A practical guide to the recovery of wavelet coefficients from Fourier measurements

TL;DR

This paper presents a computationally efficient implementation of generalized sampling for recovering wavelet coefficients from Fourier measurements, achieving near-optimal complexity and stability.

Contribution

It demonstrates how to implement generalized sampling efficiently, reducing computational complexity to O(M(N) log N) for wavelet coefficient recovery from Fourier data.

Findings

Achieves computational complexity of O(M(N) log N)

Provides a stable and optimal recovery method

Links the number of Fourier samples to wavelet coefficients linearly

Abstract

In a series of recent papers (Adcock, Hansen and Poon, 2013, Appl. Comput. Harm. Anal. 45(5):3132-3167), (Adcock, Gataric and Hansen, 2014, SIAM J. Imaging Sci. 7(3):1690-1723) and (Adcock, Hansen, Kutyniok and Ma, 2015, SIAM J. Math. Anal. 47(2):1196-1233), it was shown that one can optimally recover the wavelet coefficients of an unknown compactly supported function from pointwise evaluations of its Fourier transform via the method of generalized sampling. While these papers focused on the optimality of generalized sampling in terms of its stability and error bounds, the current paper explains how this optimal method can be implemented to yield a computationally efficient algorithm. In particular, we show that generalized sampling has a computational complexity of $\mathcal{O}(M(N)\log N)$ when recovering the first $N$ boundary-corrected wavelet coefficients of an unknown compactly supported function from $M(N)$ Fourier samples. Therefore, due to the linear correspondences between the number of samples $M$ and number of coefficients $N$ shown previously, generalized sampling offers a computationally optimal way of recovering wavelet coefficients from Fourier data.