# Free-Breathing and Ungated Cardiac MRI Reconstruction Using a Deep Kernel Representation
###### Abstract  
Free-breathing and ungated cardiac MRI is a challenging problem due to the cardiac motion and respiration motion, which are not tracked. In this work, we propose an unsupervised deep kernel method for reconstructing real-time free-breathing and ungated cardiac MRI from highly undersampled k-t space measurements. We propose implementing the feature map and kernel function in the kernel method using CNNs. The parameters of the CNNs are learned from specific-subject data directly. Comparisons with state-of-the-art kernel methods show improved performance of the proposed deep kernel method.  
convolutional neural networks; real-time cardiac MRI; kernel method; image reconstruction +
Footnote †: journal: Medical Imaging Research Center, Dallas, TX 75390,  
imaging [15; 18]. This leads to the fact that the manifold approaches are often associated with high memory demand. Here, real-time images refer to high spatial resolution images using the data acquired continuously without binning.  
To resolve the issue of high memory demand from manifold approaches and to promote a computationally efficient framework, kernel methods [19] have been introduced. The basic idea of kernel methods is to first embed the original data in the original space into a suitable feature space and then use algorithms to discover the relations for the embedded data. The solution of kernel methods usually comprises two modules: (1) Embedding the data in the original space into the feature space using some non-linear feature maps and (2) Solving (linear) equations to discover the linear patterns in the feature space. The advantage of kernel methods comes from the better data representation using features that are obtained by non-linear operations. Because of the advantages of kernel methods, they are adapted for MRI recovery [20; 21; 22]. However, in the existing kernel methods, the non-linear feature maps and the kernels (defined by the inner product of features) are usually chosen empirically, and the parameters in the associated feature maps and/or the kernel functions are manually tuned based on the problems at hand. This often leads to the sub-optimal performance of the kernel methods.  
In this work, we show that the two modules in the kernel representation can be realized by the cascade of two convolutional neural networks (CNN). CNN is a special type of artificial neural network that uses the convolution operator instead of general matrix multiplication in the layers of the neural networks. The CNNs in this work are then learned based on the available undersampled data. The main advantage of the proposed kernel method using CNN is the elimination of the manual choice for the feature maps and the kernels. Deep CNN is known to be able to provide better feature extraction [23]. In this work, we leverage the power of the CNN for feature extraction to implement the feature map in the proposed kernel method as a CNN. The CNN are then learned from the undersampled data, and hence no manual selection is needed. The inner products of the features, which are the kernels, can also be implemented using a one-layer CNN. This indicates the data representation in the proposed kernel method (linear combination of the kernels) can be implemented using the cascade of two CNNs, followed by a fully connected layer. Another advantage of the proposed deep kernel method is that this scheme is totally unsupervised. Unlike most of the current CNN-based MRI reconstruction schemes [24; 25; 26], which require a large amount of fully-sampled training data to train the network, the proposed scheme requires only undersampled data. We should note that in the free-breathing and ungated cardiac MRI setting, fully-sampled data is usually not accessible. The proposed deep kernel method is subject-specific, and hence the subject-specific CNN parameters are learned directly from the undersampled data based on the specific subject.