Two reconstruction procedures for a 3D phaseless inverse scattering problem for the generalized Helmholtz equation

Abstract

The 3-d inverse scattering problem of the reconstruction of the unknown dielectric permittivity in the generalized Helmholtz equation is considered. The main difference with the conventional inverse scattering problems is that only the modulus of the scattering wave field is measured. The phase is not measured. The initializing wave field is the incident plane wave. On the other hand, in the previous recent works of the authors about the "phaseless topic" the case of the point source was considered [20,21,22]. Two reconstruction procedures are developed for a linearized case. However, the linearization is not the Born approximation. This means that, unlike the Born approximation, our linearization does not break down when the frequency tends to the infinity. Applications are in imaging of nanostructures and biological cells.

A Numerical Method to Solve a Phaseless Coefficient Inverse Problem from a Single Measurement of Experimental Data

“…Both these theorems use ideas of the Riemannian geometry and asymptotic analysis. While Theorem 3.1 was actually proved in [43], Theorem 4.1 is completely new. As a result, our upper estimate (4.6) of |u sc (x, k)| 2 | Pmeas is a reasonable one for the given range of parameters.…”

“…3 Asymptotic behavior of the total wave as k → ∞ In this section we establish the asymptotic behavior of the function u (x, k) and at k → ∞. Although results of this section follow from [43], we need to formulate these results again here since we essentially use them in our numerical method.…”

“…For an arbitrary point x ∈ {x 3 > −a} the travel time along the geodesic line Γ(x, a) from the plane P a to the point x is (see [43])…”

“…We refer the reader to other versions of the uniqueness theorems for 3D phaseless CIPs in [31,33,34,44]. As mentioned above, the analytic reconstruction procedures in for 3D phaseless CIPs were proposed in [40,41,42,43] for the case when the intensity of the scattered rather than full wave field is measured.…”

A reference ball based iterative algorithm for imaging acoustic obstacle from phaseless far-field data

“…The advantage of introducing the reference ball is that we would be able to obtain the location of the obstacle via this iterative method, since the update information (∆c 1 , ∆c 2 ) about the location of the obstacle is contained in the term ℜ(A ∞ 2 (p 2 , ψ 2 )A ′∞ 1 [p 1 , ψ 1 ]q) of equation (16).…”