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[J]. Electronic Research Archive, 2021, 29(2): 2149-2165. doi: 10.3934/era.2020110
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.
[1] | Phased and phaseless domain reconstructions in the inverse scattering problem via scattering coefficients. SIAM J. Appl. Math. (2016) 76: 1000-1030. |
[2] | Numerical solution of an inverse diffraction grating problem from phaseless data. J. Opt. Soc. Am. A (2013) 30: 293-299. |
[3] | Imaging of local surface displacement on an infinite ground plane: The multiple frequency case. SIAM J. Appl. Math. (2011) 71: 1733-1752. |
[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 |
[5] | Uniqueness in the large of a class of multidimensional inverse problems. Dokl. Akad. Nauk SSSR (1981) 260: 269-272. |
[6] | The direct and inverse scattering problem for partially coated obstacles. Inverse Problems (2001) 17: 1997-2015. |
[7] | Phase retrieval via Wirtinger flow: Theory and algorithms. IEEE Trans. Information Theory (2015) 61: 1985-2007. |
[8] | PhaseLift: Exact and stable signal recovery from magnitude measurements via convex programming. Commun. Pure Appl. Math. (2013) 66: 1241-1274. |
[9] | A direct imaging method for the half-space inverse scattering problem with phaseless data. Inverse Probl. Imaging (2017) 11: 901-916. |
[10] | Phaseless imaging by reverse time migration: Acoustic waves. Numer. Math. Theor. Meth. Appl. (2017) 10: 1-21. |
[11] | Looking back on inverse scattering theory. SIAM Rev. (2018) 60: 779-807. |
[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 |
[13] | Inverse obstacle scattering problem for elastic waves with phased or phaseless far-field data. SIAM J. Imaging Sci. (2019) 12: 809-838. |
[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 |
[15] | A reference ball based iterative algorithm for imaging acoustic obstacle from phaseless far-field data. Inverse Problems and Imagin (2019) 13: 177-195. |
[16] | Inverse scattering via nonlinear integral equations method for a sound-soft crack from phaseless data. Applications of Mathematics (2018) 63: 149-165. |
[17] | Shape reconstruction of acoustic obstacles from the modulus of the far field pattern. Inverse Probl. Imaging (2007) 1: 609-622. |
[18] | Identification of sound-soft 3D obstacles from phaseless data. Inverse Probl. Imaging (2010) 4: 131-149. |
[19] | Inverse scattering for surface impedance from phaseless far field data. J. Comput. Phys. (2011) 230: 3443-3452. |
[20] | Target reconstruction with a reference point scatterer using phaseless far field patterns. SIAM J. Imaging Sci. (2019) 12: 372-391. |
[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 |
[22] | (2008) The Factorization Methods for Inverse Problems.Oxford Lecture Series in Mathematics and its Applications, 36. Oxford University Press. |
[23] | Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems. J. Inverse and Ⅲ-Posed Problems (2013) 21: 477-510. |
[24] | Uniqueness of two phaseless non-overdetermined inverse acoustics problems in 3-d. Applicable Analysis (2014) 93: 1135-1149. |
[25] | Phaseless inverse scattering problems in three dimensions. SIAM J. Appl. Math. (2014) 74: 392-410. |
[26] | A phaseless inverse scattering problem for the 3-D Helmholtz equation. Inverse Probl. Imaging (2017) 11: 263-276. |
[27] | Reconstruction procedures for two inverse scattering problems without the phase information. SIAM J. Appl. Math. (2016) 76: 178-196. |
[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 |
[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. |
[30] | Shape reconstructions from phaseless data. Eng. Anal. Bound. Elem. (2016) 71: 174-178. |
[31] | Recovering a polyhedral obstacle by a few backscattering measurements. J. Differential Equat. (2015) 259: 2101-2120. |
[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 |
[33] | Strengthened linear sampling method with a reference ball. SIAM J. Sci. Comput. (2009) 31: 4013-4040. |
[34] | On stability for a translated obstacle with impedance boundary condition. Nonlinear Anal. (2004) 59: 731-744. |
[35] | Phase-retrieval and intensity-only reconstruction algorithms for optical diffraction tomography. J. Opt. Soc. Am. A (1993) 10: 1086-1092. |
[36] | Stability estimates for linearized near-field phase retrieval in X-ray phase contrast imaging. SIAM J. Appl. Math. (2017) 77: 384-408. |
[37] | (2000) Strongly Elliptic Systems and Boundary Integral Equations.Cambridge University Press. |
[38] | Formulas for phase recovering from phaseless scattering data at fixed frequency. Bull. Sci. Math. (2015) 139: 923-936. |
[39] | Explicit formulas and global uniqueness for phaseless inverse scattering in multidimensions. J. Geom. Anal. (2016) 26: 346-359. |
[40] | Subspace-based optimization method for inverse scattering problems utilizing phaseless data. IEEE Trans. Geosci. Remote Sensing (2011) 49: 981-987. |
[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 |
[42] | Phaseless inverse problems for Schrödinger, Helmholtz, and Maxwell Equations. Comput. Math. Math. Phys. (2020) 60: 1045-1062. |
[43] | Phaseless inverse problems with interference waves. J. Inverse Ⅲ-Posed Probl. (2018) 26: 681-688. |
[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 |
[45] | Reconstruction algorithm of the refractive index of a cylindrical object from the intensity measurements of the total field. Microwave Opt. Tech. Lett. (1997) 14: 139-197. |
[46] | Uniqueness in inverse scattering problems with phaseless far-field data at a fixed frequency. SIAM J. Appl. Math. (2018) 78: 1737-1753. |
[47] | Uniqueness in inverse scattering problems with phaseless far-field data at a fixed frequency. Ⅱ. SIAM J. Appl. Math. (2018) 78: 3024-3039. |
[48] | Uniqueness in inverse acoustic and electromagnetic scattering with phaseless near-field data at a fixed frequency. Inverse Probl. Imaging (2020) 14: 489-510. |
[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 |
[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 |
[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 |
[52] | Unique determinations in inverse scattering problems with phaseless near-field measurements. Inverse Probl. Imaging (2020) 14: 569-582. |
[53] | D. Zhang, Y. Guo, F. Sun and X. Wang, Reconstruction of acoustic sources from multi-frequency phaseless far-field data, preprint, arXiv: 2002.03279. |
[54] | A finite element method with perfectly matched absorbing layers for the wave scattering from a cavity. J. Comput. Phys. (2008) 25: 301-308. |
[55] | D. Zhang, Y. Wang, Y. Guo and J. Li, Uniqueness in inverse cavity scattering problem with phaseless near-field data, Inverse Problems, 36 (2020), 025004, 10 pp. doi: 10.1088/1361-6420/ab53ee |
[56] | Recovering scattering obstacles by multi-frequency phaseless far-field data. J. Comput. Phys. (2017) 345: 58-73. |
[57] | J. Zheng, J. Cheng, P. Li and S. Lu, Periodic surface identification with phase or phaseless near-field data, Inverse Problems, 33 (2017), 115004, 35 pp. doi: 10.1088/1361-6420/aa8cb3 |