1 Introduction
Due to the limitations of the Magnetic Resonance Imaging (MRI), MRI scanning takes a long time, which limits the widespread use of MRI. The kspace data is usually highly undersampled to reduce the time required for MR scanning. How to reconstruct highquality images from highly undersampled kspace data is a pivotal topic in MR reconstruction. In the past, fast MRI was performed mainly through parallel imaging and compressed sensing methods (Lingala et al. [2011], Lin and Fessler [2019], Majumdar [2015]). With the development of deep learning methods, deep learning MRI reconstruction methods have achieved superior reconstruction results (Aggarwal et al. [2018], Cui et al. [2021], Cao et al. [2022], Ke et al. [2021, 2020], Cheng et al. [2020], Huang et al. [2021]).
Recently, diffusion models have achieved excellent performance in many image processing tasks, including two types of generative models, scorebased and DDPM (Song and Ermon [2019], Ho et al. [2020]), which provide new approaches to solve the inverse image reconstruction problem. In the framework of continuous SDE, Song et al. [2021] unified them and proposed three forms of SDEs, VESDE, VPSDE and subVPSDE, to further improve the performance of diffusion models. Currently, Jalal et al. [2021], Song et al. [2022], Chung and Ye [2022] have achieved impressive results in MRI reconstruction tasks using scorebased generative model or VESDE.
The initial value of the reversal process of VPSDE (corresponding to the reconstruction process of MRI) is gaussian white noise, which is not related to the original image. In MR reconstruction, the fully sampled original image is not available, while the initial value of the VESDE reverse process requires the original image, which is not available in reality. So VPSDE is more suitable to be applied to MR reconstruction tasks than VESDE. The reversetime SDE iteration rule of VPSDE is:
(1) 
where is considered to be the reconstructed MR image, is the trained scorebased model, and is a given coefficient to control the noise level. From optimization perspective, the reverse VPSDE can be viewed as the Stochastic gradient Langvien descent algorithm for the following problem:
(2) 
where are positive weights. When , the objective function may obtain the minimum value at , which means that there is a risk of divergence of the VPSDE.
1.1 Contributions
Motivated by the above mentioned problems and combined with MRI, we propose a more stable diffusion model. Specifically, the main contributions of our work are summarized as follows.

[leftmargin = *]

For MRI reconstruction, we propose a highfrequency SDE diffusion model, termed HFSSDE, in which the energy of the SDE sequence does not expand infinitely as time evolves. Meanwhile, the diffusion process only takes place at high frequencies, which can more effectively characterize the image highfrequency prior.

Since the HFSSDE diffuses only in highfrequency space, the initial value of its reverse process can be allowed to zero filling images instead of Gaussian white noise. From the optimization point of view, a good initial value facilitates a faster and more accurate algorithm convergence to the desired solution.

Experiments show that our HFSSDE consistently outperforms its competitor VPSDE and other supervised deep learning methods and conventional parallel imaging methods regarding MR reconstruction accuracy and stability.
1.2 Related Works
Several related works have used the diffusion model to accelerate MRI reconstruction due to its superior performance. Jalal et al. [2021] first applied the scorebased generative model to fast MR reconstruction. In the framework of continuous SDE, Song et al. [2022] and Chung and Ye [2022] proposed to apply VESDE to the MR reconstruction. Unlike these three works, we propose a new SDE, which performs the diffusion process only in the highfrequency space and can better reconstruct the highfrequency details of MR images.
Xie and Li [2022] proposed measurementconditioned DDPM (MCDDPM) for accelerated MRI reconstruction. MCDDPM trains the diffusion model by adding an undersampled mask to the training data. However, the diffusion process of MCDDPM is performed in the kspace domain, which leads to the case that MCDDPM cannot handle data with different number of coils. Unlike MCDDPM, the diffusion process of our method is performed in the highfrequency space of the image. The method we proposed samples only the central region of the kspace data. Moreover, our proposed method is based on the SDE continuous framework, and for the drawback of VPSDE, we propose a new form of SDE called HFSSDE.
The following sections of the paper are organized as follows: Section 2 introduces the backward of generative diffusion processes, Section 3 describes the proposed method, Section 4 provides the experimental results. Discussion and conclusion are given in Section 5 and Section 6.
2 Backward
Song et al. [2021] proposed a unified framework that generalizes two types of generative models (Song and Ermon [2019], Ho et al. [2020]) through the lens of stochastic differential equations (SDEs). The diffusion process is described as the solution of the following SDE:
and are called the drift and diffusion coefficients of x, respectively. is the standard Wiener process. The reverse diffusion process is obtained from the following reversetime SDE:
is known as score function and can be generated by sampling from . The score network estimates the score function, and the loss function of the network is
(3) 
is a positive weighting function, is uniformly sampled over .
3 Method: HighFrequency Space Diffusion Stochastic Differential Equations
Since the energy of the image is mainly concentrated in the low frequency region, we solve the instability problem of VPSDE by constructing the diffusion process in the highfrequency region of the image. We define and as highfrequency and lowfrequency operators. In particular, in the multichannel scenario, , , indicates the number of coils, and denotes the sensitivity of the th coil. is the Fourier operator, is the operator to obtain the central lowfrequency data that only takes center region of kspace and
denotes identity matrix.
The discrete Markov chain form of the forward HFSSDE is as follows
(4) 
are given coefficients to control the noise level and . By introducing the auxiliary variable , Eq. (4) can be rewritten as
When , can be written as and . Let ,
(5)  
Eq. (5) converges to
By Särkkä and Solin [2019] and let for , the perturbation kernel of the discrete form of the HFSSDE can be derived as
The highfrequency operator is exponential power, which is nonlinear and impractical to implement directly. Assuming a coefficient of before , we solve this problem with the following trick
Since , we can get
The mean and covariance matrix of the perturbation kernel can be reexpressed as
The parameters in the network is trained via the following loss function:
is the positive weighting function. The network structure in HFSSDE is DDPM++, which is improved by Song et al. [2021]. For the MRI inverse problem, we can only acquire the undersampled kspace measurements y, so we aim to reconstruct the MR image based on . Based on the trained network, we can perform the following reverse HFSSDE conditional on to obtain the reconstructed image. (Anderson [1982])
Further, we have
Hence, The reconstruction process is shown in Alg. 1.
4 Experiments
4.1 Experimental Details
We trained on the multichannel fastMRI T1weighted dataset (Zbontar et al. [2018], Knoll et al. [2020]) using 34 individuals with a total of 1002 images, and we cropped the image size to 320x320 and tested on T1weighted knee scans and T2weighted brain scans.
We compared with the traditional parallel imaging method, supervised deep learning method and DDPMdriven VPSDE (Pruessmann et al. [1999], Zhang and Ghanem [2018], Song et al. [2021]
). Each method was tuned to its best achievable result, and the VpSDE and HFSSDE were trained with 500 epochs. The parameters required in Alg.
1 are set as follows: .4.2 Experimental Results
The reconstruction results of the multichannel uniformly undersampled by 8fold with 18 ACS lines are shown in Fig. 1. The reconstruction results of HFSSDE are remarkably better than these three methods, and the highfrequency details of the image are accurately reconstructed.
Fig. 2 shows the reconstruction results in the extreme case of 10 undersampled. HFSSDE still shows no artifacts and reconstructs the image details well. Because the VPSDE takes the optimal solution when the image energy is maximized, the VPSDE is severely affected by artifacts in high acceleration scenes. The HFSSDE can suppress the artifacts well, which indicates that our method can well correct the unstable defects of VPSDE.
We conducted anatomyshift and mask shift experiments to demonstrate the generalization performance of the HFSSDE. In the anatomyshift experiment, ISTANet, VPSDE and HFSSDE were trained on T1weight knee data and tested on T2weight brain data, and the reconstruction results are shown in Fig. 3. In the maskshift experiment, the undersampling mask in the test phase was the cartesian mask. In the training process, the mask imposed in the ISTANet was a uniform 6fold mask. operator in HFSSDE captures 22 central lines in kspace, and the reconstruction results are shown in Fig. 4. When the distribution of training and test data is significantly different, the performance of the supervised training method (ISTANet) is greatly degraded, while the diffusion models method can still reconstruct the image accurately. In particular, the HFSSDE significantly outperforms the VPSDE in recovering highfrequency details.
5 Discussion
Because the diffusion process of HFSSDE is performed only in highfrequency space, the initial value of the reverse diffusion is the lowfrequency part of the image plus the Gaussian noise in the highfrequency part. When the operator selects a less lowfrequency region, it will easily lead to excessive energy instability. However, when the selected lowfrequency region is too large (greatly exceeding the number of ACS lines of the undersampling mask), the reconstruction results will have lost highfrequency details. Hence, the range selection of the operator has a large impact on the HFSSDE performance. We conducted experiments in undersampling scenarios of 10, and the ACS lines number of the undersampling mask was chosen to be 18. The experimental results are detailed in Fig. 5. When the undersampling rate is low (less than 6), the lowfrequency region selected by the operator should make consistent with the number of ACS lines in the centre of the undersampling mask. As the undersampling rate increases, the lowfrequency region of the operator should be gradually increased to suppress the generation of artifacts.
6 Conclusions
In this paper, we propose a new SDE for fast MR reconstruction from an optimization point of view, whose diffusion process is in the highfrequency part of the image. The energy concentrated lowfrequency part of the MR image is no longer amplified, and the diffusion process focuses more on acquiring highfrequency prior information. It not only improves the stability of the diffusion model but also enables more accurate recovery of highfrequency details in MR images. We conducted experiments on the multichannel fastMRI dataset, and the experiments show that our method outperforms VPSDE, supervised deep learning methods and traditional parallel imaging methods in terms of stability and reconstruction accuracy.
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