# On the stable sampling rate for binary measurements and wavelet reconstruction

## TL;DR

This paper demonstrates that binary measurements, modeled with Walsh functions and Hadamard matrices, have a linear stable sampling rate for wavelet reconstruction, making them as efficient as Fourier samples for infinite-dimensional signal recovery.

## Contribution

It establishes the linear stable sampling rate for binary measurements in wavelet reconstruction, enabling the use of binary sampling in advanced reconstruction techniques.

## Key findings

- Binary measurements have a linear stable sampling rate.
- Binary measurements are as efficient as Fourier samples with wavelets.
- The results facilitate the use of binary measurements in generalized sampling methods.

## Abstract

This paper is concerned with the problem of reconstructing an infinite-dimensional signal from a limited number of linear measurements. In particular, we show that for binary measurements (modelled with Walsh functions and Hadamard matrices) and wavelet reconstruction the stable sampling rate is linear. This implies that binary measurements are as efficient as Fourier samples when using wavelets as the reconstruction space. Powerful techniques for reconstructions include generalized sampling and its compressed versions, as well as recent methods based on data assimilation. Common to these methods is that the reconstruction quality depends highly on the subspace angle between the sampling and the reconstruction space, which is dictated by the stable sampling rate. As a result of the theory provided in this paper, these methods can now easily use binary measurements and wavelet reconstruction bases.