Inverse scattering merits numerous applications in radar, medical imaging, geophysics, non-destructive testing, and many others. In the last thirty years, sampling methods for shape reconstruction in inverse scattering problems have attracted a lot of interest. Classical examples include the linear sampling method  and the factorization method . Such methods are independent of any a priori information on the geometry and physical properties of the unknown scatterers. The theoretical foundation of these sampling methods is the factorization of the far field or the near field operator in the form
with some coercive operator and a pair of operators and (in some cases is the adjoint of ). The operators , , and may vary in different inverse scattering problems. With this factorization, one may derive a necessary and sufficient criteria to reconstruct the scatterer and design an indicator function which is large inside the underlying scatterer and relatively small outside. We refer to the monographs [12, 15, 28] for a comprehensive introduction.
There have been recent efforts on other types of sampling methods such as orthogonality sampling method [19, 21, 33], direct sampling method [4, 5, 22, 25, 30], single-shot method , reverse time migration , and other direct reconstruction methods [2, 3]. These sampling methods inherit many advantages of the classical ones. The main feature of these sampling methods is that only inner product of the measurements with some suitably chosen functions is involved in the imaging function and thus these sampling methods are robust to noises. Recently, a sampling type method is of the form [31, 32]
with suitably chosen and such that
where is the adjoint of given in (1.1);
is a point spread function that peaks when the sampling point located on the boundaries of the unknown scatterers (or inside the scatterers in different problems).
Based on the factorization of the far field or near field operator (1.1), we deduce that their exists two positive constant and such that
This theoretical foundation (1.3) implies that is qualitatively the same as a suitable norm of the point spread function , i.e.
which allows us to design a robust and fast imaging algorithm.
In this work, we apply the above framework to introduce a multi-frequency sampling method for the inverse source problems using sparse measurements. Most of the works in literature assume that the measurements are taken all around the unknown source. In this full-aperture case, uniqueness of the source function
can be proved using the Fourier transform, see for instance[7, 18]. We refer to  for an iterative method with respect to frequencies. Recently, some non-iterative methods have been proposed. In particular, [36, 37] studied a Fourier method to reconstruct the source function and [11, 38] investigated the direct sampling methods for identifying the point sources. One practical situation involves sparse measurements, i.e. measurements that are only available at a limited number of sensors. Generally, neither the source nor its support is uniquely determined by the sparse data. Some lower bounds for the support of a source have been established in  in terms of multi-frequency sparse far field measurements. If it is known a priori that the source is a combination of point sources, uniqueness for the locations and the scattering strengths can be established [23, 24, 26]. We refer to  and  for a factorization method and a direct sampling method, respectively, using multi-frequency sparse far field measurements. For the multi-frequency sparse near field measurements, the corresponding uniqueness and direct sampling method can be found in a recent work . Following the framework (1.1)–(1.3) developed in [31, 32] for inverse obstacle/medium scattering problems, this work contributes to a multi-frequency sampling method for the inverse source problems with sparse measurements.
The paper is further organized as follows. We introduce the model of the forward problem and inverse source problem with multi-frequency sparse near field measurements in Section 2. In contrast to the existing near field operator based on the integral over the space variable, we define a multi-frequency near field operator for each measurement point based on the integral over the frequency variable. We further provide a factorization of this multi-frequency near field operator in the form (1.1) in Section 3. This factorization is one of the key ingredients in our multi-frequency sampling method. In Section 4, we design an imaging function with properly chosen and . Furthermore we provide its theoretical foundation based on the factorization and a point spread function. Section 5 presents numerical examples to illustrate our multi-frequency sampling method with sparse near field measurements. Finally in Section 6 we briefly discuss how to extend the near field case to the far field case.
2 Mathematical model and the inverse problem
We consider the acoustic wave propagation due to a source in a homogeneous isotropic medium in with speed of sound . Denote by the support of the unknown source, which is a bounded Lipschitz domain in with connected complement . The mathematical model of the scattering of time-harmonic wave leads to the nonhomogeneous Helmholtz equation
where the wave number is with frequency . The scattered field is required to satisfy the Sommerfeld radiation condition
uniformly for all directions . Throughout the paper we write to emphasize the dependence on the wave number . If not otherwise stated, we consider multiple wave numbers in a bounded interval, i.e.,
with an upper bound .
being the fundamental solution to the Helmholtz equation.
A practical inverse problem is to recover the source support from the multi-frequency sparse measurements where the scattered fields are available at finitely many points in the set
Accordingly, we obtain the following multi-frequency sparse near field measurements
3 The multi-frequency near field operator and its factorization
In this section, we introduce a multi-frequency near field operator given by the measurements (2.8) and study its factorization which plays a key role in the analysis and design of our imaging algorithm. We assume that the source function is real-valued and satisfies that
for two constants .
To begin with, we define the scattered fields with negative frequencies by
Note that is real-valued, we then have the following integral representation for the scattered fields
For each , we define the integral operator by
which we will call the multi-frequency near field operator. Different from the usual near field operator which uses full-aperture measurements with a single frequency, this operator seeks to explore the relations among different frequencies. The factorization of the usual near field operator plays an important role in the modified sampling method  so that no asymptotic assumptions on the distance between the measurement surface and the scatterers were made there. In the same spirit, we seek to given a characterization of the unknown source support by the multi-frequency near field operator together with its factorization.
To factorize , we first define by
We can directly derive that its adjoint is given by
Finally we define the operator by
It directly follows that is a self-adjoint operator since is real-valued. Now we are ready to give the following factorization of the multi-frequency near field operator .
For each , the following factorization holds
Theorem 3.1 provides a symmetric factorization which allows us to study a factorization method, which is expected to be similar to the factorization method using multi-frequency far field measurements . Alternatively, our goal here is to utilize this factorization to introduce a multi-frequency sampling method. We remark that this type of sampling method may still work even in the case of non-symmetric factorization (see for instance ).
4 The multi-frequency sampling method
This section is devoted to a multi-frequency sampling method using sparse near field measurements (2.7). To begin with, we provide the following property of the middle operator .
Assume that satisfies (3.9). For each , the operator is self-adjoint and coercive, i.e.
We now prove the following theorem which plays an important role in the analysis and design of our imaging algorithm.
Assume that satisfies (3.9). For each , the following inequality holds
The above theorem implies that is qualitatively the same as . The following theorem gives a properly chosen such that is a point spread function.
For each and , we define by
It holds that
where . In particular, if .
where “” holds if and only if . The following asymptotic holds
From the definition of (3.13), we have that
Then equation (4.20) follows from a direct computation. It then follows that if .
Furthermore we derive that
where “” holds if and only if .
Finally the asymptotic of for large follows from (4.20). This proves the theorem. ∎
To illustrate the asymptotic property of as a function of , we refer to Figure 1 where we plot with respect to different . is a point spread function of . In particular, Figure 1 implies that the resolution seems better with larger wave number .
Now for given and sampling point , peaks at , i.e. at sampling point such that . In another word, as a function of sampling point peaks at a sphere centered at with radius . Furthermore, turning to as a function of sampling point , we expect the function peaks in the smallest annulus support centered at the measurement point with and given by (4.17). See Figure 2 for an illustration.
Unfortunately, it is not feasible to compute directly since is what we aim to reconstruct. However we are able to relate to an indicator function given by measurements. Precisely, letting in Theorem 4.2 yields that
i.e. is qualitatively the same as for a single measurement point . This implies that the measurement driven function peaks in the annulus support without the knowledge of . Consequently for all , we introduce the indicator function
To summarize, the indicator function peaks in the annulus support for a single measurement point . With the increase of measurement points, an approximate source support is expected to be reconstructed by plotting over a sampling region.
5 Numerical examples
In this section, we present some numerical examples to illustrate the performance of the multi-frequency sampling method.
We generate the synthetic data using the finite element computational software Netgen/NGSolve . To be more precise, the computational domain is and the measurements are on the sphere . We apply a radial Perfectly Matched Layer (PML) in the domain and choose PML absorbing coefficient . See Figure 3 for an illustration. In all of the numerical examples, we apply the second order finite element to solve for the wave field where the source is constant ; the mesh size is chosen as ; the set of wave numbers is . We further add Gaussian noise to the synthetic data to implement the indicator function given by (4.21) in Matlab; for the visualization, we plot over the sampling region and we always normalize it such that its maximum value is .
We first illustrate the performance of the multi-frequency sampling method with one measurement point. In this example, the measurement point is . The source has support given by a ball
We plot the exact ball, the three dimensional view of the reconstruction, and its iso-surface view with iso-value in Figure 4. As expected, the reconstruction is an annulus support of the exact ball.
What may we reconstruct with a little bit more measurements points? Now we still consider the reconstruction of the same ball (5.22) but with measurement points given by Table 1, where these measurement points are on the upper half sphere. We plot both the three dimensional reconstruction and its cross-section views in Figure 5. It is observed that the location of ball is clearly indicated with only measurement points.
To further shed light on the performance of our imaging algorithm, we consider the case when multiple (but still sparse) measurement points are available all around the source. In the following examples, we consider measurement points which are roughly equally distributed on the measurement sphere. We remark that only measurements (in three dimensional space and frequency) are used in this case. The polar coordinates of the measurement points are given in Table 2.
Figure 6 shows the reconstruction of the ball (5.22) using multiple but sparse measurements. Compared with the reconstruction using measurement points, it is observed that both the location and shape of the ball are better reconstructed.
We continue to consider a variety of different geometries of the support: a cube (reconstruction in Figure 7) given by
a rounded cylinder (reconstruction in Figure 8) given by
a peanut-shape support (reconstruction in Figure 9) given by
a L-shape support (reconstruction in Figure 10) given by
and two balls (reconstruction in Figure 10) given by
We observe from the numerical examples that (1) the annulus support of the source can be reconstructed from only one measurement point; (2) the location of a single source can be reconstructed from a few (sparse) measurement points (where we use measurement points with different frequencies); (3) both the shape and location of a single source can be well reconstructed from multiple (but sparse) measurement points (where we use measurement points with different frequencies) that are roughly equally distributed all around the source. It is also observed that the concave part of the support (see Figure 9 for the peanut and top of Figure 10 for the L-shape) may be reconstructed; (4) the imaging algorithm also works when there are multiple sources (see bottom of Figure 10).
The research of X. Liu is supported by the NNSF of China grant 11971471 and the Youth Innovation Promotion Association, CAS.
6 Appendix: extension to far field measurements
In this appendix we discuss briefly the indicator function defined by the multi-frequency sparse far field measurements. From the asymptotic behavior of Hankel functions, we deduce that the corresponding far field pattern of (2.7) has the form
We consider the following multi-frequency sparse far field measurements
In view of (6.23), we have
Therefore, the multi-frequency sparse far field measurements (6.24) gives the following data
We define the multi-frequency far field operator by
The far field operator has a factorization in the form
Here, is given by
and its adjoint is given by
The operator is a multiplication operator given by , where is the source with support . Moreover, it holds that
where and are two constants given in (3.9).
Motivated by the results in Theorem 6.1, we introduce the following indicator function
where is given by
Consequently we have for that
where is given by (6.28). Similarly, is a point spread function of .
We remark that the factorization (6.26) has been derived in  where a factorization method is investigated. As can be seen, we have allowed a complex-valued in the case of far field measurements. We finally remark that the results in this appendix can be directly extended to the two dimensional case.
-  A. Alzaalig, G. Hu, X. Liu and J. Sun, Fast acoustic source imaging using multi-frequency sparse data, Inverse Problems 36, (2020), 025009.
-  H. Ammari, J. Garnier, W. Jing, H. Kang, M. Lim, K. Sølna, and H. Wang, Mathematical and statistical methods for multistatic imaging, volume 2098. Springer, 2013.
-  H. Ammari, J. Garnier, V. Jugnon, and H. Kang, Direct reconstruction methods in ultrasound imaging of small anomalies. In Mathematical Modeling in Biomedical Imaging II, pages 31–55. Springer, 2012.
-  T. Arens, X. Ji and X. Liu, Inverse electromagnetic obstacle scattering problems with multi-frequency sparse backscattering far field data, Inverse Problems 36, (2020), 105007.
-  G. Bao, K. Huang, P. Li and H. Zhao, A direct imaging method for inverse scattering using the Generalized Foldy-Lax formulation, Contemp. Math. 615 (2014), 49-70.
-  G. Bao, P. Li, J. Lin and F. Triki, Inverse scattering problems with multifrequencies, Inverse Problems 31, (2015), 093001.
-  G. Bao, J. Lin and F. Triki, A multi-frequency inverse source problem, J. Differ. Equ. 249, (2010), 3443-3465.
-  G. Bao, J. Lin and F. Triki, Numerical solution of the inverse source problem for the Helmholtz equation with multiple frequency data, Mathematical and statistical methods for imaging, 45-60, Contemp. Math. 548, Amer. Math. Soc., Providence, RI, 2011.
-  G. Bao, S. Lu, W. Rundell, and B. Xu, A recursive algorithm for multi-frequency acoustic inverse source problems, SIAM J. Numer. Anal. 53, (2015), 1608-1628.
-  N. Bleistein and J. Cohen, Nonuniqueness in the inverse source problem in acoustics and electromagnetics, J. Math. Phys. 18, (1977), 194-201.
-  S. Bousba, Y. Guo, X. Wang and L. Li, Identifying multipolar acoustic sources by the direct sampling method, Applicable Analysis 99(5), (2020), 856-879.
-  F. Cakoni and D. Colton. A qualitative approach to inverse scattering theory, volume 767. Springer, 2014.
-  J. Chen, Z. Chen and G. Huang, Reverse Time Migration for Extended Obstacles: Acoustic Waves, Inverse Problems 29, (2013), 085005.
-  D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region. Inverse Problems 12 (1996), 383-393.
-  D. Colton and R. Kress. Inverse acoustic and electromagnetic scattering theory, volume 93. Springer Nature, 2019.
-  A. Devaney, E. Marengo and M. Li, The inverse source problem in nonhomogeneous background media, SIAM J. Appl. Math. 67, (2007), 1353-1378.
-  A. Devaney and G. Sherman, Nonuniqueness in inverse source and scattering problems, IEEE Trans. Antennas Propag. 30, (1982), 1034-1037.
-  M. Eller and N. Valdivia, Acoustic source identification using multiple frequency information, Inverse Problems 25, (2009), 115005.
-  R Griesmaier. Multi-frequency orthogonality sampling for inverse obstacle scattering problems. Inverse Problems 27(8), (2011), 085005.
-  R. Griesmaier and C. Schmiedecke, A factorization method for multi-frequency inverse source problems with sparse far field measurements, SIAM J. Imag. Sci. 10, (2017), 2119-2139.
-  I Harris, D-L Nguyen. Orthogonality sampling method for the electromagnetic inverse scattering problem. SIAM J. Sci. Comput. 42(3),(2020),722–737.
-  K. Ito, B. Jin, and J. Zou, A direct sampling method to an inverse medium scattering problem, Inverse Problems 28, (2012), 025003.
-  X. Ji, Reconstruction of multipolar point sources with multifrequency sparse far field data, Inverse Problems 37, (2021), 065015.
-  X. Ji and X. Liu, Identification of point like objects with multifrequency sparse data, SIAM J. Sci. Comput. 42(4), (2020), A2325-A2343.
-  X. Ji and X. Liu, Inverse electromagnetic source scattering problems with multi-frequency sparse phased and phaseless far field data, SIAM J. Sci. Comput. 41(6), 2019, B1368-B1388.
-  X. Ji and X. Liu, Source reconstruction with multi-frequency sparse scattered fields, SIAM J. Appl. Math, 2021, to appear.
-  A. Kirsch, Charaterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems 14, (1998), 1489–1512.
-  A. Kirsch and N. Grinberg. The factorization method for inverse problems. Number 36. Oxford University Press, 2008.
-  J. Li, H. Liu and J. Zou, Locating multiple multiscale acoustic scatterers, SIAM Multiscale Model. Simul. 12, (2014), 927–952.
-  J. Li and J. Zou, A direct sampling method for inverse scattering using far-field data, Inverse Problems and Imaging 7, (2013), 757-775.
-  X. Liu, A novel sampling method for multiple multiscale targets from scattering amplitudes at a fixed frequency, Inverse Problems 33, (2017), 085011.
-  X. Liu, S. Meng and B. Zhang, Modified sampling method with near field measurements, arXiv: 2107.00387, 2021.
-  R. Potthast, A study on orthogonality sampling, Inverse Problems 26, (2010), 074075.
-  J. Schöberl, Netgen an advancing front 2d/3d-mesh generator based on abstract rules, Computing and Visualization in Science 1, (1997), 41–52.
-  J. Sylvester and J. Kelly, A scattering support for broadband sparse far field measurements, Inverse Problems 21, (2005), 759-771.
-  X. Wang, Y. Guo, D. Zhang and H. Liu, Fourier method for recovering acoustic sources from multi-frequency far-field data, Inverse Problems 33, (2017), 035001.
-  D. Zhang, Y. Guo, Fourier method for solving the multi-frequency inverse source problem for the Helmholtz equation, Inverse Problems 31, (2015), 035007.
-  D. Zhang, Y. Guo, J. Li and H. Liu, Locating multiple multipolar acoustic sources using the direct sampling method, Commun. Comput. Phys. 25(5), (2019), 1328-1356.