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