The nonuniform discrete Fourier transform (NDFT) can be computed with a fast algorithm, referred to as the nonuniform fast Fourier transform (NFFT). In L dimensions, the NFFT requires O(N(-ln #) L + ( Q L #=1 M # ) P L #=1 log M # ) operations, where M # is the number of Fourier components along dimension #, N is the number of irregularly spaced samples, and # is the required accuracy. This is a dramatic improvement over the O(N Q L #=1 M # ) operations required for the direct evaluation (NDFT). The performance of the NFFT depends on the lowpass filter used in the algorithm. A truncated Gauss pulse, proposed in the literature, is optimized. A newly proposed filter, a Gauss pulse tapered with a Hanning window, performs better than the truncated Gauss pulse and the B-spline, also proposed in the literature. For small filter length, a numerically optimized filter shows the best results. Numerical experiments for 1-D and 2-D implementations confirm the theoretically predicted ...

s general reconstruction optimized such that the Fourier transform of the #lter is closest to a desired bandlimiting #lter in the least squares sense. Midpoint-o#set reconstruction along crosslines After the reposting the midpoint-o#set reconstruction can be applied per crossline. The algorithm used for this has been discussed bySchonewille and Duijndam #1996#. The starting point is the inverse transform from a twodimensional spatial Fourier domain to the midpoint-o#set domain, per temporal frequency !: P #xk;h k ;!#= #kx 2# #k h 2# P l ~ P #k x;l ;k h;l ;!#e ,j#k x;l x k +k h;l h p k # ; #3# where xk and hk denote the midpoint and o#set coordinate of spatial sample k respectively. The power p takes the value 1 for the standard Fourier transform or linear Radon transform over the o#set axis and the value 2 for the Fourier transform after quadratic stretching of the o#set axis, or, the parabolic Radon transform. Wemay de#ne kh = !p for the linear Radon transform, where ...