Sampling linear inverse problems with noise

TL;DR

This paper investigates how additive noise affects the inversion of Fourier integral operators, using microlocal defect measures to analyze noise transformation, with applications to the Radon transform and numerical demonstrations.

Contribution

It introduces a microlocal analysis framework to quantify noise propagation in inverse problems involving FIOs, including explicit formulas for noise standard deviation after inversion.

Findings

Derived formulas for noise amplification in FIO inversion

Analyzed noise effects in Radon transform in different geometries

Provided numerical examples validating theoretical results

Abstract

We study the effect of additive noise to the inversion of FIOs associated to a diffeomorphic canonical relation. We use the microlocal defect measures to measure the power spectrum of the noise and analyze how that power spectrum is transformed under the inversion. In particular, we compute the standard deviation of the noise added to the inversion as a function of the standard deviation of the noise added to the data. As an example, we study the Radon transform in the plane in parallel and fan-beam coordinates, and present numerical examples.