A numerical method to solve a phaseless coefficient inverse problem from a single measurement of experimental data

TL;DR

This paper introduces a two-stage numerical method for reconstructing the refractive index of scatterers from intensity measurements alone, combining asymptotic analysis and a globally convergent algorithm, effective on both simulated and experimental data.

Contribution

The paper presents a novel two-stage approach that reduces a phaseless inverse scattering problem to a phased one and applies a globally convergent method, avoiding the need for initial guesses.

Findings

Accurately reconstructs refractive index from noisy experimental data.

Effectively reduces phaseless problem to phased problem using asymptotic estimates.

Demonstrates robustness on both simulated and real-world measurements.

Abstract

We propose in this paper a globally numerical method to solve a phaseless coefficient inverse problem: how to reconstruct the spatially distributed refractive index of scatterers from the intensity (modulus square) of the full complex valued wave field at an array of light detectors located on a measurement board. The propagation of the wave field is governed by the 3D Helmholtz equation. Our method consists of two stages. On the first stage, we use asymptotic analysis to obtain an upper estimate for the modulus of the scattered wave field. This estimate allows us to approximately reconstruct the wave field at the measurement board using an inversion formula. This reduces the phaseless inverse scattering problem to the phased one. At the second stage, we apply a recently developed globally convergent numerical method to reconstruct the desired refractive index from the total wave obtained at the first stage. Unlike the optimization approach, the two-stage method described above is global in the sense that it does not require a good initial guess of the true solution. We test our numerical method on both computationally simulated and experimental data. Although experimental data are noisy, our method produces quite accurate numerical results.