In magnetic resonance imaging (MRI), methods that use a non-Cartesian grid in k-space are becoming increasingly important. In this paper, we use a recently proposed implicit discretisation scheme which generalises the standard approach based on gridding. While the latter succeeds for sufficiently uniform sampling sets and accurate estimated density compensation weights, the implicit method further improves the reconstruction quality when the sampling scheme or the weights are less regular. Both approaches can be solved efficiently with the nonequispaced FFT. Due to several new techniques for the storage of an involved sparse matrix, our examples include also the reconstruction of a large 3D data set. We present four case studies and report on efficient implementation of the related algorithms.
