Robust MRI Reconstruction by Smoothed Unrolling (SMUG)

TL;DR

The paper tackles the stability of deep unrolled MRI reconstructions under input perturbations and test-time variations by introducing Smoothed Unrolling (SMUG), which injects randomized smoothing into intermediate denoisers within unrolled networks.SMUG uses a pre-training and fine-tuning regime with an unrolled-stability loss, and is further enhanced by Weighted SMUG, which learns adaptive smoothing weights to preserve sharpness while improving robustness.A theoretical robustness bound is derived for SMUG, showing that the reconstruction perturbation error can be controlled by the smoothing variance and network properties, and the approach is demonstrated across MoDL, ISTA-Net, and E2E-VarNet with strong robustness gains over RS-E2E, adversarial training, and diffusion-based baselines.Empirical results on fastMRI knee and brain datasets show that SMUG and especially Weighted SMUG achieve superior robust PSNR/SSIM under random and worst-case perturbations, while maintaining competitive clean accuracy and reasonable inference times.These findings suggest SMUG as a generalizable and practically impactful strategy to enhance reliability of MRI reconstructions in the presence of noise, artifacts, and operator variations.

Abstract

As the popularity of deep learning (DL) in the field of magnetic resonance imaging (MRI) continues to rise, recent research has indicated that DL-based MRI reconstruction models might be excessively sensitive to minor input disturbances, including worst-case additive perturbations. This sensitivity often leads to unstable, aliased images. This raises the question of how to devise DL techniques for MRI reconstruction that can be robust to train-test variations. To address this problem, we propose a novel image reconstruction framework, termed Smoothed Unrolling (SMUG), which advances a deep unrolling-based MRI reconstruction model using a randomized smoothing (RS)-based robust learning approach. RS, which improves the tolerance of a model against input noises, has been widely used in the design of adversarial defense approaches for image classification tasks. Yet, we find that the conventional design that applies RS to the entire DL-based MRI model is ineffective. In this paper, we show that SMUG and its variants address the above issue by customizing the RS process based on the unrolling architecture of a DL-based MRI reconstruction model. Compared to the vanilla RS approach, we show that SMUG improves the robustness of MRI reconstruction with respect to a diverse set of instability sources, including worst-case and random noise perturbations to input measurements, varying measurement sampling rates, and different numbers of unrolling steps. Furthermore, we theoretically analyze the robustness of our method in the presence of perturbations.

Figures (16)

• Figure 1: MoDL's instabilities resulting from perturbations to input data, the measurement sampling rate, and the number of unrolling steps used at testing phase shown on an image from the fastMRI dataset zbontar2018fastmri. We refer readers to Section \ref{['sec: experiment']} for further details about the experimental settings. (a) MoDL reconstruction from benign (i.e., without additional noise/perturbation) measurements with $4\times$ acceleration (i.e., 25% sampling rate) and 8 unrolling steps. (b) MoDL reconstruction from disturbed input with perturbation strength $\epsilon = 0.02$ (see Section \ref{['sec:experiment:setup']}). (c) MoDL reconstruction from clean measurements with $2\times$ acceleration (i.e., 50% sampling), and using 8 unrolling steps. (d) MoDL reconstruction from clean or unperturbed measurements with $4\times$ acceleration and 16 unrolling steps. In (b), (c), and (d), the network trained in (a) is used.
• Figure 2: The three randomized smoothing-based architectures for reconstruction. In RS‐E2E, we generate $N$ noisy k-space versions by adding Gaussian noise to $\mathbf{y}$ and then apply the Hermitian operator $\mathbf{A}^H$ to obtain samples that are batch-processed by a neural network for initial denoising. These outputs are refined by a data consistency module using the closed-form update (\ref{['modleqn1']}), and after a few unrolled iterations, the final reconstruction is obtained by averaging the outputs. In contrast, the SMUG architecture directly adds random Gaussian noise in the image domain to create multiple noisy versions that are denoised by the neural network; their averaged output serves as a randomized smoothing step before applying the same data consistency module, yielding the final smoothed result after several iterations. Extending this framework, Weighted SMUG employs a learned weighted averaging obtained from a weighted encoder applied prior to the data consistency step—to produce the final smoothed reconstruction after a few unrolled iterations.
• Figure 3: Reconstruction accuracy box plots for the fastMRI brain dataset with 4x acceleration factor. The additive random Gaussian noise of the second column plots is obtained using standard deviation of $0.01$. The worst-case additive noise of the third column is obtained using the PGD method with $\epsilon = 0.02$.
• Figure 4: Visualization of ground truth and reconstructed images using different methods for 4x k-space undersampling, evaluated on PGD-generated worst-case inputs of perturbation strength $\epsilon = 0.02$. The reconstruction PSNRs are shown with the best values bolded.
• Figure 5: Reconstruction accuracy box plots for the fastMRI knee dataset with 4x Acceleration factor. The additive random Gaussian noise of the second column plots is obtained using a standard deviation of $0.01$. The worst-case additive noise of the third column is obtained using the PGD method with $\epsilon = 0.02$.

Theorems & Definitions (4)

• Theorem 1
• Lemma 1
• proof
• proof