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

TL;DR

This paper introduces an exact, computationally efficient method for characterizing noise in GRAPPA MRI reconstructions, accounting for arbitrary undersampling patterns, improving over approximate methods.

Contribution

Develops a novel exact noise analysis technique for GRAPPA MRI that handles non-uniform undersampling without large matrix computations.

Findings

Method matches Monte Carlo simulations accurately.

Outperforms previous approximation techniques.

Applicable to various undersampling patterns.

Abstract

Noise characterization in MRI has multiple applications, including quality assurance and protocol optimization. It is particularly important in the presence of parallel imaging acceleration, where the noise distribution can contain severe spatial heterogeneities. If the parallel imaging reconstruction is a linear process, an exact noise analysis is possible by taking into account the correlations between all the samples involved. However, for k-space based techniques like 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. Previous methods avoid this computational burden by approximating 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 k-space undersampling (e.g.: containing a calibration region). For this reason, in this work we develop an exact characterization of the noise distribution for self-calibrated parallel imaging in the presence of arbitrary Cartesian undersampling patterns. By exploiting the symmetries and separability in the noise propagation process, the proposed method is computationally efficient and does not require large matrices. In this manuscript, we present the proposed noise characterization method and compare it to previous techniques using Monte-Carlo simulations as well as phantom acquisitions.