Nonuniform DFT Applications in MRI: A Novel Algorithm and Parallel Implementation

Abstract

Typically, the discrete Fourier transform (DFT) is performed on a sequence of data points that are equidistant on a given domain; however, it may be desirable to relax this constraint. In certain types of magnetic resonance imaging (MRI), there is the opportunity to take several unequally spaced samples simultaneously. In one MRI scheme known as “pure phase encoding”, each time a point is sampled in Fourier space it is possible to sample several nearby points very cheaply, which can improve the resulting image. Reconstructing an image from these points requires an inverse non-uniform DFT.

This paper presents a novel algorithm that uses numerical cubature with Delaunay triangulation to perform a 2D inverse DFT on unequally spaced data in the spatial frequency (k-space) domain. The additional data points contribute positively to spatial resolution and provide a better image in the same scan time. The intrinsic parallelism in the algorithm is exploited to carry out efficient implementation on an IBM SP parallel computer.