Multi-Contrast Low-Field MRI Acceleration with K-Space Progressive Learning and Image-Space Hybrid Attention Fusion

Abstract

Multi-contrast MRI provides complementary tissue information for diagnosis and treatment planning but is limited by the long acquisition time and system noise, which deteriorates at low field strength. To jointly accelerate and denoise multi-contrast MRI acquired at low field strength, we present a novel dual-domain framework designed to reconstruct high-quality multi-contrast MR images from k-space data corrupted by under-sampling and system noise. Our dual-domain framework first enhances k-space data quality through a k-space Low-to-High Frequency Progressive (LHFP) learning network, and then further refines the k-space outputs with an image-space Hybrid Attention Fusion Network (HAFNet). In k-space learning, the magnitude imbalance between the low- and high-frequency components may cause the network to be dominated by low-frequency components, leading to sub-optimal recovery of high-frequency components. To tackle this challenge, the two-stage LHFP learning network first recovers low-frequency components and then emphasizes high-frequency learning through patient-specific adaptive prediction of the low-high frequency boundary. In image domain learning, the challenge of efficiently capturing long-range dependencies across the multi-contrast images is resolved through Hybrid Window-based Attention Fusion (HWAF) modules, which integrate features by alternately computing self-attention within dense and dilated windows. Extensive experiments on the BraTs MRI and M4Raw low-field MRI datasets demonstrate the superiority of our method over state-of-the-art MRI reconstruction methods. Our source code will be made publicly available upon acceptance.

1. Introduction

Magnetic Resonance Imaging (MRI) is a non-invasive and standard method for medical examination and diagnosis. In MRI systems, signals are acquired in k-space (i.e., Fourier space) and then reconstructed into images (Weishaupt et al., 2006). Recently, low-field MRI systems (< 1T) have been gaining popularity for portable imaging, pediatric imaging, and image-guidance during interventional procedures (Lyu et al., 2023b). However, the low-field strength leads to decreased signal-to-noise ratio (SNR), limiting the image quality of low-field MRI systems. Additionally, the scanning process can be time-consuming due to the physical nature of MRI. To expedite the MR scanning procedure, k-space data is often under-sampled, which can further degrade image quality (Tsao and Kozerke, 2012). Therefore, it is highly desirable to reconstruct high-quality MR images from under-sampled and noisy k-space data in low-field MRI systems.

In clinical practice, multi-contrast MR images, which contain complementary structural information, are typically acquired together for medical diagnosis and treatment planning. For instance, T1-weighted images (T1WIs) provide detailed morphological and structural information, while T2-weighted images (T2WIs) highlight edema and inflammation. Due to the physical characteristics of MR imaging systems, T2WIs are more challenging to obtain than T1WIs because they require longer repetition and echo times (Xiang et al., 2018). Recent studies have shown that using an easily obtainable modality (e.g., T1WI) as a guidance modality to accelerate and enhance the target modality (e.g., T2WI) is a promising approach, especially for modalities that require longer scanning times (Feng et al., 2021).

In recent years, deep learning-based methods have been widely used for multi-contrast MRI reconstruction (Xiang et al., 2018; Feng et al., 2021, 2022; Han et al., 2019; Ding and Zhang, 2022; Huang et al., 2023b; Pan et al., 2023; Zhou and Zhou, 2020; Lyu et al., 2022; Liu et al., 2022). These methods can be broadly categorized into image domain reconstruction, k-space reconstruction, and dual-domain reconstruction. Image domain reconstruction involves transforming corrupted k-space data into low-quality images and then denoising them in the image domain (Xiang et al., 2018; Feng et al., 2021, 2022). However, under-sampling violates the Nyquist-Shannon sampling theorem, introducing aliasing artifacts and causing loss of details in MR images before image-domain regularizers take effect. The second approach focuses on k-space reconstruction (Han et al., 2019; Ding and Zhang, 2022; Huang et al., 2023b; Pan et al., 2023), where high-quality k-space signals are recovered through interpolation and denoising in the Fourier domain, and then these refined k-space signals are reconstructed into images. However, k-space reconstruction is complicated by the intricate nature of frequency domain data, making it challenging to interpret and manipulate. Besides, k-space data is highly sensitive to noise and artifacts. Small errors in k-space can lead to widespread artifacts in the reconstructed image, making it critical to accurately denoise and correct for these errors (Singh et al., 2023). Dual-domain reconstruction methods (Zhou and Zhou, 2020; Lyu et al., 2022; Liu et al., 2022) first perform interpolation and denoising in the Fourier domain to recover high-quality k-space data. This recovered k-space data is then transformed into the image domain, where further denoising and refinement are applied to enhance the final image quality. By leveraging information from both domains, these methods effectively enhance image quality and mitigate artifacts, leading to more accurate MRI reconstructions.

While promising for accelerating multi-contrast MRI acquisition, current methods have not fully explored the acceleration of multi-contrast MRI in low-field settings, which require joint de-aliasing and de-noising. Most existing methods (Zhou and Zhou, 2020; Lyu et al., 2022; Liu et al., 2022) employ data consistency layers to improve reconstruction accuracy by enforcing consistency between model predictions and the acquired k-space data. However, in low-field MRI systems, the acquired signals have lower SNR, posing additional challenges for directly applying data consistency constraints. To effectively leverage data consistency layers in this setting, additional constraints or regularizers are required to mitigate noise and enhance signal fidelity, making the overall model design more complex. Additionally, existing methods typically use the same network structures for both k-space and image space (e.g., convolutional networks in DudoRNet (Zhou and Zhou, 2020) and Transformers in DudoCAF (Lyu et al., 2022)), without considering the domain-specific characteristics and tailoring different models for each domain. To reconstruct high-quality MRI acquired at low field strength, an approach that is tailored to domain-specific characteristics and effectively extract features from both k-space and image domain needs to be developed.

In k-space learning, low-frequency (LF) components correspond to the overall appearance and basic structure of an image, while high-frequency (HF) components contribute to its sharpness and detailed structure. Ensuring the accurate recovery of both frequency components is crucial for reconstructing high-quality images. However, the magnitude of different frequency components is highly imbalanced, with LF components having magnitudes significantly higher than HF components (Pan et al., 2023). Consequently, during network training, the dominance of high-magnitude LF components can overshadow the recovery of HF components, leading to suboptimal reconstruction outcomes. Additionally, compared to LF components, the low-magnitude HF signals are more susceptible to low-field noise and have a lower SNR, making their recovery more challenging. Therefore, instead of treating LF and HF components equally, greater emphasis should be placed on the recovery of HF components, which is essential for enhancing k-space learning networks.

For image domain denoising, as complementary features exist across different modalities and various regions of an image, it is crucial to explore complementary information from the multi-contrast images. Existing studies indicate that compared to convolutional networks (Feng et al., 2021; Li et al., 2023), transformers (Feng et al., 2022; Lyu et al., 2023a) have a stronger ability to integrate long-range correlated features, leading to improved reconstruction performance. However, the self-attention operation across entire feature maps introduces quadratic computational complexity. To mitigate this, some methods compute feature correlations within local windows (Liu et al., 2021; Liang et al., 2021), compromising their capability to capture long-range dependencies. The situation worsens in low-field scenarios, as it is challenging to accurately compute correlations between noisy features. Therefore, it is highly desirable to design a network that can comprehensively and accurately explore long-range dependencies in a computationally efficient manner.

To tackle the unique challenges for low-field multi-contrast MRI acceleration, we present a dual-domain learning framework comprising a k-space Low-to-High Frequency Progressive (LHFP) learning network and an image space Hybrid Attention Fusion Network (HAFNet). The k-space LHFP learning network includes two stages, which first recovers LF components by uniformly weighting various frequency components, then emphasizes HF learning by reducing the loss weight of LF components. The instance-specific low-high frequency boundary is adaptively estimated by a mask predictor module, which is jointly optimized with the k-space learning network. The refined k-space data from the LHFP network is then reconstructed into images and input into the image domain HAFNet for subsequent denoising processes. The HAFNet employs convolutional blocks to extract multi-level features from each modality, then utilizes shallow-level and deep-level Hybrid Window-based Attention Fusion (HWAF) modules to efficiently integrate multi-contrast features in their respective layers. By alternately computing self-attention within dense and dilated windows, the HAFNet can capture long-range dependencies efficiently and comprehensively. To reduce complexity and stabilize the training process, our framework adopts a sequential pipeline rather than an end-to-end joint training approach, with the k-space and image domain networks trained separately in a single pass. This design enables the k-space network to refine frequency components while the image domain network enhances details and suppresses artifacts, leading to improved reconstruction quality. Our main contributions are as follows:

• To address the inadequate recovery of HF component in k-space learning, we present a novel LHFP learning network that progressively recovers LF and HF components, with the low-high frequency boundary adaptively optimized for each image.

• We propose a HAFNet to efficiently integrate long-range dependent features from the multi-contrast images by alternately computing self-attention within dense and dilated windows.

• Experiments on the BraTs and low-field M4Raw datasets demonstrate the superiority of our method over state-of-the-art multi-contrast MRI reconstruction methods.

2. Related Work

2.1. K-space learning-based MRI Reconstruction

To reconstruct high-quality images from corrupted k-space data, an effective approach involves performing interpolation and denoising in the k-space, followed by transforming the recovered k-space data into high-quality images (Han et al., 2019; Ding and Zhang, 2022; Huang et al., 2023b; Zhao et al., 2022; Pan et al., 2023).

Han et al. (2019) employed a U-Net to recover high-quality k-space data from corrupted signals. Given under-sampled k-space data, Ding and Zhang (2022) first interpolated missing frequency components through a weighted summation of neighboring signals, followed by convolutional blocks to correct potential prediction errors from the interpolation stage. Subsequent works (Zhao et al., 2022; Pan et al., 2023) enhanced the long-range modeling capabilities of k-space learning networks by employing transformer architectures (Vaswani et al., 2017). More recently, Huang et al. (2023b) introduced a network based on neural implicit functions (NIF) (Chen et al., 2021; Liu et al., 2024), which directly learns a mapping from coordinates to k-space signals. Inspired by NIF (Chen et al., 2021), Zhao et al. (2022) utilized the coordinates of unsampled k-space points as inputs to a transformer decoder, interpolating unsampled signals by querying information from sampled points. For dynamic MRI reconstruction, Pan et al. (2023) proposed a transformer-based k-GIN model that uses a masked AutoEncoder (He et al., 2022) to interpolate unsampled k-space signals and refine the interpolated data with a k-space iterative refinement module. However, these methods fail to sufficiently account for the magnitude disparity between LF and HF components and their variable resistance to low-field noise, leading to sub-optimal reconstruction outcomes.

2.2. Image Domain MRI Reconstruction

Another stream of MRI reconstruction approach first transforms the corrupted k-space data into low-quality images, and then improves the image quality with a denoising network. Wang et al. (2016) pioneered the use of convolutional neural networks (CNNs) for recovering high-quality images from low-quality ones. Building on this, Quan et al. (2018) enhanced reconstruction performance through adversarial learning and perceptual loss. Schlemper et al. (2017) proposed a cascade of CNNs equipped with data consistency layers to ensure k-space data fidelity during MRI reconstruction. Sandino et al. (2021) proposed the DL-ESPIRiT model that synergistically combines a robust parallel imaging model and deep learning-based priors for accelerated cardiac cine MRI. Huang et al. (2021) introduced a method that leverages low-rank and sparse priors to effectively suppress noise and artifacts, thereby improving dynamic MRI reconstruction quality. More recently, Hammernik et al. (2023) emphasized the significance of embedding physicsbased constraints into deep learning frameworks to enhance reliability and generalizability in MRI reconstruction.

To leverage the rich information from multi-contrast MRI, Xiang et al. (2018) used a fully sampled sequence as a reference to guide the reconstruction of an under-sampled target modality. However, by simply concatenating the fully sampled T1WI and under-sampled T2WI as input, their method failed to fully exploit the complementary information between these modalities. To better explore the relationships between different contrasts, Feng et al. (2021) extracted hierarchical feature representations for each modality and then integrated the multi-contrast features based on their similarities. Despite these advancements, CNN-based methods generally struggle with modeling long-range dependencies, which are crucial for multi-contrast MRI reconstruction. To address this, Feng et al. (2022) introduced a multi-modal transformer that integrates complementary features across different modalities using multi-head cross-attention mechanisms. Huang et al. (2023a) utilized convolutional networks to extract multi-scale features and then integrated multi-contrast features with dual cross-attention modules. However, compared to CNNs, the transformer architecture introduces quadratic computational complexity due to cross-attention computation, which may limit its clinical applicability. Therefore, network design should carefully balance computational cost and the ability to model long-range dependencies.

2.3. Dual-Domain MRI Reconstruction

Compared to the aforementioned single-domain reconstruction methods, dual-domain approaches (Eo et al., 2018; Liu et al., 2022; Zhou and Zhou, 2020; Lyu et al., 2022; Gao and Zhou, 2023) demonstrate superior reconstruction performance by leveraging the strength of both k-space and image domain learning. For example, KIKI-Net (Eo et al., 2018) alternately applied a k-space CNN, an image domain CNN, and a data consistency layer for single-contrast MRI reconstruction. Liu et al. (2022) devised a dual-domain KV-Net, where a lightweight image domain network and a k-space network are fused and cascaded in parallel. To enhance dynamic MRI reconstruction, Qin et al. (2021) explored complementary information from the temporal Fourier domain and spatiotemporal image domain. Xu et al. (2025) combined low-rank, image-domain, and k-space information to enable single breath-hold cardiac cine MRI reconstruction, improving both image quality and efficiency. These methods suggest that for single-contrast MRI reconstruction, better performance can be achieved by harnessing information from both k-space and image domains.

Recent studies have explored dual-domain learning for multi-contrast MRI reconstruction. Zhou and Zhou (2020) designed DuDoRNet, embedding a deep T1 prior into recurrently cascaded k-space restoration networks and image domain denoising networks. To better capture long-range dependencies in both k-space and image domains, Lyu et al. (2022) proposed DuDoCAF which extracts features through recurrent transformer blocks and integrates multi-contrast features via cross-attention fusion in both domains. Gao and Zhou (2023) emphasized the importance of domain-specific characteristics, designing domain-specific modules to enhance MRI reconstruction performance. Despite these achievements, existing dual-domain methods (Eo et al., 2018; Liu et al., 2022; Zhou and Zhou, 2020; Lyu et al., 2022; Gao and Zhou, 2023) have insufficient investigation of the low-field MRI reconstruction. Unlike existing works, we address the challenges of low-field scenarios and analyze the unique characteristics of k-space and image domains, developing the LHFP learning strategy and HAFNet specifically for these domains.

3. Method

In MRI systems, high-quality T1-weighted (T1WI) and T2-weighted (T2WI) images, along with fully-sampled k-space signals, can be represented as $I_{T1}$, $I_{T2}\in R^{h\times w}$ and $k_{T1}$, $k_{T2}\in R^{h\times w\times 2}$, where $h$ and $w$ denote the image height and width in pixels, and the k-space data comprises two channels (real and imaginary). Due to the system noise and under-sampling, the k-space data is corrupted, denoted as $k_{T1}^{(0)}=k_{T1}+ϵ$, $k_{T2}^{(0)}=\mathbf{M}\odot \left(k_{T2}+ϵ\right)$, where $ϵ$ represents low-field noise and $\mathbf{M}$ is the under-sampling mask. The T1WI data is affected only by low-field noise, whereas the T2WI signal is both noisy and under-sampled for acceleration. In this work, we jointly reconstruct high-quality images $I_{T1}^{\prime }$ and $I_{T2}^{\prime }$ from the corrupted k-space signals $k_{T1}^{(0)}$ and $k_{T2}^{(0)}$ by sequentially performing k-space and image domain learning.

As illustrated in Fig.1, the corrupted k-space data $k_{T1}^{\left(0\right)}$ and $k_{T2}^{\left(0\right)}$ are refined by a Low-to-High Frequency Progressive (LHFP) learning network to generate higher quality $k_{T1}^{\prime }$ and $k_{T2}^{\prime }$. The LHFP network includes a low-frequency learning stage and a high-frequency enhancement stage. To further refine image details and exploit multi-modal complementary features, the refined k-space data $k_{T1}^{\prime }$ and $k_{T2}^{\prime }$ are reconstructed into images $I_{T1}^{(k)}$ and $I_{T2}^{(k)}$, and then jointly denoised by the image space Hybrid Attention Fusion Network (HAFNet). Both the k-space LHFP network and image space HAFNet are supervised by high-quality MRI images $\left(I_{T1},I_{T2}\right)$ and k-space signals $\left(k_{T1},\right.\left.k_{T2}\right)$.

Fig. 1.

Illustration of our proposed multi-contrast MRI reconstruction framework. The low-quality k-space data $k_{T1}^{(0)}$ and $k_{T2}^{(0)}$ are first refined using the (A) K-space Low-to-High Frequency Progressive (LHFP) learning network, resulting in $k_{T1}^{\prime }$ and $k_{T2}^{\prime }$. The refined k-space data is then transformed into images $I_{T1}^{(k)}$ and $I_{T2}^{(k)}$, while the original k-space data is converted into images $I_{T1}^{(in)}$ and $I_{T2}^{(in)}$. These multi-contrast images $I_{T1}^{(k)}$, $I_{T2}^{(k)}$, $I_{T1}^{(in)}$, and $I_{T2}^{(in)}$ are refined by the (B) Image space Hybrid Attention Fusion Network (HAFNet) to produce high-quality images $I_{T1}^{\prime }$ and $I_{T2}^{\prime }$.

3.1. Low-to-High Frequency Progressive Learning

The imbalanced magnitudes and SNR pose significant challenges to k-space learning, particularly in recovering HF components. To address this, we propose a two-stage Low-to-High Frequency Progressive (LHFP) learning network that initially recovers the LF components and then emphasizes the recovery of HF components in the subsequent stage. Details of the LHFP network are illustrated in Fig. 2.

Fig. 2.

Illustration of our proposed Low-to-High Frequency Progressive (LHFP) learning network, which consists of a Low-Frequency (LF) learning stage and a High-Frequency (HF) enhancement stage. In the LF learning stage, equal loss weights are applied to all frequency components. In the HF enhancement stage, the focus shifts to enhancing HF components learning, guided by the estimated low-high frequency boundaries ${\mathbf{M}}_{T1}^{(LF)}$ and ${\mathbf{M}}_{T2}^{(LF)}$. The refined k-space data $k_{T1}^{(3)}$ and $k_{T2}^{(3)}$ are reconstructed into MR images $I_{T1}^{(k)}$ and $I_{T2}^{(k)}$.

In the first stage, the low-quality k-space data $k_{T1}^{(0)}$ and $k_{T2}^{(0)}$ are jointly denoised by the model $Ne{t}^{(LF)}$, resulting in refined k-space data $k_{T1}^{(1)}$ and $k_{T2}^{(1)}$. During this stage, the same loss weight is assigned to all frequency components, and the k-space loss is defined as follows:

$${\mathcal{L}}_{LF}^{(k)}={‖k_{T1}^{(1)}-k_{T1}‖}_1+{‖k_{T2}^{(1)}-k_{T2}‖}_1.$$ (1)

Due to the imbalance in amplitude between the LF and HF components, the network naturally prioritizes learning the LF components.

In the HF enhancement stage, the low-quality k-space data ($k_{T1}^{(0)}$ and $k_{T2}^{(0)}$) and the refined k-space data ($k_{T1}^{(1)}$ and $k_{T2}^{(1)}$) are concatenated as inputs to the model $Ne{t}^{(HF)}$, producing further refined k-space data $k_{T1}^{(2)}$ and $k_{T2}^{(2)}$. To enhance the recovery of HF components, the loss computation reduces the weight of LF components. Specifically, the k-space loss is defined as follows:

$$\begin{array}{l}{\mathcal{L}}_{HF}^{(k)}=\lambda \cdot {\mathbf{M}}_{T1}^{(LF)}\cdot {‖k_{T1}^{(2)}-k_{T1}‖}_1+\left(1-{\mathbf{M}}_{T1}^{(LF)}\right)\cdot {‖k_{T1}^{(2)}-k_{T1}‖}_1\\ +\lambda \cdot {\mathbf{M}}_{T2}^{(LF)}\cdot {‖k_{T2}^{(2)}-k_{T2}‖}_1+\left(1-{\mathbf{M}}_{T2}^{(LF)}\right)\cdot {‖k_{T2}^{(2)}-k_{T2}‖}_1,\end{array}$$ (2)

where $\lambda$ is set as 0.01 to prevent the LF components loss from dominating, ${\mathbf{M}}_{T1}^{(LF)}$ and ${\mathbf{M}}_{T2}^{(LF)}$ are the estimated masks for the LF components in the T1WI and T2WI modalities, respectively. To stabilize the training process and ensure that the “Stage1: LF Learning” network provides a well-refined input for the second stage, the two stages are trained separately.

3.1.1. Low-High Frequency Boundary Prediction

One key challenge in the HF enhancement stage is determining the boundary between low and high frequencies. Since the learning pace varies for each sample, it’s crucial to adaptively optimize the low-frequency masks ${\mathbf{M}}_{T1}^{(LF)}$ and ${\mathbf{M}}_{T2}^{(LF)}$ for each image rather than keeping them constant. Taking the T1WI as an example, the input data $Cat\left(k_{T1}^{(0)},k_{T1}^{(1)}\right)$ is passed through the Mask Predictor network $P^{(T1)}$ to estimate the width ${\alpha }_1$ and heigh-tour mask predictor networks ${\beta }_1$ of the LF mask, where ${\alpha }_1$, ${\beta }_1\in (0,1]$. Specifically, our mask predictor networks $P^{(T1)}$ and $P^{(T2)}$ consist of three convolutional blocks, a global average pooling layer, and a fully connected layer. Finally, a sigmoid activation function is applied to scale the predicted values $\left({\alpha }_1,{\beta }_1\right)$ and $\left({\alpha }_2,{\beta }_2\right)$ to a range between [0,1]. The mask ${\mathbf{M}}_{T1}^{(LF)}$ for T1WI is defined as:

$${\mathbf{M}}_{T1}^{(LF)}[i,j]=\left\{\begin{array}{ll}1,\hfill & \text{if}\frac{1-{\beta }_1}{2}h\le i\le \frac{1+{\beta }_1}{2}h,\frac{1-{\alpha }_1}{2}w\le j\le \frac{1+{\alpha }_1}{2}w,\hfill \\ 0,\hfill & \text{otherwise,}\hfill \end{array}\right.$$ (3)

where $(i,j)$ denotes the spatial indices. In this way, we can define a rectangular mask ${\mathbf{M}}_{T1}^{(LF)}$ around the center of the k-space. Similarly, the mask ${\mathbf{M}}_{T2}^{(LF)}$ for the T2WI can be decided based on the $\left({\alpha }_2,{\beta }_2\right)$:

$${\mathbf{M}}_{T2}^{(LF)}[i,j]=\left\{\begin{array}{ll}1,\hfill & \text{if}\frac{1-{\beta }_2}{2}h\le i\le \frac{1+{\beta }_2}{2}h,\frac{1-{\alpha }_2}{2}w\le j\le \frac{1+{\alpha }_2}{2}w,\hfill \\ 0,\hfill & \text{otherwise}.\hfill \end{array}\right.$$ (4)

While Eqs. (3) and (4) define the low-frequency regions using binary hard masks, we implement scaled sigmoid-based soft masks as differentiable surrogates to facilitate gradient-based optimization. These soft masks closely approximate the structure of the corresponding hard masks, preserving spatial localization while enabling end-to-end training.

3.1.2. LF-HF Fusion

In the second stage, while decreasing the weight of LF components can improve HF learning, it might compromise LF reconstruction accuracy, thus hindering overall performance optimization. To optimize the reconstruction of both LF and HF components, we introduce a LF-HF fusion strategy to dynamically combine the k-space data predicted by the two stages.

As depicted in Fig. 2, we combine the outputs $k_{T1}^{(1)}$ $\left(k_{T2}^{(1)}\right)$ from Stage 1 and the outputs $k_{T1}^{(2)}$ $\left(k_{T2}^{(2)}\right)$ from Stage 2 based on the predicted masks ${\mathbf{M}}_{T1}^{(LF)}$ $\left({\mathbf{M}}_{T2}^{(LF)}\right)$ and the compensation weights $w_1$ $\left(w_2\right)$. The resulting k-space data $k_{T1}^{(3)}$ $\left(k_{T2}^{(3)}\right)$ are defined as:

$$\begin{array}{l}{k}_{T1}^{(3)}=w_1\cdot {\mathbf{M}}_{T1}^{(LF)}\odot k_{T1}^{(1)}+\left(1-w_1\cdot {\mathbf{M}}_{T1}^{(LF)}\right)\odot k_{T1}^{(2)},\\ k_{T2}^{(3)}=w_2\cdot {\mathbf{M}}_{T2}^{(LF)}\odot k_{T2}^{(1)}+\left(1-w_2\cdot {\mathbf{M}}_{T2}^{(LF)}\right)\odot k_{T2}^{(2)}.\end{array}$$ (5)

For the HF components, only the outputs $k_{T1}^{(2)}$ $\left(k_{T2}^{(2)}\right)$ from Stage 2 are considered. Meanwhile, for the LF components, the outputs from both stages are integrated according to the weights $w_1$ $\left(w_2\right)$. This integration ensures that the outputs from Stage 1 can compensate for the insufficient learning of LF components in Stage 2.

The combined k-space outputs $k_{T1}^{(3)}$ and $k_{T2}^{(3)}$ are converted to real-valued images $I_{T1}^{(k)}$ and $I_{T2}^{(k)}$ by applying the inverse 2D Fourier transform followed by a magnitude operation. For multi-coil data, coil-wise images are further combined using the root-sum-of-squares method. The resulting images are supervised using L1 loss in the image domain:

$${\mathcal{L}}^{(img)}={‖I_{T1}^{(k)}-I_{T1}‖}_1+{‖I_{T2}^{(k)}-I_{T2}‖}_1.$$ (6)

The k-space loss in Eq. (2) and the image domain loss in Eq. (6) are optimized concurrently. The parameters $\left({\alpha }_1,{\beta }_1\right)$ and $\left({\alpha }_2,\right.\left.{\beta }_2\right)$ for the LF masks, as well as the compensation weights $w_1$ and $w_2$, are predicted by the mask predictor networks $P^{(T1)}$ and $P^{(T2)}$, and are progressively refined during training to optimize the reconstruction results.

3.2. Hybrid Attention Fusion Network

To balance computational efficiency with the capability of capturing long-range dependencies across multi-contrast images, we introduce a Hybrid Attention Fusion Network (HAFNet) for image domain learning. As shown in Fig. 3 (a), our HAFNet follows a U-Net-based architecture, consisting of four convolutional blocks with down-sampling layers to extract multi-scale features from each modality. To effectively integrate multi-contrast features, we incorporate Hybrid Window-based Attention Fusion (HWAF) modules at both shallow and deep levels, applying them in the first two and last two stages, respectively.

Fig. 3.

Illustration of (a) the image-space HAFNet, which integrates multi-contrast features using (b) the Shallow-level HWAF in the first two stages and (c) the Deep-level HWAF modules in the last two stages, respectively.

3.2.1. Dense-Dilated Window Partition

To capture long-range dependencies while avoiding quadratic complexity caused by self-attention across the entire feature map, we alternately compute self-attention within dense and dilated windows. As shown in Fig. 3 (b, c), the feature $f_{T1}$ or $f_{T2}\in R^{H\times W\times C}$ is partitioned into $\frac{HW}{M^2}$ non-overlapping dense windows (Liu et al., 2021), with each window containing $N=M^2$ tokens. The window-wise features for the two modalities can be represented as:

$$\begin{array}{l}{\mathit{Z}}_{\mathit{T}\mathbf{1}}=\left[z_{T1}^1,z_{T1}^2,\dots ,z_{T1}^N\right]\in R^{N\times C},\\ {\mathit{Z}}_{\mathit{T}\mathbf{2}}=\left[z_{T2}^1,z_{T2}^2,\dots ,z_{T2}^N\right]\in R^{N\times C},\end{array}$$ (7)

where $N$ is the number of tokens and $C$ is the feature dimension of each token. Following the operations in (Liu et al., 2021; Liang et al., 2021), we alternatively use regular and shifted window partitioning to enable cross-window connections, where shifted window partitioning means shifting the feature by $\left(⌊\frac{M}{2}⌋,⌊\frac{M}{2}⌋\right)$ pixels before partitioning. By computing self-attention among the N tokens within each window, correlated features can be integrated to enhance feature representations.

Although computing self-attention within dense windows reduces computational complexity, it fails to capture long-range dependencies outside these windows. Inspired by dilated convolution (Yu and Koltun, 2015), we expand the windows to larger regions, enabling interactions between distant but correlated tokens. As depicted in Fig. 3 (b, c), the feature map $f_{T1}^{\prime }$ or $f_{T2}^{\prime }$ is split into $P^2$ dilated windows, where each window contains $N=\frac{HW}{P^2}$ tokens, and $P$ is the interval size between neighboring tokens. Therefore, window-wise features ${\mathit{Z}}_{\mathit{T}\mathbf{1}}$, ${\mathit{Z}}_{\mathit{T}\mathbf{2}}\in R^{N\times C}$ can be obtained. By alternately computing self-attention within dense and dilated windows for $L$ times, both local and global correlations can be captured without heavy computational cost.

3.2.2. Hybrid Window-based Attention Fusion (HWAF)

To comprehensively integrate complementary information from the multi-contrast features, we design two Hybrid Window-based Attention Fusion (HWAF) strategies for shallow and deep levels. The main difference between these strategies lies in how the tokens ${\mathit{Z}}_{\mathit{T}\mathbf{1}}$, ${\mathit{Z}}_{\mathit{T}\mathbf{2}}\in R^{N\times C}$ from the corresponding windows of the two modalities are fused.

A straightforward method to integrate these multi-contrast tokens is to concatenate them and then compute the correlation among the $2N$ tokens. However, in shallow layers (i.e., the first two encoder stages), the features are susceptible to noise present in the input images, which can lead to inaccurate correlation computations. Additionally, shallow-level features have a large spatial resolution, resulting in a large number of windows. Concatenating along the token dimension significantly increases the computational load for each window, thereby heavily increasing the overall computational complexity. To enhance the accuracy of correlation computation without introducing heavy computation, we combine the shallow-level features in corresponding windows using channel-wise concatenation. Specifically, for each dense or dilated window, the multi-contrast feature can be represented as:

$$\begin{array}{l}\mathit{Z}=\left[z^1,z^2,\dots ,z^N\right]\in R^{N\times 2C},\\ z^i=Concat\left(z_{T1}^i,z_{T2}^i\right)\in R^{2C},i=1,2,\dots ,N,\end{array}$$ (8)

where the number of tokens remains $N$, while the feature dimension of each token increases to $2C$. Then, we compute the correlations among the $N$ tokens in $Z$ using multi-head self-attention (MSA) (Vaswani et al., 2017) and integrate correlated tokens. Inspired by the Swin Transformer (Liu et al., 2021), we incorporate a relative position bias to encode token position information in the self-attention mechanism.

Due to the down-sampling operation, the deep-level features have a smaller spatial resolution but a higher number of channels. Compared to shallow-level features, deep features contain richer contextual information and are partitioned into fewer windows. Therefore, we concatenate the tokens from corresponding windows of the two modalities along the token dimension instead of the channel dimension. Specifically, for each dense or dilated window, the fused multi-contrast feature is defined as:

$$\mathit{Z}=\left[z_{T1}^1,z_{T1}^2,\dots ,z_{T1}^N,z_{T2}^1,z_{T2}^2,\dots ,z_{T2}^N\right]\in R^{2N\times C},$$ (9)

where the number of tokens increases to $2N$ while the dimension of each token remains unchanged as $C$. By computing MSA between the $2N$ tokens in $Z$, correlations between tokens from different modalities are explicitly captured. Consequently, the feature representations of the target modality are enhanced by integrating complementary features from the auxiliary modality.

The shallow-level and deep-level HWAF modules enhance feature representations by integrating long-range correlated information from the multi-contrast images. These enhanced features are then passed to the decoder to reconstruct high-quality images for both modalities. These enhanced features are then passed to the decoder, which consists of four convolutional blocks with up-sampling layers in between to progressively restore feature resolution. This U-shaped network architecture is designed to reconstruct high-quality MRI images for each modality.

4. Experiments

4.1. Dataset and Implementation

4.1.1. Dataset Description

We evaluate our method on two open-access MRI datasets:

1. BraTs Dataset (Menze et al., 2014): The BraTs dataset includes multi-contrast brain MRI scans, specifically T1, T1ce, T2, and PD weighted images. All MRI scans are resampled to an isotropic voxel size of 1 × 1 × 1 mm³, skull-stripped, and co-registered to a common anatomical template. The dataset is split into training and testing sets with a ratio of 3:1, including 75 subjects (3,621 slices) for training and 25 subjects (1,088 slices) for validation. We investigate two tasks: a) T1WI-guided 4× acceleration of T2WI images, where the T1WI data is fully 4× acceleration of T2WI images, where the T1WI data is fully sampled and a 4× under-sampling is applied to the k-space data of the T2WI modality; b) T1WI-guided 4× acceleration of T2WI images in a low-field scenario, where zero-mean complex Gaussian noise is added to the k-space data of both modalities to simulate degraded SNR for a low-field system. The noise level is determined based on a target SNR of 10dB. The T2WI k-space data is further corrupted by 4× under-sampling. For the BraTs dataset, single coil k-space data is simulated from the image samples.

2. M4Raw Dataset (Lyu et al., 2023b): The M4Raw dataset contains multi-contrast, multi-repetition MRI k-space data collected using a low-field (0.3T) MRI system. Imaging parameters include an in-plane (axial) resolution of approximately 0.94 × 1.23 mm² and a slice thickness of 5 mm. Following the official settings, 128 subjects (6,912 slices) are selected for training and 30 subjects (1,620 slices) are reserved for validation. We investigate T1WI-guided 4× and 8× acceleration of T2WI MRI. Both T1WI and T2WI MRI are corrupted by low-field noise, necessitating joint denoising of both modalities and acceleration of the T2WI modality. The multi-repetition averaged MRI serves as the high-quality ground truth. Since multi-coil k-space data is available for the M4Raw dataset, our k-space network processes the original multi-coil k-space data by concatenating all coils along the channel dimension. In contrast, the image-domain network operates on coil-combined images, which are obtained by applying an inverse Fourier transform and combining the coil signals using the root sum of squares method, following the same data processing procedure as the provider of the dataset.

In our experiments, we apply 1D under-sampling in the k-space domain. For the 4× acceleration, the sampling pattern is generated using a random Gaussian distribution, with the central 10% of k-space data fully sampled to preserve crucial structural information. For the 8× acceleration, we use a regular under-sampling pattern, where the central 4% of k-space data is fully sampled. To stabilize model training, we normalize the MRI images by subtracting the mean and dividing by the standard deviation, ensuring that the data has a mean of 0 and a standard deviation of 1. The images are then centrally cropped to a size of 240 × 240 pixels, ensuring a consistent input size across all training and evaluation datasets. Additionally, both datasets used in our experiments are spatially aligned across multiple contrasts to ensure consistency among the different imaging modalities.

4.1.2. Implementation Details

Our method and all comparison methods are implemented using PyTorch (Paszke et al., 2019) on an NVIDIA RTX 2080ti GPU. For the k-space LHFP learning network illustrated in Fig. 2, both $Ne{t}^{(LF)}$ and $Ne{t}^{(HF)}$ are implemented as UNet models (Ronneberger et al., 2015) with 4 encoder and decoder blocks. Specifically, each encoder/decoder block consists of two convolutional layers with a 3×3 kernel, followed by instance normalization, dropout layers, and LeakyReLU activation functions. Down-sampling operations between the four encoder blocks are implemented using 2sively reduce the spatial resolution, while up-sampling oper-× 2 max pooling with stride 2 to progresations between the four decoder blocks are performed using 2 × 2 transposed convolutions with stride 2 to gradually restore the resolution. The filter counts for each of the four encoder and decoder layers are 32, 64, 128, and 256, respectively. The mask predictors $P^{(T1)}$ and $P^{(T2)}$ are both implemented as 3 convolutional layers. Both Stage 1 and Stage 2 are trained for 100 epochs, optimized using the summation of the k-space loss and image domain loss. In HAFNet, features are progressively down-sampled across four scales: (240,240), (120,120), (60,60), (30,30), with a HWAF module applied at each scale to fuse multi-contrast features. The filter counts for the features at each of these four scales are 32, 64, 128, and 256, respectively. The window size M is set to 10, and the token interval P is 24, 12, 6, and 3 for the four scales, respectively. The k-space LHFP learning network and the image domain HAFNet are trained with an initial learning rate of 0.0002, which is halved every 20 epochs. The entire network is trained using the AdamW optimizer with ${\beta }_1=0.5$ and ${\beta }_1=0.999$, with a batch size of 4. During training, 20% of the training data is used as a validation set, and the model with the best validation performance is selected for final evaluation.

Model performance is measured using the Peak Signal-to-Noise Ratio (PSNR) and the Structural Similarity Index Measure (SSIM) on the validation sets. Higher PSNR and SSIM values indicate higher better reconstruction performance. In low-field MRI settings, direct use of data consistency constraints is suboptimal due to the low SNR of the acquired signals. To avoid introducing complicated regularization, we do not apply data consistency layers in our method across all experiments.

4.2. Comparison with State-of-the-art Methods

4.2.1. Quantitative Results

We compare our method against a k-space learning-based MRI reconstruction method K-DCNN (Han et al., 2019), five image domain MRI reconstruction methods (VarNet (Sriram et al., 2020), MINet (Feng et al., 2021), MTrans (Feng et al., 2022), MCCA (Li et al., 2023), DCAMSR (Huang et al., 2023a)), and three dual-domain MRI reconstruction methods (KV-Net (Liu et al., 2022), DudoRNet (Zhou and Zhou, 2020), DudoCAF (Lyu et al., 2022)). Table 1 presents the PSNR and SSIM metrics of our method and state-of-the-art methods on the BraTs (Menze et al., 2014) and M4Raw (Lyu et al., 2023b) datasets, demonstrating that our method consistently achieves the best reconstruction performance across various settings. For the T1WI-guided 4× acceleration of T2WI images on the BraTs dataset, our method achieves a PSNR of 40.25 dB, outperforming the state-of-the-art DudoRNet by 1.34 dB. When both T1WI and T2WI MRI are corrupted by low-field noise, our method yields PSNRs of 39.21 dB and 36.55 dB for these two modalities, surpassing existing methods. On the M4Raw dataset, our method achieves 36.88 dB, 31.87 dB, and 30.40 dB for the low-field T1WI denoising, 4× and 8× acceleration of the T2WI MRI, consistently surpassing existing single domain and dual-domain reconstruction methods.

Table 1.

For the BraTs and M4Raw datasets, comparison of our method with state-of-the-art methods, including Zero Filling, K-DCNN (Han et al., 2019), VarNet (Sriram et al., 2020), MINet (Feng et al., 2021), MTrans (Feng et al., 2022), MCCA (Li et al., 2023), DCAMSR (Huang et al., 2023a), KV-Net (Liu et al., 2022), DudoRNet (Zhou and Zhou, 2020), and DudoCAF (Lyu et al., 2022). mean ± standard deviation are reported for each method, and the best metrics are highlighted in bold font.

Task		4× Acceleration		4× Acceleration (Low-field)
BraTs		T2WI (4×)		T1WI (Low-field)		T2WI (4×, Low-field)
Domain	Method	PSNR ↑	SSIM ↑	PSNR ↑	SSIM ↑	PSNR ↑	SSIM ↑
Single domain	Zero Filling	30.04±1.51	0.748±0.034	22.27±2.08	0.226±0.020	25.39±1.15	0.217±0.017
	K-DCNN	34.72±1.56	0.929±0.013	34.30±2.14	0.925±0.015	32.60±1.43	0.904±0.013
	VarNet	35.43±1.59	0.958±0.019	37.20±1.99	0.967±0.007	32.49±1.47	0.931±0.011
	MINet	38.26±1.74	0.976±0.006	37.32±2.04	0.967±0.007	35.25±1.56	0.960±0.010
	MTrans	37.83±1.63	0.973±0.009	37.10±2.01	0.966±0.007	34.75±1.42	0.959±0.012
	MCCA	37.95±1.68	0.975±0.006	37.21±2.02	0.967±0.007	34.78±1.48	0.958±0.010
	DCAMSR	38.60±1.75	0.978±0.006	37.60±2.03	0.968±0.007	35.36±1.56	0.961±0.009
Dual domain	KV-Net	36.78±1.64	0.967±0.007	38.09±2.00	0.970±0.006	34.23±1.53	0.947±0.010
	DudoRNet	38.91±1.68	0.976±0.005	38.40±2.01	0.972±0.006	35.75±1.56	0.959±0.010
	DudoCAF	38.76±1.65	0.975±0.007	38.15±1.92	0.971±0.006	35.26±1.45	0.956±0.010
	Ours	40.25±1.71	0.983±0.005	39.21±1.93	0.977±0.005	36.55±1.54	0.968±0.008
Task		4× Acceleration (Low-field)				8× Acceleration (Low-field)
M4Raw		T1WI (Low-field)		T2WI (4×, Low-field)		T2WI (8×, Low-field)
Domain	Method	PSNR ↑	SSIM ↑	PSNR ↑	SSIM ↑	PSNR ↑	SSIM ↑
Single domain	Zero Filling	33.44±1.27	0.857±0.027	24.06±1.40	0.636±0.022	18.94±1.69	0.522±0.026
	K-DCNN	34.06±2.51	0.918±0.017	28.70±1.22	0.792±0.027	27.26±1.32	0.759±0.036
	VarNet	36.04±1.40	0.925±0.016	29.61±1.25	0.808±0.025	28.11±1.15	0.773±0.030
	MINet	36.48±1.36	0.928±0.016	30.86±1.33	0.828±0.024	29.37±1.47	0.802±0.033
	MTrans	36.24±1.53	0.928±0.017	30.16±1.23	0.815±0.025	29.38±1.49	0.805±0.032
	MCCA	35.82±1.48	0.926±0.016	30.67±1.31	0.827±0.024	29.41±1.47	0.804±0.031
	DCAMSR	36.41±1.33	0.928±0.016	31.03±1.32	0.833±0.023	29.56±1.38	0.806±0.031
Dual domain	KV-Net	35.69±1.51	0.925±0.016	30.24±1.27	0.820±0.025	29.36±1.43	0.801±0.031
	DudoRNet	36.23±1.33	0.926±0.016	30.36±1.28	0.823±0.025	29.43±1.39	0.802±0.031
	DudoCAF	36.34±1.32	0.928±0.017	30.68±1.30	0.830±0.025	29.40±1.45	0.803±0.032
	Ours	36.88±1.34	0.930±0.016	31.87±1.28	0.839±0.023	30.40±1.44	0.816±0.029

Moreover, we adopt the paired t-test (Kim, 2015) to evaluate the significant differences between our proposed method and the other methods. The p-values for all evaluation metrics are smaller than 0.05, indicating that our method demonstrates a statistically significant improvement over the other methods.

4.2.2. Qualitative Results

Fig. 4 demonstrates the reconstructed images from two examples: the 4× MRI acceleration task on the BraTs dataset (Menze et al., 2014) and the 8× low-field MRI acceleration task on the M4Raw dataset (Lyu et al., 2023b). For an intuitive comparison, error maps of different methods are also provided to highlight the differences between the ground truth and reconstructed images. As shown, the under-sampled images suffer from significant aliasing artifacts and loss of anatomical details. Compared to other methods (Han et al., 2019; Feng et al., 2021, 2022; Li et al., 2023; Huang et al., 2023a; Liu et al., 2022; Zhou and Zhou, 2020), our method preserves more detailed structures and consistently delivers superior reconstruction quality. The superior performance of our method is mainly due to its ability to recover HF components in the k-space network and capture long-range dependencies in the image domain HAFNet.

Fig. 4.

Visualization of the images reconstructed with our method and state-of-the-art methods, including K-DCNN (Han et al., 2019), MINet (Feng et al., 2021), MTrans (Feng et al., 2022), MCCA (Li et al., 2023), DCAMSR (Huang et al., 2023a), KV-Net (Liu et al., 2022), and DudoRNet (Zhou and Zhou, 2020). The first two rows show the 4× acceleration of a T2WI image from the BraTs dataset. The last two rows show the 8× acceleration of a T2WI image from the low-field M4Raw dataset.

4.3. Ablation Studies

To evaluate the contribution of each component in the proposed method, we conduct several ablation experiments on the reconstruction of the 4× accelerated T2WI, and joint reconstruction of the low-field T1WI and the 4× accelerated low-field T2WI on the BraTs dataset (Menze et al., 2014).

4.3.1. Effectiveness of the dual-domain network

As our proposed method consists of a k-space network and an image domain network, we investigate the contribution of these two domains separately. As summarized in Table 2, for the 4× acceleration of T2WI, and the joint reconstruction of the low-field T1WI and 4× under-sampled low-field T2WI, the k-space LHFP learning network achieves PSNRs of 35.45 dB, 36.68 dB, and 33.37 dB, respectively. The image domain HAFNet, using zero-filled images as inputs, achieves PSNRs of 39.95 dB, 37.52 dB, and 35.57 dB for the three tasks respectively. The image domain HAFNet demonstrates a more substantial performance improvement compared to the k-space LHFP learning network, particularly in high-field MRI scenario. This aligns with prior studies demonstrating that image domain networks excel at capturing local textures and anatomical structures, making them particularly effective for tasks like denoising and artifact correction (Zbontar et al., 2018; Pal and Rathi, 2022). However, in low-field MRI reconstruction, k-space learning also plays an essential role in improving reconstruction accuracy. The inherently lower SNR in low-field settings makes k-space refinement beneficial before transitioning to the image domain, ensuring better signal fidelity. These results highlight the complementary strengths of both domains: the image domain network is particularly effective in mitigating under-sampling artifacts, while denoising benefits from the joint contributions of both k-space and image domain learning. As shown in the last row of Table 2, our dual-domain method consistently delivers the best performance across all settings. This superior performance stems from the synergy between k-space and image domain processing, where k-space learning enhances signal quality, providing better inputs for the subsequent image domain network. These findings underscore the effectiveness of our dual-domain learning framework.

Table 2.

For the BraTs dataset, ablation of the k-space LHFP network and image domain HAFNet.

Domain	T2WI (4×)		T1WI (Low-field)		T2WI (4×, Low-field)
	PSNR ↑	SSIM ↑	PSNR ↑	SSIM ↑	PSNR ↑	SSIM ↑
Raw input	30.04±1.51	0.748±0.034	22.27±2.08	0.226±0.020	25.39±1.15	0.217±0.017
K-space LHFP only	35.45±1.60	0.957±0.008	36.68±1.87	0.956±0.009	33.37±1.45	0.937±0.026
Image domain HAFNet only	39.95±1.72	0.982±0.005	37.52±2.03	0.968±0.007	35.57±1.53	0.962±0.009
Dual domain (Ours)	40.25±1.71	0.983±0.005	39.21±1.93	0.977±0.005	36.55±1.54	0.968±0.008

4.3.2. Effectiveness of the k-space LHFP learning network

As the proposed k-space LHFP learning network includes the first stage for LF learning and the second stage for HF enhancement (HFE), we conduct ablation experiments to investigate the effect of these two stages. As shown in the 2^nd row of Table 3, removing the Stage 2 from our k-space network results in PSNRs of 34.72 dB, 34.30 dB, and 32.60 dB for the three tasks, respectively. This demonstrates that the first stage alone, which focuses on low-frequency recovery, is not sufficient to achieve optimal performance. The absence of the HFE stage results in suboptimal recovery of high-frequency components, thereby limiting the overall quality of the reconstruction. Additionally, we examine the effect of removing the HFE from the Stage 2, which equivalents to stacking the Stage 1 twice. As shown in the 3^rd row of Table 3, stacking two identical stages leads to performance gains over using just a single stage. However, compared to our method, the absence of HFE results in consistent performance drops. These findings demonstrate that the superiority of our k-space network arises not only from stacking two stages but also from the enhanced learning of high-frequency components.

Table 3.

For the BraTs dataset, ablation of the key components in the k-space progressive learning network.

K-space network	T2WI (4×)		T1WI (Low-field)		T2WI (4×, Low-field)
	PSNR ↑	SSIM ↑	PSNR ↑	SSIM ↑	PSNR ↑	SSIM ↑
Raw input	30.04±1.51	0.748±0.034	22.27±2.08	0.226±0.020	25.39±1.15	0.217±0.017
w/o Stage 2	34.72±1.56	0.929±0.013	34.30±1.82	0.925±0.017	32.60±1.41	0.904±0.015
w/o HFE	34.48±1.60	0.945±0.008	36.13±1.98	0.951±0.010	32.78±1.59	0.925±0.010
Ours	35.45±1.60	0.957±0.008	36.68±1.87	0.956±0.009	33.37±1.45	0.937±0.026

In Fig. 5, we visualize the reconstructed images and error maps obtained from different variants of the k-space network. Compared with our method (shown in the last column), the reconstructed images in the 2^nd and 3^rd columns exhibit higher errors, particularly in the detailed structural information. These results suggest that our method enables more accurate recovery of high-frequency information through the HFE in Stage 2.

Fig. 5.

For an example of the joint reconstruction of the low-field T1WI MRI and 4× accelerated low-field T2WI MRI in the BraTs dataset, visualization of the reconstructed images and error maps obtained from different versions of the k-space network.

4.3.3. Analysis of the low-frequency masks in the k-space LHFP learning network

To better understand the k-space LHFP learning process, we analyze the low-frequency masks predicted in Stage 2, as defined in Eq. (3) and Eq. (4). Fig. 6 illustrates the input k-space data in the first row and the predicted low-frequency masks in the second row. In the high-field scenario, Fig. 6 (a-b) corresponds to the joint reconstruction of fully sampled T1WI and under-sampled T2WI MRI. Since T1WI is fully sampled, a significant portion of its low-frequency signal is already well recovered in Stage 1. As a result, the predicted low-frequency mask for T1WI covers a relatively large region. In contrast, T2WI is under-sampled, the missing frequency components are not well recovered in Stage 1. Consequently, the predicted low-frequency mask for T2WI appears as a narrow strip, emphasizing substantial regions that require further refinement in Stage 2. In the low-field scenario, Fig. 6 (c-d) represents the joint reconstruction of fully-sampled T1WI and under-sampled T2WI MRI. Due to the presence of low-field noise, the low-frequency components of both contrasts are not well recovered in Stage 1. As a result, the predicted low-frequency masks are much smaller compared to the high-field setting, indicating an emphasis on the poorly recovered regions. These visualizations support an important conclusion: our LHFP learning strategy effectively identifies the high-frequency components that were not well recovered in Stage 1 and emphasizes them in Stage 2. This adaptive process ensures a more comprehensive recovery of all frequency components, ultimately improving the overall reconstruction quality.

Fig. 6.

Visualization of k-space data and predicted low-frequency masks for: (a) high-quality T1WI MRI, (b) under-sampled T2WI MRI, (c) low-field T1WI MRI, and (d) low-field under-sampled T2WI MRI.

4.3.4. Effectiveness of the multi-modal fusion methods in the image space HAFNet

The image space HAFNet integrates multi-contrast features with shallow-level and deep-level HWAF modules in their respective levels. To verify the effectiveness of these HWAF modules, we conduct ablation experiments under two settings: 1) Single-domain setting: We remove the k-space network and use raw images converted from the corrupted k-space data as model inputs; 2) Dual-domain setting: We concatenate the corrupted images and the images reconstructed by the k-space network as model inputs.

As shown in Table 4, for the single-domain reconstruction of 4× accelerated T2WI, our HAFNet significantly improves image quality from 30.04 dB to 39.95 dB in PSNR. When we replace the HWAF modules with concatenation-based fusion, the PSNR drops to 38.12 dB, indicating that our method is more effective in integrating cross-modal complementary information than simple feature concatenation. To investigate the contribution of dilated windows in the HWAF module, we replace them with dense windows. As shown in the 3^rd row of Table 4, this replacement results in a PSNR degradation to 38.34 dB. This finding suggests that attention computation in dilated windows is effective for capturing long-range dependencies and improving reconstruction performance. Furthermore, in the last two layers of the HAFNet, we replace the deep-level HWAF module with shallow-level HWAF, resulting in a PSNR decrease to 39.35 dB. This is because the deep-level HWAF module can explicitly integrate complementary information from the auxiliary modality and enhance the reconstruction quality of the T2WI modality. For the joint reconstruction of T1WI (low-field) and T2WI (4×, low-field), similar comparison results are observed in the last 4 columns of Table 4, indicating the consistent superiority of our proposed multi-modal fusion method. As shown in the example of Fig. 7, compared with other methods in the 2–4 columns, our method (in the last column) reconstructs the T2WI MRI with higher quality and produces a smaller error map.

Table 4.

For the BraTs dataset, comparison of different multi-modal fusion methods in the image domain.

Domain	Fusion method	T2WI (4×)		T1WI (Low-field)		T2WI (4×, Low-field)
		PSNR ↑	SSIM ↑	PSNR ↑	SSIM ↑	PSNR ↑	SSIM ↑
Single Domain	Raw input	30.04±1.51	0.748±0.034	22.27±2.08	0.226±0.020	25.39±1.15	0.217±0.017
	Concatenation	38.12±1.66	0.976±0.006	37.03±2.02	0.961±0.004	34.92±1.48	0.959±0.010
	w/o dilated window	38.34±1.67	0.977±0.006	37.40±2.03	0.967±0.007	35.23±1.53	0.960±0.010
	w/o deep-level HWAF	39.35±1.70	0.981±0.007	37.46±2.02	0.968±0.007	35.32±1.52	0.961±0.009
	Ours	39.95±1.72	0.982±0.005	37.52±2.03	0.968±0.007	35.57±1.53	0.962±0.009
Dual Domain	K-space refined input	35.45±1.60	0.957±0.008	36.68±1.87	0.956±0.009	33.37±1.45	0.937±0.026
	Concatenation	38.26±1.69	0.976±0.006	38.85±1.93	0.975±0.005	35.86±1.52	0.964±0.008
	w/o dilated window	38.56±1.67	0.978±0.006	39.09±1.92	0.977±0.005	36.04±1.54	0.965±0.008
	w/o deep-level HWAF	39.56±1.70	0.981±0.007	39.10±1.92	0.977±0.005	36.13±1.53	0.966±0.014
	Ours	40.25±1.71	0.983±0.005	39.21±1.93	0.977±0.005	36.55±1.54	0.968±0.008

Fig. 7.

For a 4× accelerated T2WI MRI in the BraTs dataset, visualization of the reconstructed images and error maps generated by different multi-modal fusion methods in the image domain network.

In dual-domain reconstruction, since the images refined by the k-space network provide a better prior for the subsequent image domain HAFNet, the performance of all tasks and fusion method variants surpasses that of their single-domain counterparts. This superiority is particularly pronounced in the reconstruction of low-field MRI, further validating the contribution of the k-space network. Moreover, in the dual-domain setting, our method consistently outperforms other variants (i.e., concatenation, w/o dilated window, and w/o deep-level HWAF), affirming the effectiveness of the proposed HAFNet.

4.3.5. Analysis of attention mechanism in the HWAF modules

To elucidate the reasons behind the improvements brought by the HWAF modules, we visualize the window-wise attention metrices in Fig. 8. For anchor points denoted by red or green crosses, the attention distributions assess the correlation between the anchor point and other features within the dense or dilated windows. As depicted in Fig. 8 (a), similar structural information within the dense windows is aggregated to anchor points with higher weights. Similarly, for the anchor point situated at the boundary of white matter and gray matter in Fig. 8 (b), similar boundary information within the dilated window is aggregated with higher weights. These findings suggest that the HWAF module can integrate correlated features from both local and global regions, thereby enhancing feature representation and improving reconstruction quality.

Fig. 8.

Visualization of attention matrices in the (a) dense windows and (b) dilated window of the shallow-level HWAF module, and in the (c) dense windows and (d) dilated windows in the deep-level HWAF module. The anchor points in the shallow and deep HWAF modules are indicated by red and green crosses, respectively.

Fig. 8 (c-d) illustrates the attention distribution within the dense window and dilated window in the deep-level HWAF module. For anchor points on the T2WI image, similar structural information from both T1WI and T2WI images receives higher attention weights. Consequently, the deep-level HWAF module enhances the feature representation of the T2WI modality by incorporating auxiliary information from the T1WI modality.

5. Discussion

In this study, we address the challenging task of reconstructing high-quality multi-contrast MR images from low-field MRI data, which are severely degraded by both under-sampling and significant system noise. Low-field MRI has gained increasing adoption in resource-limited environments, pediatric imaging, and portable applications due to its lower cost, improved safety, and greater accessibility (Lyu et al., 2023b). However, these benefits come at the expense of reduced SNR, which significantly complicates the reconstruction of multi-contrast images. To overcome these challenges, we propose a novel dual-domain framework that integrates a k-space LHFP learning network with an image-domain HAFNet. The k-space LHFP network sequentially reconstructs the low- and high-frequency components in two stages, effectively mitigating the inherent frequency magnitude imbalance issue in k-space. The image-domain HAFNet further refines the reconstruction by fusing multi-contrast features using specially designed hybrid window-based attention modules. This dual-domain approach improves the overall reconstruction quality by addressing both the frequency imbalance in k-space and the structural complexity across contrasts, making it particularly effective for multi-contrast reconstruction in low-field MRI settings.

We validated the effectiveness of our method on two representative datasets, BraTs (Menze et al., 2014) and M4Raw (Lyu et al., 2023b), where our dual-domain framework consistently outperforms state-of-the-art approaches in terms of both PSNR and SSIM. Since both datasets provide spatially aligned multi-contrast MRI scans, no additional alignment pre-processing was required during training or evaluation. Our framework can be effectively applied to other multi-contrast MRI reconstruction tasks in both high-field and low-field settings. Moreover, the ablation results in Table 2 show that the image-domain HAFNet consistently outperforms the k-space LHFP network when used alone, highlighting its strong capacity for structural feature integration. Nonetheless, the combination of both domains yields the best performance. This is especially evident in low-field MRI reconstruction, where the k-space network contributes more significantly due to its ability to directly suppress noise and recover frequency components under challenging acquisition conditions. These findings underscore the complementary strengths of k-space and image-domain learning, emphasizing the importance of dual-domain integration for robust and high-quality multi-contrast MRI reconstruction, particularly in low-field scenarios.

To assess the potential of clinical application, we evaluate the computational efficiency of our proposed method in terms of the number of trainable parameters, total training time, and inference speed. All metrics are measured on the BraTs dataset under a 4× acceleration setting to ensure a consistent and fair comparison across different methods. As shown in Table 5, Transformer-based methods such as MTrans, DCAMSR, and DudoCAF typically involve a significantly larger number of trainable parameters compared to CNN-based approaches, mainly due to the inclusion of multi-head self-attention modules along with their associated projection and feed-forward layers. While our method also incorporates Transformer-based operations in the image-domain HAFNet, it achieves improved computational efficiency compared to these models, while maintaining a moderate number of trainable parameters. Regarding training time, dual-domain methods typically require longer training due to the joint optimization in both k-space and image domains. Nevertheless, our method demonstrates improved training efficiency (20.0 hours) compared to other dual-domain counterparts such as DudoRNet (28.4 hours) and DudoCAF (42.6 hours). Furthermore, our inference time (79.66 ms per slice) is faster than that of VarNet and DudoCAF and remains well within the threshold required for clinical use. Overall, our method achieves state-of-the-art reconstruction accuracy with a reasonable computational cost, making it well-suited for clinical deployment.

Table 5.

Computation efficiency of different methods.

Method	Trainable param. (M)	Training time (h)	Inference time (ms)
VarNet	28.55	12.0	91.40
MINet	3.75	16.5	10.30
MTrans	187.0	16.4	7.13
MCCA	15.21	5.8	19.37
DCAMSR	43.32	17.7	67.56
KV-Net	10.54	13.7	17.76
DudoRNet	11.17	28.4	46.34
DudoCAF	86.47	42.6	96.32
Ours	40.40	20.0	79.66
K-space LHFP	19.50	3.8	21.59
Image HAFNet	20.90	16.2	58.07

Despite the strong performance of our proposed method, several limitations remain to be addressed in future work. First, our current framework does not explicitly focus on lesion-centered reconstruction. While PSNR and SSIM are valuable for assessing global image quality, they may not fully reflect the accuracy of lesion recovery, which is critical for clinical diagnosis and treatment planning. In future work, we aim to incorporate lesion-guided reconstruction strategies and adopt task-specific evaluation metrics to better assess and enhance lesion-level fidelity, along with exploring the model’s generalizability across different datasets (Wang et al., 2024). Second, we did not include data consistency layers in our network design due to the inherently low SNR of the acquired k-space signals in low-field MRI systems. As a future direction, we will investigate incorporating data consistency layers with additional regularizers to improve reconstruction fidelity (Zheng et al., 2019). Third, our current dual-domain framework operates in a sequential manner, with limited interaction between the k-space and image-space networks. Future extensions will explore iterative and interleaved dual-domain learning to enable tighter coupling and more effective integration between the two domains (Zhou and Zhou, 2020). Fourth, our current framework requires paired multi-contrast MRI images as inputs during both training and inference, limiting its applicability in clinical scenarios where one of the modalities is unavailable. To address this limitation, future work will explore developing a reconstruction framework that can generalize to missing modality scenarios by leveraging techniques such as modality hallucination or knowledge distillation, thereby enhancing its flexibility for real-world clinical deployment. Lastly, our current study is limited to retrospective under-sampling experiments, which may not fully capture the complexities and artifacts present in real-world prospective acquisitions. Although retrospective experiments enable controlled and reproducible evaluation, they do not account for factors such as hardware constraints and sampling inaccuracies inherent in prospective scenarios. To address this limitation, future work will involve validating the proposed framework on prospectively under-sampled MRI acquisitions to better assess its robustness and clinical applicability.

6. Conclusion

In this work, we explore the realm of multi-contrast low-field MRI acceleration with a novel dual-domain framework comprising a k-space LHFP learning network and an image space HAFNet. In k-space learning, our proposed LHFP network first recovers low-frequency components and then emphasizes high-frequency learning in the second stage. This progressive learning strategy effectively mitigates the challenge posed by the magnitude imbalance between low- and high-frequency components, leading to superior recovery outcomes. In image space learning, our HAFNet integrates long-range dependent features from multi-contrast images through shallow-level and deep-level HWAF modules in their respective layers. Experimental results on the BraTs dataset (with simulated low-field noise) and the M4Raw dataset (acquired from low-field MRI systems) demonstrate significant performance gains over state-of-the-art methods, both qualitatively and quantitatively. These findings highlight the promising potential of our approach for clinical applications.

Highlights:

• We present a novel dual-domain framework to jointly accelerate and denoise multi-contrast MRI acquired at low field strength.

• In k-space learning, the magnitude imbalance between the low-frequency (LF) and high-frequency (HF) components may cause the network to be dominated by LF components, leading to sub-optimal recovery of HF components. To address the inadequate recovery of HF component in k-space learning, we present a novel Low-to-High Frequency Progressive (LHFP) learning network that progressively recovers LF and HF components, with the low-high frequency boundary adaptively optimized for each image.

• In image domain learning, we propose a Hybrid Window-based Attention Fusion Network (HAFNet) to efficiently integrate long-range dependent features from the multi-contrast images by alternately computing self-attention within dense and dilated windows.

• Experiments on the BraTs and low-field M4Raw datasets demonstrate the superiority of our method over state-of-the-art multi-contrast MRI reconstruction methods.