Rapid analysis of non-uniformly sampled pulsed field gradient data for velocity estimation

Bretthorst's recent generalization of the Lomb-Scargle periodogram shows that a sufficient statistic for frequency estimation from non-uniformly, but simultaneously sampled quadrature data is equivalent to the FFT of those data with the missing samples replaced by zeros. We have applied this concept to the rapid analysis of pulsed field gradient MRI data which have been non-uniformly sampled in the velocity encoding wave vector q. For a small number of q samples, it is more computationally efficient to calculate the periodogram directly rather than using the FFT algorithm with a large number of zeros. The algorithm we have implemented for finding the peak of the generalized periodogram is simple and robust; it involves repeated apodization and grid searching of the periodogram until the desired velocity resolution is achieved. The final estimate is refined by quadratic interpolation. We have tested the method for fully developed Poiseuille flow of a Newtonian fluid and have demonstrated substantial improvement in the precision of velocity measurement achievable in a fixed acquisition time with non-uniform sampling. The method is readily extendible to multidimensional data. Analysis of a 256 by 256 pixel image with 8 q samples and an effective velocity resolution of better than 1/680 of the Nyquist range requires approximately 1 minute computation time on a 400 MHz SUN Ultrasparc II processor.