  Magnetic resonance imaging (MRI) is mainly limited by long scanning time and
vulnerable to human tissue motion artifacts, in 3D clinical scenarios. Thus,
k-space undersampling is used to accelerate the acquisition of MRI while
leading to visually poor MR images. Recently, some studies 1) use effective
undersampling patterns, or 2) design deep neural networks to improve the
quality of resulting images. However, they are considered as two separate
optimization strategies. In this paper, we propose a cross-domain network for
MR image reconstruction, in a retrospective data-driven manner, under limited
sampling rates. Our method can simultaneously obtain the optimal undersampling
pattern (in k-space) and the reconstruction model, which are customized to the
type of training data, by using an end-to-end learning strategy. We propose a
1D probabilistic undersampling layer, to obtain the optimal undersampling
pattern and its probability distribution in a differentiable way. We propose a
1D inverse Fourier transform layer, which connects the Fourier domain and the
image domain during the forward pass and the backpropagation. In addition, by
training 3D fully-sampled k-space data and MR images with the traditional
Euclidean loss, we discover the universal relationship between the probability
distribution of the optimal undersampling pattern and its corresponding
sampling rate. Experiments show that the quantitative and qualitative results
of recovered MR images by our 1D probabilistic undersampling pattern obviously
outperform those of several existing sampling strategies.
