Cramer-Rao bound for gated PET

Respiratory and cardiac motions degrade the spatial resolution in PET and SPECT imaging, with a negative impact on the diagnostic accuracy for cardiac studies and for the detection of lung tumors. The goal of this work is to determine the limit on the achievable performance in gated PET reconstruction with joint motion estimation, without external data, using the Cramer-Rao lower bound (CRLB). For a single ring scanner and a simple beating phantom we compute the CRLB with and without motion estimation. This comparison gives an insight into the increase in the number of events required to obtain in the presence of motion the same variance as with an hypothetical scan without motion. This preliminary study assumes an unbiased estimator for the image coefficients. The importance of the bias is however illustrated for a problem without motion and for the bias corresponding to a specific algorithm, OSEM with 7 subsets and 8 iterations. The empirical variance observed with that algorithm is lower than the unbiased CRLB but, as expected, higher than the biased CRLB. When the number of counts decreases below a certain threshold, the difference between the unbiased and biased CRLBs increases dramatically, and in addition the variance of the OSEM reconstruction becomes significantly higher than the lower bound given by the CRLB with the corresponding bias. Finally, we set up a direction to estimate the CRLB for larger images, by approximating the bound using a submatrix of the Fisher matrix.