Non-Uniform Fast Fourier Transformation of SPRITE MRI Data

A new algorithm has been developed which permits extremely accurate and fast computation of the Discrete Fourier Transform for certain sets of non-uniformly sampled data. Specifically, this algorithm can be used to process the data acquired during a Magnetic Resonance Imaging experiment using the SPRITE technique with Multiple Point Acquisition, which samples data non-uniformly in a spatial frequency domain. Non-uniform MRI data is generally processed with an interpolation or regridding algorithm which computes an approximate answer in O(NlogN+Nlog(1/ )) time; the accuracy depends on the parameter and is typically one part in 10. The algorithm developed herein uses a combination of Chirp Z-Transforms (CZT) to compute the Fourier Transform to within one part in 10 (which is on the order of the machine epsilon) in O(NTNlogN) time, where N is the number of points reconstructed and NT is the number of samples collected at each step of the SPRITE experiment. When MRI data is processed by the CZT algorithm, the result is an improvement in the signal-to-noise ratio of the final image, by a factor of √ NT . Though a resolution increase is also observed in some images, it can be demonstrated that this effect is due entirely to the SNR improvement. This result allows us to use a modified version of the transform which operates in O(NlogN) time for all practical purposes. This algorithm has immediate application in the rapid imaging of low-signal materials, and may also be developed for other uses, such as the numerical solution of differential equations.