It is often difficult to get complete information in the Fourier domain associated with a signal due to long data acquisition time or the presence of information that cannot be obtained from the measuring equipment. In particular, the undersampled or phaseless Fourier data causes severe artifacts or loss of contents in the spatial domain. To address this issue, the studies to reconstruct the image from the missing Fourier data have been actively carried out. Although conventional approaches such as compressed sensing (CS) and alternating projection-based methods provide suitable results, their computational complexity is relatively high due to the inevitable iterative operation required in the algorithms. Furthermore, the performance was somewhat limited and very sensitive to initialization or the presence of noise. In this paper, we propose deep learning based algorithms to deal with the reconstruction of the incomplete Fourier data in accelerated MRI and phase retrieval applications. Recently, deep convolutional neural networks (CNNs) have been widely employed to reconstruct images from missing k-space and phase information. While these approaches provide significant performance gain over traditional methods, it is not clear how to choose an appropriate network architecture to balance the trade-off between network complexity and performance. First, we propose an algorithm to reconstruct the undersampled k-space data for accelerated MRI based on the geometric analysis of an encoder-decoder CNN. Specifically, a novel attention scheme combined bootstrapping and subnetwork aggregation improves the expressivity of network with a minimal increase in complexity, resulting in a performance gain. This study is based on supervised learning in the presence of sufficient pairs of undersampled k-space data and matched fully sampled k-space data. However, in real-world situations such as time-resolved MR angiography (tMRA), where a ground-truth image with high spatio-temporal resolution is not available, it is not possible to obtain reference data to be used as labels for supervised learning. To restore the signal from the subsampled k-space data of tMRA, we propose a novel unsupervised training scheme using modified cycle-consistent generative adversarial network (cycleGAN) inspired by the optimal transport (OT) theory. Thanks to the simple and stable training, the proposed method can not only generate high quality
reconstruction results, but also provide the various trade-offs between spatial and temporal resolution depending on user’s choice. The problem of missing information in the Fourier domain is not limited to subsampling. In the fields such as optic and astronomy, phase information cannot be achievable from Fourier data, but the magnitude of the k-space can be. With regard to phase retrieval, since it is also difficult to obtain a large amount of matched dataset that consists of phaseless Fourier data and ground-truth data, we propose a neural network implementation of a modern convex relaxation method in the unsupervised learning framework. Experimental results confirmed
that the proposed algorithm for phase retrieval yields superior results compared to the conventional methods.