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Score-based diffusion models for accelerated MRI

Hyungjin Chung, Jong Chul Ye

TL;DR

This work introduces a continuous-time score-based diffusion framework for accelerated MRI that trains a magnitude-only score model and uses a data-consistency guided reverse SDE to reconstruct complex-valued, multi-coil images. The approach is sampling-pattern agnostic and supports parallel imaging, enabling posterior sampling to quantify uncertainty while achieving state-of-the-art reconstruction quality, even under distribution shift. By leveraging predictor-corrector sampling and a carefully designed forward model, the method extends to complex-valued data and PI without coil-calibration maps, and demonstrates strong performance on real and OOD data as well as pathology-detection tasks. The paper also discusses practical considerations such as inference speed, conditional generation, and broader impacts, while acknowledging limitations in extreme undersampling scenarios.

Abstract

Score-based diffusion models provide a powerful way to model images using the gradient of the data distribution. Leveraging the learned score function as a prior, here we introduce a way to sample data from a conditional distribution given the measurements, such that the model can be readily used for solving inverse problems in imaging, especially for accelerated MRI. In short, we train a continuous time-dependent score function with denoising score matching. Then, at the inference stage, we iterate between numerical SDE solver and data consistency projection step to achieve reconstruction. Our model requires magnitude images only for training, and yet is able to reconstruct complex-valued data, and even extends to parallel imaging. The proposed method is agnostic to sub-sampling patterns, and can be used with any sampling schemes. Also, due to its generative nature, our approach can quantify uncertainty, which is not possible with standard regression settings. On top of all the advantages, our method also has very strong performance, even beating the models trained with full supervision. With extensive experiments, we verify the superiority of our method in terms of quality and practicality.

Score-based diffusion models for accelerated MRI

TL;DR

This work introduces a continuous-time score-based diffusion framework for accelerated MRI that trains a magnitude-only score model and uses a data-consistency guided reverse SDE to reconstruct complex-valued, multi-coil images. The approach is sampling-pattern agnostic and supports parallel imaging, enabling posterior sampling to quantify uncertainty while achieving state-of-the-art reconstruction quality, even under distribution shift. By leveraging predictor-corrector sampling and a carefully designed forward model, the method extends to complex-valued data and PI without coil-calibration maps, and demonstrates strong performance on real and OOD data as well as pathology-detection tasks. The paper also discusses practical considerations such as inference speed, conditional generation, and broader impacts, while acknowledging limitations in extreme undersampling scenarios.

Abstract

Score-based diffusion models provide a powerful way to model images using the gradient of the data distribution. Leveraging the learned score function as a prior, here we introduce a way to sample data from a conditional distribution given the measurements, such that the model can be readily used for solving inverse problems in imaging, especially for accelerated MRI. In short, we train a continuous time-dependent score function with denoising score matching. Then, at the inference stage, we iterate between numerical SDE solver and data consistency projection step to achieve reconstruction. Our model requires magnitude images only for training, and yet is able to reconstruct complex-valued data, and even extends to parallel imaging. The proposed method is agnostic to sub-sampling patterns, and can be used with any sampling schemes. Also, due to its generative nature, our approach can quantify uncertainty, which is not possible with standard regression settings. On top of all the advantages, our method also has very strong performance, even beating the models trained with full supervision. With extensive experiments, we verify the superiority of our method in terms of quality and practicality.

Paper Structure

This paper contains 30 sections, 1 theorem, 25 equations, 15 figures, 3 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Score-based SDE
  4. Main Contributions
  5. Forward Measurement Model
  6. Reverse SDE for Accelerated MR Reconstruction
  7. Diffusion model meets Parallel Imaging (PI)
  8. Methods
  9. Experimental data
  10. Implementation details
  11. Model Architecture
  12. Comparison study
  13. Measurement of Reconstruction Quality
  14. Pathology Detection
  15. Generalization to different anatomy and contrast
  16. ...and 15 more sections

Key Result

Proposition 1

With the sensitivity normalization in eq:normalization, $(I-\lambda A^*A)$ is non-expansive for $\lambda \in [0,1]$.

Figures (15)

  • Figure 1: Overview of the proposed method. Starting from ${\bm{x}\xspace}_T$, sampled from the prior distribution, ${\bm{x}\xspace}_0$ is reached by solving the reverse SDE with score-based sampling, alternating between the update step, and the data consistency step.
  • Figure 2: Illustration of parallel imaging applications. (a) SSOS-type sampling. Coil images are reconstructed separately, then merged with the SSOS operation. (b) Hybrid-type sampling. Dependency between the coil images is injected every $m$ steps of iteration.
  • Figure 3: Reconstructions of the real-valued simulation study. (a) Sub-sampling mask used to generate under-sampled image, (b) TV, (c) supervised learning (U-Net) (d) DuDoRNet zhou2020dudornet, (e) proposed method, and (f) ground truth. 1$^\text{st}$ row: 2D $\times 8$ Gaussian random sampling, 2$^\text{nd}$ row: 1D $\times 4$ uniform random sampling, 3$^\text{rd}$ row: 1D $\times 8$ Gaussian random sampling, 4$^\text{th}$ row: $\times 15$ variable density poisson disk sampling. Green box: Zoom in version of the indicated yellow box, Blue box: Difference magnitude of the inset (in Viridis colormap). Yellow numbers in the upper right corner indicate PSNR [db], and SSIM, respectively.
  • Figure 4: Single-coil complex-valued image reconstruction results. (a) Sub-sampling mask used to generate under-sampled image, (b) TV, (c) supervised learning (U-Net) (d) DuDoRNet zhou2020dudornet, (e) proposed method, and (f) ground truth. 1$^\text{st}$ row: 2D $\times 8$ Gaussian random sampling, 2$^\text{nd}$ row: 1D $\times 4$ Gaussian random sampling, 3$^\text{rd}$ row: 1D $\times 8$ uniform random sampling, 4$^\text{th}$ row: $\times 15$ variable density poisson disk sampling. Green box: Zoom in version of the indicated yellow box, Blue box: Difference magnitude of the inset (in Viridis colormap). Yellow numbers in the upper right corner indicate PSNR [db], and SSIM, respectively.
  • Figure 5: Multi-coil reconstruction results. (a) Sub-sampling mask used to generate under-sampled image, (b) TV, (c) supervised learning (U-Net), (d) E2E-varnet sriram2020end, (e) the proposed method, and (f) the ground truth. 1$^{\text{st}}$ row: 2D $\times$8 Gaussian random sampling, 2$^{\text{nd}}$ row: 1D $\times$4 Gaussian random sampling, 3$^{\text{rd}}$ row: 1D $\times$ 4 uniform random sampling, 4$^{\text{th}}$ row: $\times$8 variable density poisson disk sampling. Green box: Zoom in version of the indicated yellow box, Blue box: Difference magnitude of the inset (in Viridis colormap). Yellow numbers in the upper right corner indiate PSNR [db], and SSIM, respectively.
  • ...and 10 more figures

Theorems & Definitions (2)

  • Proposition 1
  • proof