Reconstruction Procedures for Two Inverse Scattering Problems Without the Phase Information

Abstract

This is a continuation of two recent publications of the authors [17,18] about reconstruction procedures for 3-d phaseless inverse scattering problems. The main novelty of this paper is that, unlike [18], the Born approximation for the case of the wave-like equation is not considered. It is shown here that the phaseless inverse scattering problem for the 3-d wave-like equation in the frequency domain leads to the well known Inverse Kinematic Problem. Uniqueness theorem follows. Still, since the Inverse Kinematic Problem is very hard to solve, a linearization is applied. More precisely, geodesic lines are replaced with straight lines. As a result, an approximate explicit reconstruction formula is obtained via the inverse Radon transform. The second reconstruction method is via solving a problem of the integral geometry using integral equations of the Abel type.

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

“…However, there is no translation invariance property for phaseless near-field data. Therefore, many numerical algorithms for inverse scattering problems with phaseless near-field data have been developed (see, e.g., [3,4,6,18,32,36] for the acoustic case and [5] for the electromagnetic case). 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.…”

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

“…[12,22] and the references therein). Recently, a great deal of effort has been devoted to phaseless inverse scattering problems [1,4,5,23,24,25,26]. The motivation for investigating phaseless inverse problems is mainly due to the fact that such phase information is extremely difficult to be measured accurately or even completely unavailable in a rich variety of realistic scenarios.…”

Travel time tomography with formally determined incomplete data in 3D

“…The TTTP has important applications in geophysics [7,24,33,34]. In addition, it was established in [15] that the TTTP arises in the phaseless inverse problem of scattering of electromagnetic waves at high frequencies. The specific TTTP considered here has potential applications in geophysics, checking the bulky baggage in airports, search for possible defects inside the walls, etc..…”