Fast Computation of Partial Fourier Transforms
Provides faster computation for partial Fourier transforms, benefiting seismic imaging and related fields, but the improvement is incremental over existing fast algorithms.
The paper introduces two algorithms for partial Fourier transforms, achieving O(N log^2 N) complexity in 1D and O(N^2 log^2 N) in 2D, motivated by wave extrapolation in seismology.
We introduce two efficient algorithms for computing the partial Fourier transforms in one and two dimensions. Our study is motivated by the wave extrapolation procedure in reflection seismology. In both algorithms, the main idea is to decompose the summation domain of into simpler components in a multiscale way. Existing fast algorithms are then applied to each component to obtain optimal complexity. The algorithm in 1D is exact and takes $O(N\log^2 N)$ steps. Our solution in 2D is an approximate but accurate algorithm that takes $O(N^2 \log^2 N)$ steps. In both cases, the complexities are almost linear in terms of the degree of freedom. We provide numerical results on several test examples.