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August  2022, 16(4): 925-942. doi: 10.3934/ipi.2022005

## An iterative scheme for imaging acoustic obstacle from phaseless total-field data

 School of Mathematics, Jilin University, Changchun, 130012, China

* Corresponding author: Deyue Zhang

Received  July 2021 Revised  December 2021 Published  August 2022 Early access  February 2022

Fund Project: The first author is supported by NSFC grants 12171201 and the National Key Research and Development Program of China (grant No. 2020YFA0713602). The second author is supported by NSFC grants 12171200

In this paper, we consider the inverse problem of determining the location and the shape of a sound-soft or sound-hard obstacle from the modulus of the total-field collected on a measured curve for an incident point source. We propose a system of nonlinear integral equations based iterative scheme to reconstruct both the location and the shape 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, Yingwei Chi. An iterative scheme for imaging acoustic obstacle from phaseless total-field data. Inverse Problems and Imaging, 2022, 16 (4) : 925-942. doi: 10.3934/ipi.2022005
##### 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, 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.  doi: 10.1364/JOSAA.30.000293. [3] 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. [4] 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. [5] H. Dong, J. Lai and P. Li, An inverse acoustic-elastic interaction problem with phased or phaseless far-field data, Inverse Probl., 36 (2020), 35014, 36 pp. doi: 10.1088/1361-6420/ab693e. [6] H. Dong, J. Lai and P. Li, Inverse obstacle scattering for elastic waves with phased or phaseless far-field data, SIAM J.Imaging Science, 12 (2019), 809-838.  doi: 10.1137/18M1227263. [7] H. Dong, D. 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. [8] P. Gao, H. 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. [9] 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. [10] 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. [11] O. Ivanyshyn and T. Johansson, Boundary integral eqautions for acoustical inverse sound-soft scattering, J. Inverse Ill-Posed Probl., 16 (2008), 65-78.  doi: 10.1515/jiip.2008.005. [12] 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. [13] X. Ji, X. 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. [14] 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. [15] 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. [16] 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. [17] R. Kress, Newton's method for inverse obstacle scattering meets the method of least squares, Inverse Probl., 19 (2003), 91-104.  doi: 10.1088/0266-5611/19/6/056. [18] 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, (1997), 75–92. [19] K. M. Lee, Shape reconstructions from phaseless data, Eng. Anal. Bound. Elem., 71 (2016), 174-178.  doi: 10.1016/j.enganabound.2016.08.001. [20] 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. [21] X. Xu, B. Zhang and H. Zhang, Uniqueness in inverse acoustic and electromagnetic scattering with phaseless near-field data at a fixed frequency, Inverse Probl. Imaging, 14 (2020), 489-510.  doi: 10.3934/ipi.2020023. [22] N. Ye and J. Liu, Uniqueness in determining a sound-hard ball with the modulus of a far-field datum, Appl. Anal., 98 (2019), 969-980.  doi: 10.1080/00036811.2017.1408078. [23] D. Zhang, Y. Guo, F. Sun and H. Liu, Unique determinations in inverse scattering problems with phaseless near-field measurements, Inverse Probl. Imaging, 14 (2020), 569-582.  doi: 10.3934/ipi.2020026. [24] 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. [25] B. Zhang and H. Zhang, An approximate factorization method for inverse acoustic scattering with phaseless total-field data, SIAM J. Appl. Math., 80 (2020), 2271-2298.  doi: 10.1137/19M1280612.

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, 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.  doi: 10.1364/JOSAA.30.000293. [3] 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. [4] 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. [5] H. Dong, J. Lai and P. Li, An inverse acoustic-elastic interaction problem with phased or phaseless far-field data, Inverse Probl., 36 (2020), 35014, 36 pp. doi: 10.1088/1361-6420/ab693e. [6] H. Dong, J. Lai and P. Li, Inverse obstacle scattering for elastic waves with phased or phaseless far-field data, SIAM J.Imaging Science, 12 (2019), 809-838.  doi: 10.1137/18M1227263. [7] H. Dong, D. 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. [8] P. Gao, H. 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. [9] 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. [10] 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. [11] O. Ivanyshyn and T. Johansson, Boundary integral eqautions for acoustical inverse sound-soft scattering, J. Inverse Ill-Posed Probl., 16 (2008), 65-78.  doi: 10.1515/jiip.2008.005. [12] 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. [13] X. Ji, X. 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. [14] 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. [15] 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. [16] 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. [17] R. Kress, Newton's method for inverse obstacle scattering meets the method of least squares, Inverse Probl., 19 (2003), 91-104.  doi: 10.1088/0266-5611/19/6/056. [18] 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, (1997), 75–92. [19] K. M. Lee, Shape reconstructions from phaseless data, Eng. Anal. Bound. Elem., 71 (2016), 174-178.  doi: 10.1016/j.enganabound.2016.08.001. [20] 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. [21] X. Xu, B. Zhang and H. Zhang, Uniqueness in inverse acoustic and electromagnetic scattering with phaseless near-field data at a fixed frequency, Inverse Probl. Imaging, 14 (2020), 489-510.  doi: 10.3934/ipi.2020023. [22] N. Ye and J. Liu, Uniqueness in determining a sound-hard ball with the modulus of a far-field datum, Appl. Anal., 98 (2019), 969-980.  doi: 10.1080/00036811.2017.1408078. [23] D. Zhang, Y. Guo, F. Sun and H. Liu, Unique determinations in inverse scattering problems with phaseless near-field measurements, Inverse Probl. Imaging, 14 (2020), 569-582.  doi: 10.3934/ipi.2020026. [24] 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. [25] B. Zhang and H. Zhang, An approximate factorization method for inverse acoustic scattering with phaseless total-field data, SIAM J. Appl. Math., 80 (2020), 2271-2298.  doi: 10.1137/19M1280612.
The geometry of model problem
Reconstructions of an apple-shaped domain with $1\%$ noise and $\epsilon = 0.008$
Reconstructions of an apple-shaped domain with $5\%$ noise and $\epsilon = 0.023$
Reconstructions of an apple-shaped domain with different initial guesses for $1\%$ noise. The source location is $z_0 = (-1.80, 1.15)$ and (a) $\epsilon = 0.009$, (b) $\epsilon = 0.008$
Reconstructions of an apple-shaped domain with different source locations for $1\%$ noise. The initial guess is $(c_1^{(0)}, c_2^{(0)}) = (-0.65, -0.15), r^{(0)} = 0.2$
Reconstructions of a peanut-shaped domain with $1\%$ noise and $\epsilon = 0.015$
Reconstructions of a peanut-shaped domain with $5\%$ noise and $\epsilon = 0.02$
Reconstructions of a peanut-shaped domain with different initial guesses for $1\%$ noise. The source location is $(-0.01, 1.65)$ and (a) $\epsilon = 0.01$, (b) $\epsilon = 0.009$
Reconstructions of a peanut-shaped domain with different source locations for $1\%$ noise. The initial guess is $(c_1^{(0)}, c_2^{(0)}) = (0.65, 0.6)$, $r^{(0)} = 0.2$
Reconstructions of an apple-shaped domain with $1\%$ noise and $\epsilon = 0.06$
Reconstructions of an apple-shaped domain with $5\%$ noise and $\epsilon = 0.06$
Reconstructions of an apple-shaped domain with different initial guesses for $1\%$ noise. The source location is $z_0 = (-1.50, 1.85)$ and (a) $\epsilon = 0.055$, (b) $\epsilon = 0.048$
Reconstructions of an apple-shaped domain with different source locations and $1\%$ noise. The initial guess $(c_1^{(0)}, c_2^{(0)}) = (-0.85, 0.01), r^{(0)} = 0.37$
Reconstructions of a peanut-shaped domain with $1\%$ noise and $\epsilon = 0.05$
Reconstructions of a peanut-shaped domain with $5\%$ noise and $\epsilon = 0.05$
Reconstructions of a peanut-shaped domain with different initial guess for $1\%$ noise. The source location is $(-0.01, 1.05)$ and $\epsilon = 0.05$
Reconstructions of a peanut-shaped domain with different source location $z_0$ for $1\%$ noise. The initial guess is $(c_1^{(0)}, c_2^{(0)}) = (-0.55, -0.65), r^{(0)} = 0.2$
Parametrization of the exact boundary curves
 Type Parametrization apple-shaped $p_D(t)= \frac{0.55(1+0.9\cos{t}+0.1\sin{2t})}{1+0.75\cos{t}}(\cos{t}, \sin{t}), \quad t\in [0, 2\pi]$ peanut-shaped $p_D(t)=0.25\sqrt{3\cos^2{t}+1}(\cos{t}, \sin{t}), \quad t\in[0, 2\pi]$
 Type Parametrization apple-shaped $p_D(t)= \frac{0.55(1+0.9\cos{t}+0.1\sin{2t})}{1+0.75\cos{t}}(\cos{t}, \sin{t}), \quad t\in [0, 2\pi]$ peanut-shaped $p_D(t)=0.25\sqrt{3\cos^2{t}+1}(\cos{t}, \sin{t}), \quad t\in[0, 2\pi]$
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