Generalized sampling reconstruction from Fourier measurements using compactly supported shearlets

TL;DR

This paper investigates the stable reconstruction of compactly supported functions from finite Fourier samples using shearlet systems, demonstrating near-linear sampling rates and comparing them to wavelet-based methods.

Contribution

It introduces a generalized sampling framework for shearlet-based reconstruction from Fourier measurements and analyzes its stability and efficiency.

Findings

Stable recovery is achievable with almost linear sampling rates.

Shearlet-based reconstruction outperforms wavelet-based methods in certain scenarios.

The method provides a practical approach for Fourier-based signal reconstruction.

Abstract

In this paper we study the general reconstruction of a compactly supported function from its Fourier coefficients using compactly supported shearlet systems. We assume that only finitely many Fourier samples of the function are accessible and based on this finite collection of measurements an approximation is sought in a finite dimensional shearlet reconstruction space. We analyse this sampling and reconstruction process by a recently introduced method called generalized sampling. In particular by studying the stable sampling rate of generalized sampling we then show stable recovery of the signal is possible using an almost linear rate. Furthermore, we compare the result to the previously obtained rates for wavelets.