# American Institute of Mathematical Sciences

doi: 10.3934/ipi.2022025
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## A direct imaging method for the exterior and interior inverse scattering problems

 1 School of Mathematics, Jilin University, Changchun 130012, China 2 School of Mathematics, Harbin Institute of Technology, Harbin 150001, China

*Corresponding author: Yukun Guo

Received  January 2022 Revised  April 2022 Early access May 2022

Fund Project: The work of D. Zhang, Y. Wu and Y. Wang are supported by NSFC grant 12171200. The work of Y. Guo is supported by NSFC grant 11971133 and the Fundamental Research Funds for the Central Universities

This paper is concerned with the inverse acoustic scattering problems by an obstacle or a cavity with a sound-soft or a sound-hard boundary. A direct imaging method relying on the boundary conditions is proposed for reconstructing the shape of the obstacle or cavity. First, the scattered fields are approximated by the Fourier-Bessel functions with the measurements on a closed curve. Then, the indicator functions are established by the superposition of the total fields or their derivatives to the incident point sources. We prove that the indicator functions vanish only on the boundary of the obstacle or cavity. Numerical examples are also included to demonstrate the effectiveness of the method.

Citation: Deyue Zhang, Yue Wu, Yinglin Wang, Yukun Guo. A direct imaging method for the exterior and interior inverse scattering problems. Inverse Problems and Imaging, doi: 10.3934/ipi.2022025
##### References:
 [1] G. Bao, P. Li, J. Lin and F. Triki, Inverse scattering problems with multi-frequencies, Inverse Problems, 31 (2015), 093001, 21 pp. doi: 10.1088/0266-5611/31/9/093001. [2] G. Bao and L. Zhang, Shape reconstruction of the multi-scale rough surface from multi-frequency phaseless data, Inverse Problems, 32 (2016), 085002, 16 pp. doi: 10.1088/0266-5611/32/8/085002. [3] E. Blåsten and H. Liu, Recovering piecewise constant refractive indices by a single far-field pattern, Inverse Problems, 36 (2020), 085005.  doi: 10.1088/1361-6420/ab958f. [4] E. Blåsten, H. Liu and J. Xiao, On an electromagnetic problem in a corner and its applications, Anal. PDE, 14 (2021), 2207-2224.  doi: 10.2140/apde.2021.14.2207. [5] F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory, Springer-Verlag, Berlin, 2006. [6] X. Cao, H. Diao, H. Liu and J. Zou, On nodal and generalized singular structures of Laplacian eigenfunctions and applications to inverse scattering problems, J. Math. Pures Appl., 143 (2020), 116-161.  doi: 10.1016/j.matpur.2020.09.011. [7] Z. Chen and G. Huang, Phaseless imaging by reverse time migration: Acoustic waves, Numer. Math. Theor. Meth. Appl., 10 (2017), 1-21.  doi: 10.4208/nmtma.2017.m1617. [8] Y. Chow, Y. Deng, Y. He, H. Liu and X. Wang, Surface-localized transmission eigenstates, super-resolution imaging, and pseudo surface plasmon modes, SIAM J. Imaging Sci., 14 (2021), 946-975.  doi: 10.1137/20M1388498. [9] D. Colton and R. Kress, Looking back on inverse scattering theory, SIAM Review, 60 (2018), 779-807.  doi: 10.1137/17M1144763. [10] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 4$^{th}$ edition, Springer-Verlag, Cham, 2019. doi: 10.1007/978-3-030-30351-8. [11] H. Diao, X. Cao and H. Liu, On the geometric structures of transmission eigenfunctions with a conductive boundary condition and applications, Comm. Partial Differential Equations, 46 (2021), 630-679.  doi: 10.1080/03605302.2020.1857397. [12] H. Dong, D. Zhang and Y. Guo, A reference ball based iterative algorithm for imaging acoustic obstacle from phaseless far-field data, Inverse Probl. Imaging, 13 (2019), 177-195.  doi: 10.3934/ipi.2019010. [13] Y. Hu, F. Cakoni and J. Liu, The inverse scattering problem for a partially coated cavity with interior measurements, Appl. Anal., 93 (2014), 936-956.  doi: 10.1080/00036811.2013.801458. [14] P. Jakubik and R. Potthast, Testing the integrity of some cavity-the Cauchy problem and the range test, Appl. Numer. Math., 58 (2008), 899-914.  doi: 10.1016/j.apnum.2007.04.007. [15] M. V. Klibanov and V. G. Romanov, Uniqueness of a 3-D coefficient inverse scattering problem without the phase information, Inverse Problems, 33 (2017), 095007.  doi: 10.1088/1361-6420/aa7a18. [16] J. Li and H. Liu, Recovering a polyhedral obstacle by a few backscattering measurements, J. Differential Equat., 259 (2015), 2101-2120.  doi: 10.1016/j.jde.2015.03.030. [17] J. Li, H. Liu and Y. Wang, Recovering an electromagnetic obstacle by a few phaseless backscattering measurements, Inverse Problems, 33 (2017), 035001.  doi: 10.1088/1361-6420/aa5bf3. [18] J. Li, H. Liu and J. Zou, Strengthened linear sampling method with a reference ball, SIAM J. Sci. Comput., 31 (2009/10), 4013-4040.  doi: 10.1137/080734170. [19] H. Liu and J. Zou, Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers, Inverse Problems, 22 (2006), 515-524.  doi: 10.1088/0266-5611/22/2/008. [20] M. Liu, D. Zhang, X. Zhou and F. Liu, The Fourier-Bessel method for solving the Cauchy problem connected with the Helmholtz equation, J. Comput. Appl. Math., 311 (2017), 183-193.  doi: 10.1016/j.cam.2016.07.023. [21] X. Liu, The factorization method for cavities, Inverse Problems, 30 (2014), 015006.  doi: 10.1088/0266-5611/30/1/015006. [22] H. Qin and F. Cakoni, Nonlinear integral equations for shape reconstruction in the inverse interior scattering problem, Inverse Problems, 27 (2011), 035005.  doi: 10.1088/0266-5611/27/3/035005. [23] H. Qin and D. Colton, The inverse scattering problem for cavities, Appl. Numer. Math., 62 (2012), 699-708.  doi: 10.1016/j.apnum.2010.10.011. [24] H. Qin and D. Colton, The inverse scattering problem for cavities with impedance boundary condition, Adv. Comput. Math., 36 (2012), 157-174.  doi: 10.1007/s10444-011-9179-2. [25] H. Qin and X. Liu, The interior inverse scattering problem for cavities with an artificial obstacle, Appl. Numer. Math., 88 (2015), 18-30.  doi: 10.1016/j.apnum.2014.10.002. [26] Y. Sun, Y. Guo and F. Ma, The reciprocity gap functional method for the inverse scattering problem for cavities, Appl. Anal., 95 (2016), 1327-1346.  doi: 10.1080/00036811.2015.1064519. [27] F. Zeng, F. Cakoni and J. Sun, An inverse electromagnetic scattering problem for a cavity, Inverse Problems, 27 (2011), 125002.  doi: 10.1088/0266-5611/27/12/125002. [28] F. Zeng, P. Suarez and J. Sun, A decomposition method for an interior inverse scattering problem, Inverse Probl. Imaging, 7 (2013), 291-303.  doi: 10.3934/ipi.2013.7.291. [29] D. Zhang and Y. Guo, Fourier method for solving the multifrequency inverse source problem for the Helmholtz equation, Inverse Problems, 31 (2015), 035007.  doi: 10.1088/0266-5611/31/3/035007. [30] D. Zhang and Y. Guo, Uniqueness results on phaseless inverse scattering with a reference ball, Inverse Problems, 34 (2018), 085002.  doi: 10.1088/1361-6420/aac53c. [31] D. Zhang and Y. Guo, Some recent developments in the unique determinations in phaseless inverse acoustic scattering theory, Electronic Res. Arch., 29 (2021), 2149-2165.  doi: 10.3934/era.2020110. [32] D. Zhang, Y. Guo, J. Li and H. Liu, Retrieval of acoustic sources from multi-frequency phaseless data, Inverse Problems, 34 (2018), 094001.  doi: 10.1088/1361-6420/aaccda. [33] D. Zhang, F. Sun, Y. Ma and Y. Guo, A Fourier-Bessel method with a regularization strategy for the boundary value problems of the Helmholtz equation, J. Comput. Appl. Math., 368 (2020), 112562.  doi: 10.1016/j.cam.2019.112562. [34] D. Zhang and W. Sun, Stability analysis of the Fourier-Bessel method for the Cauchy problem of the Helmholtz equation, Inverse Probl. Sci. Eng., 24 (2016), 583-603.  doi: 10.1080/17415977.2015.1051531. [35] D. Zhang, Y. Wang, Y. Guo and J. Li, Uniqueness in inverse cavity scattering problems with phaseless near-field data, Inverse Problems, 36 (2020), 025004.  doi: 10.1088/1361-6420/ab53ee.

show all references

##### References:
 [1] G. Bao, P. Li, J. Lin and F. Triki, Inverse scattering problems with multi-frequencies, Inverse Problems, 31 (2015), 093001, 21 pp. doi: 10.1088/0266-5611/31/9/093001. [2] G. Bao and L. Zhang, Shape reconstruction of the multi-scale rough surface from multi-frequency phaseless data, Inverse Problems, 32 (2016), 085002, 16 pp. doi: 10.1088/0266-5611/32/8/085002. [3] E. Blåsten and H. Liu, Recovering piecewise constant refractive indices by a single far-field pattern, Inverse Problems, 36 (2020), 085005.  doi: 10.1088/1361-6420/ab958f. [4] E. Blåsten, H. Liu and J. Xiao, On an electromagnetic problem in a corner and its applications, Anal. PDE, 14 (2021), 2207-2224.  doi: 10.2140/apde.2021.14.2207. [5] F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory, Springer-Verlag, Berlin, 2006. [6] X. Cao, H. Diao, H. Liu and J. Zou, On nodal and generalized singular structures of Laplacian eigenfunctions and applications to inverse scattering problems, J. Math. Pures Appl., 143 (2020), 116-161.  doi: 10.1016/j.matpur.2020.09.011. [7] Z. Chen and G. Huang, Phaseless imaging by reverse time migration: Acoustic waves, Numer. Math. Theor. Meth. Appl., 10 (2017), 1-21.  doi: 10.4208/nmtma.2017.m1617. [8] Y. Chow, Y. Deng, Y. He, H. Liu and X. Wang, Surface-localized transmission eigenstates, super-resolution imaging, and pseudo surface plasmon modes, SIAM J. Imaging Sci., 14 (2021), 946-975.  doi: 10.1137/20M1388498. [9] D. Colton and R. Kress, Looking back on inverse scattering theory, SIAM Review, 60 (2018), 779-807.  doi: 10.1137/17M1144763. [10] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 4$^{th}$ edition, Springer-Verlag, Cham, 2019. doi: 10.1007/978-3-030-30351-8. [11] H. Diao, X. Cao and H. Liu, On the geometric structures of transmission eigenfunctions with a conductive boundary condition and applications, Comm. Partial Differential Equations, 46 (2021), 630-679.  doi: 10.1080/03605302.2020.1857397. [12] H. Dong, D. Zhang and Y. Guo, A reference ball based iterative algorithm for imaging acoustic obstacle from phaseless far-field data, Inverse Probl. Imaging, 13 (2019), 177-195.  doi: 10.3934/ipi.2019010. [13] Y. Hu, F. Cakoni and J. Liu, The inverse scattering problem for a partially coated cavity with interior measurements, Appl. Anal., 93 (2014), 936-956.  doi: 10.1080/00036811.2013.801458. [14] P. Jakubik and R. Potthast, Testing the integrity of some cavity-the Cauchy problem and the range test, Appl. Numer. Math., 58 (2008), 899-914.  doi: 10.1016/j.apnum.2007.04.007. [15] M. V. Klibanov and V. G. Romanov, Uniqueness of a 3-D coefficient inverse scattering problem without the phase information, Inverse Problems, 33 (2017), 095007.  doi: 10.1088/1361-6420/aa7a18. [16] J. Li and H. Liu, Recovering a polyhedral obstacle by a few backscattering measurements, J. Differential Equat., 259 (2015), 2101-2120.  doi: 10.1016/j.jde.2015.03.030. [17] J. Li, H. Liu and Y. Wang, Recovering an electromagnetic obstacle by a few phaseless backscattering measurements, Inverse Problems, 33 (2017), 035001.  doi: 10.1088/1361-6420/aa5bf3. [18] J. Li, H. Liu and J. Zou, Strengthened linear sampling method with a reference ball, SIAM J. Sci. Comput., 31 (2009/10), 4013-4040.  doi: 10.1137/080734170. [19] H. Liu and J. Zou, Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers, Inverse Problems, 22 (2006), 515-524.  doi: 10.1088/0266-5611/22/2/008. [20] M. Liu, D. Zhang, X. Zhou and F. Liu, The Fourier-Bessel method for solving the Cauchy problem connected with the Helmholtz equation, J. Comput. Appl. Math., 311 (2017), 183-193.  doi: 10.1016/j.cam.2016.07.023. [21] X. Liu, The factorization method for cavities, Inverse Problems, 30 (2014), 015006.  doi: 10.1088/0266-5611/30/1/015006. [22] H. Qin and F. Cakoni, Nonlinear integral equations for shape reconstruction in the inverse interior scattering problem, Inverse Problems, 27 (2011), 035005.  doi: 10.1088/0266-5611/27/3/035005. [23] H. Qin and D. Colton, The inverse scattering problem for cavities, Appl. Numer. Math., 62 (2012), 699-708.  doi: 10.1016/j.apnum.2010.10.011. [24] H. Qin and D. Colton, The inverse scattering problem for cavities with impedance boundary condition, Adv. Comput. Math., 36 (2012), 157-174.  doi: 10.1007/s10444-011-9179-2. [25] H. Qin and X. Liu, The interior inverse scattering problem for cavities with an artificial obstacle, Appl. Numer. Math., 88 (2015), 18-30.  doi: 10.1016/j.apnum.2014.10.002. [26] Y. Sun, Y. Guo and F. Ma, The reciprocity gap functional method for the inverse scattering problem for cavities, Appl. Anal., 95 (2016), 1327-1346.  doi: 10.1080/00036811.2015.1064519. [27] F. Zeng, F. Cakoni and J. Sun, An inverse electromagnetic scattering problem for a cavity, Inverse Problems, 27 (2011), 125002.  doi: 10.1088/0266-5611/27/12/125002. [28] F. Zeng, P. Suarez and J. Sun, A decomposition method for an interior inverse scattering problem, Inverse Probl. Imaging, 7 (2013), 291-303.  doi: 10.3934/ipi.2013.7.291. [29] D. Zhang and Y. Guo, Fourier method for solving the multifrequency inverse source problem for the Helmholtz equation, Inverse Problems, 31 (2015), 035007.  doi: 10.1088/0266-5611/31/3/035007. [30] D. Zhang and Y. Guo, Uniqueness results on phaseless inverse scattering with a reference ball, Inverse Problems, 34 (2018), 085002.  doi: 10.1088/1361-6420/aac53c. [31] D. Zhang and Y. Guo, Some recent developments in the unique determinations in phaseless inverse acoustic scattering theory, Electronic Res. Arch., 29 (2021), 2149-2165.  doi: 10.3934/era.2020110. [32] D. Zhang, Y. Guo, J. Li and H. Liu, Retrieval of acoustic sources from multi-frequency phaseless data, Inverse Problems, 34 (2018), 094001.  doi: 10.1088/1361-6420/aaccda. [33] D. Zhang, F. Sun, Y. Ma and Y. Guo, A Fourier-Bessel method with a regularization strategy for the boundary value problems of the Helmholtz equation, J. Comput. Appl. Math., 368 (2020), 112562.  doi: 10.1016/j.cam.2019.112562. [34] D. Zhang and W. Sun, Stability analysis of the Fourier-Bessel method for the Cauchy problem of the Helmholtz equation, Inverse Probl. Sci. Eng., 24 (2016), 583-603.  doi: 10.1080/17415977.2015.1051531. [35] D. Zhang, Y. Wang, Y. Guo and J. Li, Uniqueness in inverse cavity scattering problems with phaseless near-field data, Inverse Problems, 36 (2020), 025004.  doi: 10.1088/1361-6420/ab53ee.
An illustration of the inverse obstacle scattering problem
An illustration of the interior inverse scattering problem
The model scatterers for testing the reconstructions. (a) circle: $(\cos t, \sin t)$; (b) kite: $(\cos t+0.6\cos 2t-0.3, 1.3\sin t)$; (c) starfish: $(1+0.2\cos 5t)(\cos t, \sin t)$
Reconstruction of sound-soft obstacles with 5% noise. Top row: $k = 3$; Bottom row: $k = 6$
Reconstruction of sound-hard obstacles with 2% noise. Top row: $k = 4$; Middle row: $k = 5$; Bottom row: superposition with multiple frequencies $k = 3, 3.5, \cdots, 6$
Reconstruction of sound-soft cavities with 5% noise. Top row: $k = 3$; Bottom row: $k = 5$
Reconstruction of sound-hard cavities with 2% noise. Top row: $k = 4$; Middle row: $k = 5$; Bottom row: superposition with multiple frequencies $k = 3, 3.5, \cdots, 6$
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