Optimal Measurement of Magnitude and Phase from MR Data

General expressions of magnitude- and phase-probability distributions of MRI signals are presented and discussed. From these expressions, maximum-likelihood estimators are derived that allow optimal measurement of true magnitude and true phase from a set of noisy samples. Bias and uncertainty of estimates are determined by Monte Carlo simulations. It is demonstrated that with 100 samples and for signal-to-noise ratios above 1, bias and uncertainty of ML estimates are approximately 1 and 3% for magnitude and 0.1 and 5% for phase, respectively.