Noise Estimation in the Complex Domain

Abstract

This is the first of several chapters that deal with the problem of noise estimation in MRI. Here, we focus on the case of stationary additive Gaussian noise. In MRI acquisitions, noise is assumed to be a Gaussian process that equally affects to all the frequencies in the k-space, independent and identically distributed for both the real and imaginary components. Thus, the stationary Gaussian assumption is also valid for each coil in the x-space, before the magnitude is considered. In addition, The Gaussian model can also be used as a simplification of more complex models of noise in the magnitude signal for high SNR. In this chapter we review some of the methods proposed in literature to estimate the variance of noise $\sigma ^{2}$ in univariate and multivariate Gaussian models, and the covariance between coils $\sigma _\mathrm{xy}$ when multiple correlated coils are considered. The analysis done over the complex Gaussian data shows many advantages when compared to other models, since it is supported by highly validated techniques and well-grounded assumptions. All the methods reviewed are defined in the x-space, but results can be easily extrapolated to k-space. In the second part of the chapter the proposed estimators are compared and a performance analysis is carried out over synthetic and real data.