Motion-compensated image reconstruction (MCIR) methods incorporate motion models to improve image quality in the presence of motion. MCIR methods differ in terms of how they use motion information and they have been well-studied separately. However, there have been less theoretical comparisions of different MCIR methods. This paper compares the theoretical noise properties of three popular MCIR methods assuming known nonrigid motion.We show the relationship among three MCIR methods - motion-compensated temporal regularization (MTR), the parametric motion model (PMM), and post-reconstruction motion correction (PMC) - for penalized weighted least square cases. These analyses show that PMM and MTR are matrix-weighted sums of all registered image frames, while PMC is a scalar-weighted sum.We further investigate the noise properties of MCIR methods with Poisson models and quadratic regularizers by deriving accurate and fast variance prediction formulas using an “analytical approach”. These theoretical noise analyses show that the variances of PMM and MTR are lower than or comparable to the variance of PMC due to the statistical weighting. These analyses also facilitate comparisons of the noise properties of different MCIR methods, including the effects of different quadratic regularizers, the influence of the motion through its Jacobian determinant, and the effect of assuming that total activity is preserved. 2D PET simulations demonstrate the theoretical results.
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