8 research outputs found
Target reconstruction with a reference point scatterer using phaseless far field patterns
An important property of the phaseless far field patterns with incident plane
waves is the translation invariance. Thus it is impossible to reconstruct the
location of the underlying scatterers. By adding a reference point scatterer
into the model, we design a novel direct sampling method using the phaseless
data directly. The reference point technique not only overcomes the translation
invariance, but also brings a practical phase retrieval algorithm. Based on
this, we propose a hybrid method combining the novel phase retrieval algorithm
and the classical direct sampling methods. Numerical examples in two dimensions
are presented to demonstrate their effectiveness and robustness
Uniqueness in inverse electromagnetic scattering problem with phaseless far-field data at a fixed frequency
This paper is concerned with uniqueness in inverse electromagnetic scattering
with phaseless far-field pattern at a fixed frequency. In our previous work
[{\em SIAM J. Appl. Math.} {\bf 78} (2018), 3024-3039], by adding a known
reference ball into the acoustic scattering system, it was proved that the
impenetrable obstacle and the index of refraction of an inhomogeneous medium
can be uniquely determined by the acoustic phaseless far-field patterns
generated by infinitely many sets of superpositions of two plane waves with
different directions at a fixed frequency. In this paper, we extend these
uniqueness results to the inverse electromagnetic scattering case. The
phaseless far-field data are the modulus of the tangential component in the
orientations and , respectively, of the
electric far-field pattern measured on the unit sphere and generated by
infinitely many sets of superpositions of two electromagnetic plane waves with
different directions and polarizations. Our proof is mainly based on Rellich's
lemma and the Stratton--Chu formula for radiating solutions to the Maxwell
equations.Comment: arXiv admin note: text overlap with arXiv:1806.0912
Uniqueness and direct imaging method for inverse scattering by locally rough surfaces with phaseless near-field data
This paper is concerned with inverse scattering of plane waves by a locally
perturbed infinite plane (which is called a locally rough surface) with the
modulus of the total-field data (also called the phaseless near-field data) at
a fixed frequency in two dimensions. We consider the case where a Dirichlet
boundary condition is imposed on the locally rough surface. This problem models
inverse scattering of plane acoustic waves by a one-dimensional sound-soft,
locally rough surface; it also models inverse scattering of plane
electromagnetic waves by a locally perturbed, perfectly reflecting, infinite
plane in the TE polarization case. We prove that the locally rough surface is
uniquely determined by the phaseless near-field data generated by a countably
infinite number of plane waves and measured on an open domain above the locally
rough surface. Further, a direct imaging method is proposed to reconstruct the
locally rough surface from the phaseless near-field data generated by plane
waves and measured on the upper part of the circle with a sufficiently large
radius. Theoretical analysis of the imaging algorithm is derived by making use
of properties of the scattering solution and results from the theory of
oscillatory integrals (especially the method of stationary phase). Moreover, as
a by-product of the theoretical analysis, a similar direct imaging method with
full far-field data is also proposed to reconstruct the locally rough surface.
Finally, numerical experiments are carried out to demonstrate that the imaging
algorithm with phaseless near-field data and full far-field data are fast,
accurate and very robust with respect to noise in the data
Imaging of buried obstacles in a two-layered medium with phaseless far-field data
The inverse problem we consider is to reconstruct the location and shape of
buried obstacles in the lower half-space of an unbounded two-layered medium in
two dimensions from phaseless far-field data. A main difficulty of this problem
is that the translation invariance property of the modulus of the far field
pattern is unavoidable, which is similar to the homogenous background medium
case. Based on the idea of using superpositions of two plane waves with
different directions as the incident fields, we first develop a direct imaging
method to locate the position of small anomalies and give a theoretical
analysis of the algorithm. Then a recursive Newton-type iteration algorithm in
frequencies is proposed to reconstruct extended obstacles. Finally, numerical
experiments are presented to illustrate the feasibility of our algorithms
Inverse elastic scattering problems with phaseless far field data
This paper is concerned with uniqueness, phase retrieval and shape
reconstruction methods for inverse elastic scattering problems with phaseless
far field data. Systematically, we study two basic models, i.e., inverse
scattering of plane waves by rigid bodies and inverse scattering of sources
with compact support. For both models, we show that the location of the objects
can not be uniquely recovered by the data. To solve this problem, we consider
simultaneously the incident point sources with one fixed source point and at
most three scattering strengths. We then establish some uniqueness results for
source scattering problem with multi-frequency phaseless far field data.
Furthermore, a fast and stable phase retrieval approach is proposed based on a
simple geometric result which provides a stable reconstruction of a point in
the plane from three distances to given points. Difficulties arise for inverse
scattering by rigid bodies due to the additional unknown far field pattern of
the point sources. To overcome this difficulty, we introduce an artificial
rigid body into the system and show that the underlying rigid bodies can be
uniquely determined by the corresponding phaseless far field data at a fixed
frequency. Noting that the far field pattern of the scattered field
corresponding to point sources is very small if the source point is far away
from the scatterers, we propose an appropriate phase retrieval method for
obstacle scattering problems, without using the artificial rigid body. Finally,
we propose several sampling methods for shape reconstruction with phaseless far
field data. Extended numerical examples in two dimensions are conducted with
noisy data, and the results further verify the effectiveness and robustness of
the proposed phase retrieval techniques and sampling methods.Comment: 37 page
Uniqueness in inverse acoustic and electromagnetic scattering with phaseless near-field data at a fixed frequency
This paper is concerned with uniqueness results in inverse acoustic and
electromagnetic scattering problems with phaseless total-field data at a fixed
frequency. Motivated by our previous work ({\em SIAM J. Appl. Math. \bf78}
(2018), 1737-1753), where uniqueness results were proved for inverse acoustic
scattering with phaseless far-field data generated by superpositions of two
plane waves as the incident waves at a fixed frequency, in this paper, we use
superpositions of two point sources as the incident fields at a fixed frequency
and measure the modulus of the acoustic total-field (called phaseless acoustic
near-field data) on two spheres enclosing the scatterers generated by such
incident fields on the two spheres. Based on this idea, we prove that the
impenetrable bounded obstacle or the index of refraction of an inhomogeneous
medium can be uniquely determined from the phaseless acoustic near-field data
at a fixed frequency. Moreover, the idea is also extended to the
electromagnetic case, and it is proved that the impenetrable bounded obstacle
or the index of refraction of an inhomogeneous medium can be uniquely
determined by the phaseless electric near-field data at a fixed frequency, that
is, the modulus of the tangential component with the orientations and , respectively, of the electric total-field
measured on a sphere enclosing the scatters and generated by superpositions of
two electric dipoles at a fixed frequency located on the measurement sphere and
another bigger sphere with the polarization vectors and
, respectively. As far as we know, this is the first
uniqueness result for three-dimensional inverse electromagnetic scattering with
phaseless near-field data
An approximate factorization method for inverse acoustic scattering with phaseless near-field data
This paper is concerned with the inverse acoustic scattering problem with
phaseless near-field data at a fixed frequency. An approximate factorization
method is developed to numerically reconstruct both the location and shape of
the unknown scatterer from the phaseless near-field data generated by incident
plane waves at a fixed frequency and measured on the circle with
a sufficiently large radius . The theoretical analysis of our method is
based on the asymptotic property in the operator norm from to of the phaseless near-field operator
defined in terms of the phaseless near-field data measured on
with large enough , where is a Sobolev space on the
unit circle for real number , together with the
factorization of a modified far-field operator. The asymptotic property of the
phaseless near-field operator is also established in this paper with the theory
of oscillatory integrals. The unknown scatterer can be either an impenetrable
obstacle of sound-soft, sound-hard or impedance type or an inhomogeneous medium
with a compact support, and the proposed inversion algorithm does not need to
know the boundary condition of the unknown obstacle in advance. Numerical
examples are also carried out to demonstrate the effectiveness of our inversion
method. To the best of our knowledge, it is the first attempt to develop a
factorization type method for inverse scattering problems with phaseless data
An inverse acoustic-elastic interaction problem with phased or phaseless far-field data
Consider the scattering of a time-harmonic acoustic plane wave by a bounded
elastic obstacle which is immersed in a homogeneous acoustic medium. This paper
concerns an inverse acoustic-elastic interaction problem, which is to determine
the location and shape of the elastic obstacle by using either the phased or
phaseless far-field data. By introducing the Helmholtz decomposition, the model
problem is reduced to a coupled boundary value problem of the Helmholtz
equations. The jump relations are studied for the second derivatives of the
single-layer potential in order to establish the corresponding boundary
integral equations. The well-posedness is discussed for the solution of the
coupled boundary integral equations. An efficient and high order
Nystr\"{o}m-type discretization method is proposed for the integral system. A
numerical method of nonlinear integral equations is developed for the inverse
problem. For the case of phaseless data, we show that the modulus of the
far-field pattern is invariant under a translation of the obstacle. To break
the translation invariance, an elastic reference ball technique is introduced.
We prove that the inverse problem with phaseless far-field pattern has a unique
solution under certain conditions. In addition, a numerical method of the
reference ball technique based nonlinear integral equations is also proposed
for the phaseless inverse problem. Numerical experiments are provided to
demonstrate the effectiveness and robustness of the proposed methods.Comment: arXiv admin note: text overlap with arXiv:1811.1257