# American Institute of Mathematical Sciences

doi: 10.3934/ipi.2022015
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## Direct sampling methods for isotropic and anisotropic scatterers with point source measurements

 1 Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA 2 Department of Mathematics, Kansas State University, Manhattan, KS 66506, USA

*Corresponding author: Isaac Harris

Received  July 2021 Revised  January 2022 Early access April 2022

Fund Project: The author I. Harris is supported in part by the NSF DMS grant 2107891 and the author D.-L. Nguyen is supported in part by the NSF DMS grant 1812693

In this paper, we consider the inverse scattering problem for recovering either an isotropic or anisotropic scatterer from the measured scattered field initiated by a point source. We propose two new imaging functionals for solving the inverse problem. The first one employs a 'far-field' transform to the data which we then use to derive and provide an explicit decay rate for the imaging functional. In order to analyze the behavior of this imaging functional we use the factorization of the near field operator as well as the Funk-Hecke integral identity. For the second imaging functional the Cauchy data is used to define the functional and its behavior is analyzed using the Green's identities. Numerical experiments are given in two dimensions for both isotropic and anisotropic scatterers.

Citation: Isaac Harris, Dinh-Liem Nguyen, Thi-Phong Nguyen. Direct sampling methods for isotropic and anisotropic scatterers with point source measurements. Inverse Problems and Imaging, doi: 10.3934/ipi.2022015
##### References:
 [1] F. Cakoni, D. Colton and H. Haddar, Inverse Scattering Theory and Transmission Eigenvalues, CBMS-NSF Regional Conference Series in Applied Mathematics, 88. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016. doi: 10.1137/1.9781611974461.ch1. [2] F. Cakoni, H. Haddar and A. Lechleiter, On the factorization method for a far field inverse scattering in the time domain, SIAM J. Math. Anal., 51 (2019), 854-872.  doi: 10.1137/18M1214809. [3] J. Chen, Z. Chen and G. Huang, Reverse time migration for extended obstacles: Acoustic waves, Inverse Problems, 29 (2015), 085005, 17 pp. doi: 10.1088/0266-5611/29/8/085005. [4] Y.-T. Chow, F. Han and J. Zou, A direct sampling method for simultaneously recovering inhomogeneous inclusions of different nature, SIAM J. Sci. Comput., 43 (2021), A2161-A2189.  doi: 10.1137/20M133628X. [5] Y.-T. Chow, K. Ito, K. Liu and J. Zou, Direct sampling method for diffusive optical tomography, SIAM J. Sci. Comput., 37 (2015), A1658-A1684.  doi: 10.1137/14097519X. [6] Y.-T. Chow, K. Ito and J. Zou, A direct sampling method for electrical impedance tomography, Inverse Problems, 30 (2014), 095003, 25 pp. doi: 10.1088/0266-5611/30/9/095003. [7] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Third edition, Applied Mathematical Sciences, 93. Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3. [8] L. C. Evans, Partial Differential Equation, Second edition, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019. [9] J. Guo, G. Nakamura and H. Wang, The factorization method for recovering cavities in a heat conductor, preprint, (2019), arXiv: 1912.11590. [10] P. Hähner, On the uniqueness of the shape of a penetrable, anisotropic obstacle, J. Comput. Appl. Math., 116 (2000), 167-180.  doi: 10.1016/S0377-0427(99)00323-4. [11] I. Harris and A. Kleefeld., Analysis of new direct sampling indicators for far-field measurements, Inverse Problems, 35 (2019), 054002, 18 pp. doi: 10.1088/1361-6420/ab08be. [12] I. Harris and D.-L. Nguyen, Orthogonality sampling method for the electromagnetic inverse scattering problem, SIAM J. Sci. Comput., 42 (2020), B722-B737.  doi: 10.1137/19M129783X. [13] I. Harris and S. Rome, Near field imaging of small isotropic and extended anisotropic scatterers, Applicable Analysis, 96 (2017), 1713-1736.  doi: 10.1080/00036811.2017.1284312. [14] G. Hu, J. Yang, B. Zhang and H. Zhang, Near-field imaging of scattering obstacles with the factorization method, Inverse Problems, 30 (2014), 095005, 25 pp. doi: 10.1088/0266-5611/30/9/095005. [15] K. Ito, B. Jin and J. Zou, A two-stage method for inverse medium scattering, J. Comput. Phys., 237 (2013), 211-223.  doi: 10.1016/j.jcp.2012.12.004. [16] K. Ito, B. Jin and J. Zou, A direct sampling method to an inverse medium scattering problem, Inverse Problems, 28 (2012), 025003, 11 pp. doi: 10.1088/0266-5611/28/2/025003. [17] K. Ito, B. Jin and J. Zou, A direct sampling method for inverse electromagnetic medium scattering, Inverse Problems, 29 (2013), 095018, 19 pp. doi: 10.1088/0266-5611/29/9/095018. [18] S. Kang, M. Lambert, C. Y. Ahn, T. Ha and W.-K. Park, Single- and multi-frequency direct sampling methods in a limited-aperture inverse scattering problem, IEEE Access, 8 (2020), 121637-121649.  doi: 10.1109/ACCESS.2020.3006341. [19] A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems, 14 (1998), 1489-1512.  doi: 10.1088/0266-5611/14/6/009. [20] A. Kirsch, The factorization method for Maxwell's equations, Inverse Problems, 20 (2004), S117-S134.  doi: 10.1088/0266-5611/20/6/S08. [21] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford Lecture Series in Mathematics and its Applications, 36. Oxford University Press, Oxford, 2008. [22] A. Lechleiter and D.-L. Nguyen, A trigonometric Galerkin method for volume integral equations arising in TM grating scattering, Advances in Comput. Math., 40 (2014), 1-25.  doi: 10.1007/s10444-013-9295-2. [23] K. H. Leem, J. Liu and G. Pelekanos, Two direct factorization methods for inverse scattering problems, Inverse Problems, 34 (2018), 125004, 26 pp. doi: 10.1088/1361-6420/aae15e. [24] J. Li and J. Zou, A direct sampling method for inverse scattering using far-field data, Inverse Problems and Imaging, 7 (2013), 757-775.  doi: 10.3934/ipi.2013.7.757. [25] X. Liu, A novel sampling method for multiple multiscale targets from scattering amplitudes at a fixed frequency, Inverse Problems, 33 (2017), 085011, 20 pp. doi: 10.1088/1361-6420/aa777d. [26] X. Liu and J. Sun, Data recovery in inverse scattering: From limited-aperture to full-aperture, Journal of Computational Physics, 386 (2019), 350-364.  doi: 10.1016/j.jcp.2018.10.036. [27] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. [28] S. Meng, H. Haddar and F. Cakoni., The factorization method for a cavity in an inhomogeneous medium, Inverse Problems, 30 (2014), 045008, 20 pp. doi: 10.1088/0266-5611/30/4/045008. [29] D.-L. Nguyen, Direct and inverse electromagnetic scattering problems for bi-anisotropic media, Inverse Problems, 35 (2019), 124001, 27 pp. doi: 10.1088/1361-6420/ab382d. [30] R. Potthast, A study on orthogonality sampling, Inverse Problems, 26 (2010), 074015.  doi: 10.1088/0266-5611/26/7/074015. [31] Asymptotic Expansions for Large Order, Digital Library of Mathematical Functions, NIST, 2021, Available from: https://dlmf.nist.gov/10.19.

show all references

##### References:
 [1] F. Cakoni, D. Colton and H. Haddar, Inverse Scattering Theory and Transmission Eigenvalues, CBMS-NSF Regional Conference Series in Applied Mathematics, 88. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016. doi: 10.1137/1.9781611974461.ch1. [2] F. Cakoni, H. Haddar and A. Lechleiter, On the factorization method for a far field inverse scattering in the time domain, SIAM J. Math. Anal., 51 (2019), 854-872.  doi: 10.1137/18M1214809. [3] J. Chen, Z. Chen and G. Huang, Reverse time migration for extended obstacles: Acoustic waves, Inverse Problems, 29 (2015), 085005, 17 pp. doi: 10.1088/0266-5611/29/8/085005. [4] Y.-T. Chow, F. Han and J. Zou, A direct sampling method for simultaneously recovering inhomogeneous inclusions of different nature, SIAM J. Sci. Comput., 43 (2021), A2161-A2189.  doi: 10.1137/20M133628X. [5] Y.-T. Chow, K. Ito, K. Liu and J. Zou, Direct sampling method for diffusive optical tomography, SIAM J. Sci. Comput., 37 (2015), A1658-A1684.  doi: 10.1137/14097519X. [6] Y.-T. Chow, K. Ito and J. Zou, A direct sampling method for electrical impedance tomography, Inverse Problems, 30 (2014), 095003, 25 pp. doi: 10.1088/0266-5611/30/9/095003. [7] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Third edition, Applied Mathematical Sciences, 93. Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3. [8] L. C. Evans, Partial Differential Equation, Second edition, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019. [9] J. Guo, G. Nakamura and H. Wang, The factorization method for recovering cavities in a heat conductor, preprint, (2019), arXiv: 1912.11590. [10] P. Hähner, On the uniqueness of the shape of a penetrable, anisotropic obstacle, J. Comput. Appl. Math., 116 (2000), 167-180.  doi: 10.1016/S0377-0427(99)00323-4. [11] I. Harris and A. Kleefeld., Analysis of new direct sampling indicators for far-field measurements, Inverse Problems, 35 (2019), 054002, 18 pp. doi: 10.1088/1361-6420/ab08be. [12] I. Harris and D.-L. Nguyen, Orthogonality sampling method for the electromagnetic inverse scattering problem, SIAM J. Sci. Comput., 42 (2020), B722-B737.  doi: 10.1137/19M129783X. [13] I. Harris and S. Rome, Near field imaging of small isotropic and extended anisotropic scatterers, Applicable Analysis, 96 (2017), 1713-1736.  doi: 10.1080/00036811.2017.1284312. [14] G. Hu, J. Yang, B. Zhang and H. Zhang, Near-field imaging of scattering obstacles with the factorization method, Inverse Problems, 30 (2014), 095005, 25 pp. doi: 10.1088/0266-5611/30/9/095005. [15] K. Ito, B. Jin and J. Zou, A two-stage method for inverse medium scattering, J. Comput. Phys., 237 (2013), 211-223.  doi: 10.1016/j.jcp.2012.12.004. [16] K. Ito, B. Jin and J. Zou, A direct sampling method to an inverse medium scattering problem, Inverse Problems, 28 (2012), 025003, 11 pp. doi: 10.1088/0266-5611/28/2/025003. [17] K. Ito, B. Jin and J. Zou, A direct sampling method for inverse electromagnetic medium scattering, Inverse Problems, 29 (2013), 095018, 19 pp. doi: 10.1088/0266-5611/29/9/095018. [18] S. Kang, M. Lambert, C. Y. Ahn, T. Ha and W.-K. Park, Single- and multi-frequency direct sampling methods in a limited-aperture inverse scattering problem, IEEE Access, 8 (2020), 121637-121649.  doi: 10.1109/ACCESS.2020.3006341. [19] A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems, 14 (1998), 1489-1512.  doi: 10.1088/0266-5611/14/6/009. [20] A. Kirsch, The factorization method for Maxwell's equations, Inverse Problems, 20 (2004), S117-S134.  doi: 10.1088/0266-5611/20/6/S08. [21] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford Lecture Series in Mathematics and its Applications, 36. Oxford University Press, Oxford, 2008. [22] A. Lechleiter and D.-L. Nguyen, A trigonometric Galerkin method for volume integral equations arising in TM grating scattering, Advances in Comput. Math., 40 (2014), 1-25.  doi: 10.1007/s10444-013-9295-2. [23] K. H. Leem, J. Liu and G. Pelekanos, Two direct factorization methods for inverse scattering problems, Inverse Problems, 34 (2018), 125004, 26 pp. doi: 10.1088/1361-6420/aae15e. [24] J. Li and J. Zou, A direct sampling method for inverse scattering using far-field data, Inverse Problems and Imaging, 7 (2013), 757-775.  doi: 10.3934/ipi.2013.7.757. [25] X. Liu, A novel sampling method for multiple multiscale targets from scattering amplitudes at a fixed frequency, Inverse Problems, 33 (2017), 085011, 20 pp. doi: 10.1088/1361-6420/aa777d. [26] X. Liu and J. Sun, Data recovery in inverse scattering: From limited-aperture to full-aperture, Journal of Computational Physics, 386 (2019), 350-364.  doi: 10.1016/j.jcp.2018.10.036. [27] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. [28] S. Meng, H. Haddar and F. Cakoni., The factorization method for a cavity in an inhomogeneous medium, Inverse Problems, 30 (2014), 045008, 20 pp. doi: 10.1088/0266-5611/30/4/045008. [29] D.-L. Nguyen, Direct and inverse electromagnetic scattering problems for bi-anisotropic media, Inverse Problems, 35 (2019), 124001, 27 pp. doi: 10.1088/1361-6420/ab382d. [30] R. Potthast, A study on orthogonality sampling, Inverse Problems, 26 (2010), 074015.  doi: 10.1088/0266-5611/26/7/074015. [31] Asymptotic Expansions for Large Order, Digital Library of Mathematical Functions, NIST, 2021, Available from: https://dlmf.nist.gov/10.19.
The exact geometry of the scatterer(s)
The reconstruction of the kite-shaped scatterer $D$ where the boundary $\partial D =$kite by the direct sampling method with 'Far-Field' transformation
The reconstruction of the kite-shaped scatterer $D$ where the boundary $\partial D =$kite by the direct sampling method with Cauchy data
The reconstruction of the scatterer with two disjoint components $\partial D = {\rm{disk}} \cup {\rm{rectangle}}$ by the direct sampling method with 'Far-Field' transformation
The reconstruction of the scatterer with two disjoint components $\partial D = {\rm{disk}} \cup {\rm{rectangle}}$ by the direct sampling method with Cauchy data
The reconstruction of the scatterer with three disjoint components $\partial D = {\rm{kite}} \cup {\rm{ellipse}} \cup {\rm{peanut}}$ by the direct sampling method with 'Far-Field' transformation
The reconstruction of the scatterer with two disjoint components $\partial D = {\rm{kite}} \cup {\rm{ellipse}} \cup {\rm{peanut}}$ by the direct sampling method with Cauchy data
The reconstruction of the kite shaped scatterer by the direct sampling method with 'Far-Field' transformation for different amounts of measurements
The reconstruction of the kite shaped scatterer by the direct sampling method with Cauchy data for different amounts of measurements
The reconstruction of the kite shaped scatterer when there are more receivers than sources with 25 receivers
The reconstruction of the kite shaped scatterer by the direct sampling method with 'Far-Field' transformation for different wave numbers $k$
The reconstruction of the kite shaped scatterer by the direct sampling method with Cauchy data for different wave numbers $k$
The reconstruction of the kite shaped scatterer by the direct sampling method with Cauchy data for different values of parameter $\rho$
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