Exact Calculation of Noise Maps and ${g}$ -Factor in GRAPPA Using a ${k}$ -Space Analysis

Characterization of the noise distribution in magnetic resonance images has multiple applications, including quality assurance and protocol optimization. Noise characterization is particularly important in the presence of parallel imaging acceleration with multi-coil acquisitions, where the noise distribution can contain severe spatial heterogeneities. If the parallel imaging reconstruction is a linear process, an accurate noise analysis can be carried out by taking into account the correlations between all the samples involved. However, for k-space-based techniques such as generalized autocalibrating partially parallel acquisition (GRAPPA), the exact analysis has been considered computationally-prohibitive due to the very large size of the noise covariance matrices required to characterize the noise propagation from k-space to image space. Previously proposed methods avoid this computational burden by formulating the GRAPPA reconstruction as a pixel-wise linear operation performed in the image space. However, these methods are not exact in the presence of non-uniform sampling of k-space (e.g., containing a calibration region). For this reason, in this paper, we develop an accurate characterization of the noise distribution for self-calibrated parallel imaging in the presence of arbitrary Cartesian sampling patterns. By exploiting the symmetries and separability in the noise propagation process, the proposed method is computationally efficient and does not require large matrices. Under the assumption of a fixed reconstruction kernel, this method provides the precise distribution of the noise variance for each coil's image. These coil-by-coil noise maps are subsequently combined according to the coil combination approach used in image reconstruction, and therefore can be applied with both complex coil combination and root-sum-of-squares-approaches. In this paper, we present the proposed noise characterization method and compare it to previous techniques using Monte Carlo simulations as well as phantom acquisitions.