June  2020, 14(3): 569-582. doi: 10.3934/ipi.2020026

Unique determinations in inverse scattering problems with phaseless near-field measurements

1. 

School of Mathematics, Jilin University, Changchun 130012, China

2. 

School of Mathematics, Harbin Institute of Technology, Harbin 150001, China

3. 

Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong SAR, China

* Corresponding authors: Yukun Guo and Hongyu Liu

Received  November 2019 Published  March 2020

Fund Project: The first and third authors are supported by NSFC grant 11671170. The second author is supported by NSFC grants 11971133, 11601107, 11671111 and 41474102. The fourth author is supported by Hong Kong RGC General Research Funds, 12302017, 12301218, 12302919

In this paper, we establish the unique determination results for several inverse acoustic scattering problems using the modulus of the near-field data. By utilizing the superpositions of point sources as the incident waves, we rigorously prove that the phaseless near-fields collected on an admissible surface can uniquely determine the location and shape of the obstacle as well as its boundary condition and the refractive index of a medium inclusion, respectively. We also establish the uniqueness in determining a locally rough surface from the phaseless near-field data due to superpositions of point sources. These are novel uniqueness results in inverse scattering with phaseless near-field data.

Citation: Deyue Zhang, Yukun Guo, Fenglin Sun, Hongyu Liu. Unique determinations in inverse scattering problems with phaseless near-field measurements. Inverse Problems and Imaging, 2020, 14 (3) : 569-582. doi: 10.3934/ipi.2020026
References:
[1]

H. Ammari and H. Kang, Polarization and Moment Tensors, With Applications to Inverse Problems and Effective Medium Theory, Springer, New York, 2007.

[2]

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.

[3]

C. AthanasiadisP. A. MartinA. Spyropoulos and I. G. Stratis, Scattering relations for point sources: Acoustic and electromagnetic waves, J. Math. Phys., 43 (2002), 5683-5697.  doi: 10.1063/1.1509089.

[4]

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.

[5]

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.

[6]

G. Bao and L. Zhang, Shape reconstruction of the multi-scale rough surface from multi-frequency phaseless data, Inverse Problems, 32 (2016), 085002. doi: 10.1088/0266-5611/32/8/085002.

[7]

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.

[8]

E. J. CandèsT. 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.

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E. J. CandèsX. 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.

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S. CaorsiA. MassaM. Pastorino and A. Randazzo, Electromagnetic detection of dielectric scatterers using phaseless synthetic and real data and the memetic algorithm, IEEE Trans. Geoscience Remote Sensing, 41 (2003), 2745-2753.  doi: 10.1109/TGRS.2003.815676.

[11]

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.

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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 Probl. Imaging, 13 (2019), 177-195.  doi: 10.3934/ipi.2019010.

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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.

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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.

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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.

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M. V. KlibanovN. A. KoshevD.-L. NguyenL. H. NguyenA. Brettin and V. Astratov, A numerical method to solve a phaseless coefficient inverse problem from a single measurement of experimental data, SIAM J. Imaging Sci., 11 (2018), 2339-2367.  doi: 10.1137/18M1179560.

[27]

M. V. KlibanovD.-L. Nguyen and L. H. Nguyen, A coefficient inverse problem with a single measurement of phaseless scattering data, SIAM J. Appl. Math., 79 (2019), 1-27.  doi: 10.1137/18M1168303.

[28]

M. V. KlibanovL.-H. Nguyen and K. Pan, Nanostructures imaging via numerical solution of a 3-D inverse scattering problem without the phase information, Appl. Numer. Math., 110 (2016), 190-203.  doi: 10.1016/j.apnum.2016.08.014.

[29]

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.

[30]

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.

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

[32]

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

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J. Li and H. Liu, Recovering a polyhedral obstacle by a few backscattering measurements, J. Differential Equations, 259 (2015), 2101-2120.  doi: 10.1016/j.jde.2015.03.030.

[34]

J. LiH. Liu and H. Sun, On a gesture-computing technique using electromagnetic waves, Inverse Probl. Imaging, 12 (2018), 677-696.  doi: 10.3934/ipi.2018029.

[35]

J. LiH. LiuW.-Y. Tsui and X. Wang, An inverse scattering approach for geometric body generation: A machine learning perspective, Mathematics in Engineering, 1 (2019), 800-823.  doi: 10.3934/mine.2019.4.800.

[36]

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.

[37]

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C. Lines, Inverse Scattering by Unbounded Obstacles, Ph.D thesis, Brunel University, 2003.

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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.

[40]

M. H. MalekiA. J. Devaney and A. Schatzberg, Tomographic reconstruction from optical scattered intensities, J. Opt. Soc. Am. A, 9 (1992), 1356-1363.  doi: 10.1364/JOSAA.9.001356.

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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.

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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.

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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.

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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.

[46]

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

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R. Potthast, Point Sources and Multipoles in Inverse Scattering Theory, Chapman & Hall/CRC, Boca Raton, 2001. doi: 10.1201/9781420035483.

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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, arXiv: 1901.08703v1, 2019. doi: 10.1137/19M1240745.

[49]

V. G. Romanov, A problem on determining the permittivity coefficient in a stationary system of Maxwell equations, Dokl. Math., 95 (2017), 230-234.  doi: 10.1134/s1064562417030164.

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

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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. doi: 10.1088/1361-6420/ab3373.

[52]

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), 182-188.  doi: 10.1002/(SICI)1098-2760(19970220)14:3<182::AID-MOP15>3.0.CO;2-A.

[53]

X. Wang, Y. Guo, J. Li and H. Liu, Mathematical design of a novel input/instruction device using a moving acoustic emitter, Inverse Problems, 33 (2017), 105009. doi: 10.1088/1361-6420/aa873f.

[54]

X. WangY. GuoJ. Li and H. Liu, Two gesture-computing approaches by using electromagnetic waves, Inverse Probl. Imaging, 13 (2019), 879-901.  doi: 10.3934/ipi.2019040.

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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.

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show all references

References:
[1]

H. Ammari and H. Kang, Polarization and Moment Tensors, With Applications to Inverse Problems and Effective Medium Theory, Springer, New York, 2007.

[2]

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.

[3]

C. AthanasiadisP. A. MartinA. Spyropoulos and I. G. Stratis, Scattering relations for point sources: Acoustic and electromagnetic waves, J. Math. Phys., 43 (2002), 5683-5697.  doi: 10.1063/1.1509089.

[4]

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.

[5]

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.

[6]

G. Bao and L. Zhang, Shape reconstruction of the multi-scale rough surface from multi-frequency phaseless data, Inverse Problems, 32 (2016), 085002. doi: 10.1088/0266-5611/32/8/085002.

[7]

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.

[8]

E. J. CandèsT. 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.

[9]

E. J. CandèsX. 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.

[10]

S. CaorsiA. MassaM. Pastorino and A. Randazzo, Electromagnetic detection of dielectric scatterers using phaseless synthetic and real data and the memetic algorithm, IEEE Trans. Geoscience Remote Sensing, 41 (2003), 2745-2753.  doi: 10.1109/TGRS.2003.815676.

[11]

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.

[12]

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.

[13]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3$^rd$ edition, Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3.

[14]

Y. DengJ. Li and H. Liu, On identifying magnetized anomalies using geomagnetic monitoring, Arch. Ration. Mech. Anal., 231 (2019), 153-187.  doi: 10.1007/s00205-018-1276-7.

[15]

Y. DengH. Liu and G. Uhlmann, On an inverse boundary problem arising in brain imaging, J. Differential Equations, 267 (2019), 2471-2502.  doi: 10.1016/j.jde.2019.03.019.

[16]

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.

[17]

H. DongD. 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.

[18]

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

[19]

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.

[20]

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.

[21]

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

[22]

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.

[23] A. Kirsch and N. Grinberg, The Factorization Methods for Inverse Problems, Oxford University Press, Oxford, 2008. 
[24]

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

[25]

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.

[26]

M. V. KlibanovN. A. KoshevD.-L. NguyenL. H. NguyenA. Brettin and V. Astratov, A numerical method to solve a phaseless coefficient inverse problem from a single measurement of experimental data, SIAM J. Imaging Sci., 11 (2018), 2339-2367.  doi: 10.1137/18M1179560.

[27]

M. V. KlibanovD.-L. Nguyen and L. H. Nguyen, A coefficient inverse problem with a single measurement of phaseless scattering data, SIAM J. Appl. Math., 79 (2019), 1-27.  doi: 10.1137/18M1168303.

[28]

M. V. KlibanovL.-H. Nguyen and K. Pan, Nanostructures imaging via numerical solution of a 3-D inverse scattering problem without the phase information, Appl. Numer. Math., 110 (2016), 190-203.  doi: 10.1016/j.apnum.2016.08.014.

[29]

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.

[30]

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.

[31]

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

[32]

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

[33]

J. Li and H. Liu, Recovering a polyhedral obstacle by a few backscattering measurements, J. Differential Equations, 259 (2015), 2101-2120.  doi: 10.1016/j.jde.2015.03.030.

[34]

J. LiH. Liu and H. Sun, On a gesture-computing technique using electromagnetic waves, Inverse Probl. Imaging, 12 (2018), 677-696.  doi: 10.3934/ipi.2018029.

[35]

J. LiH. LiuW.-Y. Tsui and X. Wang, An inverse scattering approach for geometric body generation: A machine learning perspective, Mathematics in Engineering, 1 (2019), 800-823.  doi: 10.3934/mine.2019.4.800.

[36]

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.

[37]

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.

[38]

C. Lines, Inverse Scattering by Unbounded Obstacles, Ph.D thesis, Brunel University, 2003.

[39]

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.

[40]

M. H. MalekiA. J. Devaney and A. Schatzberg, Tomographic reconstruction from optical scattered intensities, J. Opt. Soc. Am. A, 9 (1992), 1356-1363.  doi: 10.1364/JOSAA.9.001356.

[41]

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.

[42]

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.

[43] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. 
[44]

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.

[45]

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.

[46]

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

[47]

R. Potthast, Point Sources and Multipoles in Inverse Scattering Theory, Chapman & Hall/CRC, Boca Raton, 2001. doi: 10.1201/9781420035483.

[48]

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, arXiv: 1901.08703v1, 2019. doi: 10.1137/19M1240745.

[49]

V. G. Romanov, A problem on determining the permittivity coefficient in a stationary system of Maxwell equations, Dokl. Math., 95 (2017), 230-234.  doi: 10.1134/s1064562417030164.

[50]

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

[51]

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. doi: 10.1088/1361-6420/ab3373.

[52]

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), 182-188.  doi: 10.1002/(SICI)1098-2760(19970220)14:3<182::AID-MOP15>3.0.CO;2-A.

[53]

X. Wang, Y. Guo, J. Li and H. Liu, Mathematical design of a novel input/instruction device using a moving acoustic emitter, Inverse Problems, 33 (2017), 105009. doi: 10.1088/1361-6420/aa873f.

[54]

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Figure 1.  An illustration of the phaseless inverse scattering by a bounded scatterer
Figure 2.  An illustration of the phaseless inverse scattering by a locally perturbed half-plane
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