20,758 research outputs found
Theoretical and Experimental Analysis of a Randomized Algorithm for Sparse Fourier Transform Analysis
We analyze a sublinear RAlSFA (Randomized Algorithm for Sparse Fourier
Analysis) that finds a near-optimal B-term Sparse Representation R for a given
discrete signal S of length N, in time and space poly(B,log(N)), following the
approach given in \cite{GGIMS}. Its time cost poly(log(N)) should be compared
with the superlinear O(N log N) time requirement of the Fast Fourier Transform
(FFT). A straightforward implementation of the RAlSFA, as presented in the
theoretical paper \cite{GGIMS}, turns out to be very slow in practice. Our main
result is a greatly improved and practical RAlSFA. We introduce several new
ideas and techniques that speed up the algorithm. Both rigorous and heuristic
arguments for parameter choices are presented. Our RAlSFA constructs, with
probability at least 1-delta, a near-optimal B-term representation R in time
poly(B)log(N)log(1/delta)/ epsilon^{2} log(M) such that
||S-R||^{2}<=(1+epsilon)||S-R_{opt}||^{2}. Furthermore, this RAlSFA
implementation already beats the FFTW for not unreasonably large N. We extend
the algorithm to higher dimensional cases both theoretically and numerically.
The crossover point lies at N=70000 in one dimension, and at N=900 for data on
a N*N grid in two dimensions for small B signals where there is noise.Comment: 21 pages, 8 figures, submitted to Journal of Computational Physic
FPS-SFT: A Multi-dimensional Sparse Fourier Transform Based on the Fourier Projection-slice Theorem
We propose a multi-dimensional (M-D) sparse Fourier transform inspired by the
idea of the Fourier projection-slice theorem, called FPS-SFT. FPS-SFT extracts
samples along lines (1-dimensional slices from an M-D data cube), which are
parameterized by random slopes and offsets. The discrete Fourier transform
(DFT) along those lines represents projections of M-D DFT of the M-D data onto
those lines. The M-D sinusoids that are contained in the signal can be
reconstructed from the DFT along lines with a low sample and computational
complexity provided that the signal is sparse in the frequency domain and the
lines are appropriately designed. The performance of FPS-SFT is demonstrated
both theoretically and numerically. A sparse image reconstruction application
is illustrated, which shows the capability of the FPS-SFT in solving practical
problems
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