# Linear reconstructions and the analysis of the stable sampling rate

## TL;DR

This paper analyzes the stable sampling rate (SSR) in linear reconstruction methods, showing its linearity with Walsh functions and wavelets, and providing exact SSR calculations for certain wavelets to improve reconstruction accuracy.

## Contribution

It demonstrates the linearity of SSR in binary Walsh sampling and wavelet reconstruction, and derives exact SSR values for specific wavelets, enhancing understanding of sampling efficiency.

## Key findings

- SSR is linear with Walsh functions and wavelet reconstruction.
- Exact SSR can be determined for certain wavelets.
- Results enable sharper performance estimates for reconstruction methods.

## Abstract

The theory of sampling and the reconstruction of data has a wide range of applications and a rich collection of techniques. For many methods a core problem is the estimation of the number of samples needed in order to secure a stable and accurate reconstruction. This can often be controlled by the Stable Sampling Rate (SSR). In this paper we discuss the SSR and how it is crucial for two key linear methods in sampling theory: generalized sampling and the recently developed Parametrized Background Data Weak (PBDW) method. Both of these approaches rely on estimates of the SSR in order to be accurate. In many areas of signal and image processing binary samples are crucial and such samples, which can be modelled by Walsh functions, are the core of our analysis. As we show, the SSR is linear when considering binary sampling with Walsh functions and wavelet reconstruction. Moreover, for certain wavelets it is possible to determine the SSR exactly, allowing sharp estimates for the performance of the methods.