Statistical Properties of SPECT MAP Reconstruction Incorporating Window Based Scatter Correction

In SPECT, the sinogram contains scatter counts that degrade the reconstructed image quality. We develop theoretical expressions to predict the reconstructed mean, covariance, and local point-spread function for SPECT MAP (maximum a posteriori) reconstructions. These expressions apply to window-based scatter correction methods, such as the triple-energy-window method, where the scatter correction is incorporated directly into the forward imaging model as an affine term. In addition this forward imaging model is incorporated directly in the MAP objective. We model a scatter estimate as a noisy quantity so that the reconstruction is driven by both photon noise in the photopeak and the noise in the scatter estimate. We use sample reconstruction methods to validate our theoretical expressions. We compare the speed of our theoretical methods to that of methods based on sample reconstructions. Such theoretical formulae could be used to rapidly assess the impact of different scatter correction strategies on image quality.