Theoretical Analysis of Resolution and Noise Properties of PET Image Reconstruction with and without Sinogram Blurring Modeling

The detector response due to inter-crystal scatter, penetration and non-colinearity in PET can be modeled using blurring point spread functions (PSFs) in sinogram or image space. Incorporating PSFs into image reconstruction is expected to provide better image quality because of accurate system modeling. It has been widely observed that PSF-based reconstruction produces better spatial resolution with more pronounced overshoot/undershoot near edges and yields lower variance and more correlated image noise than non-PSF reconstruction. These findings have been mostly empirically made and there has not been much of rigorous explanation about the reason for the behaviors. In this study, we investigate the underlying mathematical structure of PSF-based and non-PSF image reconstruction and attempt to provide a mathematical explanation about the distinct behaviors they show. The local impulse response and the covariance of PSF-based and non-PSF penalized-likelihood image reconstruction are analyzed in continuous space to obtain useful insights. From the continuous-space analysis, we address why PSF-based reconstruction produces pronounced overshoot/undershoot near edges and what is the limitation of non-PSF reconstruction in achievable image resolution. In addition, it is analytically shown that PSF-based reconstruction results in spatially more correlated noise than non-PSF reconstruction at matched ensemble pixel variance. Finally, it is demonstrated that PSF-based reconstruction not only produces a lower pixel variance than non-PSF reconstruction at matched spatial resolution but also nearly achieves the minimum pixel variance for given spatial resolution.