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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[47]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[55]

A. Willers, The Helmholtz equation in disturbed half-spaces, Math. Methods Appl. Sci., 9 (1987), 312-323.  doi: 10.1002/mma.1670090124.  Google Scholar

[56]

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

[57]

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

[58]

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.  Google Scholar

[59]

D. ZhangF. Ma and M. Fang, A finite element method with perfectly matched absorbing layers for the wave scattering from a cavity, Chinese Journal of Computational Physics, 25 (2008), 301-308.   Google Scholar

[60]

D. Zhang, Y. Wang, Y. Guo and J. Li, Uniqueness in inverse cavity scattering problems with phaseless near-field data, Inverse Problems, 35 (2019), 39 pp. doi: 10.1088/1361-6420/ab2a35.  Google Scholar

[61]

B. Zhang and H. Zhang, Recovering scattering obstacles by multi-frequency phaseless far-field data, J. Comput. Phys., 345 (2017), 58-73.  doi: 10.1016/j.jcp.2017.05.022.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[47]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[55]

A. Willers, The Helmholtz equation in disturbed half-spaces, Math. Methods Appl. Sci., 9 (1987), 312-323.  doi: 10.1002/mma.1670090124.  Google Scholar

[56]

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

[57]

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

[58]

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.  Google Scholar

[59]

D. ZhangF. Ma and M. Fang, A finite element method with perfectly matched absorbing layers for the wave scattering from a cavity, Chinese Journal of Computational Physics, 25 (2008), 301-308.   Google Scholar

[60]

D. Zhang, Y. Wang, Y. Guo and J. Li, Uniqueness in inverse cavity scattering problems with phaseless near-field data, Inverse Problems, 35 (2019), 39 pp. doi: 10.1088/1361-6420/ab2a35.  Google Scholar

[61]

B. Zhang and H. Zhang, Recovering scattering obstacles by multi-frequency phaseless far-field data, J. Comput. Phys., 345 (2017), 58-73.  doi: 10.1016/j.jcp.2017.05.022.  Google Scholar

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
[1]

Xiaoxu Xu, Bo Zhang, Haiwen Zhang. Uniqueness in inverse acoustic and electromagnetic scattering with phaseless near-field data at a fixed frequency. Inverse Problems & Imaging, 2020, 14 (3) : 489-510. doi: 10.3934/ipi.2020023

[2]

Fenglong Qu, Jiaqing Yang. On recovery of an inhomogeneous cavity in inverse acoustic scattering. Inverse Problems & Imaging, 2018, 12 (2) : 281-291. doi: 10.3934/ipi.2018012

[3]

Michael V. Klibanov. A phaseless inverse scattering problem for the 3-D Helmholtz equation. Inverse Problems & Imaging, 2017, 11 (2) : 263-276. doi: 10.3934/ipi.2017013

[4]

Heping Dong, Deyue Zhang, Yukun Guo. A reference ball based iterative algorithm for imaging acoustic obstacle from phaseless far-field data. Inverse Problems & Imaging, 2019, 13 (1) : 177-195. doi: 10.3934/ipi.2019010

[5]

Brian Sleeman. The inverse acoustic obstacle scattering problem and its interior dual. Inverse Problems & Imaging, 2009, 3 (2) : 211-229. doi: 10.3934/ipi.2009.3.211

[6]

Mourad Sini, Nguyen Trung Thành. Inverse acoustic obstacle scattering problems using multifrequency measurements. Inverse Problems & Imaging, 2012, 6 (4) : 749-773. doi: 10.3934/ipi.2012.6.749

[7]

Johannes Elschner, Guanghui Hu. Uniqueness in inverse transmission scattering problems for multilayered obstacles. Inverse Problems & Imaging, 2011, 5 (4) : 793-813. doi: 10.3934/ipi.2011.5.793

[8]

Zhiming Chen, Shaofeng Fang, Guanghui Huang. A direct imaging method for the half-space inverse scattering problem with phaseless data. Inverse Problems & Imaging, 2017, 11 (5) : 901-916. doi: 10.3934/ipi.2017042

[9]

Peijun Li, Ganghua Yuan. Increasing stability for the inverse source scattering problem with multi-frequencies. Inverse Problems & Imaging, 2017, 11 (4) : 745-759. doi: 10.3934/ipi.2017035

[10]

Frederic Weidling, Thorsten Hohage. Variational source conditions and stability estimates for inverse electromagnetic medium scattering problems. Inverse Problems & Imaging, 2017, 11 (1) : 203-220. doi: 10.3934/ipi.2017010

[11]

Marc Bonnet. Inverse acoustic scattering using high-order small-inclusion expansion of misfit function. Inverse Problems & Imaging, 2018, 12 (4) : 921-953. doi: 10.3934/ipi.2018039

[12]

Johannes Elschner, Guanghui Hu, Masahiro Yamamoto. Uniqueness in inverse elastic scattering from unbounded rigid surfaces of rectangular type. Inverse Problems & Imaging, 2015, 9 (1) : 127-141. doi: 10.3934/ipi.2015.9.127

[13]

Jingzhi Li, Jun Zou. A direct sampling method for inverse scattering using far-field data. Inverse Problems & Imaging, 2013, 7 (3) : 757-775. doi: 10.3934/ipi.2013.7.757

[14]

Huai-An Diao, Peijun Li, Xiaokai Yuan. Inverse elastic surface scattering with far-field data. Inverse Problems & Imaging, 2019, 13 (4) : 721-744. doi: 10.3934/ipi.2019033

[15]

Peter Monk, Virginia Selgas. Near field sampling type methods for the inverse fluid--solid interaction problem. Inverse Problems & Imaging, 2011, 5 (2) : 465-483. doi: 10.3934/ipi.2011.5.465

[16]

Jun Lai, Ming Li, Peijun Li, Wei Li. A fast direct imaging method for the inverse obstacle scattering problem with nonlinear point scatterers. Inverse Problems & Imaging, 2018, 12 (3) : 635-665. doi: 10.3934/ipi.2018027

[17]

Yi-Hsuan Lin. Reconstruction of penetrable obstacles in the anisotropic acoustic scattering. Inverse Problems & Imaging, 2016, 10 (3) : 765-780. doi: 10.3934/ipi.2016020

[18]

Guanghui Hu, Peijun Li, Xiaodong Liu, Yue Zhao. Inverse source problems in electrodynamics. Inverse Problems & Imaging, 2018, 12 (6) : 1411-1428. doi: 10.3934/ipi.2018059

[19]

Jingzhi Li, Hongyu Liu, Hongpeng Sun, Jun Zou. Imaging acoustic obstacles by singular and hypersingular point sources. Inverse Problems & Imaging, 2013, 7 (2) : 545-563. doi: 10.3934/ipi.2013.7.545

[20]

Masaru Ikehata. The enclosure method for inverse obstacle scattering using a single electromagnetic wave in time domain. Inverse Problems & Imaging, 2016, 10 (1) : 131-163. doi: 10.3934/ipi.2016.10.131

2018 Impact Factor: 1.469

Article outline

Figures and Tables

[Back to Top]