Fast imaging of scattering obstacles from phaseless far-field measurements at a fixed frequency

TL;DR

This paper introduces a fast and robust direct imaging algorithm for locating and shaping scattering obstacles using phaseless far-field data generated by superpositions of two plane waves at a fixed frequency, overcoming translation invariance issues.

Contribution

The paper develops a novel direct imaging method that efficiently reconstructs obstacle location and shape from phaseless data without prior boundary condition knowledge, applicable to multiple obstacles.

Findings

The algorithm is fast and easy to implement.

It is stable and robust to noise.

It can reconstruct multiple obstacles with different boundary conditions.

Abstract

This paper is concerned with the inverse obstacle scattering problem with phaseless far-field data at a fixed frequency. The main difficulty of this problem is the so-called translation invariance property of the modulus of the far-field pattern or the phaseless far-field pattern generated by one plane wave as the incident field, which means that the location of the obstacle can not be recovered from such phaseless far-field data at a fixed frequency. It was recently proved in our previous work \cite{XZZ18} that the obstacle can be uniquely determined by the phaseless far-field patterns generated by infinitely many sets of superpositions of two plane waves with different directions at a fixed frequency if the obstacle is a priori known to be a sound-soft or an impedance obstacle with real-valued impedance function. The purpose of this paper is to develop a direct imaging algorithm to reconstruct the location and shape of the obstacle from the phaseless far-field data corresponding to infinitely many sets of superpositions of two plane waves with a fixed frequency as the incident fields. Our imaging algorithm only involves the calculation of the products of the measurement data with two exponential functions at each sampling point and is thus fast and easy to implement. Further, the proposed imaging algorithm does not need to know the type of boundary conditions on the obstacle in advance and is capable to reconstruct multiple obstacles with different boundary conditions. Numerical experiments are also carried out to illustrate that our imaging method is stable, accurate and robust to noise.