The Regular Fourier Matrices and Nonuniform Fast Fourier Transforms

Abstract

For any triple of positive integers (m, N, q), the matrix F (m, N, q), called the (m, N, q)-regular Fourier matrix, is defined. The regular Fourier matrices F (m, N, q) are then applied to set up new algorithms for nonuniform fast Fourier transforms. Numerical results show that the accuracies obtained by our algorithms are much better than previously reported results with the same computation complexity. The algorithms require O(N · log 2 N ) arithmetic operations, where N is the number of data points.

On NUFFT-based gridding for non-Cartesian MRI

“…But in the absence of such optimization, the conventional KB approach for gridding is preferable to the LS_NUFFT approach with conventional scaling factors including cosine scaling factors [1,7]. Interpolator cores for the LS_NUFFT interpolator with cosine scaling factors, and for the conventional Kaiser-Bessel interpolator, normalized to unity maximum.…”

“…Furthermore, the frequency ω enters the expressions above only in the form ω − γ k 0 (ω). So due to (11), the LS_NUFFT optimal interpolator satisfies exactly the same type of shift invariance seen in the ideal interpolator (7).…”

“…Unfortunately there is no apparent analytical optimizer for s. Therefore, one must use numerical methods to optimize s. In the literature, several choices for s have been proposed, all of which have the form (22) for various choices for the function Ψ(t), such as uniform factors, i.e., Ψ(t) = 1, cosine factors [7]: Ψ(t) = cos(πt), and Gaussian factors [7,8]: with σ chosen to minimize E max. For the "type 2" NUFFT we have recommended Kaiser-Bessel factors previously [4].…”

“…The (positive) s n values are called scaling factors [7]. The design problem is to choose the scaling factors s = (s 0 , …, s N−1 ) and the interpolator φ k (·) to minimize the approximation error in (5).…”

“…If one imposed no constraints on φ k , then (5) could be made to be exact by choosing uniform scaling factors s n = 1, and by using the following ideal interpolator: (6) for k = 0, …, K − 1, where (7) where η 0 ≜ (N − 1)/2 and δ N denotes the following K-periodic Dirichlet kernel: (8) Note that δ N (κ), the "core" of the ideal interpolator, is real; the only complex aspect of (7) is the linear phase term. Furthermore, the κ − k argument in (6) indicates an (integer) shift invariance.…”