Spiral FFT: An efficient method for 3-D FFTS on spiral MRI contours

The Fast Fourier Transform (FFT) allows the Discrete Time Fourier Transform (DTFT) to be efficiently sampled on a uniform grid in frequency. In many applications, including Magnetic Resonance Imaging (MRI), uniform measurements are undesirable or impractical. Non-equispaced measurements in the Fourier domain are typically obtained through methods that use FFT values to interpolate the DTFT at off-grid locations. These algorithms, known as NUFFTs, are prohibitively expensive for large data sets in 3-D because of the interpolation cost. This paper proposes an exact transform called the SpiralFFT capable of sampling the DTFT on spiral patterns in 3-D frequency space. The SpiralFFT uses spiral structure to replace 3-D calculations with 1-D FFTs and chirp Z-transforms (CZTs). Simulations compare the SpiralFFT with a NUFFT algorithm on a realistic 3-D MRI data set. Results show that the SpiralFFT exhibits a factor of 8 increase in speed for comparable accuracy, and 8 orders of magnitude improvement in accuracy for comparable execution time. These results demonstrate the potential use of the SpiralFFT in spiral MRI to improve reconstruction speed and quality.