# Benchmarking MRI Reconstruction Neural Networks on Large Public Datasets
## 3 Models
### Neural Networks
#### 3.3.2 Cross-Domains Networks  
The second class of networks we introduce, we term cross-domain networks. The key intuitive idea is that they correct the data in both the k-space and the image space alternatively, using the Fourier transform to go from one space to the other. They are derived from the optimization algorithms used to solve the optimization problems introduced before, using the idea of \"unrolling\" introduced in [31]. An illustration of this class of networks is presented in Figure 2.  
Because these networks work directly on the input data (and not on a primarily reconstructed version of it), they need to handle complex-valued data. In particular, the classical deep learning frameworks (TensorFlow and Pytorch) do not feature the ability to perform complex convolutions off-the-shelf. The way convolution is performed in the original papers is, therefore, to concatenate the real and imaginary of the image (respectively the k-space), making it a two-channel image, performing the series of convolutions, and having the output be a two-channel image then transformed back in a complex image (respectively k-space). The **Cascade-net**[11] is based on the dictionary learning optimization Problem (4). The idea is to replace the dictionary learning step by convolutional neural networks and still keep the data consistency step in the k-space. The optimization algorithm is then unrolled to allow us to perform back-propagation. The authors of [11] show that we can perform back-propagation through the data consistency step (which is linear) and derive the corresponding Jacobian. The parameters used here for the implementation are the same as those in the original paper, except the number of filters, which was decreased from 64 to 48 to fit on a single GPU. This network is illustrated in Figure 3.  
Figure 1: Illustration of the U-net from [28]. In our case, the output is not a segmentation map but a reconstructed image of the same size (we perform zero-padding to prevent decreasing sizes in convolutions). Figure 3: Illustration of the Cascade-net from [11]. Here, each \(C_{i}\) is a convolutional block of 64 filters (48 in our implementation) followed by a ReLU non-linearity, \(n_{d}\) is the number of such convolutional blocks forming a convolutional subnetwork between each data consistency layer \(DC\), and \(n_{c}\) is the number of convolutional subnetworks. Figure 2: The common backbone between the Cascade net, the KIKI-net, and the PD-net. US mask stands for under-sampling mask. DC stands for data consistency. (I)FFT stands for (Inverse) Fast Fourier Transform. \(N_{k,d}\) is the number of convolution layers applied in the k-space. \(N_{i,d}\) is the number of convolution layers applied in the image space. \(N_{C}\) is the total number of alternations between the k-space and the image-space. It is to be noted that in the case of PD-net, the data consistency step is not performed, the Fourier operators are performed with the original under-sampling mask, and a buffer is concatenated along with the current iteration to allow for some memory between iterations and to learn the acceleration (in the k-space net—dual net—it is also concatenated with the original k-space input). In the case of the Cascade net, \(N_{k,d}=0\), only the data consistency is performed in the k-space. In the case of the KIKI-net, there is no residual connection in the k-space. However, the k-space nets and image space nets could potentially be any kind of image-to-image neural network. The **KIKI-net**[10] is an extension of the Cascade-net where they additionally perform convolutions after the data consistency step in the k-space. The parameters used here for the implementation are the same as those in the original paper. This network is illustrated in Figure 4.  
The **Primal-Dual-net (PD-net)** was introduced by [9] and applied to MRI by [32], is based on the wavelet-based denoising (3), and in particular, the resolution of the corresponding optimization problem with the PDHG [23] algorithm.