Imaging of locally rough surfaces from intensity-only far-field or near-field data

Abstract

This paper is concerned with a nonlinear imaging problem, which aims to reconstruct a locally perturbed, perfectly reflecting, infinite plane from intensity-only (or phaseless) far-field or near-field data. A recursive Newton iteration algorithm in frequencies is developed to reconstruct the locally rough surface from multi-frequency intensity-only far-field or near-field data, where the fast integral equation solver developed in [39] is used to solve the direct scattering problem in each iteration. For the case with far-field data, a main feature of our work is that the incident field is taken as a superposition of two plane waves with different directions rather than one plane wave, so the location and shape of the local perturbation of the infinite plane can be reconstructed simultaneously from intensity-only far-field data with multiple wave numbers. This is different from previous work on inverse scattering from phaseless far-field data, where only the shape reconstruction was considered due to the translation invariance property of the phaseless far-field pattern corresponding to one plane wave as the incident field. Finally, numerical examples are carried out to demonstrate that our reconstruction algorithm is stable and accurate even for the case of multiple-scale profiles.

Uniqueness in Inverse Scattering Problems with Phaseless Far-Field Data at a Fixed Frequency

“…Uniqueness results and stability have also been established for inverse scattering problems with phaseless near-field data (see [16,17,19,24,26,27,30,37,42,43] for the acoustic and potential scattering case and [29,37] for the electromagnetic scattering case).Recently in [38], it was proved that the translation invariance property of the phaseless far-field pattern can be broken by using superpositions of two plane waves as the incident fields with an interval of frequencies. Following this idea, several algorithms have been developed for inverse acoustic scattering problems with phaseless far-field data, based on using the superposition of two plane waves as the incident field (see [38,39,40]). Further, by using the spectral properties of the far-field operator, rigorous uniqueness results have also been established in [34] for inverse acoustic scattering problems with phaseless far-field data generated by infinitely many sets of superpositions of two plane waves with different directions at a fixed frequency, under certain a priori assumptions on the property of the scatterers.…”

Uniqueness in inverse cavity scattering problems with phaseless near-field data

“…An effective attempt in this direction is the superposition of distinct incident plane waves proposed in [49]. This idea leads to the multi-frequency Newton iteration algorithm [49,50] and the fast imaging algorithm at a fixed frequency [51]. Further, by the superposition of two incident plane waves, uniqueness results were established in [46] under some a priori assumptions.…”

Rough surface reconstruction from phaseless single frequency data at grazing angles

“…The surfaces studied consisted of superposition of typically 5 or 6 sinusoidal spatial components, and Landweber iteration was employed. In another study [26] a recursive Newton iteration was used to recover two-scale and piecewise linear surfaces from intensities of multi-frequency far-field data. A somewhat related problem of two-dimensional target characterization from diffracted intensity was tackled in [27].…”