On optimal wavelet reconstructions from Fourier samples: linearity and universality of the stable sampling rate

TL;DR

This paper demonstrates that generalized sampling allows for stable, accurate wavelet coefficient reconstruction from Fourier samples with a linearly growing sample size, establishing near-optimality and fundamental limits of the method.

Contribution

It proves the linear relationship between Fourier samples and wavelet coefficients, derives the exact proportionality constant for Daubechies wavelets, and shows the near-optimality and limitations of generalized sampling.

Findings

Stable and accurate reconstruction with linearly growing Fourier samples.

Exact proportionality constant for Daubechies wavelets.

Generalized sampling is nearly optimal and cannot be significantly improved.

Abstract

In this paper we study the problem of computing wavelet coefficients of compactly supported functions from their Fourier samples. For this, we use the recently introduced framework of generalized sampling. Our first result demonstrates that using generalized sampling one obtains a stable and accurate reconstruction, provided the number of Fourier samples grows linearly in the number of wavelet coefficients recovered. For the class of Daubechies wavelets we derive the exact constant of proportionality. Our second result concerns the optimality of generalized sampling for this problem. Under some mild assumptions we show that generalized sampling cannot be outperformed in terms of approximation quality by more than a constant factor. Moreover, for the class of so-called perfect methods, any attempt to lower the sampling ratio below a certain critical threshold necessarily results in exponential ill-conditioning. Thus generalized sampling provides a nearly-optimal solution to this problem.