  Purpose: To develop a deep learning method on a nonlinear manifold to explore
the temporal redundancy of dynamic signals to reconstruct cardiac MRI data from
highly undersampled measurements.
  Methods: Cardiac MR image reconstruction is modeled as general compressed
sensing (CS) based optimization on a low-rank tensor manifold. The nonlinear
manifold is designed to characterize the temporal correlation of dynamic
signals. Iterative procedures can be obtained by solving the optimization model
on the manifold, including gradient calculation, projection of the gradient to
tangent space, and retraction of the tangent space to the manifold. The
iterative procedures on the manifold are unrolled to a neural network, dubbed
as Manifold-Net. The Manifold-Net is trained using in vivo data with a
retrospective electrocardiogram (ECG)-gated segmented bSSFP sequence.
  Results: Experimental results at high accelerations demonstrate that the
proposed method can obtain improved reconstruction compared with a compressed
sensing (CS) method k-t SLR and two state-of-the-art deep learning-based
methods, DC-CNN and CRNN.
  Conclusion: This work represents the first study unrolling the optimization
on manifolds into neural networks. Specifically, the designed low-rank manifold
provides a new technical route for applying low-rank priors in dynamic MR
imaging.
