum likelihood objective for emission tomography; (2) joint maximum likelihood estimation of emission intensity and attenuation; and (3) a “robust” maximum likelihood model where the additional variance introduced by the attenuation measurements is compensated through the use of a shifted Poisson model. Resulting limiting expressions for the covariance are shown to be the same for all methods as the measurement noise approaches zero