  Optimizing k-space sampling trajectories is a promising yet challenging topic
for fast magnetic resonance imaging (MRI). This work proposes to optimize a
reconstruction method and sampling trajectories jointly concerning image
reconstruction quality in a supervised learning manner. We parameterize
trajectories with quadratic B-spline kernels to reduce the number of parameters
and apply multi-scale optimization, which may help to avoid sub-optimal local
minima. The algorithm includes an efficient non-Cartesian unrolled neural
network-based reconstruction and an accurate approximation for backpropagation
through the non-uniform fast Fourier transform (NUFFT) operator to accurately
reconstruct and back-propagate multi-coil non-Cartesian data. Penalties on slew
rate and gradient amplitude enforce hardware constraints. Sampling and
reconstruction are trained jointly using large public datasets. To correct for
possible eddy-current effects introduced by the curved trajectory, we use a
pencil-beam trajectory mapping technique. In both simulations and in-vivo
experiments, the learned trajectory demonstrates significantly improved image
quality compared to previous model-based and learning-based trajectory
optimization methods for 10 times acceleration factors. Though trained with
neural network-based reconstruction, the proposed trajectory also leads to
improved image quality with compressed sensing-based reconstruction.
