1 Introduction
Compressed sensing magnetic resonance imaging (CSMRI) [1] has been proposed for accelerating MRI process. This technique uses a small fraction of data to reconstruct images from subNyquist sampling. Assuming the raw data is compressible, CSMRI performs nonlinear optimisations on the undersampling data without sacrificing the quality of the reconstructed images significantly.
However, it is still very challenging to consolidate the speed of reconstruction and robustness of image quality maintenance in one CSMRI based framework. On the one hand, CSMRI tries to solve underdetermined equations to perceive the original signals from the limited undersampled data. This requires nonlinear optimisation solvers for a common nonconvex system that usually involve iterative computations, which can result in prolonged reconstruction time [2]
. On the other hand, CSMRI may produce images with degraded image quality and low signaltonoise ratio (SNR) from randomly highly undersampled
kspace data [3]. Moreover, in addition to a large amount of computation needed for the nonlinear optimisation, CSMRI also requires that the acquisition matrix and the sparse transformation matrix are unrelated. Based on the above limitations, the acceleration factor of CSMRI is generally between 2 and 6.Recently, deep learningbased CSMRI methods have emerged as an effective way to solve the problems of slow and unstable MRI reconstruction [4, 5, 6, 7, 8, 9, 10, 17, 18, 19, 20, 21, 22, 23, 24, 25]. For example, a conditional Generative Adversarial Networksbased model (DAGAN) was proposed to achieve fast CSMRI [5], but still, this endtoend training neglected the correlation between adjacent 2D slices. Thus, although DAGAN can achieve fast MRI reconstruction, it may lose image quality without using a priori information. For another example, DCCNN [6]
applied cascades of convolutional neural networks with a residual connection for CSMRI. Besides, DCCNN also used a data consistency (DC) step to ensure that the output of each cascade was consistent with the original
kspace information. However, DCCNN approach was not able to effectively utilise the full temporal domain information. In contrast, a convolutional recurrent neural network (CRNN) method was proposed to incorporate a bidirectional convolutional recurrent unit for a faster and more stable reconstruction [7]. However, such an approach was not able to effectively exploit the kspace information from individual images.In this study, we propose a GAN based architecture that works on continuous sequential data for CSMRI. This intuitively mimics the way reporting clinicians scrutinise the 3D data by scrolling up and down to fully sense the information above and below the current 2D slice. Our method can not only overcome the shortcomings of slow reconstruction but can also maintain higher reconstructed image quality by combining the characteristics in time and frequency domains. To the best of our knowledge, this is the first study to combine Recurrent Neural Networks (RNN) with GAN in the field of MRI reconstruction. In particular, we design a novel generator with bidirectional convolutional long shortterm memory (BiConvLSTM) that can encode the a priori frequency and timedomain information. Besides, another significant contribution of our work is that we propose a spatial attentionbased model that the attention unit in our model can distinguish between significant and nonsignificant features in terms of the MRI reconstruction task. In addition, we utilise WGAN with gradient penalty (WGANGP) as a critic function, which can significantly improve the stability of GAN. We also couple the adversarial loss with pixelwise mean square error (MSE) and the perceptual loss
[11] to achieve better reconstruction details with superior perceptual image quality.2 Method
2.1 Problem formulation of CSMRI
2.1.1 Deep Learningbased CSMRI Reconstruction.
Let represents the slice of 2D images to be reconstructed, which consists of pixels for one image, and let denotes the undersampled measurements in kspace. For deep learningbased methods, previous studies such as [4] and [6] incorporated a CNN into CSMRI reconstruction, transformed the unconstrained optimisation problem into:
(1) 
in which denotes the forward propagation of data through the CNN parametrised by , and is a regularisation parameter. is the reconstruction from the zerofilled undersampled kspace measurements.
In the reconstruction network selection, many previous studies, e.g., [5, 10, 17, 18], relied on an encoderdecoder structure. Nevertheless, our preliminary experiments indicated that these single structures performed poorly in the PSNR. Moreover, there are also methods [6, 7] that developed for the dynamic MR reconstruction, but they did not perform well at higher kspace undersampling.
2.2 DAWGAN for CSMRI
In this study, we propose a Deep Attentive Wasserstein Generative Adversarial Networks (DAWGAN) method to reconstruct MRI images from highly undersampled data with continuous sequential data. It contains three key components: a BiConvLSTM block, a spatial attention block (SAB) and a WGANGP as the critic function. The workflow of our DAWGAN is summarised in Figure 1.
2.2.1 Image Domain Feature Extraction via a Sequential Learning.
To achieve more aggressive undersampling, one way is to encode the a priori frequency and timedomain information in sequential data, e.g., 2D MRI slices of a 3D volumetric data. We assumed as the feature representation of our 2D sequential MRI data slices throughout the 3D volume. Here denoted the representation at slice and iteration . We needed to take into account and in the reconstruction process to provide information for . To that end, we proposed a BiConvLSTM subnetwork to exploit both temporal and iteration dependencies jointly. The BiConvLSTM subnetwork can be formulated as:
(2) 
where denoted the forward direction and denoted the backward direction. Through BiConvLSTM layer, our model can learn the differences and correlations of successive MRI data slices. The output of the BiConvLSTM layer then took a refinement connection to prevent data shifting.
2.2.2 Spatial Attention Block (SAB).
The main aim of the designed SAB was to increase representation power by using attention mechanism: focusing on important features and suppressing unnecessary ones. Details about the SAB are shown in Figure 1. Inspired by [12]
, we set the SAB after the first convolution block, which was also propagated to the upsampling layers with the skipconnections. We conducted averagepooling and maxpooling operations on the feature map obtained from the upper layer to generate an efficient feature descriptor. Then we utilised a convolution layer to generate a feature map that could encode where to emphasize or suppress. The SAB took all the features extracted by the upper layer to calculate the attention map.
We assumed that the 2D maps generated by pooling operations were and . Each denoted averagepooled features and maxpooled features across the feature map. The two maps were then stacked and convolved by a standard convolution layer to produce the 2D spatial attention map. Hence, our spatial attention map was computed as
(3) 
where represented the convolution operation with the filter size of 77 and
denoted the sigmoid function according to
[12]. The spatial attention calculated the feature correlation across the channel domain to find the cardinal features across the entire spatial domain.2.2.3 Loss Function and Training.
Our loss function consisted of content loss and adversarial loss. The content loss function was basically made up of three parts, i.e., a pixelwise image domain mean square error (MSE) loss, a frequency domain MSE loss and a perceptual VGG loss. The whole loss function could be formulated as
(4) 
where represented the hyperparameters.
Most of the GAN based CSMRI studies used vanilla GAN objective [13], which applied the KullbackLeibler (KL) divergence, as the adversarial loss function. However, during the training of the generator, when the generator deviated from the optimal solution, the parameters of the generator might not be updated continuously, which could then lead to complicated training process and model collapse [14]. In this study, we introduced WGANGP [15] as an alternative strategy of using Wasserstein distance to displace the KL divergence for solving the potential complications in training the GAN. WGANGP also introduced the gradient penalty to better solve the common gradient vanishing problem. We used a loss that was calculated as the following
(5) 
In order to improve the perceptual quality, we also incorporated the content loss with three different combinations of the loss functions:
(6)  
We used normalised MSE (NMSE) as the optimisation cost function. However, the use of NMSE as content loss alone might lead to perceptually uneven reconstruction, resulting in a lack of coherent image details. Therefore, to consider the perceptual similarity of images, we also added NMSE of the frequency domain data and VGG loss () as additional constraints.
3 Experiments and Results
Brain MRI Data  

Methods  10%  30%  50%  
PSNR  MOS  PSNR  MOS  PSNR  MOS  
Zerofilling  28.16(3.33)  1.02(0.13)  34.83(2.78)  1.12(0.21)  39.36(2.61)  1.09(0.34) 
ADMM  28.20(3.36)  1.21(0.35)  35.21(4.03)  1.22(0.31)  39.99(4.08)  1.28(0.37) 
DAGAN  33.25(4.10)  2.52(0.88)  38.12(3.56)  3.08(0.68)  45.41(4.13)  3.27(0.73) 
CRNN  33.57(3.16)  2.78(0.62)  38.26(3.86)  2.98(0.43)  46.10(2.29)  3.58(0.61) 
DAWGAN  34.31(3.01)  3.01(0.69)  40.74(3.57)  3.23(0.69)  46.43(2.19)  3.98(0.72) 
Cardiac MRI Data  
Zerofilling  27.08(0.84)  1.02(0.13)  31.49(0.88)  1.12(0.21)  35.13(0.92)  1.09(0.34) 
ADMM  27.20(1.64)  1.25(0.28)  31.88(1.72)  1.38(0.21)  35.54(1.73)  1.98(0.45) 
DAGAN  29.35(1.33)  2.21(0.54)  33.85(1.62)  2.49(0.41)  37.86(1.22)  2.81(0.53) 
CRNN  29.62(2.15)  2.68(0.61)  34.29(2.33)  2.83(0.71)  38.12(2.29)  2.96(0.67) 
DAWGAN  31.06(1.71)  2.92(0.72)  35.97(1.77)  3.06(0.59)  39.66(1.79)  3.42(0.56) 
3.1 Experiments
3.1.1 Datasets.
Our experiments were performed on two datasets (1) Brain MRI dataset: We trained and tested our model using a MICCAI 2013 grand challenge dataset. In total, we included 726 3D data for our study. We randomly used 503 data for training, 173 for validation and 50 for testing. Each 2D slice had a shape of 256256, and we normalised the intensities into a range of [1 1]. (2) Cardiac MRI dataset: A population of 100 3D LGE MRI patient data, which were made available through the 2018 Atrial Segmentation Challenge, were used in this work. The scanner used for this clinical study was a wholebody MRI scanner, within an image acquisition resolution of 0.625mm. The studied data were randomly divided into 80 for training, 10 for validation and 10 for testing. Similarly, we normalised each slice into a range of [1 1].
3.1.2 Experiments Setup.
For all the input data, we applied data augmentation on the input 2D image slices. Besides, we used raw kspace data with different undersampling ratios to simulate the corresponding acceleration factors. In particular, and retained raw kspace data were simulated representing accelerations assuming that the preparation time of MRI scanning is insignificant. All our comparison studies were carried out using different CSMRI reconstruction algorithms using these three levels of undersampling ratios. Our studies were mainly divided into three experiments: First, We compared the performance of our method with that of other SOTA at three different acceleration factors. In addition to the traditional metrics of PSNR, we also introduced the mean opinion scores (MOS) to take human perception into account, which was the results of domain experts evaluating the reconstructions and averaging their perceptual quality. Then, at different acceleration factors, we tested the noise reduction effect of all models at different noise level to prove that our model could significantly suppress the residual noise. To test the noise tolerance of different CSMRI methods, we added white Gaussian noises to the kspace data before applying the undersampling. Inspired by [16]
, we conducted a noise level estimation for all the reconstruction results. Finally, we tested the effectiveness of various network configurations of our proposed framework. In the final quantification, we used PSNR, SSIM and NMSE as the evaluation metrics.
3.2 Results
3.2.1 Image Quality Comparison.
Our method has demonstrated the best performance by comparing with four SATO methods (on both brain and cardiac MRI datasets). Table 1 shows that the results of our proposed DAWGAN performed best in PSNR and MOS. At the and acceleration factors, the PSNR and MOS achieved by our DAWGAN were significantly higher than the other methods. In addition, Figure 2 shows that DAWGAN produced less noiseinduced artefacts in all the simulation studies, while the other methods had more noiseinduced artefacts. Although CRNN and DAGAN could also suppress some artefacts, the reconstructions of the brain area were less detailed than those DAWGAN reconstructed. Moreover, ADMM and zerofilling could not effectively inhibit remaining aliasing artefacts.
3.2.2 Noise Suppression Comparison.
In terms of reconstruction details, we demonstrated that DAWGAN could effectively reduce residual noise in the reconstructed images. As shown in Figure 3, DAWGAN suppressed the noise effectively at different noise levels. Figure 4 shows the PSNR results with respect to different noise levels and various undersampling patterns. Our proposed DAWGAN also demonstrated considerable noise tolerance at different noise levels, and the mean value of the PSNR was higher than other methods.
3.2.3 Ablation Studies.
The performance of our framework with various network components was shown by our ablation studies. The submodels we compared were WGANGP+RNN, WGANGP+Attention and Attention+RNN. Table 2 shows that the DAWGAN full model was superior to other submodel variations using all three metrics, which indicated that the current configurations in our proposed network architecture are effective.
Brain MRI Data  

Comparison Study  10%  30%  50%  
PSNR  NMSE  SSIM  PSNR  NMSE  SSIM  PSNR  NMSE  SSIM  
WGANGP+RNN  34.11(3.91)  0.16(0.03)  0.96(0.02)  40.95(4.13)  0.07(0.02)  0.98(0.01)  46.08(4.37)  0.03(0.01)  0.99(0.00) 
WGANGP+Attention  34.22(2.01)  0.17(0.03)  0.96(0.02)  39.90(3.57)  0.08(0.02)  0.98(0.01)  46.29(3.81)  0.03(0.01)  0.99(0.00) 
Attention+RNN  33.71(3.30)  0.16(0.03)  0.96(0.02)  40.74(3.57)  0.08(0.02)  0.98(0.01)  46.29(3.81)  0.03(0.01)  0.99(0.00) 
DAWGAN  34.31(3.01)  0.16(0.03)  0.96(0.02)  40.74(3.57)  0.07(0.02)  0.98(0.01)  46.43(2.19)  0.03(0.01)  0.99(0.00) 
Cardiac MRI Data  
WGANGP+RNN  30.62(2.33)  0.24(0.03)  0.89(0.02)  34.51(2.40)  0.15(0.02)  0.94(0.01)  38.44(2.51)  0.09(0.01)  0.97(0.00) 
WGANGP+Attention  30.68(2.61)  0.23(0.03)  0.89(0.02)  34.81(2.13)  0.14(0.02)  0.94(0.01)  37.87(2.54)  0.10(0.01)  0.96(0.00) 
Attention+RNN  30.51(2.85)  0.23(0.03)  0.87(0.02)  34.35(2.70)  0.15(0.02)  0.93(0.01)  37.67(2.10)  0.10(0.01)  0.96(0.00) 
DAWGAN  31.06(1.71)  0.23(0.02)  0.90(0.02)  35.97(1.77)  0.13(0.01)  0.96(0.01)  39.66(1.79)  0.09(0.01)  0.97(0.00) 
4 Conclusion
In this paper, we proposed the DAWGAN to reconstruct MRI images from highly undersampled kspace data. Our DAWGAN employed WGANGP to improve the stability of vanilla GAN. The incorporated BiConvLSTM block can make full use of the relationships among successive MRI slices to improve the reconstruction results. In addition, the proposed SAB can distinguish between significant and nonsignificant features for our MRI reconstruction task. Our ablation studies have demonstrated the effectiveness of the key components of our framework. The comprehensive comparison studies on both brain and cardiac MRI datasets have corroborated that our method can not only achieve better reconstruction results but can also effectively reduce residual noise generated during the reconstruction process.
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