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Efficient Noise Calculation in Deep Learning-based MRI Reconstructions

Onat Dalmaz, Arjun D. Desai, Reinhard Heckel, Tolga Çukur, Akshay S. Chaudhari, Brian A. Hargreaves

TL;DR

The paper tackles the challenge of characterizing noise propagation in DL-based MRI reconstructions by deriving a theoretically grounded, memory-efficient Jacobian sketching approach to estimate voxel-wise image variance due to acquisition noise. It shows how to obtain an unbiased diagonal estimate of the image covariance using Jacobian-vector products and a vectorized random-phase sketch, achieving near Monte Carlo accuracy with orders-of-magnitude reductions in computation and memory. Across knee and brain data and multiple reconstruction architectures, the method yields high fidelity variance maps (PCC ≈ 99.8% with MC references) that are robust to noise levels, undersampling schemes, and acceleration factors. This enables practical uncertainty quantification to guide acquisition and deployment of DL-based MRI, with potential extensions to off-diagonal covariances and broader imaging applications.

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.

Efficient Noise Calculation in Deep Learning-based MRI Reconstructions

TL;DR

The paper tackles the challenge of characterizing noise propagation in DL-based MRI reconstructions by deriving a theoretically grounded, memory-efficient Jacobian sketching approach to estimate voxel-wise image variance due to acquisition noise. It shows how to obtain an unbiased diagonal estimate of the image covariance using Jacobian-vector products and a vectorized random-phase sketch, achieving near Monte Carlo accuracy with orders-of-magnitude reductions in computation and memory. Across knee and brain data and multiple reconstruction architectures, the method yields high fidelity variance maps (PCC ≈ 99.8% with MC references) that are robust to noise levels, undersampling schemes, and acceleration factors. This enables practical uncertainty quantification to guide acquisition and deployment of DL-based MRI, with potential extensions to off-diagonal covariances and broader imaging applications.

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.
Paper Structure (58 sections, 2 theorems, 66 equations, 13 figures, 10 tables, 2 algorithms)

This paper contains 58 sections, 2 theorems, 66 equations, 13 figures, 10 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Theory
  4. Accelerated MRI
  5. Neural Network MRI Reconstruction
  6. Noise Propagation
  7. Estimating Noise via Jacobian Sketching
  8. Vectorized Implementation: Jacobian Sketching
  9. Choice of Random Vectors
  10. Experiments and Results
  11. Datasets and Experimental Setup
  12. Generalizability Across Reconstruction Methods
  13. Computational and Memory Benefits
  14. Generalization to Various Undersampling Schemes
  15. Robustness to k-space Noise Level
  16. ...and 43 more sections

Key Result

Theorem 3.1

Let $\bm{\Sigma}\in \mathbb{C}^{n\times n}$ be Hermitian, and let $\bm{v}\in\mathbb{C}^n$ satisfy Define $\bm{y} = (\bm{\Sigma}\,\bm{v}) \odot \bm{v}^{*}$, where $\odot$ is the Hadamard product, and $*$ is scalar complex conjugation. Then:

Figures (13)

  • Figure 1: Each column corresponds to a distinct deep reconstruction method at $R=8,\alpha=1$ on knee data. In each column: (top row) shows ZF and reconstructed images; (middle row) displays noise variance maps derived by the proposed method and empirical simulations; (bottom row) presents difference and amplified (x10) difference maps between the proposed and empirical variance maps to highlight spatial discrepancies. Color bars indicate each panel's relative noise map display window.
  • Figure 2: Each column corresponds to a different undersampling pattern (E2E-VarNet on knee data at $R=8, \alpha=1$). In each column: (top row) shows ZF and reconstructed images; (middle row) displays noise variance maps derived by the proposed method and empirical simulations; (bottom row) presents difference and amplified (x10) difference maps between the proposed and empirical variance maps to highlight spatial discrepancies. Color bars indicate each panel's relative noise map display window.
  • Figure 3: Each column corresponds to a different noise scaling factor $\alpha$ resulting in varying SNR scenarios (E2E-VarNet on knee data at $R=8$). In each column: (top row) shows ZF and reconstructed images; (middle row) displays noise variance maps derived by the proposed method and empirical simulations; (bottom row) presents difference and amplified (x10) difference maps between the proposed and empirical variance maps to highlight spatial discrepancies. Color bars indicate each panel's relative noise map display window. Note that $\alpha=50,200$ corresponds to SNR values well under a 10dBs, which is often cited as a threshold for diagnostic utility brown2014magnetic.
  • Figure 4: NRMSE of the proposed method over test slices vs. size of the sketching matrix ($S$) (E2E-VarNet on brain data at $R=8, \alpha=1$). As the size $S$ of the sketching matrix $\bm{V}_S$ grows from 100 to 1,900, the Normalized Root-Mean-Square Error (NRMSE) of the variance estimator decreases. While additional probing vectors generally improve accuracy by capturing more of the Jacobian’s structure, the marginal benefit tapers off beyond about 1,000--1,200 vectors, suggesting a practical trade-off between improved precision and increased computation time.
  • Figure 5: Mean NRMSE of empirical baseline over test slices vs. the number of MC Trials (N) (E2E-VarNet on brain data at $R=8,\alpha=1$). We vary the total number of MC realizations from 100 up to 5,000 and compare the resulting variance maps to a high-sample reference (e.g., 10,000 trials). As more trials are included, the empirical variance estimation converges around $N=3000$, shown here by the decreasing NRMSE.
  • ...and 8 more figures

Theorems & Definitions (6)

  • Theorem 3.1: Unbiased Diagonal Estimator
  • proof
  • Lemma 3.2
  • proof
  • Definition 3.1
  • Definition 3.2