Maximum likelihood estimation of signal amplitude and noise variance from MR data

In MRI, the raw data, which are acquired in spatial frequency space, are intrinsically complex valued and corrupted by Gaussian-distributed noise. After applying an inverse Fourier transform, the data remain complex valued and Gaussian distributed. If the signal amplitude is to be estimated, one has two options. It can be estimated directly from the complex valued data set, or one can first perform a magnitude operation on this data set, which changes the distribution of the data from Gaussian to Rician, and estimate the signal amplitude from the obtained magnitude image. Similarly, the noise variance can be estimated from both the complex and magnitude data sets. This article addresses the question whether it is better to use complex valued data or magnitude data for the estimation of these parameters using the maximum likelihood method. As a performance criterion, the mean-squared error (MSE) is used.