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 ihe resulting attenuation-corrected emission reconstruction PET. 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 an attenuation correction onto the PET reconstructions, but here we theoretically analyze two cases: (I) A linear (on log data) transmission reconstruction is reprojected to get an ACF (Attenuation Correction Factor) estimate which is then used in a linear emission estimate. (2) PET reconstruction is a nonlinear estimate based on maximizing a regularized likelihood objective, and attenuation is modeled directly in the objective function. A validation study is presented for the mean and covariance expressions for case (1).