Validity of the noncentral chi model in multiple-coil systems with noise correlations

Noise in Multiple-Coil Systems(2) Noise is one of the main sources of quality deterioration in Magnetic Resonance (MR) data. It is usually modeled attending to the scanner coil architecture. The complex spatial MR data is typically assumed to be a complex Gaussian process, where the real and imaginary parts of the original signal are corrupted with uncorrelated Gaussian noise with zero mean and equal variance σn. Thus, the magnitude signal in single-coil systems is the Rician distributed envelope of the complex signal. In multiple-coil MR acquisition systems, the process is repeated for each receiving coil. As a consequence, the noise in the complex signal in the x-space for coil l-th (l=1,..,L) will also be Gaussian. If the composite magnitude signal (CMS) is obtained using sum-of-squares (SoS) it can be modeled as a noncentral chi (nc-χ) distribution [Const97]. However, the CMS will only follow a nc-χ distribution if the variance of noise is the same for all coils, and no correlation exists between them. Although it is well known that in phased array systems noise correlations do exist [Hayes90], the effect of noise correlations is usually left aside due to their minimal effect and practical considerations. In [Const97] it is stated that the effect of such correlations is minimal over humans and phantoms. However, these correlations may affect the final statistical model and the effective level of noise in the image. Let us assume the simple covariance matrix between coils in eq. (1). Following a similar reasoning to the one in [Aja10] it is evident that the CMS after SoS reconstruction will not strictly follow a nc-χ. However, our hypothesis is that for low values of the coefficient of correlation (ρ) the nc-χ distribution can be used as a good approximation of the actual one. The correlation between coils will translate in a decrease of the number of Degrees of Freedom (DoF) of the distribution. Thus, the resulting distribution will show a (reduced) effective number of coils and an (increased) effective variance of noise that can be calculated using the method of the moments, see eq. (2), with