# Free-Breathing and Ungated Cardiac MRI Reconstruction Using a Deep Kernel Representation
## 3 Deep Kernel Method for Dynamic MRI Recovery
### Deep Kernel Representation  
In kernel representation, the choice of the kernel function is important after determining the non-linear feature map. The kernel function is an inner product of the features. Once it is a well-defined inner product, we can get a valid kernel function. In most of the current kernel methods, the kernel functions are chosen empirically. Popular choices of the kernel functions include the radial Gaussian kernel [31]:  
\[\mathcal{K}_{m,n}=\exp\left(-\frac{||\varphi_{m}-\varphi_{n}||^{2}}{2\sigma^{ 2}}\right),\]and the Dirichlet kernel with bandwidth \(d\)[32]:  
\[\mathcal{K}_{m,n}=\sum_{\mathbf{k}\in\mathbb{Z}^{n},i\|\mathbf{k}\|\leq d}\exp \big{(}j2\pi\mathbf{k}^{\mathrm{T}}(\varphi_{m}-\varphi_{n})\big{)}.\]  
Besides the empirical choice of the kernel functions, the parameters in the kernel functions, such as \(\sigma\) in the radial Gaussian kernel and \(d\) in the Dirichlet kernel, need manual tuning. This significantly limits the performance of the kernel methods. In this work, we implement the kernel function as a convolutional layer. In other word, we define  
\[\mathcal{K}_{m,n}=\varphi_{m}*\bar{\varphi_{n}}, \tag{2}\]  
where \(\bar{\varphi_{n}}\) is the intrinsic features in the convolutional layer. \(\bar{\varphi_{n}}\) is learned directly from the data. We use the word \"intrinsic features\" to distinguish the features \(\varphi_{m}\) learned from \(\mathcal{A}^{H}\mathbf{b}\) using the feature extraction CNN as described in Section 3.1. The convolution of the feature \(\varphi_{m}\) with the learned intrinsic feature \(\bar{\varphi_{n}}\) is an inner product [33]. So we have a valid kernel function. The advantage of implementing the kernel function as a convolutional layer is the elimination of the empirical choice of the kernel function and the manual tuning of the kernel parameters. Once we have the kernel matrix \(\mathbf{K}=[\mathcal{K}_{m,n}]\) with \(\mathcal{K}_{m,n}\) defined as (2), we can have the following image representation:  
\[\mathbf{x}_{i}=\mathbf{K}\mathbf{\alpha}_{i},\quad i=1,\cdots,M.\]  
The multiplication \(\mathbf{K}\mathbf{\alpha}_{i}\) can be implemented using a linear layer and hence \(\mathbf{\alpha}_{i},i=1,\cdots,M\) can be learned from the data.  
With the aforementioned description of kernel representation using CNNs, we can have the following deep kernel representation for the free-breathing and ungated cardiac MRI representation:  
\[\mathbf{x}_{i}=\Big{(}\Phi_{\theta}(\mathcal{A}^{H}\mathbf{b})*\bar{\mathbf{ \varphi}}\Big{)}\mathbf{\alpha}_{i},\quad i=1,\cdots,M.\]  
Here \(\Phi_{\theta}\) is the feature extraction network with parameters \(\theta\), \(\bar{\mathbf{\varphi}}=[\bar{\varphi_{n}}]\) is the features in the one-layer CNN for the kernel matrix calculation, and \(\mathbf{\alpha}_{i}\) is the parameters in the linear  
Figure 1: Illustration of the deep kernel representation. The upper part shows the feature extraction CNN. A U-net without skip connection is used to implement the feature extraction operator in this work. The lower part gives the illustration of the deep kernel representation. The feature maps, which are obtained using the CNN \(\Phi_{\theta}\), are first reshaped into the size \((1\times)n_{x}\cdot n_{y}\times\ell\), where \(n_{x}\times n_{y}\) denotes the image size. Then a convolutional layer is used to obtain the kernel matrix \(\mathbf{K}\). The final reconstructed images in the time series are obtained using a fully-connect (FC) layer and images reshape.  
layer. The parameters \(\theta,\bar{\mathbf{\varphi}},\mathbf{\alpha}_{i}\) are learned using back-propagation [34]. The illustration of the deep kernel representation is depicted in Figure 1.