An analysis of noise propagation in computed T2, pseudodensity, and synthetic spin‐echo images

Methods are reviewed for estimating the transverse relaxation time T2 and the pseudodensity (PD) from spin-echo measurements acquired at an arbitrary set of echo times TEi. Least-squares fitting is applied to the logarithmically processed signals for the case in which the weights are proportional to the inverse of the logarithmically transformed signal variances (the minimum variance case). General formulas are derived for the estimated noise levels in the PD and T2 estimates due to the propagation of uncertainties in the original measurements. It is shown that the T2 and PD estimates are anticorrelated. Additionally, an expression is derived for the variance in a synthetic spin-echo signal subsequently formed from the PD and T2 estimates. It is shown that under many circumstances a signal synthesized at some echo time can have a signal-to-noise ratio superior to that in a signal directly acquired at that time. Experimental measurements made on phantoms match the theoretical predictions to a high degree.