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High-Frequency Space Diffusion Models for Accelerated MRI

Chentao Cao, Zhuo-Xu Cui, Yue Wang, Shaonan Liu, Taijin Chen, Hairong Zheng, Dong Liang, Yanjie Zhu

TL;DR

This work proposes a novel SDE tailored specifically for MR reconstruction with the diffusion process in high-frequency space (referred to as HFS-SDE), which ensures determinism in the fully sampled low-frequency regions and accelerates the sampling procedure of reverse diffusion.

Abstract

Diffusion models with continuous stochastic differential equations (SDEs) have shown superior performances in image generation. It can serve as a deep generative prior to solving the inverse problem in magnetic resonance (MR) reconstruction. However, low-frequency regions of $k$-space data are typically fully sampled in fast MR imaging, while existing diffusion models are performed throughout the entire image or $k$-space, inevitably introducing uncertainty in the reconstruction of low-frequency regions. Additionally, existing diffusion models often demand substantial iterations to converge, resulting in time-consuming reconstructions. To address these challenges, we propose a novel SDE tailored specifically for MR reconstruction with the diffusion process in high-frequency space (referred to as HFS-SDE). This approach ensures determinism in the fully sampled low-frequency regions and accelerates the sampling procedure of reverse diffusion. Experiments conducted on the publicly available fastMRI dataset demonstrate that the proposed HFS-SDE method outperforms traditional parallel imaging methods, supervised deep learning, and existing diffusion models in terms of reconstruction accuracy and stability. The fast convergence properties are also confirmed through theoretical and experimental validation. Our code and weights are available at https://github.com/Aboriginer/HFS-SDE.

High-Frequency Space Diffusion Models for Accelerated MRI

TL;DR

This work proposes a novel SDE tailored specifically for MR reconstruction with the diffusion process in high-frequency space (referred to as HFS-SDE), which ensures determinism in the fully sampled low-frequency regions and accelerates the sampling procedure of reverse diffusion.

Abstract

Diffusion models with continuous stochastic differential equations (SDEs) have shown superior performances in image generation. It can serve as a deep generative prior to solving the inverse problem in magnetic resonance (MR) reconstruction. However, low-frequency regions of -space data are typically fully sampled in fast MR imaging, while existing diffusion models are performed throughout the entire image or -space, inevitably introducing uncertainty in the reconstruction of low-frequency regions. Additionally, existing diffusion models often demand substantial iterations to converge, resulting in time-consuming reconstructions. To address these challenges, we propose a novel SDE tailored specifically for MR reconstruction with the diffusion process in high-frequency space (referred to as HFS-SDE). This approach ensures determinism in the fully sampled low-frequency regions and accelerates the sampling procedure of reverse diffusion. Experiments conducted on the publicly available fastMRI dataset demonstrate that the proposed HFS-SDE method outperforms traditional parallel imaging methods, supervised deep learning, and existing diffusion models in terms of reconstruction accuracy and stability. The fast convergence properties are also confirmed through theoretical and experimental validation. Our code and weights are available at https://github.com/Aboriginer/HFS-SDE.
Paper Structure (33 sections, 1 theorem, 36 equations, 10 figures, 7 tables, 1 algorithm)

This paper contains 33 sections, 1 theorem, 36 equations, 10 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions
  3. Background
  4. Problem Formulation
  5. Score-based SDE
  6. VE-SDE
  7. VP-SDE
  8. Score-based SDEs for MRI Reconstruction
  9. Theory and methods
  10. The High-Frequency Space SDE
  11. Estimate Score for HFS-SDE
  12. Reverse HFS-SDE for MR Reconstruction
  13. Experiments
  14. Experimental Setup
  15. Experimental Data
  16. ...and 18 more sections

Key Result

Theorem III.1

Suppose Assumption ass:1 holds. The discrete HFS-SDE exhibits a smaller weak convergence upper bound compared to the SDE in the full space, which is achieved by replacing the operator $\boldsymbol{\mathcal{F}}_h$ with the identity operator $\boldsymbol{\mathcal{I}}$.

Figures (10)

  • Figure 1: Illustrate the proposed high-frequency space diffusion model. (a) In the forward process, high-frequency noises with multiple scales are added to the data (first row). The second row shows the $k$-space corresponding to the perturbed data (no noise is added to the central region of $k$-space). (b) Demonstrate the steps to acquire high-frequency noise. High-frequency noise is added to the data in the image domain.
  • Figure 2: Predictor corrector sampling. The predictor includes the original predictorscore-based-SDE and data consistency priors, while the corrector comprises the original correctorscore-based-SDE and data consistency priors. Zero-filling represents undersampled images with $k$-space zero-filling.
  • Figure 3: The reconstruction results of fastMRI multi-coil knee data at uniform undersampling of 10-fold. The first row shows the ground truth and the reconstruction of SENSE, ISTA-Net, DeepCascade, VarNet,cycleGAN, VE-, VP-, and HFS-SDE (ours). The second row shows the enlarged view of the ROI , and the third row shows the error map of the reconstruction. The undersampling mask used for the test is shown in the lower left corner.
  • Figure 4: The reconstruction results of fastMRI multi-coil knee data at uniform undersampling of 12-fold. The first row shows the ground truth and the reconstruction of SENSE, ISTA-Net, DeepCascade, VarNet, CycleGAN, VE-, VP-, and HFS-SDE (ours). The second row shows the enlarged view of the ROI, and the third row shows the error map of the reconstruction. The undersampling mask used for the test is shown in the lower left corner.
  • Figure 5: Out-of-Distribution results. The reconstruction results of T2-weighted brain data at uniform undersampling of 12-fold. The first row shows the reconstruction results of SENSE, ISTA-Net, DeepCascade, VE-, VP-, and HFS-SDE. The second row shows the error map of the ROI. The undersampling mask is shown in the lower left corner.
  • ...and 5 more figures

Theorems & Definitions (1)

  • Theorem III.1