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February  2019, 13(1): 177-195. doi: 10.3934/ipi.2019010

## A reference ball based iterative algorithm for imaging acoustic obstacle from phaseless far-field data

 1 School of Mathematics, Jilin University, Changchun 130012, China 2 Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

* Corresponding author: Deyue Zhang

Received  April 2018 Revised  August 2018 Published  December 2018

Fund Project: The first author is supported by NSFC grants 11801213 and 11771180. The second author is supported by NSFC grant 11671170. The third author is supported by NSFC grants 11601107, 41474102 and 11671111.

In this paper, we consider the inverse problem of determining the location and the shape of a sound-soft obstacle from the modulus of the far-field data for a single incident plane wave. By adding a reference ball artificially to the inverse scattering system, we propose a system of nonlinear integral equations based iterative scheme to reconstruct both the location and the shape of the obstacle. The reference ball technique causes few extra computational costs, but breaks the translation invariance and brings information about the location of the obstacle. Several validating numerical examples are provided to illustrate the effectiveness and robustness of the proposed inversion algorithm.

Citation: Heping Dong, Deyue Zhang, Yukun Guo. A reference ball based iterative algorithm for imaging acoustic obstacle from phaseless far-field data. Inverse Problems and Imaging, 2019, 13 (1) : 177-195. doi: 10.3934/ipi.2019010
##### References:
 [1] H. Ammari, Y. T. Tin and J. Zou, Phased and phaseless domain reconstruction in inverse scattering problem via scattering coefficients, SIAM J. Appl. Math., 76 (2016), 1000-1030.  doi: 10.1137/15M1043959. [2] G. Bao and L. Zhang, Shape reconstruction of the multi-scale rough surface from multi-frequency phaseless data, Inverse Problems, 32 (2016), 085002, 16pp. doi: 10.1088/0266-5611/32/8/085002. [3] G. Bao, P. Li and J. Lv, Numerical solution of an inverse diffraction grating problem from phaseless data, J. Opt. Soc. Am. A, 30 (2013), 293-299. [4] Z. Chen and G. Huang, A direct imaging method for electromagnetic scattering data without phase information, SIAM J. Imaging Sci., 9 (2016), 1273-1297.  doi: 10.1137/15M1053475. [5] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3$^{nd}$ edition, Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3. [6] P. Gao, H. 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. [7] 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. [8] O. Ivanyshyn and T. Johansson, Nonlinear integral eqaution methods for the reconstruction of an acoustically sound-soft obstacle, J. Integral Equations Appl., 19 (2007), 289-308.  doi: 10.1216/jiea/1190905488. [9] 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. [10] X. Ji, X. Liu and B. Zhang, Target reconstruction with a reference point scatterer using phaseless far field patterns, preprint, 2018, arXiv: 1805.08035v3 [11] X. Ji, X. Liu and B. Zhang, Phaseless inverse source scattering problem: phase retrieval, uniqueness and sampling methods, preprint, 2018, arXiv: 1808.02385v1 [12] T. Johansson and B. D. Sleeman, Reconstruction of an acoustically sound-soft obstacle from one incident field and the far-field pattern, IMA J. Appl. Math., 72 (2007), 96-112.  doi: 10.1093/imamat/hxl026. [13] A. Karageorghis, B. T. Johansson and D. Lesnic, The method of fundamental solutions for the identification of a sound-soft obstacle in inverse acoustic scattering, Appl. Numer. Math., 62 (2012), 1767-1780.  doi: 10.1016/j.apnum.2012.05.011. [14] 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. [15] M. V. Klibanov, Phaseless inverse scattering problems in three dimensions, SIAM J. Appl. Math., 74 (2014), 392-410.  doi: 10.1137/130926250. [16] M. V. Klibanov, N. A. Koshev, D. Nguyen, L. H. Nguyen, A. Brettin and V. N. 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, arXiv: 1803.01374v1 doi: 10.1137/18M1179560. [17] M. V. Klibanov, L. H. Nguyen and K. Pan, Nanostructures imaging via numerical solution of a 3-D inverse scattering problem without the phase information, Applied Numerical Mathematics, 110 (2016), 190-203.  doi: 10.1016/j.apnum.2016.08.014. [18] M. V. Klibanov, D. Nguyen and L. Nguyen, A coefficient inverse problem with a single measurement of phaseless scattering data, preprint, 2017, arXiv: 1710.04804v1 [19] 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. [20] M. V. Klibanov and V. G. Romanov, Two reconstruction procedures for a 3D phaseless inverse scattering problem for the generalized Helmholtz equation, Inverse Problems, 32 (2016), 015005, 16pp. doi: 10.1088/0266-5611/32/1/015005. [21] R. Kress, Newton's method for inverse obstacle scattering meets the method of least squares, Inverse Problems, 19 (2003), S91-S104.  doi: 10.1088/0266-5611/19/6/056. [22] 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, (1997), 75–92. [23] K. M. Lee, Shape reconstructions from phaseless data, Eng. Anal. Bound. Elem., 71 (2016), 174-178.  doi: 10.1016/j.enganabound.2016.08.001. [24] 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. [25] J. Li, P. Li, H. Liu and X. Liu, Recovering multiscale buried anomalies in a two-layered medium, Inverse Problems, 31 (2015), 105006, 28pp. doi: 10.1088/0266-5611/31/10/105006. [26] J. Li, H. Liu, Z. 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. [27] J. Li, H. Liu, H. Sun and J. Zou, Imaging obstacles by hypersingular point sources, Inverse Problems and Imaging, 7 (2013), 545-563.  doi: 10.3934/ipi.2013.7.545. [28] J. Li, H. Liu and Q. Wang, Locating multiple multi-scale electromagnetic scatterers by a single far-field measurement, SIAM J. Imaging Sci., 6 (2013), 2285-2309. [29] J. Li, H. Liu and Y. Wang, Recovering an electromagnetic obstacle by a few phaseless backscattering measurements, Inverse Problems, 33 (2017), 035011, 20pp. doi: 10.1088/1361-6420/aa5bf3. [30] J. Li, H. Liu and J. Zou, Strengthened linear sampling method with a reference ball, SIAM J. Sci. Comput., 31 (2009), 4013-4040.  doi: 10.1137/080734170. [31] J. Li, H. Liu and J. Zou, Locating multiple multiscale acoustic scatterers, SIAM Mutliscale Model. Simul., 12 (2014), 927-952.  doi: 10.1137/13093409X. [32] H. Liu and J. Zou, On unique determination of partially coated polyhedral scatterers with far field measurements, Inverse Problems, 23 (2007), 297-308.  doi: 10.1088/0266-5611/23/1/016. [33] H. Liu and J. Zou, Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers, Inverse Problems, 22 (2006), 515-524.  doi: 10.1088/0266-5611/22/2/008. [34] X. Liu and B. Zhang, Unique determination of a sound-soft ball by the modulus of a single far field datum, J. Math. Anal. Appl., 365 (2010), 619-624.  doi: 10.1016/j.jmaa.2009.11.031. [35] X. Wang, Y. Guo, D. Zhang and H. Liu, Fourier method for recovering acoustic sources from multi-frequency far-field data, Inverse Problems, 33 (2017), 035001, 18pp. doi: 10.1088/1361-6420/aa573c. [36] X. Xu, B. 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. [37] 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. [38] D. Zhang and Y. Guo, Uniqueness results on phaseless inverse scattering with a reference ball, Inverse Problems, 34 (2018), 085002, 12pp. doi: 10.1088/1361-6420/aac53c. [39] D. Zhang, Y. Guo, J. Li and H. Liu, Retrieval of acoustic sources from multi-frequency phaseless data, Inverse Problems, 34 (2018), 094001, 21pp. doi: 10.1088/1361-6420/aaccda. [40] J. Zheng, J. Cheng, P. Li and S. Lu, Periodic surface identification with phase or phaseless near-field data, Inverse Problems, 33 (2017), 115004, 35pp. doi: 10.1088/1361-6420/aa8cb3.

show all references

##### References:
 [1] H. Ammari, Y. T. Tin and J. Zou, Phased and phaseless domain reconstruction in inverse scattering problem via scattering coefficients, SIAM J. Appl. Math., 76 (2016), 1000-1030.  doi: 10.1137/15M1043959. [2] G. Bao and L. Zhang, Shape reconstruction of the multi-scale rough surface from multi-frequency phaseless data, Inverse Problems, 32 (2016), 085002, 16pp. doi: 10.1088/0266-5611/32/8/085002. [3] G. Bao, P. Li and J. Lv, Numerical solution of an inverse diffraction grating problem from phaseless data, J. Opt. Soc. Am. A, 30 (2013), 293-299. [4] Z. Chen and G. Huang, A direct imaging method for electromagnetic scattering data without phase information, SIAM J. Imaging Sci., 9 (2016), 1273-1297.  doi: 10.1137/15M1053475. [5] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3$^{nd}$ edition, Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3. [6] P. Gao, H. 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. [7] 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. [8] O. Ivanyshyn and T. Johansson, Nonlinear integral eqaution methods for the reconstruction of an acoustically sound-soft obstacle, J. Integral Equations Appl., 19 (2007), 289-308.  doi: 10.1216/jiea/1190905488. [9] 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. [10] X. Ji, X. Liu and B. Zhang, Target reconstruction with a reference point scatterer using phaseless far field patterns, preprint, 2018, arXiv: 1805.08035v3 [11] X. Ji, X. Liu and B. Zhang, Phaseless inverse source scattering problem: phase retrieval, uniqueness and sampling methods, preprint, 2018, arXiv: 1808.02385v1 [12] T. Johansson and B. D. Sleeman, Reconstruction of an acoustically sound-soft obstacle from one incident field and the far-field pattern, IMA J. Appl. Math., 72 (2007), 96-112.  doi: 10.1093/imamat/hxl026. [13] A. Karageorghis, B. T. Johansson and D. Lesnic, The method of fundamental solutions for the identification of a sound-soft obstacle in inverse acoustic scattering, Appl. Numer. Math., 62 (2012), 1767-1780.  doi: 10.1016/j.apnum.2012.05.011. [14] 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. [15] M. V. Klibanov, Phaseless inverse scattering problems in three dimensions, SIAM J. Appl. Math., 74 (2014), 392-410.  doi: 10.1137/130926250. [16] M. V. Klibanov, N. A. Koshev, D. Nguyen, L. H. Nguyen, A. Brettin and V. N. 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, arXiv: 1803.01374v1 doi: 10.1137/18M1179560. [17] M. V. Klibanov, L. H. Nguyen and K. Pan, Nanostructures imaging via numerical solution of a 3-D inverse scattering problem without the phase information, Applied Numerical Mathematics, 110 (2016), 190-203.  doi: 10.1016/j.apnum.2016.08.014. [18] M. V. Klibanov, D. Nguyen and L. Nguyen, A coefficient inverse problem with a single measurement of phaseless scattering data, preprint, 2017, arXiv: 1710.04804v1 [19] 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. [20] M. V. Klibanov and V. G. Romanov, Two reconstruction procedures for a 3D phaseless inverse scattering problem for the generalized Helmholtz equation, Inverse Problems, 32 (2016), 015005, 16pp. doi: 10.1088/0266-5611/32/1/015005. [21] R. Kress, Newton's method for inverse obstacle scattering meets the method of least squares, Inverse Problems, 19 (2003), S91-S104.  doi: 10.1088/0266-5611/19/6/056. [22] 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, (1997), 75–92. [23] K. M. Lee, Shape reconstructions from phaseless data, Eng. Anal. Bound. Elem., 71 (2016), 174-178.  doi: 10.1016/j.enganabound.2016.08.001. [24] 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. [25] J. Li, P. Li, H. Liu and X. Liu, Recovering multiscale buried anomalies in a two-layered medium, Inverse Problems, 31 (2015), 105006, 28pp. doi: 10.1088/0266-5611/31/10/105006. [26] J. Li, H. Liu, Z. 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. [27] J. Li, H. Liu, H. Sun and J. Zou, Imaging obstacles by hypersingular point sources, Inverse Problems and Imaging, 7 (2013), 545-563.  doi: 10.3934/ipi.2013.7.545. [28] J. Li, H. Liu and Q. Wang, Locating multiple multi-scale electromagnetic scatterers by a single far-field measurement, SIAM J. Imaging Sci., 6 (2013), 2285-2309. [29] J. Li, H. Liu and Y. Wang, Recovering an electromagnetic obstacle by a few phaseless backscattering measurements, Inverse Problems, 33 (2017), 035011, 20pp. doi: 10.1088/1361-6420/aa5bf3. [30] J. Li, H. Liu and J. Zou, Strengthened linear sampling method with a reference ball, SIAM J. Sci. Comput., 31 (2009), 4013-4040.  doi: 10.1137/080734170. [31] J. Li, H. Liu and J. Zou, Locating multiple multiscale acoustic scatterers, SIAM Mutliscale Model. Simul., 12 (2014), 927-952.  doi: 10.1137/13093409X. [32] H. Liu and J. Zou, On unique determination of partially coated polyhedral scatterers with far field measurements, Inverse Problems, 23 (2007), 297-308.  doi: 10.1088/0266-5611/23/1/016. [33] H. Liu and J. Zou, Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers, Inverse Problems, 22 (2006), 515-524.  doi: 10.1088/0266-5611/22/2/008. [34] X. Liu and B. Zhang, Unique determination of a sound-soft ball by the modulus of a single far field datum, J. Math. Anal. Appl., 365 (2010), 619-624.  doi: 10.1016/j.jmaa.2009.11.031. [35] X. Wang, Y. Guo, D. Zhang and H. Liu, Fourier method for recovering acoustic sources from multi-frequency far-field data, Inverse Problems, 33 (2017), 035001, 18pp. doi: 10.1088/1361-6420/aa573c. [36] X. Xu, B. 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. [37] 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. [38] D. Zhang and Y. Guo, Uniqueness results on phaseless inverse scattering with a reference ball, Inverse Problems, 34 (2018), 085002, 12pp. doi: 10.1088/1361-6420/aac53c. [39] D. Zhang, Y. Guo, J. Li and H. Liu, Retrieval of acoustic sources from multi-frequency phaseless data, Inverse Problems, 34 (2018), 094001, 21pp. doi: 10.1088/1361-6420/aaccda. [40] J. Zheng, J. Cheng, P. Li and S. Lu, Periodic surface identification with phase or phaseless near-field data, Inverse Problems, 33 (2017), 115004, 35pp. doi: 10.1088/1361-6420/aa8cb3.
An illustration of the reference ball technique
Reconstructions of an apple-shaped domain with $1\%$ noise and $\epsilon = 0.015$
Reconstructions of an apple-shaped domain with $5\%$ noise and $\epsilon = 0.035$
Reconstructions of an apple-shaped domain with different reference balls and $1\%$ noise. Here we are using the initial guess $(c_1^{(0)}, c_2^{(0)}) = (-0.4, -0.8), r^{(0)} = 0.1$, and (a) $\epsilon = 0.011$, (b) $\epsilon = 0.02$, (c) $\epsilon = 0.03$, (d) $\epsilon = 0.011$
Reconstructions of an apple-shaped domain with different incoming wave directions, $1\%$ noise is added. Here, the initial guess $(c_1^{(0)}, c_2^{(0)}) = (0.7, 0.7), r^{(0)} = 0.1$, the reference ball $(b_1, b_2) = (4, 0), R = 0.4$, and $\epsilon = 0.015$
Reconstructions of an apple-shaped domain with different locations of the reference ball, $1\%$ noise is added. Here, the initial guess $(c_1^{(0)}, c_2^{(0)}) = (0.7, 0.7), r^{(0)} = 0.1$ and $\epsilon = 0.015$
Reconstructions of a peanut-shaped domain with $1\%$ noise and $\epsilon = 0.015$
{Reconstructions of a peanut-shaped domain with $5\%$ noise and $\epsilon = 0.035$
Reconstructions of a peanut-shaped domain with different reference balls, $1\%$ noise is added. Here, the initial guess $(c_1^{(0)}, c_2^{(0)}) = (0.3, -0.6), r^{(0)} = 0.1$ and $\epsilon = 0.015$. (a) $(b_1, b_2) = (3, 0), R = 0.2$. (b) $(b_1, b_2) = (4, 0), R = 0.4$. (c) $(b_1, b_2) = (5.5, 0), R = 0.8$
Reconstructions of a peanut shaped domain with different initial guesses and $1\%$ noise. Here, we are using the reference ball $(b_1, b_2) = (4, 0), R = 0.4$ and $\epsilon = 0.015$
Reconstructions of a rectangle shaped domain with $1\%$ and $5\%$ noise data respectively. Here, we are using the initial guess $(c_1^{(0)}, c_2^{(0)}) = (0.4, -0.8), r^{(0)} = 0.1$ and the reference ball $(b_1, b_2) = (4, 0), R = 0.5$
Reconstructions of a rectangle shaped domain with different initial guesses ($1\%$ noise is added). Here, we are using the incoming wave direction $d = (\cos(2\pi/3), \sin(2\pi/3))$, the reference ball $(b_1, b_2) = (4, 0), R = 0.5$ and $\epsilon = 0.015$
Reconstructions of a rectangle-shaped domain with different reference balls, $1\%$ noise is added. Here, the incoming wave direction $d = (\cos(\pi/6), \sin(\pi/6))$, the initial guess $(c_1^{(0)}, c_2^{(0)}) = (-0.3, -0.9), r^{(0)} = 0.1$, and $\epsilon = 0.015$
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