  There is growing interest in learning Fourier domain sampling strategies
(particularly for magnetic resonance imaging, MRI) using optimization
approaches. For non-Cartesian sampling patterns, the system models typically
involve non-uniform FFT (NUFFT) operations. Commonly used NUFFT algorithms
contain frequency domain interpolation, which is not differentiable with
respect to the sampling pattern, complicating the use of gradient methods. This
paper describes an efficient and accurate approach for computing approximate
gradients involving NUFFTs. Multiple numerical experiments validated the
improved accuracy and efficiency of the proposed approximation. As an
application to computational imaging, the NUFFT Jacobians were used to optimize
non-Cartesian MRI sampling trajectories via data-driven stochastic
optimization. Specifically, the sampling patterns were learned with respect to
various model-based reconstruction algorithms, including quadratic regularized
reconstruction and compressed sensing-based reconstruction. The proposed
approach enables sampling optimization for image sizes that are infeasible with
standard auto-differentiation methods due to memory limits. The synergistic
acquisition and reconstruction lead to remarkably improved image quality. In
fact, we show that model-based image reconstruction (MBIR) methods with
suitably optimized imaging parameters can perform nearly as well as CNN-based
methods.
