Efficient Noise Calculation in Deep Learning-based MRI Reconstructions

TL;DR

This paper introduces a memory-efficient, theoretically grounded method to accurately compute voxel-wise noise variance in deep learning-based MRI reconstructions, enabling better uncertainty quantification and evaluation.

Contribution

It develops an unbiased estimator and Jacobian sketching technique to efficiently approximate noise covariance, significantly reducing computational costs compared to Monte Carlo methods.

Findings

Achieves near Monte Carlo accuracy in noise variance estimation

Reduces computational and memory demands by over an order of magnitude

Robust across different datasets, noise levels, and undersampling schemes

Abstract

Accelerated MRI reconstruction involves solving an ill-posed inverse problem where noise in acquired data propagates to the reconstructed images. Noise analyses are central to MRI reconstruction for providing an explicit measure of solution fidelity and for guiding the design and deployment of novel reconstruction methods. However, deep learning (DL)-based reconstruction methods have often overlooked noise propagation due to inherent analytical and computational challenges, despite its critical importance. This work proposes a theoretically grounded, memory-efficient technique to calculate voxel-wise variance for quantifying uncertainty due to acquisition noise in accelerated MRI reconstructions. Our approach approximates noise covariance using the DL network's Jacobian, which is intractable to calculate. To circumvent this, we derive an unbiased estimator for the diagonal of this covariance matrix (voxel-wise variance) and introduce a Jacobian sketching technique to efficiently implement it. We evaluate our method on knee and brain MRI datasets for both data- and physics-driven networks trained in supervised and unsupervised manners. Compared to empirical references obtained via Monte Carlo simulations, our technique achieves near-equivalent performance while reducing computational and memory demands by an order of magnitude or more. Furthermore, our method is robust across varying input noise levels, acceleration factors, and diverse undersampling schemes, highlighting its broad applicability. Our work reintroduces accurate and efficient noise analysis as a central tenet of reconstruction algorithms, holding promise to reshape how we evaluate and deploy DL-based MRI. Our code will be made publicly available upon acceptance.