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The assessment of positron emission tomography (PET) and single photon emission computed tomography (SPECT) image reconstructions by image quality metrics is typically time consuming, even if methods employing model observers and samples of reconstructions are used to replace human testing. We consider a detection task where the background is known exactly and the signal is known except for location. We develop theoretical formulas to rapidly evaluate two relevant figures of merit: the area under the localization receiver operating characteristic (LROC) curve and the probability of correct localization. The formulas can accommodate different forms of model observer. The theory hinges on the fact that we are able to rapidly compute the mean and covariance of the reconstruction. For four forms of model observer, the theoretical expressions are validated by Monte Carlo studies for the case of MAP (maximum a posteriori) reconstruction. The theory method affords a 102-103 speedup relative to methods in which model observers are applied to sample reconstructions.