October  2017, 11(5): 901-916. doi: 10.3934/ipi.2017042

A direct imaging method for the half-space inverse scattering problem with phaseless data

1. 

LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, School of Mathematical Science, University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100190, China

2. 

School of Mathematical Science, University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100190, China

3. 

The Rice Inversion Project, Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005-1892, USA

* Corresponding author: Guanghui Huang

Received  December 2016 Published  July 2017

Fund Project: The first author was supported in part by the China NSF under the grant 113211061

We propose a direct imaging method based on the reverse time migration method for finding extended obstacles with phaseless total field data in the half space. We prove that the imaging resolution of the method is essentially the same as the imaging results using the scattering data with full phase information when the obstacle is far away from the surface of the half-space where the measurement is taken. Numerical experiments are included to illustrate the powerful imaging quality.

Citation: 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
References:
[1]

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

[2]

P. Bardsley and F. Vasquez, Kirchhoff migration without phases Inverse Problems, 32 (2016), 105006, 17 pp. doi: 10.1088/0266-5611/32/10/105006. Google Scholar

[3]

N. Bleistein, J. Cohen and J. Stockwell, Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion Springer, New York, 2001. doi: 10.1007/978-1-4613-0001-4. Google Scholar

[4]

F. Cakoni and D. Colton, A Qualitative Approach to Inverse Scattering Theory Applied Mathematical Sciences 188, Springer, New York, 2014. doi: 10.1007/978-1-4614-8827-9. Google Scholar

[5]

A. Chai, M. Moscoso and G. Papanicolaou, Array imaging using intensity-only measurements Inverse Problems 27 (2010), 015005, 16pp. doi: 10.1088/0266-5611/27/1/015005. Google Scholar

[6]

S. N. Chandler-WildeI. G. GrahamS. Langdon and M. Lindner, Condition number estimates for combined potential boundary integral operators in acoustic scattering, J. Integral Equa. Appli., 21 (2009), 229-279. doi: 10.1216/JIE-2009-21-2-229. Google Scholar

[7]

J. Chen, Z. Chen and G. Huang, Reverse time migration for extended obstacles: Acoustic waves Inverse Problems 29 (2013), 085005, 17pp. doi: 10.1088/0266-5611/29/8/085005. Google Scholar

[8]

J. Chen, Z. Chen and G. Huang, Reverse time migration for extended obstacles: Electromagnetic waves Inverse Problems 29 (2013), 085006, 17pp. doi: 10.1088/0266-5611/29/8/085006. Google Scholar

[9]

Z. Chen and G. Huang, Reverse time migration for extended obstacles: Elastic waves (in Chinese), Sci. Sin. Math., 58 (2015), 1811-1834. doi: 10.1007/s11425-015-5037-x. Google Scholar

[10]

Z. Chen and G. Huang, Reverse time migration for reconstructing extended obstacles in the half space Inverse Problems 31 (2015), 055007, 19pp. doi: 10.1088/0266-5611/31/5/055007. Google Scholar

[11]

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

[12]

Z. Chen and G. Huang, A direct imaging method for electromagnetic scattering data without phase information, SIAM J. Imag. Sci., 9 (2016), 1273-1297. doi: 10.1137/15M1053475. Google Scholar

[13]

A. J. Devaney, Structure determination from intensity measurements in scattering experiments, Physical Review Letters, 62 (1989), 2385-2388. doi: 10.1103/PhysRevLett.62.2385. Google Scholar

[14]

M. $\acute{\text{D}}$ursoK. BelkebirL. CroccoT. Isernia and A. Litman, Phaseless imaging with experimental data: Facts and challenges, J. Opt. Soc. Am. A, 25 (2008), 271-281. Google Scholar

[15]

G. FranceschiniM. DonelliR. Azaro and A. Massa, Inversion of phaseless total field data using a two-step strategy based on the iterative multiscaling approach, IEEE Trans. Geosci. Remote Sens., 44 (2006), 3527-3539. doi: 10.1109/TGRS.2006.881753. Google Scholar

[16]

L. Grafakos, Classical and Modern Fourier Analysis Pearson, London, 2004. Google Scholar

[17]

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

[18]

O. Ivanyshyn and R. Kress, Inverse scattering for surface impedance from phase-less far field data, Journal of Computational Physics, 230 (2011), 3443-3452. doi: 10.1016/j.jcp.2011.01.038. Google Scholar

[19]

R. Kress and W. Rundell, Inverse obstacle scattering with modulus of the far field pattern as data, H. W. Engl et al. (eds. ), Inverse Problems in Medical Imaging and Nondestructive Testing, Springer, Vienna, (1997), 75–92. Google Scholar

[20]

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. Google Scholar

[21]

J LiH. LiuZ. Shang and H. Sun, Two single-shot methods for locating multiple electromagnetic scatterers, SIAM J. Appl. Math., 73 (2013), 1721-1746. doi: 10.1137/130907690. Google Scholar

[22]

J Li,H. Liu and Q. Wang, Locating multiple multiscale electromagnetic scatterers by a single far-field measurement, SIAM J. Imag. Sci., 6 (2013), 2285-2309. doi: 10.1137/130920356. Google Scholar

[23]

J LiH. Liu and J. Zou, Locating multiple multiscale acoustic scatterers, Multiscale Modeling and Simulation, 12 (2014), 927-952. doi: 10.1137/13093409X. Google Scholar

[24]

A. NovikovM. Moscoso and G. Papanicolaou, Illumination strategies for intensity-only imaging, SIAM J. Imag. Sci., 8 (2015), 1547-1573. doi: 10.1137/140994617. Google Scholar

[25]

G. N. Watson, A Treatise on the Theory of Bessel Functions Cambridge University Press, Cambridge, 1995. Google Scholar

[26]

Y. Zhang and J. Sun, Practicle issues in reverse time migration: True amplitude gathers, noise removal and harmonic source encoding, First Break, 26 (2009), 29-35. Google Scholar

[27]

Y. ZhangS. XuN. Bleistein and G. Zhang, True-amplitude, angle-domain, common-image gathers from one-way wave-equation migration, Geophysics, 72 (2007), S49-S58. doi: 10.1190/1.2399371. Google Scholar

show all references

References:
[1]

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

[2]

P. Bardsley and F. Vasquez, Kirchhoff migration without phases Inverse Problems, 32 (2016), 105006, 17 pp. doi: 10.1088/0266-5611/32/10/105006. Google Scholar

[3]

N. Bleistein, J. Cohen and J. Stockwell, Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion Springer, New York, 2001. doi: 10.1007/978-1-4613-0001-4. Google Scholar

[4]

F. Cakoni and D. Colton, A Qualitative Approach to Inverse Scattering Theory Applied Mathematical Sciences 188, Springer, New York, 2014. doi: 10.1007/978-1-4614-8827-9. Google Scholar

[5]

A. Chai, M. Moscoso and G. Papanicolaou, Array imaging using intensity-only measurements Inverse Problems 27 (2010), 015005, 16pp. doi: 10.1088/0266-5611/27/1/015005. Google Scholar

[6]

S. N. Chandler-WildeI. G. GrahamS. Langdon and M. Lindner, Condition number estimates for combined potential boundary integral operators in acoustic scattering, J. Integral Equa. Appli., 21 (2009), 229-279. doi: 10.1216/JIE-2009-21-2-229. Google Scholar

[7]

J. Chen, Z. Chen and G. Huang, Reverse time migration for extended obstacles: Acoustic waves Inverse Problems 29 (2013), 085005, 17pp. doi: 10.1088/0266-5611/29/8/085005. Google Scholar

[8]

J. Chen, Z. Chen and G. Huang, Reverse time migration for extended obstacles: Electromagnetic waves Inverse Problems 29 (2013), 085006, 17pp. doi: 10.1088/0266-5611/29/8/085006. Google Scholar

[9]

Z. Chen and G. Huang, Reverse time migration for extended obstacles: Elastic waves (in Chinese), Sci. Sin. Math., 58 (2015), 1811-1834. doi: 10.1007/s11425-015-5037-x. Google Scholar

[10]

Z. Chen and G. Huang, Reverse time migration for reconstructing extended obstacles in the half space Inverse Problems 31 (2015), 055007, 19pp. doi: 10.1088/0266-5611/31/5/055007. Google Scholar

[11]

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

[12]

Z. Chen and G. Huang, A direct imaging method for electromagnetic scattering data without phase information, SIAM J. Imag. Sci., 9 (2016), 1273-1297. doi: 10.1137/15M1053475. Google Scholar

[13]

A. J. Devaney, Structure determination from intensity measurements in scattering experiments, Physical Review Letters, 62 (1989), 2385-2388. doi: 10.1103/PhysRevLett.62.2385. Google Scholar

[14]

M. $\acute{\text{D}}$ursoK. BelkebirL. CroccoT. Isernia and A. Litman, Phaseless imaging with experimental data: Facts and challenges, J. Opt. Soc. Am. A, 25 (2008), 271-281. Google Scholar

[15]

G. FranceschiniM. DonelliR. Azaro and A. Massa, Inversion of phaseless total field data using a two-step strategy based on the iterative multiscaling approach, IEEE Trans. Geosci. Remote Sens., 44 (2006), 3527-3539. doi: 10.1109/TGRS.2006.881753. Google Scholar

[16]

L. Grafakos, Classical and Modern Fourier Analysis Pearson, London, 2004. Google Scholar

[17]

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

[18]

O. Ivanyshyn and R. Kress, Inverse scattering for surface impedance from phase-less far field data, Journal of Computational Physics, 230 (2011), 3443-3452. doi: 10.1016/j.jcp.2011.01.038. Google Scholar

[19]

R. Kress and W. Rundell, Inverse obstacle scattering with modulus of the far field pattern as data, H. W. Engl et al. (eds. ), Inverse Problems in Medical Imaging and Nondestructive Testing, Springer, Vienna, (1997), 75–92. Google Scholar

[20]

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. Google Scholar

[21]

J LiH. LiuZ. Shang and H. Sun, Two single-shot methods for locating multiple electromagnetic scatterers, SIAM J. Appl. Math., 73 (2013), 1721-1746. doi: 10.1137/130907690. Google Scholar

[22]

J Li,H. Liu and Q. Wang, Locating multiple multiscale electromagnetic scatterers by a single far-field measurement, SIAM J. Imag. Sci., 6 (2013), 2285-2309. doi: 10.1137/130920356. Google Scholar

[23]

J LiH. Liu and J. Zou, Locating multiple multiscale acoustic scatterers, Multiscale Modeling and Simulation, 12 (2014), 927-952. doi: 10.1137/13093409X. Google Scholar

[24]

A. NovikovM. Moscoso and G. Papanicolaou, Illumination strategies for intensity-only imaging, SIAM J. Imag. Sci., 8 (2015), 1547-1573. doi: 10.1137/140994617. Google Scholar

[25]

G. N. Watson, A Treatise on the Theory of Bessel Functions Cambridge University Press, Cambridge, 1995. Google Scholar

[26]

Y. Zhang and J. Sun, Practicle issues in reverse time migration: True amplitude gathers, noise removal and harmonic source encoding, First Break, 26 (2009), 29-35. Google Scholar

[27]

Y. ZhangS. XuN. Bleistein and G. Zhang, True-amplitude, angle-domain, common-image gathers from one-way wave-equation migration, Geophysics, 72 (2007), S49-S58. doi: 10.1190/1.2399371. Google Scholar

Figure 1.  Examples 6.1: Imaging results of penetrable obstacles with different shapes from left to right. The top results are imaged with phaseless data by our RTM algorithm, and the bottom one are imaing results with full phase data. The probe wave number is $k=4\pi$, and $N_s=512$, $N_r=512$
Figure 2.  Example 6.2: Imaging results of an elliptic obstacle with boundary conditions as sound soft, sound hard and impedance boundary with $\lambda=1$, respectively. The probe wave number $k=4\pi$, and $N_s=512$, $N_r=512$
Figure 3.  Examples 6.3 (first test): Imaging results of a penetrable obstacle with noise levels $\mu=0.1, 0.2, 0.3, 0.4$ (from left to right). The top row is imaged with single frequency data, and the bottom row is imaged with multi-frequency data
Figure 4.  Example 6.3 (second test): Imaging results of two sound soft obstacles. All the parameters are the same as that in Figure 3
Figure 5.  Example 6.4: Imaging results of two sound soft obstacles overlaid with the true obstacle model. The left column is imaged with the single frequency data, and the right one is imaged with the multi-frequency data
[1]

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

[2]

Chérif Amrouche, Yves Raudin. Singular boundary conditions and regularity for the biharmonic problem in the half-space. Communications on Pure & Applied Analysis, 2007, 6 (4) : 957-982. doi: 10.3934/cpaa.2007.6.957

[3]

Weiwei Zhao, Jinge Yang, Sining Zheng. Liouville type theorem to an integral system in the half-space. Communications on Pure & Applied Analysis, 2014, 13 (2) : 511-525. doi: 10.3934/cpaa.2014.13.511

[4]

Chérif Amrouche, Huy Hoang Nguyen. Elliptic problems with $L^1$-data in the half-space. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 369-397. doi: 10.3934/dcdss.2012.5.369

[5]

Gershon Kresin, Vladimir Maz’ya. Optimal estimates for the gradient of harmonic functions in the multidimensional half-space. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 425-440. doi: 10.3934/dcds.2010.28.425

[6]

Nicola Abatangelo, Serena Dipierro, Mouhamed Moustapha Fall, Sven Jarohs, Alberto Saldaña. Positive powers of the Laplacian in the half-space under Dirichlet boundary conditions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1205-1235. doi: 10.3934/dcds.2019052

[7]

Giovanni Bozza, Massimo Brignone, Matteo Pastorino, Andrea Randazzo, Michele Piana. Imaging of unknown targets inside inhomogeneous backgrounds by means of qualitative inverse scattering. Inverse Problems & Imaging, 2009, 3 (2) : 231-241. doi: 10.3934/ipi.2009.3.231

[8]

Niclas Bernhoff. On half-space problems for the weakly non-linear discrete Boltzmann equation. Kinetic & Related Models, 2010, 3 (2) : 195-222. doi: 10.3934/krm.2010.3.195

[9]

Vasily Denisov and Andrey Muravnik. On asymptotic behavior of solutions of the Dirichlet problem in half-space for linear and quasi-linear elliptic equations. Electronic Research Announcements, 2003, 9: 88-93.

[10]

Yanqin Fang, Jihui Zhang. Nonexistence of positive solution for an integral equation on a Half-Space $R_+^n$. Communications on Pure & Applied Analysis, 2013, 12 (2) : 663-678. doi: 10.3934/cpaa.2013.12.663

[11]

Valter Pohjola. An inverse problem for the magnetic Schrödinger operator on a half space with partial data. Inverse Problems & Imaging, 2014, 8 (4) : 1169-1189. doi: 10.3934/ipi.2014.8.1169

[12]

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

[13]

Roland Griesmaier. Reciprocity gap music imaging for an inverse scattering problem in two-layered media. Inverse Problems & Imaging, 2009, 3 (3) : 389-403. doi: 10.3934/ipi.2009.3.389

[14]

Guangsheng Wei, Hong-Kun Xu. On the missing bound state data of inverse spectral-scattering problems on the half-line. Inverse Problems & Imaging, 2015, 9 (1) : 239-255. doi: 10.3934/ipi.2015.9.239

[15]

Kaitlyn (Voccola) Muller. SAR correlation imaging and anisotropic scattering. Inverse Problems & Imaging, 2018, 12 (3) : 697-731. doi: 10.3934/ipi.2018030

[16]

Atsushi Kawamoto. Hölder stability estimate in an inverse source problem for a first and half order time fractional diffusion equation. Inverse Problems & Imaging, 2018, 12 (2) : 315-330. doi: 10.3934/ipi.2018014

[17]

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

[18]

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

[19]

Masaya Maeda, Hironobu Sasaki, Etsuo Segawa, Akito Suzuki, Kanako Suzuki. Scattering and inverse scattering for nonlinear quantum walks. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3687-3703. doi: 10.3934/dcds.2018159

[20]

Francesco Demontis, Cornelis Van der Mee. Novel formulation of inverse scattering and characterization of scattering data. Conference Publications, 2011, 2011 (Special) : 343-350. doi: 10.3934/proc.2011.2011.343

2018 Impact Factor: 1.469

Metrics

  • PDF downloads (15)
  • HTML views (6)
  • Cited by (0)

Other articles
by authors

[Back to Top]