Abstract
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.
Highlights
The raw data for magnetic resonance imaging (MRI) is measured in k-space, the domain of spatial frequencies, where non-Cartesian sampling schemes like spiral or radial scans have received much attention
For readers not familiar with the nonequispaced fast Fourier transform (NFFT), we suggest to read the appendix of this paper first
We suggest the conjugate gradient method for the reconstruction problem and show that the solution is efficiently computed by the iterative use of the nonequispaced fast Fourier transform (FFT)
Summary
The raw data for magnetic resonance imaging (MRI) is measured in k-space, the domain of spatial frequencies, where non-Cartesian sampling schemes like spiral or radial scans have received much attention. In contrast to the use of the computationally efficient fast Fourier transform (FFT) for the reconstruction from Cartesian grids, the more general sampling trajectories ask for the so-called nonequispaced FFTs. On the other hand, iterative image reconstruction algorithms play an important role in modern tomographic systems [1]. Iterative image reconstruction in combination with the nonequispaced FFT has been applied to data on spiral k-space trajectories [2] and in the presence of field inhomogeneities [3]. Efficient reconstruction procedures for sensitivity encoding with arbitrary kspace trajectories were proposed in [4]. Its authors present methods that combine the gridding principles with the conjugate gradient scheme, but mention the long computation times due to their nonoptimised preliminary software
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