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Noise characterization in MRI has multiple applications, including qualityassurance and protocol optimization. It is particularly important in thepresence of parallel imaging acceleration, where the noise distribution cancontain severe spatial heterogeneities. If the parallel imaging reconstructionis a linear process, an exact noise analysis is possible by taking into accountthe correlations between all the samples involved. However, for k-space basedtechniques like GRAPPA, the exact analysis has been considered computationallyprohibitive due to the very large size of the noise covariance matricesrequired to characterize the noise propagation from k-space to image-space.Previous methods avoid this computational burden by approximating the GRAPPAreconstruction as a pixel-wise linear operation performed in the image-space.However, these methods are not exact in the presence of non-uniform k-spaceundersampling (e.g.: containing a calibration region). For this reason, in thiswork we develop an exact characterization of the noise distribution forself-calibrated parallel imaging in the presence of arbitrary Cartesianundersampling patterns. By exploiting the symmetries and separability in thenoise propagation process, the proposed method is computationally efficient anddoes not require large matrices. In this manuscript, we present the proposednoise characterization method and compare it to previous techniques usingMonte-Carlo simulations as well as phantom acquisitions.

Exact Calculation of Noise Maps and g-Factor in GRAPPA using a k-space Analysis