Statistical noise model in GRAPPA-reconstructed images

Noise in Parallel Imaging Parallel MRI (pMRI) techniques extended the applicability of multiple-coil systems by increasing the acquisition rate via subsampled acquisitions of the k-space data. Many reconstruction methods have been proposed in order to suppress the aliasing and underlying artifacts created by the subsampling. Dominant among these are SENSE and GRAPPA. One of the effects in accelerated pMRI acquisitions is a significant change in the noise model, which depends on reconstruction scheme. From a statistical point of view, it is known that, if no subsampling of the k-space is done, the composite magnitude signal from several coils can be modeled as a non-central Chi distribution [Const97]. When k-space is subsampled, the noise power in-fact varies across the reconstructed image, and yet (maybe?) different for each receiving coil. Depending on the way the information from each coil is combined (i.e, depending on the reconstruction method), the statistics of the image may no longer follow the non-central Chi statistics. In SENSE, for instance, noise in the x-space is known to be Rician but non-stationary, i.e. the noise power varies from point to point [Thumb07]. Some authors suggest that GRAPPA-reconstructed images may follow a non-central Chi distribution [Thumb07], but no thorough study has been done. The great importance of the characterization of noise statistics in MRI has been extensively reported in literature. Many filtering, noise estimation or diffusion tensor estimation methods for conventional (non-parallel) MRI rely on the Rician noise model. However, during the last years some techniques have been developed assuming a non-central Chi distribution of noise. As an example, in [TV09] authors prove that the Weighted Least Squares method used in DTI to estimate the diffusion tensor is also valid under the non-central Chi assumption. In [Breuer09] authors present a novel g-factor derivation for GRAPPA, and the variance of noise in each coil calculated as