1 Introduction
Magnetic Resonance Imaging (MRI) is a powerful diagnostic tool for a variety of disorders, but its utility is often limited by its slow speed compared to competing modalities like CT or XRays. Reducing the time required for a scan would decrease the cost of MR imaging, and allow for acquiring scans in situations where a patient cannot stay still for the current minimum scan duration. One approach to accelerating MRI acquisition, called Parallel Imaging (PI) [13, 8, 3], utilizes multiple receiver coils to simultaneously acquire multiple views of the underlying anatomy, which are then combined in software. Multicoil imaging is widely used in current clinical practice. A complementary approach to accelerating MRIs acquires only a subset of measurements and utilizes Compressed Sensing (CS) [1, 7]
methods to reconstruct the final image from these undersampled measurements. The combination of PI and CS, which involves acquiring undersampled measurements from multiple views of the anatomy, has the potential to allow faster scans than is possible by either method alone. Reconstructing MRIs from such undersampled multicoil measurements has remained an active area of research. MRI reconstruction can be viewed as an inverse problem and previous research has proposed neural networks whose design is inspired by the optimization procedure to solve such a problem
[4, 9, 10, 6]. A limitation of such an approach is that it assumes the forward process is completely known, which is an unrealistic assumption for the multicoil reconstruction problem. In this paper, we present a novel technique for reconstructing MRI images from undersampled multicoil data that does not make this assumption. We extend previously proposed variational methods by learning the forward process in conjunction with reconstruction, alleviating this limitation. We show through experiments on the fastMRI dataset that such an approach yields higher fidelity reconstructions. Our contributions are as follows: 1) we extend the previously proposed variational network model by learning completely endtoend; 2) we explore the design space for the variational networks to determine the optimal intermediate representations and neural network architectures for better reconstruction quality; and 3) we perform extensive experiments using our model on the fastMRI dataset and obtain new stateoftheart results for both the knee and the brain MRIs.2 Background and Related Work
2.1 Accelerated MRI acquisition
An MR scanner images a patient’s anatomy by acquiring measurements in the frequency domain, called
kspace, using a measuring instrument called a receiver coil. The image can then be obtained by applying an inverse multidimensional Fourier transform
to the measured kspace samples. The underlying image is related to the measured kspace samples as(1) 
where is the measurement noise and is the fourier transform operator. Most modern scanners contain multiple receiver coils. Each coil acquires kspace samples that are modulated by the sensitivity of the coil to the MR signal arising from different regions of the anatomy. Thus, the th coil measures:
(2) 
where is a complexvalued diagonal matrix encoding the position dependent sensitivity map of the th coil and is the number of coils. The sensitivity maps are normalized to satisfy [15]:
(3) 
The speed of MRI acquisition is limited by the number of kspace samples obtained. This acquisition process can be accelerated by obtaining undersampled kspace data, , where is a binary mask operator that selects a subset of the kspace points and denotes the measured kspace data. The same mask is used for all coils. Applying an inverse Fourier transform naively to this undersampled kspace data results in aliasing artifacts. Parallel Imaging can be used to accelerate imaging by exploiting redundancies in kspace samples measured by different coils. The sensitivity maps
can be estimated using the central region of kspace corresponding to low frequencies, called the
AutoCalibration Signal (ACS), which is typically fully sampled. To accurately estimate these sensitivity maps, the ACS must be sufficiently large, which limits the maximum possible acceleration.2.2 Compressed Sensing for Parallel MRI Reconstruction
Compressed Sensing [2] enables reconstruction of images by using fewer kspace measurements than is possible with classical signal processing methods by enforcing suitable priors. Classical compressed sensing methods solve the following optimization problem:
(4)  
(5) 
where is a regularization function that enforces a sparsity constraint, is the linear forward operator that multiplies by the sensitivity maps, applies 2D fourier transform and then undersamples the data, and
is the vector of masked kspace data from all coils. This problem can be solved by iterative gradient descent methods. In the
th step the image is updated from to using:(6) 
where is the learning rate, is the gradient of with respect to , and is the hermitian of the forward operator .
2.3 Deep Learning for Parallel MRI Reconstruction
In the past few years, there has been rapid development of deep learning based approaches to MRI reconstruction [4, 10, 9, 5, 6, 12]. A comprehensive survey of recent developments in using deep learning for parallel MRI reconstruction can be found in [5]. Our work builds upon the Variational Network (VarNet) [4], which consists of multiple layers, each modeled after a single gradient update step in equation 6. Thus, the th layer of the VarNet takes as input and computes using:
(7) 
where
is a small convolutional neural network that maps complexvalued images to complexvalued images of the same shape. The
values as well as the parameters of the s are learned from data. The and operators involve the use of sensitivity maps which are computed using a traditional PI method and fed in as additional inputs. As noted in section 2.1, these sensitivity maps cannot be estimated accurately when the number of autocalibration lines is small, which is necessary to achieve higher acceleration factors. As a result, the performance of the VarNet degrades significantly at higher accelerations. We alleviate this problem in our model by learning to predict the sensitivity maps from data as part of the network.3 EndtoEnd Variational Network
Let be the vector of masked multicoil kspace data. Similar to the VarNet, our model takes this masked kspace data as input and applies a number of refinement steps of the same form. We refer to each of these steps as a cascade (following [12]), to avoid overloading the term "layer" which is already heavily used. Unlike the VN, however, our model uses kspace intermediate quantities rather than imagespace quantities. We call the resulting method the EndtoEnd Variational Network or E2EVarNet.
3.1 Preliminaries
To simplify notation, we first define two operators: the expand operator () and the reduce operator (). The expand operator () takes the image and sensitivity maps as input and computes the corresponding image seen by each coil in the idealized noisefree case:
(8) 
where is the sensitivity map of coil . We do not explicitly represent the sensitivity maps as inputs for the sake of readability. The inverse operator, called the reduce operator () combines the individual coil images:
(9) 
Using the expand and reduce operators, and can be written succinctly as and .
3.2 Cascades
Each cascade in our model applies a refinement step similar to the gradient descent step in equation 7, except that the intermediate quantities are in kspace. Applying to both sides of 7 gives the corresponding update equation in kspace:
(10) 
where is the refinement module given by:
(11) 
Here, we use the fact that ). can be any parametric function that takes a complex image as input and maps it to another complex image. Since the CNN is applied after combining all coils into a single complex image, the same network can be used for MRIs with different number of coils. Each cascade applies the function represented by equation 10 to refine the kspace. In our experiments, we use a UNet [11] for the .
3.3 Learned sensitivity maps
The expand and reduce operators in equation 11 take sensitivity maps as inputs. In the original VarNet model, these sensitivity maps are computed using the ESPIRiT algorithm [15] and fed in to the model as additional inputs. In our model, however, we estimate the sensitivity maps as part of the reconstruction network using a Sensitivity Map Estimation (SME) module:
(12) 
The operator zeros out all lines except for the autocalibration or ACS lines (described in Section 2.1). This is similar to classical parallel imaging approaches which estimate sensitivity maps from the ACS lines. The CNN follows the same architecture as the CNN in the cascades, except with fewer channels and thus fewer parameters in intermediate layers. This CNN is applied to each coil image independently. Finally, the dSS operator normalizes the estimated sensitivity maps to ensure that the property in equation 3 is satisfied.
3.4 E2EVarNet model architecture
As previously described, our model takes the masked multicoil kspace as input. First, we apply the SME module to to compute the sensitivity maps. Next we apply a series of cascades, each of which applies the function in equation 10, to the input kspace to obtain the final kspace representation . This final kspace representation is converted to image space by applying an inverse Fourier transform followed by a rootsumsquares (RSS) reduction for each pixel:
(13) 
where and is the kspace representation for coil . The model is illustrated in figure 1. All of the parameters of the network, including the parameters of the CNN model in SME, the parameters of the CNN in each cascade along with the s, are estimated from the training data by minimizing the structural similarity loss, , where SSIM is the Structural Similarity index [17] and , are the reconstruction and ground truth images, respectively.
4 Experiments
4.1 Experimental setup
We designed and validated our method using the multicoil track of the fastMRI dataset [18] which is a large and open dataset of knee and brain MRIs. To validate the various design choices we made, we evaluated the following models on the knee dataset:

Variational network [4] ()

Variational network with the shallow CNNs replaced with UNets ()

Similar to , but with kspace intermediate quantities ()

Our proposed endtoend variational network model ()
The model employs shallow convolutional networks with RBF kernels that have about 150K parameters in total. replaces these shallow networks with UNets to ensure a fair comparison with our model. is similar to our proposed model but uses fixed sensitivity maps computed using classical parallel imaging methods. The difference in reconstruction quality between and shows the value of using kspace intermediate quantities for reconstruction, while the difference between and shows the importance of learning sensitivity maps as part of the network. We used the same model architecture and training procedure for the model as in the original VarNet [4] paper. For each of the other models, we used cascades, containing a total of about 29.5M parameters. The
model contained an additional 0.5M parameters in the SME module, taking the total number of parameters to 30M. We trained these models using the Adam optimizer with a learning rate of 0.0003 for 50 epochs, without using any regularization or data augmentation techniques. We used two types of undersampling masks:
equispaced masks , which sample lowfrequency lines from the center of kspace and every th line from the remaining kspace; and random masks , which sample a fraction of the full width of kspace for the ACS lines in addition to a subset of higher frequency lines, selected uniformly at random, to make the overall acceleration equal to . These random masks are identical to those used in [18]. We also use equispaced masks as they are easier to implement in MRI machines.4.2 Results
Tables 2 and 2 show the results of our experiments for equispaced and random masks respectively, over a range of downsampling mask parameters. The model outperforms the baseline model by a large margin due to its larger capacity and the multiscale modeling ability of the UNets. outperforms demonstrating the value of using kspace intermediate quantities. outperforms showing the importance of learning sensitivity maps as part of the network. It is worth noting that the relative performance does not depend on the type of mask or the mask parameters. Some example reconstructions are shown in figure 2.
Accel()  Num ACS()  Model  SSIM 

0.818  
4  30  0.919  
0.922  
0.923  
0.795  
6  22  0.904  
0.907  
0.907  
0.782  
8  16  0.889  
0.891  
0.893 
Accel()  Frac ACS()  Model  SSIM 

0.813  
4  0.08  0.906  
0.907  
0.910  
0.767  
6  0.06  0.886  
0.887  
0.892  
0.762  
8  0.04  0.870  
0.871  
0.878 
4.2.1 Significance of learning sensitivity maps
Figure 3 shows the SSIM values for each model with various equispaced mask parameters. In all cases, learning the sensitivity maps improves the SSIM score. Notably, this improvement in SSIM is larger when the number of low frequency lines is smaller. As previously stated, the quality of the estimated sensitivity maps tends to be poor when there are few ACS lines, which leads to a degradation in the final reconstruction quality. The model is able to overcome this limitation and generate good reconstructions even with a small number of ACS lines.
4.2.2 Experiments on test data
Dataset  Model  4 Acceleration  8 Acceleration  
SSIM  NMSE  PSNR  SSIM  NMSE  PSNR  
E2EVN  0.930  0.005  40  0.890  0.009  37  
SubtleMR  0.929  0.005  40  0.888  0.009  37  
Knee  AIRS Medical  0.929  0.005  40  0.888  0.009  37 
SigmaNet  0.928  0.005  40  0.888  0.009  37  
iRIM  0.928  0.005  40  0.888  0.009  37  
Brain  E2EVN  0.959  0.004  41  0.943  0.008  38 
UNet  0.945  0.011  38  0.915  0.023  35 
Table 3 shows our results on the test datasets for both the brain and knee MRIs compared with the best models on the fastMRI leaderboard^{2}^{2}2http://fastmri.org/leaderboards. To obtain these results, we used the same training procedure as our previous experiments, except that we trained on both the training and validation sets for 100 epochs. We used the same type of masks that are used for the fastMRI paper [4]. Our model outperforms all other models published on the fastMRI leaderboard for both anatomies.
5 Conclusion
In this paper, we introduced EndtoEnd Variational Networks for multicoil MRI reconstruction. While MRI reconstruction can be posed as an inverse problem, multicoil MRI reconstruction is particularly challenging because the forward process (which is determined by the sensitivity maps) is not completely known. We alleviate this problem by estimating the sensitivity maps within the network, and learning fully endtoend. Further, we explored the architecture space to identify the best neural network layers and intermediate representation for this problem, which allowed our model to obtain new stateofthe art results on both brain and knee MRIs. The quantitative measures we have used only provide a rough estimate for the quality of the reconstructions. Many clinically important details tend to be subtle and limited to small regions of the MR image. Rigorous clinical validation needs to be performed before such methods can be used in clinical practice to ensure that there is no degradation in the quality of diagnosis.
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6 Supplementary Materials
6.1 Dithering as postprocessing
The Structural Similarity (SSIM) loss [16]
we used to train our models has a tendency to produce overly smooth reconstructions even when all of the diagnostic content is preserved. We noticed a similar behavior with other frequently used loss functions like mean squared error, mean absolute error, etc. Sriram et al.
[14] found that dithering the image by adding a small amount of random gaussian noise helped enhance the perceived sharpness of their reconstructions. We found that the same kind of dithering helped improve the sharpness of our reconstructions, but we tuned the scale of the noise by manual inspection. Similar to [14], we adjusted the scale of the added noise to the brightness of the image around each pixel to avoid obscuring dark areas of the reconstruction. Specifically, we first normalize the image by dividing each pixel by the maximum pixel intensity. Then we add zeromean random gaussian noise to each pixel. The standard deviation of the noise at a given pixel location is equal to
times the square root of the local median computed over a patch of pixels around that pixel location. We set for the brain images and non fat suppressed knee images, and for the fat suppressed knee images. Example reconstructions with and without noise are shown in creftypeplural 7654. The dithered images look more natural, especially at higher accelerations.







