Reconstructing Small Perturbations of an Obstacle for Acoustic Waves from Boundary Measurements on the Perturbed Shape Itself

TL;DR

This paper develops rigorous mathematical relationships linking small shape deformations of an obstacle to boundary measurements of acoustic scattering fields, enabling effective algorithms for reconstructing shape perturbations from boundary data.

Contribution

It introduces a systematic derivation of asymptotic expansions for boundary operators in terms of shape perturbations using layer potential techniques and the field expansion method.

Findings

Derived asymptotic expansions of DNO and NDO operators for small shape perturbations.

Established relationships between shape deformation and boundary measurements.

Developed algorithms to determine Fourier coefficients of shape perturbations.

Abstract

We derive relationships between the shape deformation of an impenetrable obstacle and boundary measurements of scattering fields on the perturbed shape itself. Our derivation is rigourous by using systematic way, based on layer potential techniques and the field expansion (FE) method (formal derivation). We extend these techniques to derive asymptotic expansions of the Dirichlet-to-Neumann (DNO) and Neumann-to-Dirichlet (NDO) operators in terms of the small perturbations of the obstacle as well as relationships between the shape deformation of an obstacle and boundary measurements of DNO or NDO on the perturbed shape itself. All relationships lead us to very effective algorithms for determining lower-order Fourier coefficients of the shape perturbation of the obstacle.