用于加速MRI重建的端到端变分网络

Abstract. The slow acquisition speed of magnetic resonance imaging (MRI) has led to the development of two complementary methods: acquiring multiple views of the anatomy simultaneously (parallel imaging) and acquiring fewer samples than necessary for traditional signal processing methods (compressed sensing). While the combination of these methods has the potential to allow much faster scan times, reconstruction from such under sampled multi-coil data has remained an open problem. In this paper, we present a new approach to this problem that extends previously proposed variation al methods by learning fully end-to-end. Our method obtains new state-of-the-art results on the fastMRI dataset [18] for both brain and knee MRIs.3

摘要。磁共振成像(MRI)的缓慢采集速度催生了两类互补方法：同步采集多视角解剖结构(并行成像)和采集少于传统信号处理方法所需样本量(压缩感知)。虽然这些方法的结合有望实现更快的扫描速度，但从这种欠采样的多线圈数据中重建图像仍是一个悬而未决的问题。本文提出一种新方法，通过端到端学习扩展了先前提出的变分方法。我们的方法在fastMRI数据集[18]上取得了脑部和膝部MRI的最新最优结果。3

Keywords: MRI Acceleration · End-to-end learning · Deep learning.

关键词: MRI加速·端到端学习·深度学习

1 Introduction

1 引言

Magnetic Resonance Imaging (MRI) is a powerful diagnostic tool for a variety of disorders, but its utility is often limited by its slow speed compared to competing modalities like CT or X-Rays. Reducing the time required for a scan would decrease the cost of MR imaging, and allow for acquiring scans in situations where a patient cannot stay still for the current minimum scan duration.

磁共振成像 (MRI) 是诊断多种疾病的重要工具，但与CT或X射线等竞争技术相比，其速度较慢往往限制了其实用性。缩短扫描时间既能降低核磁共振成像成本，也能为无法保持静止达到当前最短扫描时长的患者完成检查。

One approach to accelerating MRI acquisition, called Parallel Imaging (PI) [3, 8, 13], utilizes multiple receiver coils to simultaneously acquire multiple views of the underlying anatomy, which are then combined in software. Multi-coil imaging is widely used in current clinical practice. A complementary approach to accelerating MRIs acquires only a subset of measurements and utilizes Compressed Sensing (CS) [1, 7] methods to reconstruct the final image from these under sampled measurements. The combination of PI and CS, which involves acquiring under sampled measurements from multiple views of the anatomy, has the potential to allow faster scans than is possible by either method alone. Reconstructing MRIs from such under sampled multi-coil measurements has remained an active area of research.

一种加速MRI采集的方法称为并行成像 (Parallel Imaging, PI) [3, 8, 13]，它利用多个接收线圈同时获取解剖结构的多个视角，随后通过软件进行组合。多线圈成像在当前的临床实践中得到广泛应用。另一种互补的MRI加速方法仅采集部分测量数据，并利用压缩感知 (Compressed Sensing, CS) [1, 7] 技术从这些欠采样测量中重建最终图像。结合PI和CS的方法（即从解剖结构的多个视角采集欠采样测量数据）有望实现比单独使用任一方法更快的扫描速度。从这类欠采样的多线圈测量数据中重建MRI图像，至今仍是活跃的研究领域。

MRI reconstruction can be viewed as an inverse problem and previous research has proposed neural networks whose design is inspired by the optimization procedure to solve such a problem [4, 6, 9, 10]. A limitation of such an approach is that it assumes the forward process is completely known, which is an unrealistic assumption for the multi-coil reconstruction problem. In this paper, we present a novel technique for reconstructing MRI images from under sampled multi-coil data that does not make this assumption. We extend previously proposed variation al methods by learning the forward process in conjunction with reconstruction, alleviating this limitation. We show through experiments on the fastMRI dataset that such an approach yields higher fidelity reconstructions.

MRI重建可视为一个逆问题，先前研究提出的神经网络设计灵感来源于解决此类问题的优化过程 [4, 6, 9, 10]。这类方法的局限性在于假设前向过程完全已知，这对多线圈重建问题是不现实的假设。本文提出一种无需此假设的创新技术，用于从欠采样的多线圈数据重建MRI图像。我们通过联合学习前向过程与重建任务，扩展了先前提出的变分方法，从而缓解了这一限制。在fastMRI数据集上的实验表明，该方法能获得更高保真度的重建结果。

Our contributions are as follows: 1) we extend the previously proposed variation al network model by learning completely end-to-end; 2) we explore the design space for the variation al networks to determine the optimal intermediate representations and neural network architectures for better reconstruction quality; and 3) we perform extensive experiments using our model on the fastMRI dataset and obtain new state-of-the-art results for both the knee and the brain MRIs.

我们的贡献如下：1) 我们通过完全端到端学习扩展了先前提出的变分网络模型；2) 我们探索了变分网络的设计空间，以确定最佳的中间表示和神经网络架构，从而提高重建质量；3) 我们在fastMRI数据集上使用我们的模型进行了大量实验，并在膝盖和脑部MRI上获得了新的最先进结果。

2 Background and Related Work

2 背景与相关工作

2.1 Accelerated MRI acquisition

2.1 加速MRI采集

An MR scanner images a patient’s anatomy by acquiring measurements in the frequency domain, called $k$ -space, using a measuring instrument called a receiver coil. The image can then be obtained by applying an inverse multidimensional Fourier transform ${\mathcal{F}}^{-1}$ to the measured k-space samples. The underlying image $\mathbf{x}\in\mathbb{C}^{M}$ is related to the measured k-space samples $\mathbf{k}\in\mathbb{C}^{M}$ as

MR扫描仪通过使用称为接收线圈的测量仪器在频域(称为$k$空间)中采集测量数据来对患者的解剖结构进行成像。然后可以通过对测量的k空间样本应用逆多维傅里叶变换${\mathcal{F}}^{-1}$来获得图像。底层图像$\mathbf{x}\in\mathbb{C}^{M}$与测量的k空间样本$\mathbf{k}\in\mathbb{C}^{M}$之间的关系为

$$
\begin{array}{r}{\mathbf{k}=\mathcal{F}(\mathbf{x})+\boldsymbol{\epsilon},}\end{array}
$$

$$
\begin{array}{r}{\mathbf{k}=\mathcal{F}(\mathbf{x})+\boldsymbol{\epsilon},}\end{array}
$$

where $\epsilon$ is the measurement noise and $\mathcal{F}$ is the fourier transform operator.

其中 $\epsilon$ 是测量噪声，$\mathcal{F}$ 是傅里叶变换 (Fourier transform) 算子。

Most modern scanners contain multiple receiver coils. Each coil acquires k-space samples that are modulated by the sensitivity of the coil to the MR signal arising from different regions of the anatomy. Thus, the $i$ -th coil measures:

现代大多数扫描仪都配备多个接收线圈。每个线圈采集的k空间样本会受到该线圈对解剖结构不同区域产生的MR信号敏感度的调制。因此，第$i$个线圈的测量值为：

$$
{\bf k}{i}={\mathcal F}(S_{i}{\bf x})+\epsilon_{i},i=1,2,\ldots,N,
$$

$$
{\bf k}{i}={\mathcal F}(S_{i}{\bf x})+\epsilon_{i},i=1,2,\ldots,N,
$$

where $S_{i}$ is a complex-valued diagonal matrix encoding the position dependent sensitivity map of the $i$ -th coil and $N$ is the number of coils. The sensitivity maps are normalized to satisfy [15]:

其中 $S_{i}$ 是编码第 $i$ 个线圈位置相关灵敏度图的复值对角矩阵，$N$ 是线圈数量。灵敏度图经过归一化以满足 [15]:

$$
\sum_{i=1}^{N}S_{i}^{*}S_{i}=1
$$

$$
\sum_{i=1}^{N}S_{i}^{*}S_{i}=1
$$

The speed of MRI acquisition is limited by the number of k-space samples obtained. This acquisition process can be accelerated by obtaining under sampled k-space data, $\tilde{\mathbf{k}{i}}=M\mathbf{k}{i}$ , where $M$ is a binary mask operator that selects a subset of the k-space points and $\tilde{\mathbf{k}_{i}}$ denotes the measured k-space data. The same mask is used for all coils. Applying an inverse Fourier transform naively to this under-sampled k-space data results in aliasing artifacts.

MRI采集速度受限于获取的k空间样本数量。通过获取欠采样的k空间数据可以加速这一采集过程，$\tilde{\mathbf{k}{i}}=M\mathbf{k}{i}$，其中$M$是选择k空间点子集的二元掩码算子，$\tilde{\mathbf{k}_{i}}$表示测量到的k空间数据。所有线圈使用相同的掩码。若直接对欠采样k空间数据应用逆傅里叶变换，会导致混叠伪影。

Parallel Imaging can be used to accelerate imaging by exploiting redundancies in k-space samples measured by different coils. The sensitivity maps $S_{i}$ can be estimated using the central region of k-space corresponding to low frequencies, called the Auto-Calibration Signal (ACS), which is typically fully sampled. To accurately estimate these sensitivity maps, the ACS must be sufficiently large, which limits the maximum possible acceleration.

并行成像 (Parallel Imaging) 可通过利用不同线圈采集的k空间样本冗余性来加速成像。灵敏度图 $S_{i}$ 可通过k空间中心对应的低频区域（称为自动校准信号 (Auto-Calibration Signal, ACS) ）进行估计，该区域通常为全采样。为准确估计灵敏度图，ACS区域需足够大，这会限制最大可实现加速度。

2.2 Compressed Sensing for Parallel MRI Reconstruction

2.2 并行MRI重建中的压缩感知

Compressed Sensing [2] enables reconstruction of images by using fewer k-space measurements than is possible with classical signal processing methods by enforcing suitable priors. Classical compressed sensing methods solve the following optimization problem:

压缩感知 [2] 通过施加合适的先验条件，能够使用比经典信号处理方法更少的k空间测量值重建图像。经典压缩感知方法求解以下优化问题:

$$
\begin{array}{l}{\displaystyle\hat{\mathbf{x}}=\arg\operatorname*{min}{\mathbf{x}}\frac{1}{2}\sum_{i}\left|{\cal M}\mathcal{F}(S_{i}\mathbf{x})-\tilde{\mathbf{k}{i}}\right|^{2}+\lambda\varPsi(\mathbf{x})}\ {\displaystyle=\arg\operatorname*{min}_{\mathbf{x}}\frac{1}{2}\left|{\cal A}(\mathbf{x})-\tilde{\mathbf{k}}\right|^{2}+\lambda\varPsi(\mathbf{x}),}\end{array}
$$

$$
\begin{array}{l}{\displaystyle\hat{\mathbf{x}}=\arg\operatorname*{min}{\mathbf{x}}\frac{1}{2}\sum_{i}\left|{\cal M}\mathcal{F}(S_{i}\mathbf{x})-\tilde{\mathbf{k}{i}}\right|^{2}+\lambda\varPsi(\mathbf{x})}\ {\displaystyle=\arg\operatorname*{min}_{\mathbf{x}}\frac{1}{2}\left|{\cal A}(\mathbf{x})-\tilde{\mathbf{k}}\right|^{2}+\lambda\varPsi(\mathbf{x}),}\end{array}
$$

where $\psi$ is a regular iz ation function that enforces a sparsity constraint, $A$ is the linear forward operator that multiplies by the sensitivity maps, applies 2D fourier transform and then under-samples the data, and $\tilde{\mathbf{k}}$ is the vector of masked k-space data from all coils. This problem can be solved by iterative gradient descent methods. In the $t$ -th step the image is updated from $\mathbf{x}^{t}$ to $\mathbf{x}^{t+1}$ using:

其中 $\psi$ 是一个施加稀疏约束的正则化函数，$A$ 是线性前向算子，通过乘以灵敏度图、应用二维傅里叶变换然后对数据进行欠采样，$\tilde{\mathbf{k}}$ 是来自所有线圈的掩码k空间数据向量。该问题可以通过迭代梯度下降方法求解。在第 $t$ 步中，图像从 $\mathbf{x}^{t}$ 更新到 $\mathbf{x}^{t+1}$，使用：

$$
\mathbf{x}^{t+1}=\mathbf{x}^{t}-\eta^{t}\left(A^{*}(A(\mathbf{x})-\tilde{\mathbf{k}})+\lambda\varPhi(\mathbf{x}^{t})\right),
$$

$$
\mathbf{x}^{t+1}=\mathbf{x}^{t}-\eta^{t}\left(A^{*}(A(\mathbf{x})-\tilde{\mathbf{k}})+\lambda\varPhi(\mathbf{x}^{t})\right),
$$

where $\eta^{t}$ is the learning rate, $\varPhi(\mathbf{x})$ is the gradient of $\psi$ with respect to $\mathbf{x}$ , and $A^{*}$ is the hermitian of the forward operator $A$ .

其中 $\eta^{t}$ 是学习率，$\varPhi(\mathbf{x})$ 是 $\psi$ 关于 $\mathbf{x}$ 的梯度，$A^{*}$ 是前向算子 $A$ 的共轭转置。

2.3 Deep Learning for Parallel MRI Reconstruction

2.3 并行MRI重建的深度学习

In the past few years, there has been rapid development of deep learning based approaches to MRI reconstruction [4–6, 9, 10, 12]. A comprehensive survey of recent developments in using deep learning for parallel MRI reconstruction can be found in [5]. Our work builds upon the Variation al Network (VarNet) [4], which consists of multiple layers, each modeled after a single gradient update step in equation 6. Thus, the $t$ -th layer of the VarNet takes $\mathbf{x}^{t}$ as input and computes $\mathbf{x}^{t+1}$ using:

过去几年，基于深度学习的MRI重建方法发展迅速[4–6, 9, 10, 12]。文献[5]对深度学习在并行MRI重建中的最新进展进行了全面综述。我们的工作基于变分网络(VarNet)[4]，该网络由多层结构组成，每层对应方程6中的单次梯度更新步骤。因此VarNet的第$t$层以$\mathbf{x}^{t}$为输入，通过下式计算$\mathbf{x}^{t+1}$：

$$
\mathbf{x}^{t+1}=\mathbf{x}^{t}-\eta^{t}A^{*}(A(\mathbf{x}^{t})-\tilde{\mathbf{k}})+\mathrm{CNN}(\mathbf{x}^{t}),
$$

$$
\mathbf{x}^{t+1}=\mathbf{x}^{t}-\eta^{t}A^{*}(A(\mathbf{x}^{t})-\tilde{\mathbf{k}})+\mathrm{CNN}(\mathbf{x}^{t}),
$$

where CNN is a small convolutional neural network that maps complex-valued images to complex-valued images of the same shape. The $\eta^{t}$ values as well as the parameters of the CNNs are learned from data.

其中CNN是一个小型卷积神经网络，它将复数值图像映射为相同形状的复数值图像。参数$\eta^{t}$以及CNN的参数均从数据中学习得到。

The $A$ and $A^{*}$ operators involve the use of sensitivity maps which are computed using a traditional PI method and fed in as additional inputs. As noted in section 2.1, these sensitivity maps cannot be estimated accurately when the number of auto-calibration lines is small, which is necessary to achieve higher acceleration factors. As a result, the performance of the VarNet degrades significantly at higher accelerations. We alleviate this problem in our model by learning to predict the sensitivity maps from data as part of the network.

$A$ 和 $A^{*}$ 算子需要使用通过传统并行成像 (PI) 方法计算出的灵敏度图作为额外输入。如第 2.1 节所述，当自动校准线数量较少时（这是实现更高加速因子的必要条件），这些灵敏度图无法被准确估算。因此，VarNet 在更高加速因子下的性能会显著下降。我们通过让网络学习从数据中预测灵敏度图来缓解这个问题。

3 End-to-End Variation al Network

3 端到端变分网络

Let $\mathbf{k}{0}=\tilde{\mathbf{k}}$ be the vector of masked multi-coil k-space data. Similar to the VarNet, our model takes this masked k-space data $\mathbf{k}_{0}$ as input and applies a number of refinement steps of the same form. We refer to each of these steps as a cascade (following [12]), to avoid overloading the term "layer" which is already heavily used. Unlike the VN, however, our model uses k-space intermediate quantities rather than image-space quantities. We call the resulting method the End-to-End Variation al Network or E2E-VarNet.

令 $\mathbf{k}{0}=\tilde{\mathbf{k}}$ 为掩码多线圈k空间数据向量。与VarNet类似，我们的模型以该掩码k空间数据 $\mathbf{k}_{0}$ 作为输入，并应用多个相同形式的细化步骤。为避免与已广泛使用的术语"层"混淆，我们称每个步骤为一个级联 (cascade) (遵循[12])。然而与VN不同的是，我们的模型使用k空间中间量而非图像空间量。我们将最终方法称为端到端变分网络 (End-to-End Variational Network) 或E2E-VarNet。

3.1 Preliminaries

3.1 预备知识

To simplify notation, we first define two operators: the expand operator ( $\boldsymbol{\mathcal{E}}$ ) and the reduce operator ( $\mathcal{R}$ ). The expand operator ( $\boldsymbol{\mathcal{E}}$ ) takes the image $\mathbf{x}$ and sensitivity maps as input and computes the corresponding image seen by each coil in the idealized noise-free case:

为简化表示法，我们首先定义两个算子：扩展算子 ( $\boldsymbol{\mathcal{E}}$ ) 和缩减算子 ( $\mathcal{R}$ )。扩展算子 ( $\boldsymbol{\mathcal{E}}$ ) 以图像 $\mathbf{x}$ 和灵敏度图为输入，计算理想无噪声情况下每个线圈对应的观测图像：

$$
\pmb{\mathcal{E}}(\mathbf{x})=(\mathbf{x}{1},...,\mathbf{x}{N})=(S_{1}\mathbf{x},...,S_{N}\mathbf{x}).
$$

$$
\pmb{\mathcal{E}}(\mathbf{x})=(\mathbf{x}{1},...,\mathbf{x}{N})=(S_{1}\mathbf{x},...,S_{N}\mathbf{x}).
$$

where $S_{i}$ is the sensitivity map of coil $i$ . We do not explicitly represent the sensitivity maps as inputs for the sake of readability. The inverse operator, called the reduce operator ( $\mathcal{R}$ ) combines the individual coil images:

其中 $S_{i}$ 是线圈 $i$ 的灵敏度图。为了可读性，我们没有明确将灵敏度图表示为输入。逆操作称为归约算子 ( $\mathcal{R}$ )，用于组合各线圈图像：

$$
\mathcal{R}(\mathbf{x}{1},...\mathbf{x}{N})=\sum_{i=1}^{N}S_{i}^{*}\mathbf{x}_{i}.
$$

$$
\mathcal{R}(\mathbf{x}{1},...\mathbf{x}{N})=\sum_{i=1}^{N}S_{i}^{*}\mathbf{x}_{i}.
$$

Using the expand and reduce operators, $A$ and $A^{}$ can be written succinctly as $A=M\circ{\mathcal{F}}\circ{\mathcal{E}}$ and $A^{*}=\mathcal{R}\circ\mathcal{F}^{-1}\circ M$ .

使用扩展和缩减运算符，$A$ 和 $A^{}$ 可以简洁地表示为 $A=M\circ{\mathcal{F}}\circ{\mathcal{E}}$ 和 $A^{*}=\mathcal{R}\circ\mathcal{F}^{-1}\circ M$。

3.2 Cascades

3.2 级联

Each cascade in our model applies a refinement step similar to the gradient descent step in equation 7, except that the intermediate quantities are in k-space. Applying ${\mathcal{F}}\circ{\mathcal{E}}$ to both sides of 7 gives the corresponding update equation in k-space:

我们模型中的每个级联都应用了一个类似于方程7中梯度下降步骤的细化步骤，不同之处在于中间量位于k空间。对7的两边应用${\mathcal{F}}\circ{\mathcal{E}}$，得到k空间中的对应更新方程:

$$
\mathbf{k}^{t+1}=\mathbf{k}^{t}-\eta^{t}M(\mathbf{k}^{t}-\tilde{\mathbf{k}})+G(\mathbf{k}^{t})
$$

$$
\mathbf{k}^{t+1}=\mathbf{k}^{t}-\eta^{t}M(\mathbf{k}^{t}-\tilde{\mathbf{k}})+G(\mathbf{k}^{t})
$$

where $G$ is the refinement module given by:

其中 $G$ 是由以下公式给出的细化模块：

$$
G(\mathbf{k}^{t})=\mathcal{F}\circ\mathcal{E}\circ\mathrm{CNN}(\mathcal{R}\circ\mathcal{F}^{-1}(\mathbf{k}^{t})).
$$

$$
G(\mathbf{k}^{t})=\mathcal{F}\circ\mathcal{E}\circ\mathrm{CNN}(\mathcal{R}\circ\mathcal{F}^{-1}(\mathbf{k}^{t})).
$$

Here, we use the fact that $\mathbf{x}^{t}=\mathcal{R}\circ\mathcal{F}^{-1}(\mathbf{k}^{t})$ . CNN can be any parametric function that takes a complex image as input and maps it to another complex image. Since the CNN is applied after combining all coils into a single complex image, the same network can be used for MRIs with different number of coils.

这里，我们利用$\mathbf{x}^{t}=\mathcal{R}\circ\mathcal{F}^{-1}(\mathbf{k}^{t})$这一事实。CNN可以是任何以复图像为输入并映射到另一个复图像的参数化函数。由于CNN是在将所有线圈组合成单个复图像后应用的，因此相同的网络可用于具有不同线圈数量的MRI。

Each cascade applies the function represented by equation 10 to refine the k-space. In our experiments, we use a U-Net [11] for the CNN.

每个级联都应用公式10表示的函数来细化k空间。在我们的实验中，我们使用U-Net [11]作为CNN。

Fig. 1: Top: Block diagram of our model which takes under-sampled $\mathrm{k}$ -space as input and applies several cascades, followed by an inverse Fourier transform (IFT) and an RSS transform. The Data Consistency (DC) module computes a correction map that brings the intermediate k-space closer to the measured k-space values. The Refinement (R) module maps multi-coil k-space data into one image, applies a U-Net, and then back to multi-coil k-space data. The Sensitivity Map Estimation (SME) module estimates the sensitivity maps used in the Refinement module.

图 1: 上: 我们的模型框图以欠采样的 $\mathrm{k}$ 空间作为输入，应用多个级联模块，随后进行逆傅里叶变换 (IFT) 和平方和根 (RSS) 变换。数据一致性 (DC) 模块计算校正图，使中间 k 空间更接近测量值。精炼 (R) 模块将多线圈 k 空间数据映射为单幅图像，通过 U-Net 处理后再转换回多线圈 k 空间数据。灵敏度图估计 (SME) 模块为精炼模块提供估计的灵敏度图。

3.3 Learned sensitivity maps

3.3 学习型敏感度图

The expand and reduce operators in equation 11 take sensitivity maps $(S_{1},...,S_{N})$ as inputs. In the original VarNet model, these sensitivity maps are computed using the ESPIRiT algorithm [15] and fed in to the model as additional inputs. In our model, however, we estimate the sensitivity maps as part of the reconstruction network using a Sensitivity Map Estimation (SME) module:

公式11中的扩展和缩减操作以灵敏度图$(S_{1},...,S_{N})$作为输入。在原始VarNet模型中，这些灵敏度图通过ESPIRiT算法[15]计算并作为额外输入馈入模型。而在我们的模型中，我们使用灵敏度图估计(Sensitivity Map Estimation, SME)模块将灵敏度图作为重建网络的一部分进行估计：

$$
H=\mathrm{d}\mathrm{S}\mathrm{S}\circ\mathrm{C}\mathrm{N}\mathrm{N}\circ\mathcal{F}^{-1}\circ M_{\mathrm{center}}.
$$

$$
H=\mathrm{d}\mathrm{S}\mathrm{S}\circ\mathrm{C}\mathrm{N}\mathrm{N}\circ\mathcal{F}^{-1}\circ M_{\mathrm{center}}.
$$

The $M_{\mathrm{center}}$ operator zeros out all lines except for the auto calibration or ACS lines (described in Section 2.1). This is similar to classical parallel imaging approaches which estimate sensitivity maps from the ACS lines. The CNN follows the same architecture as the CNN in the cascades, except with fewer channels and thus fewer parameters in intermediate layers. This CNN is applied to each coil image independently. Finally, the dSS operator normalizes the estimated sensitivity maps to ensure that the property in equation 3 is satisfied.

$M_{\mathrm{center}}$ 运算符会清零除自动校准线或 ACS 线 (详见第 2.1 节) 外的所有线。这与传统并行成像方法类似，后者通过 ACS 线估算灵敏度图。该 CNN 采用与级联中 CNN 相同的架构，但通道数更少，因此中间层的参数也更少。该 CNN 会独立应用于每个线圈图像。最后，dSS 运算符会对估算的灵敏度图进行归一化处理，以确保满足公式 3 中的特性。

3.4 E2E-VarNet model architecture

3.4 E2E-VarNet 模型架构

As previously described, our model takes the masked multi-coil k-space $\mathbf{k}{0}$ as input. First, we apply the SME module to $\mathbf{k}_{0}$ to compute the sensitivity maps. Next we apply a series of cascades, each of which applies the function in equation 10, to the input k-space to obtain the final k-space representation $K^{T}$ . This final k-space representation is converted to image space by applying an inverse Fourier transform followed by a root-sum-squares (RSS) reduction for each pixel:

如前所述，我们的模型以掩码多线圈k空间$\mathbf{k}{0}$作为输入。首先，我们对$\mathbf{k}_{0}$应用SME模块来计算灵敏度图。接着，我们对输入k空间应用一系列级联操作（每个级联都执行公式10的函数）以获得最终k空间表示$K^{T}$。最后通过逆傅里叶变换和逐像素的平方根求和(RSS)缩减，将该k空间表示转换到图像空间：

$$
\hat{\mathbf{x}}=R S S(\mathbf{x}{1},...,\mathbf{x}{N})=\sqrt{\sum_{i=1}^{N}|\mathbf{x}_{i}|^{2}}
$$

$$
\hat{\mathbf{x}}=R S S(\mathbf{x}{1},...,\mathbf{x}{N})=\sqrt{\sum_{i=1}^{N}|\mathbf{x}_{i}|^{2}}
$$

where $\mathbf{x}{i}=\mathcal{F}^{-1}(K_{i}^{T})$ and $K_{i}^{T}$ is the $\mathrm{k\Omega}$ -space representation for coil $i$ . The model is illustrated in figure 1.

其中 $\mathbf{x}{i}=\mathcal{F}^{-1}(K_{i}^{T})$ ，且 $K_{i}^{T}$ 是线圈 $i$ 的 $