  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.
