Least-Square NUFFT Methods Applied to 2-D and 3-D Radially Encoded MR Image Reconstruction

Radially encoded MRI has gained increasing attention due to its motion insensitivity and reduced artifacts. However, because its samples are collected nonuniformly in the $k$-space, multidimensional (especially 3-D) radially sampled MRI image reconstruction is challenging. The objective of this paper is to develop a reconstruction technique in high dimensions with on-the-fly kernel calculation. It implements general multidimensional nonuniform fast Fourier transform (NUFFT) algorithms and incorporates them into a $k$-space image reconstruction framework. The method is then applied to reconstruct from the radially encoded $k$-space data, although the method is applicable to any non-Cartesian patterns. Performance comparisons are made against the conventional Kaiser–Bessel (KB) gridding method for 2-D and 3-D radially encoded computer-simulated phantoms and physically scanned phantoms. The results show that the NUFFT reconstruction method has better accuracy–efficiency tradeoff than the KB gridding method when the kernel weights are calculated on the fly. It is found that for a particular conventional kernel function, using its corresponding deapodization function as a scaling factor in the NUFFT framework has the potential to improve accuracy. In particular, when a cosine scaling factor is used, the NUFFT method is faster than KB gridding method since a closed-form solution is available and is less computationally expensive than the KB kernel (KB griding requires computation of Bessel functions). The NUFFT method has been successfully applied to 2-D and 3-D in vivo studies on small animals.