  Magnetic resonance imaging serves as an essential tool for clinical
diagnosis. However, it suffers from a long acquisition time. The utilization of
deep learning, especially the deep generative models, offers aggressive
acceleration and better reconstruction in magnetic resonance imaging.
Nevertheless, learning the data distribution as prior knowledge and
reconstructing the image from limited data remains challenging. In this work,
we propose a novel Hankel-k-space generative model (HKGM), which can generate
samples from a training set of as little as one k-space data. At the prior
learning stage, we first construct a large Hankel matrix from k-space data,
then extract multiple structured k-space patches from the large Hankel matrix
to capture the internal distribution among different patches. Extracting
patches from a Hankel matrix enables the generative model to be learned from
redundant and low-rank data space. At the iterative reconstruction stage, it is
observed that the desired solution obeys the learned prior knowledge. The
intermediate reconstruction solution is updated by taking it as the input of
the generative model. The updated result is then alternatively operated by
imposing low-rank penalty on its Hankel matrix and data consistency con-strain
on the measurement data. Experimental results confirmed that the internal
statistics of patches within a single k-space data carry enough information for
learning a powerful generative model and provide state-of-the-art
reconstruction.
