Linear Stable Sampling Rate: Optimality of 2D Wavelet Reconstructions from Fourier Measurements

TL;DR

This paper proves that the number of Fourier samples needed for stable 2D wavelet reconstructions scales linearly with the number of wavelet functions, supported by theoretical analysis and numerical experiments.

Contribution

It establishes the optimal linear sampling rate for 2D wavelet reconstructions from Fourier data, including boundary wavelets, within the generalized sampling framework.

Findings

Sampling rate scales linearly with wavelet basis size

Numerical experiments confirm theoretical predictions

Applicable to both separable and boundary wavelets

Abstract

In this paper we analyze two-dimensional wavelet reconstructions from Fourier samples within the framework of generalized sampling. For this, we consider both separable compactly-supported wavelets and boundary wavelets. We prove that the number of samples that must be acquired to ensure a stable and accurate reconstruction scales linearly with the number of reconstructing wavelet functions. We also provide numerical experiments that corroborate our theoretical results.