We are concerned with the inverse medium scattering problem for the Helmholtz equation in ( or 3). This inverse problem can be considered as a model problem for the inverse scattering of time-harmonic acoustic waves or time-harmonic TE-polarized electromagnetic waves from bounded inhomogeneous media. It has been one of the central problems in inverse scattering theory and has a wide range of applications including nondestructive testing, radar imaging, medical imaging, and geophysical exploration [Colto2013]. Needless to say, there has been a large body of literature on both theoretical and numerical studies on this inverse problem, see [Colto2000c, Colto2013] and references therein.
In the present paper, we are interested in determining the location and shape of scattering objects from (near-field or far-field) multi-static data of the scattered field. Since we study sampling methods to numerically solve this inverse problem, we will mainly discuss related results in this direction. The Linear Sampling Method (LSM) can be considered as the first sampling method developed to solve the inverse problem under consideration [Colto1996]. The LSM aims to construct an indicator function for unknown scattering objects. This indicator function is evaluated on sampling points obtained by discretizing some domain in which the unknown target is searched for. The evaluation of this indicator function is typically fast, non-iterative and its construction does not require advanced a priori information about the unknown target. These are also the main advantages of the LSM over nonlinear optimization-based methods in solving inverse scattering problems. Shortly after the finding of the LSM, other sampling type methods for inverse problems including the point source method [Potth1996], the Factorization method (FM) [Kirsc1998], the probe method [Ikeha1998b] were also developed. We refer to [Potth2006] for a discussion on sampling and probe methods studied until 2006. These methods have been later extended to solve various inverse problems, see [Potth2006, Kirsc2008, Cakon2011] and references therein.
Our work in this paper is inspired by a class of sampling methods that have been studied more recently. We are particularly interested in the orthogonality sampling method (OSM) proposed in [Potth2010]. While inheriting the advantages of the classical sampling methods mentioned above, the OSM is particularly attractive thanks to its simplicity and efficiency. For instance, the implementation of the OSM only involves an evaluation of an inner product or some double integral (no regularization is needed). The method is extremely robust with respect to noise in the data and its stability can be easily justified. However, the theoretical analysis of the OSM is far less developed compared with that of the classical sampling methods, especially the FM and LSM. The original paper [Potth2010] justifies the OSM for small scattering objects using one-wave data. Its multifrequency version has been theoretically investigated in [Gries2008]. The theoretical analysis of the OSM for extended scatterers using multi-static data has been studied in [Liu2017]. The OSM has been recently extended to Maxwell’s equations in [Nguye2019, Harri2020]. We also refer to [Liu2017, Ji2019, Ito2012, Ito2013, Harri2019, Kang2018, Park2018] for studies on direct sampling methods (DSM) which are closely related to the OSM. To our knowledge, most of the works for the OSM and DSM have been studied for inverse scattering problems with far-field data. The data is either the scattering amplitude of the scattered field or the scattered field measured on some boundary that is far away from unknown targets. The latter is typically the case when the well known Helmholtz-Kirchhoff identity is used to justify the sampling method, see for instance [Ito2012, Ito2013, Kang2018].
We propose in this paper two novel imaging functionals for solving the inverse problem of interest. These imaging functionals are inspired by the OSM. They have the advantages of the OSM and are applicable to both near-field and far-field data. More precisely, our imaging functionals use (near-field or far-field) boundary Cauchy data. Moreover, they can also be easily modified to use only the scattered field data in the case of far-field measurements. We study the theoretical analysis of the first imaging functional using the analysis of Factorization method and the Funk-Hecke formula. The idea is to use a relation between the Cauchy data and the scattering amplitude of the scattered field. This relation allows us to rewrite our imaging functional as where is the far-field operator and is some special test function. Then using a factorization of , analytical properties of the operators in the factorization and the Funk-Hecke formula it can be justified that the imaging functional can work as an approximate indicator function for the unknown scattering object. We also refer to [Liu2017] for using the Factorization method to justify a direct sampling method.
The theoretical analysis of the second imaging functional relies on the Helmholtz integral representation of the imaginary part of the Green’s function of the direct scattering problem. The representation allows us to show that the second imaging functional is equal to a functional that involves where is a sampling point and the point is inside the unknown scatterer. This function is the main ingredient for justifying the behavior of the imaging functional. Transforming an imaging functional into some functional involving the function using the Helmholtz-Kirchhoff identity has been done for the justification of the direct sampling method, see, e.g. [Ito2012, Ito2013, Kang2018], and also the time reverse migration technique [Garni2016]. However, our approach in this paper for the second imaging functional does not rely on the Helmholtz-Kirchhoff identity and therefore avoids the assumption that the measurement has to be taken far away from the unknown scattering object. Another way to avoid using the Helmholtz-Kirchhoff identity for the OSM can be found in [Akinc2016] for a 2D inverse scattering problem.
It can be seen from our numerical study that the reconstruction results obtained from the two imaging functionals are essentially the same. The justification of the first imaging functional is more satisfactory compared with that of the second one. However, we have to assume that the wave number is not an interior transmission eigenvalue in order to apply the analysis of the Factorization method (see in addition Assumption1).
The paper is organized as follows. We will formulate the inverse scattering problem of interest and an analysis of the Factorization method in Section 2
. The formulae, theoretical analysis and stability estimates of the first and second imaging functionals are respectively presented in Sections3 and 4. Section 5 is dedicated to a numerical study of the imaging functionals.
2 The inverse scattering problem and Factorization method
In this section we formulate the inverse problem we want to solve and recall some necessary ingredients of the Factorization method for the theoretical analysis of our imaging functionals. Consider a penetrable inhomogeneous medium that occupies a bounded Lipschitz domain ( or 3). Assume that this medium is characterized by the bounded function and that in . Consider the incident plane wave
where is the wave number and
is the direction vector of propagation. We consider the following model problem for the scattering ofby the inhomogeneous medium
where is the total field, is the scattered field, and the Sommerfeld radiation condition (3) holds uniformly for all directions . If is connected and , this scattering problem is known to have a unique weak solution , see [Colto2013].
Inverse problem. Consider a domain such that and denote by the outward normal unit vector to at . We aim to determine from and for all .
and that the scattered field has the asymptotic behavior
for all . The function is called the scattering amplitude or the far-field pattern of the scattered field . Let be the far-field operator defined by
where is the scattering amplitude of the unique solution to
Note that this problem is just problem (1)–(3) rewritten for the scattered field with incident field replaced by . By linearity of problem (1)–(3), is just the scattering amplitude of solution to problem (6)–(7) with , defined by
Now we define the compact operator as Then obviously the far-field operator can be factorized as
Let be the adjoint of given by
and we define as
which leads to the factorizations
To proceed further with the analysis of the Factorization method we need to briefly discuss the interior transmission eigenvalues. We call an interior transmission eigenvalue if the problem
has a nontrivial solution such that .
It is known that the set of real transmission eigenvalues is at most discrete. See [Cakon2016] and the references therein for the analysis of transmission eigenvalue problems. For the next results, we assume that is not an interior transmission eigenvalue. The following assumption is also important for the Factorization method analysis.
We assume that , and that there exists a constant such that for almost all .
The following theorem of the Factorization method is important to the theoretical analysis of the imaging functional studied in this section, see [Audib2014] for a proof of the theorem.
3 The imaging functional
We define the imaging functional as
where is given in (9) and is the scattering amplitude of the Green’s function , given by
The next theorem justifies the behavior of the imaging functional .
Assume that is not an interior transmission eigenvalue and that Assumption 1 holds true. Then the imaging functional satisfies
where is the distance from to .
From the Helmholtz integral representation for we have
which implies that
Then substituting this formula of in the far-field operator implies that
Since (the surface area of ), we have
Using the coercivity of in Theorem 2 and we obtain
Now let . Then by Theorem 2 again we have which implies that for some . We estimate
This deduces that for with .
Now for , using the Funk-Hecke formula (see [Colto2013]) we obtain
Therefore, using the asymptotic behavior of and as we obtain that
This completes the proof. ∎
In practice the data are always perturbed with some noise. We assume the noisy data and satisfy
for some positive constants . We now give a stability estimate for the imaging functional .
Theorem 4 (stability estimate).
Denote by the imaging functional corresponding to noisy data and . Then
Let and be the scattering amplitude and the far-field operator for noisy Cauchy data. That means
Using the Cauchy-Schwarz inequality we have
Let . This leads to
which implies that
Similarly we also have
Using and the triangle inequality we have
proving the theorem. ∎
We note that if the measurements are taken far away from the scattering medium, by the radiation condition we can replace by in the imaging functional . Then the modified imaging functional
only needs the scattered field data and approximates the imaging functional .
4 The imaging functional
We define our second imaging functional as
where is the Green’s function given in (4).
The imaging functional satisfies
Using the asymptotic behavior of and as we can easily estimate, for , that
For , is supposed to have finite positive values because and peak when .
Like in the stability study of the first imaging functional we can estimate
This estimate implies the stability of the imaging functional with respect to noise in the data.
As in the case of imaging functional , we can also modify to handle only the scattered field data that is measured far away from the scattering medium as follows
Furthermore, to use point sources instead of incident plane waves we can just replace in the first integral of or by some boundary or surface where the sources are located.
5 Numerical examples
We present in this section several numerical examples in 2D to illustrate the performance of the proposed imaging functionals and . More precisely, we will examine the performance of these functionals for data with different wave numbers (Figure 1), highly noisy data (Figure 2), far-field data (Figure 3), and partial data (Figure 4). Reconstruction results using the imaging functionals and are also presented in the case of far-field data. As we can see in the pictures below, the reconstruction results obtained from and are very similar. Therefore, we will present more results of and skip some of the results of to avoid repetition of similar pictures. For the pictures in this section, the imaging functionals are scaled by dividing by their maximal values.
The following common parameters and notations are used in the numerical examples
The following scattering objects are considered in the numerical examples.
a) Kite-shaped object
b) Disk-and-rectangle object
c) Square-shaped object with cavity
To generate the scattering data for the numerical examples, we solve the Lippmann-Schwinger equation (5) using a spectral Galerkin method developed in [Lechl2014]. Using incident plane waves and measuring the data at points on , the Cauchy data , where , are then matrices. We add artificial noise to these data as follows. Two complex-valued noise matrices
containing random numbers that are uniformly distributed in the complex square
are added to the data matrices. For simplicity we consider the same noise level for both and . The noisy data and are given by