Noise propagation from attenuation correction into PET reconstructions

It is useful to have the ability to analyze the propagation of errors from transmission data into the resulting attenuation-corrected PET reconstruction. We develop theoretical expressions for the mean and covariance of the emission reconstruction when the only noise source is that in the transmission data. There are several ways to impose art attenuation correction onto the PET reconstructions, but here we theoretically analyze the case of a linear (on log data) transmission reconstruction that is reprojected to get an attenuation correction factor (ACF) estimate which is then used in a linear emission estimate. We validate the expressions for mean and covariance of the emission reconstruction by using sample methods. In addition, we extend the theory to the case where the PET reconstruction is a nonlinear emission estimate based on maximizing a regularized likelihood objective, and attenuation is modeled directly in the objective function.