  Purpose: To rapidly reconstruct undersampled 3D non-Cartesian image-based
navigators (iNAVs) using an unrolled deep learning (DL) model for non-rigid
motion correction in coronary magnetic resonance angiography (CMRA).
  Methods: An unrolled network is trained to reconstruct beat-to-beat 3D iNAVs
acquired as part of a CMRA sequence. The unrolled model incorporates a
non-uniform FFT operator to perform the data consistency operation, and the
regularization term is learned by a convolutional neural network (CNN) based on
the proximal gradient descent algorithm. The training set includes 6,000 3D
iNAVs acquired from 7 different subjects and 11 scans using a variable-density
(VD) cones trajectory. For testing, 3D iNAVs from 4 additional subjects are
reconstructed using the unrolled model. To validate reconstruction accuracy,
global and localized motion estimates from DL model-based 3D iNAVs are compared
with those extracted from 3D iNAVs reconstructed with $\textit{l}_{1}$-ESPIRiT.
Then, the high-resolution coronary MRA images motion corrected with
autofocusing using the $\textit{l}_{1}$-ESPIRiT and DL model-based 3D iNAVs are
assessed for differences.
  Results: 3D iNAVs reconstructed using the DL model-based approach and
conventional $\textit{l}_{1}$-ESPIRiT generate similar global and localized
motion estimates and provide equivalent coronary image quality. Reconstruction
with the unrolled network completes in a fraction of the time compared to CPU
and GPU implementations of $\textit{l}_{1}$-ESPIRiT (20x and 3x speed
increases, respectively).
  Conclusion: We have developed a deep neural network architecture to
reconstruct undersampled 3D non-Cartesian VD cones iNAVs. Our approach
decreases reconstruction time for 3D iNAVs, while preserving the accuracy of
non-rigid motion information offered by them for correction.
