I Introduction
Cardiac magnetic resonance (CMR) imaging is a noninvasive imaging technique that can be used to evaluate cardiac function and ventricular wall motion abnormalities, providing rich information for the clinical diagnosis of heart conditions [9]. However, cardiac motion adversely affects the quality of MR images and therefore limits the temporal and spatial resolution of cardiac MR imaging [22], especially in some cardiac diseases, such as tachycardia. Therefore, it is important to accelerate cardiac MR imaging without sacrificing image quality.
Usually, fast CMR approach requires priori information to remove the aliasing artifacts caused by the violation of the Nyquist sampling theorem [14]. The advanced approach is compressed sensing (CS)/Lowrank(LR) [3, 24]. Amounts of CS/LRbased methods has been proposed to accelerate cardiac MR imaging [15, 33, 21, 37, 26, 23, 25, 30]. For example, kt FOCUSS [15] took advantage of the sparsity of xf support to reconstruct xf images from the undersampled kt space. And kt ISD [21] incorporated additional information on the support of the dynamic image in xf space based on the theory of CS with partially known support. Wang et al [37] employed a 3D patchbased spatiotemporal dictionary for sparse representations of dynamic image sequences. A typical example of lowrank is L+S [26], in which the nuclear norm was used to enforce low rank in L and the L1 norm was used to enforce sparsity in S. And kt SLR [23] exploited the correlations in a dynamic imaging dataset by modeling the data to have a compact representation in the KarhunenLouve transform (KLT) domain. These methods have greatly improved the spatiotemporal resolution of dynamic MR imaging. However, these methods take a relatively long time during the iterative solution procedure to achieve highquality reconstruction, and the selection of the regularization parameter is empirical. Additionally, most of these approaches exploit a priori information only from the tobereconstructed images or from very few reference images [20].
Recently, deep learning based methods have been proposed and successfully applied to MR imaging [36, 18, 13, 41, 7, 32, 28, 29, 27, 38, 31, 12, 5, 2, 6]. There are mainly two categories of deep learningbased fast MRI: (1) endtoend learning methods [36, 18, 13, 41, 7, 32, 28, 29, 27, 38] and (2) modelbased unrolling methods [31, 12, 5, 2]. The first category utilizes the information from big data to train a universal network for learning the mapping between the undersampled and fully sampled data pairs in an endtoend manner. For example, in [36], a plain convolutional neural network was trained to learn the mapping relationship between undersampled brain MR images and fully sampled brain MR images. AUTOMAP [41]
used a combination of fully connected networks and CNNs to learn the mapping from undersampled kspace to reconstructed image. The modelbased methods unroll the iterations of the optimization algorithm to the neural network so that the network can automatically learn the hyperparameters or transformations in the optimization algorithm. Typical modelbased networks includes ADMMNet
[31], VNNet [12], Learned PD [5], etc. Modelbased unrolling methods often achieve better reconstruction quality with less data than endtoend learning methods [6]. There are mainly three works for dynamic MR imaging, namely DCCNN [29], CRNN [27], DIMENSION [38]. DCCNN proposed a deep cascade of convolutional neural networks to accelerate the data acquisition process by combining data consistency and data sharing approaches. CRNN simultaneously learned the spatiotemporal dependencies of cardiac image series by exploiting bidirectional recurrent hidden connections across time sequences. DIMENSION developed a multisupervised network training technique to simultaneously constrain both the frequency and the spatial domain information to improve the reconstruction accuracy. However, all three of these methods require a large amount of fully sampled cardiac MR images as the ground truth. Collections of these fully sampled images are always difficult to obtain, especially breathholding and regular heart rhythms are required in the acquisition.In addition to priori information, the spatial variance coil sensitivity provided by the phase array coil also plays an important role in fast MRI. Parallel MRI utilizes the coil sensitivity to accelerate MR image acquisition and has been widely used in clinical scans. However, most CMR deep learning reconstruction studies utilize simulated singlechannel kspace data to train the network. These approaches may lose exploration of coil correlation and more importantly, cannot be applied to real MR scans.
To solve the above issues, we propose an unsupervised deep learning framework for parallel cardiac MRI in this paper. A timeinterleaved acquisition scheme is designed that can build a set of fully encoded reference data by merging adjacent time frames. These fully encoded data can be used to train a parallel network for reconstructing the image of each coil separately. Our developed modelbased reconstruction network (ADMMNetIII) was employed in our study. Finally, the coil correlations are explored, and the coil images are combined together via another CNN. Our contributions could be summarized as follows:

We propose an unsupervised framework for dynamic MRI. In our framework, the acquisition of fully sampled data for network training is no longer needed, which is one of the greatest difficulties in deep learningbased cardiac imaging, especially breathholding conditions and regular heart rhythms are required in the acquisition scheme. This is the first time that an unsupervised approach has been applied to dynamic MR imaging.

Dynamic MR images have a lot of redundancy in the temporal direction. In the proposed unsupervised framework, a timeinterleaved acquisition scheme is used, and the signals from directly adjacent time frames can be merged to build a set of fully encoded reference data for network training. In this way, temporal redundancy can be effectively utilized.

Different from other deep learning methods for dynamic MRI, which focus on singlechannel MRI, we propose a parallel imaging strategy. The proposed parallel imaging technique has three advantages. First, a modelbased unrolling method has been employed in our study, which can achieve better reconstruction quality with less data. To the best of our knowledge, this is the first time to apply modelbased unrolling method to dynamic MR imaging. Second, unlike other deep learningbased methods using singlechannel signals as input and output, our parallel network focuses on multichannel scenario and could explore coil correlations. Third, multichannel data have a less complex distribution than singlechannel data, which undoubtedly decreases the difficulty of network learning. To the best of our knowledge, this is the first time that a parallel imaging network has been applied to dynamic MRI.

Although our proposed framework is based on the timeinterleaved sampling scheme, network training and testing can be performed to any sampling patterns. The timeinterleaved sampling scheme is used only during the data preparation phase. Once the fully encoded training data are constructed, retrospective undersampling is no longer dependent on the timeinterleaved sampling pattern. Moreover, the model trained on one sampling pattern can be well generalized to other sampling patterns.

The experimental results show that the proposed method is superior to conventional CSbased methods such as kt FOCUSS, kt SLR and L+S with a much shorter runtime. These findings demonstrate the effectiveness of the unsupervised learning and the parallel network in cardiac MRI.
The rest of this paper is organized as follows. Section II states the problem and the proposed methods. Section III summarizes experimental details and the results to demonstrate the effectiveness of the proposed method, while the discussion and conclusions are presented in Section IV and Section V, respectively.
Ii Methodology
Iia Problem Formulation
The goal of our work is to estimate an unknown image from undersampled kspace data. Specifically for 2D cardiac imaging, reconstruction was performed by solving the following optimization problem:
(1) 
Here is the encoding operator. is the 2D dynamic image series and is the corresponding multichannel kspace data. is a prior regularization of . is a regularization parameter. The first term is the data fidelity, which ensures that the kspace of reconstructed data is consistent with the actual measurements. The second term is often referred to as prior regularization. In CSbased methods, is usually a sparse prior of in the temporal dimension.
In CNNbased methods, is a CNN prior, which forces to match the output of the networks:
(2) 
where is the output of the networks under the parameters . The purpose of the network training process is to find the optimal parameters . Once the network is trained, the networks’ output is the reconstruction we want. The data fidelity term is important to achieve high quality reconstruction. Therefore, data consistency (DC) layers are often introduced in the CNNbased methods.
IiB The Proposed Unsupervised Framework
The proposed unsupervised learning framework for cardiac MRI is shown in Fig.1. To simplify the symbols, we omit the channel dimension in this section, and it should be noted that Fig.1 is specific to each coil. The whole unsupervised framework can be divided into three components:

Data preparation: Undersampled kspace data are acquired according to a timeinterleaved acquisition scheme. The timeinterleaved acquisition scheme is shown in Fig.2. Adjacent time frames can be merged to build a complete set of kspace data, which called fully encoded kspace. Once the fully encoded dataset is built, the data pairs of the network input and output can be obtained by retrospectively undersampling the fully encoded data with a designed sampling mask. Although the timeinterleaved sampling scheme is not limited to uniform or random sampling, for the convenience of the experiments, we focus on uniform time interleaved sampling. Also, more neighboring frames can be averaged to increase the SNR of the fullencoded data.

Network training: The proposed parallel neural network will be described in the next section. The training datasets obtained above can be fed into the networks for network training. The input of the network is multichannel underencoded kspace data and the output is the coilcombined fully encoded image. Although the training datasets are synthetic in this stage, they can effectively represent true fully sampled data. More importantly, the temporal redundancies have been utilized through the construction of this dataset.

Online test: In the test stage, the true in vivo undersampled kspace data are fed into the trained network to reconstruct the 2D dynamic MR images.
In summary, we use timeinterleaved sampling data to synthesize fully encoded data as references to realize unsupervised learning. This framework has many advantages: (a) there is no need for a fully sampled dynamic MR dataset; (b) coil correlations can be explored; (c) there is no need for coil sensitivity maps; (d) the time redundancies are utilized. In Section III and Section IV, we will demonstrate the effectiveness of this framework with abundant experiments.
IiC The Proposed Parallel Network
For the parallel reconstruction of underencoded multichannel kspace data, we propose a novel parallel neural network, as shown in Fig.3. The underencoded multichannel kspace data are fed into the parallel network. For each coil, there is a separate network to reconstruct the coil data. We applied ADMMNetIII [6] as the reconstruction network because it is a modelbased unrolling method that can obtain highquality reconstruction results with less data. ADMMNetIII is the generalized version of ADMMNet [21]. The solution iterations of ADMMNetIII can be written as:
(3) 
In ADMMNetIII, the operators , , and the parameter are all learned by the network, while only the priori regularization and the parameter are learned in ADMMNet [31]. The generalization process and implementation results of ADMMNetIII could be found in [6]. An excessive explanation of ADMMNetIII is beyond the scope of this article. Different from the original ADMMNetIII model, each ADMMNetIII model is embedded with a data consistency (DC) layer in this paper because our previous exploration [35] showed that the data consistency layer could effectively improve the reconstruction quality. Then all of these network reconstructed coil images are concatenated and fed into another CNN to explore the coil correlations and implement coil combination.
Overall, the proposed parallel network has two components: a reconstruction network to reconstruct each coil image, and a coilcombination network to explore coil correlations and combine all the coil images together. Specifically, unlike other methods [29, 27, 38] using singlechannel signals as input and output, our parallel network focuses on multichannel scenario and could explore coil correlations. Another advantage to deal with coil images is that singlechannel data have a complex distribution [8], which undoubtedly increases the difficulty of network learning. To visually display the statistical distribution of the singlechannel and multichannel images, statistical histograms of both are also provided in Fig.4, from which we can see that the coil images have a simpler statistical distribution than the combined singlechannel image. In Section III.B, we compare a singlechannel model with a multichannel model under the proposed unsupervised framework and find that the multichannel model achieves better reconstruction results.
Iii Experimental Results
Iiia Setup
IiiA1 Data acquisition
We collected 386 2D dynamic (2Dt) fully sampled cardiac MR data from 30 healthy volunteers using a 3T scanner (SIEMENS MAGNETOM Trio) with a balanced steadystate free precession (bSSFP) sequence. Written informed consent was obtained from all the subjects. Each scan contains a singleslice bSSFP acquisition with 25 temporal frames. Retrospectively electrocardiogram ECGgated segmented imaging was conducted, and each slice was acquired in one breathhold of 1520 sec. The following parameters were used for the bSSFP scans: FOV mm, acquisition matrix , slice thickness = 6 mm, TR/TE = 3.0 ms/1.5 ms and 20 receiving coils. We randomly selected 25 volunteers for training and the rest for testing. Deep learning typically require a large amount of data for training [19]. Therefore, some data augmentation strategies were applied. We sheared the original images along the x, y and t directions. The sheared size was (
), and the stride along the three directions is 12, 12 and 5 respectively. Finally, we obtained 2149 2Dt multichannel cardiac MR data of size
() for data preparation and 603 data for testing.In this work, we focus on a timeinterleaved acquisition scheme with a uniform sampling pattern, which is shown in Fig.2 (left). For each original kspace data, we retrospectively undersampled it with 16 ACS lines. Specifically, we fully sampled the frequencyencodes (along ) and uniformly undersampled the phase encodes ( along ) according to the timeinterleaved scheme. Although fully sampled kspace data are available in our acquisition, they are unseen to the proposed unsupervised learning framework, which are only used to obtain the undersampled data retrospectively. We merged all the frames of the undersampled kspace data rather than the adjacent frames and averaged them to obtain the high SNR fully encoded training data. This also has brought some other benefits, such as the elimination of temporal redundancy, and the requirement of GPU memory is reduced.
IiiA2 Network training
For network training, we divided each data into two channels for storing real and imaginary parts of the data respectively. Therefore, the inputs of the network are underencoded multichannel kspace data , and the outputs are the coilcombined reconstructed images . The hyperparameters in the network are set as follows: for each ADMMNetIII, the number of iterations is , the numbers of convolution kernels are set as shown in Fig.3 and the size of each convolution kernel is . Xavier initialization [10]
was used to initialize the network weights. Rectifier linear units (ReLU)
[11]was selected as the nonlinear activation functions. The minibatch size was 4. The exponential decay learning rate
[40]was used in all the CNNbased experiments, and the initial learning rate was set to 0.001 with a decay of 0.98. The loss function used in this work was the mean squared error (MSE). All the models were trained by the Adam optimizer
[16] with parameters , and .The models were implemented on an Ubuntu 16.04 LTS (64bit) operating system equipped with an Intel Xeon E52640 Central Processing Unit (CPU) and Tesla TITAN Xp Graphics Processing Unit (GPU, 12 GB memory) in the open framework TensorFlow
[1]with CUDA and CUDNN support. The network training took approximately 56 hours and 100 epochs.
IiiA3 Model configuration
There are many alternative options in our proposed unsupervised framework, for example: 1. In the data preparation stage, the timeinterleaved sampling scheme is not limited to uniform or random sampling patterns; 2. In the training stage, the retrospectively undersampling mask is not limited to uniform or random sampling patterns; 3. In the test stage, the trained model can have a good generalization ability to other sampling patterns. Discussing all the cases will make the article very lengthy. Therefore, for the convenience of the experiments, we only experimented on typical cases without loss of generality. The model configurations (sampling patterns and acceleration) are arranged in TABLE I. A 1D Gaussian random undersampling pattern [15], which is one of the most common protocols in the CS/LR based methods, was applied in this paper. Specifically, we fully sampled frequency encodes (along ) and randomly undersampled the phase encodes (along ) according to a zeromean Gaussian variable density function.
Timeinterleaved  Training  Testing  Acceleration  

Section III.B  Uniform  Random  Random  4 
Section III.C  Uniform  Random  Random/Uniform  4/8 
Section IV.A  Uniform  Random  Random  4 
Section IV.B  Uniform  Random  Random  4 
Section IV.C  Uniform  Uniform  Random/Uniform  4 
IiiB Does the Proposed Parallel Network Work?
Currently, all three deep learning methods [29, 27, 38] for cardiac MRI use singlechannel data for network training and testing. In this section, we refer to them as singlechannel methods and correspondingly refer to the proposed parallel imaging method as the multichannel method. We will explore whether the multichannel method exhibits superior reconstruction performance. To ensure a fair comparison, we explored the effectiveness of the singlechannel method and multichannel method based on the same network structure (ADMMNetIII) under the unsupervised framework proposed in this paper. Although we give only the unsupervised scheme in the multichannel case in Fig.1, this unsupervised scheme can be conveniently changed to the singlechannel case, because the operations in Fig.1 focus on the temporal dimension and have nothing to do with the coil dimension. The differences between the two models exist only in two respects. First, the raw materials for data preparation are different: one is multichannel fully sampled kspace data, while the other is singlechannel fully sampled kspace data by the adaptively combining the above multichannel kspace data [34]. Second, the singlechannel model no longer requires the network for combining the coils, so it contains only the reconstruction part.
We trained the model with a 1D random Gauss sampling mask at 4fold acceleration, and the reconstruction results of the three subjects are shown in Fig.5, which clearly shows that the multichannel model can restore more details (as shown by red arrows) than our singlechannel model. The singlechannel model not only loses more detail than the multichannel model, but the reconstruction results are also more blurring.
IiiC Comparisons to the Stateoftheart Methods
To demonstrate the efficacy of the proposed unsupervised learning method, we compared it with several stateoftheart CS/LR methods including kt FOCUSS [15], kt SLR [23], and L+S [26]. We adjusted the parameters of the competing methods to their best performance. A 1D random Gauss mask was used for training and testing. And to explore the generalization performance of the proposed method with different sampling patterns, we also tested the trained model with a 1D uniform undersampling pattern. The reconstruction results of these methods at 4fold acceleration are shown in Fig.6. The reconstruction results of three CSbased methods contain fewer structural details and more artifacts than the reconstruction results of the proposed method. We also enlarged the error maps of the cardiac region for demonstration, which show that our method has the best reconstruction performance in the cardiac region, especially the details marked by the red arrow. The yt images, which were extracted from the 124th slice along the y and temporal dimensions, also clearly illustrate the comparable performance of the proposed method. The red numbers represent the reconstruction time of these methods. Our method has the shortest reconstruction time, which is hundreds of times shorter than that of the other methods. Longitudinal comparisons (comparisons of the reconstruction results of the two masks) show that the reconstruction results under the random mask are better than those under the uniform mask. This conclusion is in line with expectations, because CSbased methods perform better at random mask than uniform masks, and for the deep learning approach, training and testing with the same type of mask could yield better reconstruction results than training and testing with different types of masks. The reconstruction results of the different methods at 8fold acceleration are shown in Fig.7. At 8 fold acceleration, the same conclusion is reached as with 4fold acceleration.
Though we only show two Cartesian sampling patterns (uniform and random Gauss) in this paper, our method can be easily adopted to other sampling patterns. This finding also reflects one of the advantages of deep learningbased approaches: they have no such strict requirements for sampling masks. In another example, Knoll et al [17] also applied a network that was trained from regular undersampling mask to random undersampling with good reconstruction performance. Further exploration will be carried out in future studies.
Iv Discussion
Iva Options of the Coil Reconstruction Network
Although we introduced the proposed parallel network in Section II.C, where ADMMNetIII was selected as the reconstruction network, the whole unsupervised framework is not limited to a specific reconstruction network. The reason to choose ADMMNetIII is that it is a deep learning modelbased unrolling method, which requires less data than vanilla endtoend methods and usually exhibits superior reconstruction performance to other methods. In this section, we compared the reconstruction results of ADMMNetIII with those of DCCNN [29], which is one of the stateoftheart deep learning methods for dynamic MRI. We focused on a D5C5 model, which works well for the DCCNN model and consists of five blocks (C5) with each block containing five convolutional layers (D5). We trained two models under the proposed unsupervised framework. The reconstruction results are shown in Fig.8. Both subjects have consistent observations: the coil reconstruction network using the ADMMNetIII model has smaller artifacts and more details, especially in the heart region (as can be seen from the red and yellow arrows in the error maps, and for obvious comparisons, the display range was narrowed down to [0, 0.07]) than that using the DCCNN model.
IvB The Importance of the Data Consistency Layer in the Parallel Network
As shown in Fig.3, each ADMMNetIII model is followed by a data consistency (DC) layer. In [35], we have noted that the DC layer is very important for reconstruction in the singlechannel case. Aggarwal et al [2] noted that the system function in the singlechannel undersampled MRI acquisition is simple, where
is a sampling matrix obtained by keeping only the relevant rows of an identity matrix and
is the Fourier matrix. In this case, the DC can be analytically computed, as in, for example, [7, 29, 27, 38]. However, is not analytically invertible for complex operators such as multichannel MRI. In this case, an iterative optimization algorithm is needed to approximate DC. For example, [2] proposed to solve DC using a conjugate gradient optimization scheme. In this paper, we developed another method. Although the proposed method is for multichannel MRI, due to the coilbycoil implementation, DC still can be analytically computed for every channel. So in this method, is DC still effective and important, especially followed by a coilcombination network? To answer this question, we built two models under the proposed framework. The only difference between the two models is whether they contain DC layers. The training configurations of these two models remain the same. The reconstruction results are shown in Fig.9. All three subjects have consistent observations: the model including DC layers has smaller artifacts and more details than the other, especially in the heart region (as can be seen from the red and yellow arrows in the error maps). Therefore, we can conclude that the DC layer still plays a very important role in reconstruction in multichannel MRI with coilbycoil implementation.IvC Training the Model Under Other Sampling Patterns
The models in the above sections are all trained under the timeinterleaved acquisition scheme with a 1D random sampling pattern. Although our proposed framework is based on the timeinterleaved sampling scheme, the network training and testing can be applied to any sampling patterns. The timeinterleaved sampling scheme is used only during the data preparation phase. Once the fully encoded training data are constructed, retrospectively undersampling is no longer dependent on the timeinterleaved sampling pattern. Moreover, the model trained on one sampling pattern can be well generalized to other sampling patterns. In Section III.C, the 1D Gaussian random undersampling pattern is adopted. In particular, we trained the model under a 1D Gaussian random undersampling pattern and tested it under 1D random and 1D uniform undersampling patterns. In this section, we trained the model under a 1D uniform undersampling pattern and tested it under 1D random and 1D uniform undersampling patterns. The reconstruction results at 4fold acceleration are shown in Fig.10. Our method achieves superior reconstruction results with both undersampling patterns, especially in the heart region, which is marked by the red arrow.
IvD The Limitations of the Proposed Work
Although our method has many advantages (i.e., superior reconstruction results and the shortest reconstruction time) compared with other stateoftheart methods, there is still a certain degree of smoothness in the reconstructed images. The reason may be that the loss function we chose is the MSE. The MSE loss has a limited ability to perceive image structure information because it indicates only the mean square information between the reconstructed image and the ground truth. DAGAN [39] couples an adversarial loss with an innovative content loss to reconstruct CSMRI images, which could preserve perceptual image details. This property motivates us to use more detailfriendly loss functions in future works.
In TGRAPPA [4], more neighboring frames could be averaged to increase the SNR of the fully encoded data. Inspired by this finding, we average all the frames to obtain the highest SNR. This method yielded some benefits, such as the elimination of temporal redundancies and the GPU memory requirement is reduced. However, there are some inconveniences. For example, the temporal correlations are underutilized, and many timedependent network configurations are not available. In the current GPU condition (12 GB memory), averaging all the frames to obtain only one frame is necessary because the GPU resources cannot meet the requirements of exploring the temporal and coil correlations at the same time. In the future, with the improvement in the hardware conditions, more highdimensional exploration is expected to further improve the reconstruction of dynamic MR images.
V Conclusion and Outlook
In this paper, we propose an unsupervised deep learning method for parallel MR cardiac imaging via timeinterleaved sampling. In our framework, fully sampled reference data are no longer required for network training. The temporal redundancies can be effectively utilized via the proposed data preparation process. We also propose a coilbycoil parallel imaging technology with many advantages. To the best of our knowledge, this is the first time that a parallel imaging network has been applied to dynamic MR imaging. Although our proposed framework is based on the timeinterleaved sampling scheme, the model can be applied to any sampling patterns. The experimental results show that the proposed method is superior to conventional CSbased methods such as kt FOCUSS, kt SLR and L+S in an extremely short amount of time. These findings demonstrate the effectiveness of the unsupervised learning and the parallel network in cardiac MR imaging.
Acknowledgment
This research was partly supported by the National Natural Science Foundation of China (61771463, 81830056, U1805261, 81971611, 61871373, 81729003, 81901736); National Key RD Program of China (2017YFC0108802 and 2017YFC0112903); Natural Science Foundation of Guangdong Province (2018A0303130132); Shenzhen Key Laboratory of Ultrasound Imaging and Therapy (ZDSYS20180206180631473); Shenzhen Peacock Plan Team Program (KQTD20180413181834876); Innovation and Technology Commission of the government of Hong Kong SAR (MRP/001/18X); Strategic Priority Research Program of Chinese Academy of Sciences (XDB25000000).
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