June  2021, 29(2): 2149-2165. doi: 10.3934/era.2020110

Some recent developments in the unique determinations in phaseless inverse acoustic scattering theory

1. 

School of Mathematics, Jilin University, Changchun, China

2. 

School of Mathematics, Harbin Institute of Technology, Harbin, China

* Corresponding author: Yukun Guo

Received  May 2020 Revised  September 2020 Published  October 2020

Fund Project: The first author is supported by NSFC grant 11671170. The second author is supported by NSFC grants 11971133, 11601107, 11671111 and the Fundamental Research Funds for the Central Universities

This article is an overview on some recent advances in the inverse scattering problems with phaseless data. Based upon our previous studies on the uniqueness issues in phaseless inverse acoustic scattering theory, this survey aims to briefly summarize the relevant rudiments comprising prototypical model problems, major results therein, as well as the rationale behind the basic techniques. We hope to sort out the essential ideas and shed further lights on this intriguing field.

Citation: Deyue Zhang, Yukun Guo. Some recent developments in the unique determinations in phaseless inverse acoustic scattering theory. Electronic Research Archive, 2021, 29 (2) : 2149-2165. doi: 10.3934/era.2020110
References:
[1]

H. AmmariY. T. Chow and J. Zou, Phased and phaseless domain reconstructions in the inverse scattering problem via scattering coefficients, SIAM J. Appl. Math., 76 (2016), 1000-1030.  doi: 10.1137/15M1043959.  Google Scholar

[2]

G. BaoP. Li and J. Lv, Numerical solution of an inverse diffraction grating problem from phaseless data, J. Opt. Soc. Am. A, 30 (2013), 293-299.  doi: 10.1364/JOSAA.30.000293.  Google Scholar

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G. Bao and J. Lin, Imaging of local surface displacement on an infinite ground plane: The multiple frequency case, SIAM J. Appl. Math., 71 (2011), 1733-1752.  doi: 10.1137/110824644.  Google Scholar

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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.  Google Scholar

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A. L. Bukhgeĭm and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Dokl. Akad. Nauk SSSR, 260 (1981), 269-272.   Google Scholar

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F. CakoniD. Colton and P. Monk, The direct and inverse scattering problem for partially coated obstacles, Inverse Problems, 17 (2001), 1997-2015.  doi: 10.1088/0266-5611/17/6/327.  Google Scholar

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E. J. CandesX. Li and M. Soltanolkotabi, Phase retrieval via Wirtinger flow: Theory and algorithms, IEEE Trans. Information Theory, 61 (2015), 1985-2007.  doi: 10.1109/TIT.2015.2399924.  Google Scholar

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E. J. CandésT. Strohmer and V. Voroninski, PhaseLift: Exact and stable signal recovery from magnitude measurements via convex programming, Commun. Pure Appl. Math., 66 (2013), 1241-1274.  doi: 10.1002/cpa.21432.  Google Scholar

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Z. ChenS. Fang and G. Huang, A direct imaging method for the half-space inverse scattering problem with phaseless data, Inverse Probl. Imaging, 11 (2017), 901-916.  doi: 10.3934/ipi.2017042.  Google Scholar

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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.  Google Scholar

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D. Colton and R. Kress, Looking back on inverse scattering theory, SIAM Rev., 60 (2018), 779-807.  doi: 10.1137/17M1144763.  Google Scholar

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D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 4$^th$ edition, Applied Mathematical Sciences, 93. Springer, Cham, 2019. doi: 10.1007/978-3-030-30351-8.  Google Scholar

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H. DongJ. Lai and P. Li, Inverse obstacle scattering problem for elastic waves with phased or phaseless far-field data, SIAM J. Imaging Sci., 12 (2019), 809-838.  doi: 10.1137/18M1227263.  Google Scholar

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H. Dong, J. Lai and P. Li, An inverse acoustic-elastic interaction problem with phased or phaseless far-field data, Inverse Problems, 36 (2020), 035014, 36 pp. doi: 10.1088/1361-6420/ab693e.  Google Scholar

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H. DongD. Zhang and Y. Guo, A reference ball based iterative algorithm for imaging acoustic obstacle from phaseless far-field data, Inverse Problems and Imagin, 13 (2019), 177-195.  doi: 10.3934/ipi.2019010.  Google Scholar

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P. GaoH. Dong and F. Ma, Inverse scattering via nonlinear integral equations method for a sound-soft crack from phaseless data, Applications of Mathematics, 63 (2018), 149-165.  doi: 10.21136/AM.2018.0154-17.  Google Scholar

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O. Ivanyshyn and R. Kress, Inverse scattering for surface impedance from phaseless far field data, J. Comput. Phys., 230 (2011), 3443-3452.  doi: 10.1016/j.jcp.2011.01.038.  Google Scholar

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X. JiX. Liu and B. Zhang, Target reconstruction with a reference point scatterer using phaseless far field patterns, SIAM J. Imaging Sci., 12 (2019), 372-391.  doi: 10.1137/18M1205789.  Google Scholar

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X. Ji, X. Liu and B. Zhang, Phaseless inverse source scattering problem: Phase retrieval, uniqueness and direct sampling methods, J. Comput. Phys. X, 1 (2019), 100003, 15 pp. doi: 10.1016/j.jcpx.2019.100003.  Google Scholar

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M. V. Klibanov, Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems, J. Inverse and Ⅲ-Posed Problems, 21 (2013), 477-510.  doi: 10.1515/jip-2012-0072.  Google Scholar

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M. V. Klibanov, Uniqueness of two phaseless non-overdetermined inverse acoustics problems in 3-d, Applicable Analysis, 93 (2014), 1135-1149.  doi: 10.1080/00036811.2013.818136.  Google Scholar

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M. V. Klibanov, Phaseless inverse scattering problems in three dimensions, SIAM J. Appl. Math., 74 (2014), 392-410.  doi: 10.1137/130926250.  Google Scholar

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M. V. Klibanov, A phaseless inverse scattering problem for the 3-D Helmholtz equation, Inverse Probl. Imaging, 11 (2017), 263-276.  doi: 10.3934/ipi.2017013.  Google Scholar

[27]

M. V. Klibanov and V. G. Romanov, Reconstruction procedures for two inverse scattering problems without the phase information, SIAM J. Appl. Math., 76 (2016), 178-196.  doi: 10.1137/15M1022367.  Google Scholar

[28]

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, 10 pp. doi: 10.1088/1361-6420/aa7a18.  Google Scholar

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[30]

K.-M. Lee, Shape reconstructions from phaseless data, Eng. Anal. Bound. Elem., 71 (2016), 174-178.  doi: 10.1016/j.enganabound.2016.08.001.  Google Scholar

[31]

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.  Google Scholar

[32]

J. Li, H. Liu and Y. Wang, Recovering an electromagnetic obstacle by a few phaseless backscattering measurements, Inverse Problems, 33 (2017), 035001, 20 pp. doi: 10.1088/1361-6420/aa5bf3.  Google Scholar

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J. LiH. Liu and J. Zou, Strengthened linear sampling method with a reference ball, SIAM J. Sci. Comput., 31 (2009), 4013-4040.  doi: 10.1137/080734170.  Google Scholar

[34]

J. Liu and J. Seo, On stability for a translated obstacle with impedance boundary condition, Nonlinear Anal., 59 (2004), 731-744.  doi: 10.1016/j.na.2004.07.033.  Google Scholar

[35]

M. H. Maleki and A. J. Devaney, Phase-retrieval and intensity-only reconstruction algorithms for optical diffraction tomography, J. Opt. Soc. Am. A, 10 (1993), 1086-1092.  doi: 10.1364/JOSAA.10.001086.  Google Scholar

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[39]

R. G. Novikov, Explicit formulas and global uniqueness for phaseless inverse scattering in multidimensions, J. Geom. Anal., 26 (2016), 346-359.  doi: 10.1007/s12220-014-9553-7.  Google Scholar

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L. PanY. ZhongX. Chen and S. P. Yeo, Subspace-based optimization method for inverse scattering problems utilizing phaseless data, IEEE Trans. Geosci. Remote Sensing, 49 (2011), 981-987.  doi: 10.1109/TGRS.2010.2070512.  Google Scholar

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F. Qu, B. Zhang and H. Zhang, A novel integral equation for scattering by locally rough surfaces and application to the inverse problem: The Neumann case, SIAM J. Sci. Comput., 41 (2019), A3673–A3702. doi: 10.1137/19M1240745.  Google Scholar

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V. G. Romanov and M. Yamamoto, Phaseless inverse problems with interference waves, J. Inverse Ⅲ-Posed Probl., 26 (2018), 681-688.  doi: 10.1515/jiip-2018-0037.  Google Scholar

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show all references

References:
[1]

H. AmmariY. T. Chow and J. Zou, Phased and phaseless domain reconstructions in the inverse scattering problem via scattering coefficients, SIAM J. Appl. Math., 76 (2016), 1000-1030.  doi: 10.1137/15M1043959.  Google Scholar

[2]

G. BaoP. Li and J. Lv, Numerical solution of an inverse diffraction grating problem from phaseless data, J. Opt. Soc. Am. A, 30 (2013), 293-299.  doi: 10.1364/JOSAA.30.000293.  Google Scholar

[3]

G. Bao and J. Lin, Imaging of local surface displacement on an infinite ground plane: The multiple frequency case, SIAM J. Appl. Math., 71 (2011), 1733-1752.  doi: 10.1137/110824644.  Google Scholar

[4]

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.  Google Scholar

[5]

A. L. Bukhgeĭm and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Dokl. Akad. Nauk SSSR, 260 (1981), 269-272.   Google Scholar

[6]

F. CakoniD. Colton and P. Monk, The direct and inverse scattering problem for partially coated obstacles, Inverse Problems, 17 (2001), 1997-2015.  doi: 10.1088/0266-5611/17/6/327.  Google Scholar

[7]

E. J. CandesX. Li and M. Soltanolkotabi, Phase retrieval via Wirtinger flow: Theory and algorithms, IEEE Trans. Information Theory, 61 (2015), 1985-2007.  doi: 10.1109/TIT.2015.2399924.  Google Scholar

[8]

E. J. CandésT. Strohmer and V. Voroninski, PhaseLift: Exact and stable signal recovery from magnitude measurements via convex programming, Commun. Pure Appl. Math., 66 (2013), 1241-1274.  doi: 10.1002/cpa.21432.  Google Scholar

[9]

Z. ChenS. Fang and G. Huang, A direct imaging method for the half-space inverse scattering problem with phaseless data, Inverse Probl. Imaging, 11 (2017), 901-916.  doi: 10.3934/ipi.2017042.  Google Scholar

[10]

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.  Google Scholar

[11]

D. Colton and R. Kress, Looking back on inverse scattering theory, SIAM Rev., 60 (2018), 779-807.  doi: 10.1137/17M1144763.  Google Scholar

[12]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 4$^th$ edition, Applied Mathematical Sciences, 93. Springer, Cham, 2019. doi: 10.1007/978-3-030-30351-8.  Google Scholar

[13]

H. DongJ. Lai and P. Li, Inverse obstacle scattering problem for elastic waves with phased or phaseless far-field data, SIAM J. Imaging Sci., 12 (2019), 809-838.  doi: 10.1137/18M1227263.  Google Scholar

[14]

H. Dong, J. Lai and P. Li, An inverse acoustic-elastic interaction problem with phased or phaseless far-field data, Inverse Problems, 36 (2020), 035014, 36 pp. doi: 10.1088/1361-6420/ab693e.  Google Scholar

[15]

H. DongD. Zhang and Y. Guo, A reference ball based iterative algorithm for imaging acoustic obstacle from phaseless far-field data, Inverse Problems and Imagin, 13 (2019), 177-195.  doi: 10.3934/ipi.2019010.  Google Scholar

[16]

P. GaoH. Dong and F. Ma, Inverse scattering via nonlinear integral equations method for a sound-soft crack from phaseless data, Applications of Mathematics, 63 (2018), 149-165.  doi: 10.21136/AM.2018.0154-17.  Google Scholar

[17]

O. Ivanyshyn, Shape reconstruction of acoustic obstacles from the modulus of the far field pattern, Inverse Probl. Imaging, 1 (2007), 609-622.  doi: 10.3934/ipi.2007.1.609.  Google Scholar

[18]

O. Ivanyshyn and R. Kress, Identification of sound-soft 3D obstacles from phaseless data, Inverse Probl. Imaging, 4 (2010), 131-149.  doi: 10.3934/ipi.2010.4.131.  Google Scholar

[19]

O. Ivanyshyn and R. Kress, Inverse scattering for surface impedance from phaseless far field data, J. Comput. Phys., 230 (2011), 3443-3452.  doi: 10.1016/j.jcp.2011.01.038.  Google Scholar

[20]

X. JiX. Liu and B. Zhang, Target reconstruction with a reference point scatterer using phaseless far field patterns, SIAM J. Imaging Sci., 12 (2019), 372-391.  doi: 10.1137/18M1205789.  Google Scholar

[21]

X. Ji, X. Liu and B. Zhang, Phaseless inverse source scattering problem: Phase retrieval, uniqueness and direct sampling methods, J. Comput. Phys. X, 1 (2019), 100003, 15 pp. doi: 10.1016/j.jcpx.2019.100003.  Google Scholar

[22] A. Kirsch and N. Grinberg, The Factorization Methods for Inverse Problems, Oxford Lecture Series in Mathematics and its Applications, 36. Oxford University Press, Oxford, 2008.   Google Scholar
[23]

M. V. Klibanov, Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems, J. Inverse and Ⅲ-Posed Problems, 21 (2013), 477-510.  doi: 10.1515/jip-2012-0072.  Google Scholar

[24]

M. V. Klibanov, Uniqueness of two phaseless non-overdetermined inverse acoustics problems in 3-d, Applicable Analysis, 93 (2014), 1135-1149.  doi: 10.1080/00036811.2013.818136.  Google Scholar

[25]

M. V. Klibanov, Phaseless inverse scattering problems in three dimensions, SIAM J. Appl. Math., 74 (2014), 392-410.  doi: 10.1137/130926250.  Google Scholar

[26]

M. V. Klibanov, A phaseless inverse scattering problem for the 3-D Helmholtz equation, Inverse Probl. Imaging, 11 (2017), 263-276.  doi: 10.3934/ipi.2017013.  Google Scholar

[27]

M. V. Klibanov and V. G. Romanov, Reconstruction procedures for two inverse scattering problems without the phase information, SIAM J. Appl. Math., 76 (2016), 178-196.  doi: 10.1137/15M1022367.  Google Scholar

[28]

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, 10 pp. doi: 10.1088/1361-6420/aa7a18.  Google Scholar

[29]

R. Kress and W. Rundell, Inverse obstacle scattering with modulus of the far field pattern as data, Inverse Problems in Medical Imaging and Nondestructive Testing (Oberwolfach, 1996), (1997), 75–92.  Google Scholar

[30]

K.-M. Lee, Shape reconstructions from phaseless data, Eng. Anal. Bound. Elem., 71 (2016), 174-178.  doi: 10.1016/j.enganabound.2016.08.001.  Google Scholar

[31]

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.  Google Scholar

[32]

J. Li, H. Liu and Y. Wang, Recovering an electromagnetic obstacle by a few phaseless backscattering measurements, Inverse Problems, 33 (2017), 035001, 20 pp. doi: 10.1088/1361-6420/aa5bf3.  Google Scholar

[33]

J. LiH. Liu and J. Zou, Strengthened linear sampling method with a reference ball, SIAM J. Sci. Comput., 31 (2009), 4013-4040.  doi: 10.1137/080734170.  Google Scholar

[34]

J. Liu and J. Seo, On stability for a translated obstacle with impedance boundary condition, Nonlinear Anal., 59 (2004), 731-744.  doi: 10.1016/j.na.2004.07.033.  Google Scholar

[35]

M. H. Maleki and A. J. Devaney, Phase-retrieval and intensity-only reconstruction algorithms for optical diffraction tomography, J. Opt. Soc. Am. A, 10 (1993), 1086-1092.  doi: 10.1364/JOSAA.10.001086.  Google Scholar

[36]

S. Maretzke and T. Hohage, Stability estimates for linearized near-field phase retrieval in X-ray phase contrast imaging, SIAM J. Appl. Math., 77 (2017), 384-408.  doi: 10.1137/16M1086170.  Google Scholar

[37] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.   Google Scholar
[38]

R. G. Novikov, Formulas for phase recovering from phaseless scattering data at fixed frequency, Bull. Sci. Math., 139 (2015), 923-936.  doi: 10.1016/j.bulsci.2015.04.005.  Google Scholar

[39]

R. G. Novikov, Explicit formulas and global uniqueness for phaseless inverse scattering in multidimensions, J. Geom. Anal., 26 (2016), 346-359.  doi: 10.1007/s12220-014-9553-7.  Google Scholar

[40]

L. PanY. ZhongX. Chen and S. P. Yeo, Subspace-based optimization method for inverse scattering problems utilizing phaseless data, IEEE Trans. Geosci. Remote Sensing, 49 (2011), 981-987.  doi: 10.1109/TGRS.2010.2070512.  Google Scholar

[41]

F. Qu, B. Zhang and H. Zhang, A novel integral equation for scattering by locally rough surfaces and application to the inverse problem: The Neumann case, SIAM J. Sci. Comput., 41 (2019), A3673–A3702. doi: 10.1137/19M1240745.  Google Scholar

[42]

V. G. Romanov, Phaseless inverse problems for Schrödinger, Helmholtz, and Maxwell Equations, Comput. Math. Math. Phys., 60 (2020), 1045-1062.  doi: 10.1134/S0965542520060093.  Google Scholar

[43]

V. G. Romanov and M. Yamamoto, Phaseless inverse problems with interference waves, J. Inverse Ⅲ-Posed Probl., 26 (2018), 681-688.  doi: 10.1515/jiip-2018-0037.  Google Scholar

[44]

F. Sun, D. Zhang and Y. Guo, Uniqueness in phaseless inverse scattering problems with known superposition of incident point sources, Inverse Problems, 35 (2019), 105007, 10 pp. doi: 10.1088/1361-6420/ab3373.  Google Scholar

[45]

T. TakenakaD. J. N. WallH. Harada and M. Tanaka, Reconstruction algorithm of the refractive index of a cylindrical object from the intensity measurements of the total field, Microwave Opt. Tech. Lett., 14 (1997), 139-197.  doi: 10.1002/(SICI)1098-2760(19970220)14:3<182::AID-MOP15>3.0.CO;2-A.  Google Scholar

[46]

X. XuB. Zhang and H. Zhang, Uniqueness in inverse scattering problems with phaseless far-field data at a fixed frequency, SIAM J. Appl. Math., 78 (2018), 1737-1753.  doi: 10.1137/17M1149699.  Google Scholar

[47]

X. XuB. Zhang and H. Zhang, Uniqueness in inverse scattering problems with phaseless far-field data at a fixed frequency. Ⅱ, SIAM J. Appl. Math., 78 (2018), 3024-3039.  doi: 10.1137/18M1196820.  Google Scholar

[48]

X. XuB. Zhang and H. Zhang, Uniqueness in inverse acoustic and electromagnetic scattering with phaseless near-field data at a fixed frequency, Inverse Probl. Imaging, 14 (2020), 489-510.  doi: 10.3934/ipi.2020023.  Google Scholar

[49]

W. Yin, W. Yang and H. Liu, A neural network scheme for recovering scattering obstacles with limited phaseless far-field data, J. Comput. Phys., 417 (2020), 109594, 18 pp. doi: 10.1016/j.jcp.2020.109594.  Google Scholar

[50]

D. Zhang and Y. Guo, Uniqueness results on phaseless inverse scattering with a reference ball, Inverse Problems, 34 (2018), 085002, 12 pp. doi: 10.1088/1361-6420/aac53c.  Google Scholar

[51]

D. Zhang, Y. Guo, J. Li and H. Liu, Retrieval of acoustic sources from multi-frequency phaseless data, Inverse Problems, 34 (2018), 094001, 21 pp. doi: 10.1088/1361-6420/aaccda.  Google Scholar

[52]

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Figure 1.  An illustration of the reference ball technique
Figure 2.  An illustration of the inverse scattering with a bounded scatterer and far-field measurements
Figure 3.  An illustration of the inverse scattering with a bounded scatterer and near-field measurements
Figure 4.  An illustration of the phaseless inverse scattering by a locally perturbed half-plane
Figure 5.  An illustration of the interior inverse scattering problem
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