Fast Padé transform for magnetic resonance imaging and computerized tomography

This paper illustrates the implementation of the multidimensional fast Padé transform (FPT), which we have recently devised for magnetic resonance imaging and computerized tomography. The FPT uses the nonlinear epsilon (ε) algorithm of Wynn to accelerate the sequence of fast Fourier transforms generated with signals of gradually increasing length N=2m where m is a non-negative integer. Several illustrative computations are presently performed for one- two- and three-dimensional (1D, 2D, 3D) numerical quadratures whose accuracy is controlled solely by 2m. The unprecedented numerical precision is obtained to within 12 decimal places using m=10 (N=1024=1K). Convergence of FPT to this level of spectacular accuracy is extremely fast, since barely 64 and 256 equidistantly sampled points can secure four and eight decimal places, respectively. This is expected to introduce major improvements into Fourier-based spectroscopy, image reconstructions and computerized tomography, since merely post processing Padé–Wynn acceleration of Fourier sequences of varying length 2m (m⩽10) is capable of extracting more information from experimentally recorded data than any advance in hardware would ever be able to accomplish.