October  2021, 15(5): 1269-1286. doi: 10.3934/ipi.2021037

Inverse obstacle scattering for acoustic waves in the time domain

School of Mathematics, Jilin University, Changchun 130012, China

* Corresponding author: Heping Dong

Received  December 2020 Revised  February 2021 Published  October 2021 Early access  April 2021

This paper concerns an inverse acoustic scattering problem which is to determine the location and shape of a rigid obstacle from time domain scattered field data. An efficient convolution quadrature method combined with nonlinear integral equation method is proposed to solve the inverse problem. In particular, replacing the classic Fourier transform with the convolution quadrature method for time discretization, the boundary integral equations for the Helmholtz equation with complex wave numbers can be obtained to guarantee the numerically approximate causality property of the scattered field under some condition. Numerical experiments are presented to demonstrate the effectiveness and robustness of the proposed method.

Citation: Lu Zhao, Heping Dong, Fuming Ma. Inverse obstacle scattering for acoustic waves in the time domain. Inverse Problems and Imaging, 2021, 15 (5) : 1269-1286. doi: 10.3934/ipi.2021037
References:
[1]

L. Banjai and S. Sauter, Rapid solution of the wave equation in unbounded domains, SIAM J. Numer. Anal., 47 (2008), 227-249.  doi: 10.1137/070690754.

[2]

G. BaoY. Gao and P. Li, Time domain analysis of an acoustic-elastic interaction problem, Arch. Ration. Mech. Anal., 229 (2018), 835-884.  doi: 10.1007/s00205-018-1228-2.

[3]

G. BaoB. HuP. Li and J. Wang, Analysis of time-domain Maxwell's equations in biperiodic structures, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 259-286.  doi: 10.3934/dcdsb.2019181.

[4]

C. Burkard and R. Potthast, A time-domain probe method for three-dimensional rough surface reconstructions, Inverse Probl. Imaging, 3 (2009), 259-274.  doi: 10.3934/ipi.2009.3.259.

[5]

F. CakoniH. Haddar and A. Lechleiter, On the factorization method for a far field inverse scattering problem in the time domain, SIAM J. Math. Anal., 51 (2019), 854-872.  doi: 10.1137/18M1214809.

[6]

Z. Chen, Convergence of the time-domain perfectly matched layer method for acoustic scattering problems, Int. J. Numer. Anal. Model., 6 (2009), 124-146. 

[7]

Q. Chen, H. Haddar, A. Lechleiter and P. Monk, A sampling method for inverse scattering in the time domain, Inverse Problems, 26 (2010), 085001, 17 pp. doi: 10.1088/0266-5611/26/8/085001.

[8]

B. Chen, Y. Guo, F. Ma and Y. Sun, Numerical schemes to reconstruct three dimensional time-dependent point sources of acoustic waves, Inverse Problems, 36 (2020), 075009, 21 pp. doi: 10.1088/1361-6420/ab8f85.

[9]

B. ChenF. Ma and Y. Guo, Time domain scattering and inverse scattering problems in a locally perturbed half-plane, Appl. Anal., 96 (2017), 1303-1325.  doi: 10.1080/00036811.2016.1188288.

[10]

Z. Chen and J.-C. N$\acute{e}$d$\acute{e}$lec, On Maxwell equations with the transparent boundary condition, J. Comput. Math., 26 (2008), 284-296. 

[11]

Z. Chen and X. Wu, Long-time stability and convergence of the uniaxial perfectly matched layer method for time-domain acoustic scattering problems, SIAM J. Numer. Anal., 50 (2012), 2632-2655.  doi: 10.1137/110835268.

[12]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3$^{nd}$ edition, Springer, New York, 2013. doi: 10.1007/978-3-030-30351-8.

[13]

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.

[14]

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.

[15]

H. DongJ. Lai and P. Li, Inverse obstacle scattering for elastic waves with phased orphaseless far-field data, SIAM J. Imaging Sci., 12 (2019), 809-838.  doi: 10.1137/18M1227263.

[16]

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.

[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.

[18]

Y. Gao and P. Li, Analysis of time-domain scattering by periodic structures, J. Differential Equations, 261 (2016), 5094-5118.  doi: 10.1016/j.jde.2016.07.020.

[19]

Y. Gao and P. Li, Electromagnetic scattering for time-domain Maxwell's equations in an unbounded structure, Math. Models Methods Appl. Sci., 27 (2017), 1843-1870.  doi: 10.1142/S0218202517500336.

[20]

Y. GaoP. Li and Y. Li, Analysis of time-domain elastic scattering by an unbounded structure, Math. Methods Appl. Sci., 41 (2018), 7032-7054.  doi: 10.1002/mma.5214.

[21]

Y. GaoP. Li and B. Zhang, Analysis of transient acoustic-elastic interaction in an unbounded structure, SIAM J. Math. Anal., 49 (2017), 3951-3972.  doi: 10.1137/16M1090326.

[22]

Y. GuoD. HömbergG. HuJ. Li and H. Liu, A time domain sampling method for inverse acoustic scattering problems, J. Comput. Phys., 314 (2016), 647-660.  doi: 10.1016/j.jcp.2016.03.046.

[23]

Y. Guo, P. Monk and D. Colton, Toward a time domain approach to the linear sampling method, Inverse Problems, 29 (2013), 095016, 17 pp. doi: 10.1088/0266-5611/29/9/095016.

[24]

Y. GuoP. Monk and D. Colton, The linear sampling method for sparse small aperture data, Appl. Anal., 95 (2016), 1599-1615.  doi: 10.1080/00036811.2015.1065317.

[25]

H. HaddarA. Lechleiter and S. Marmorat, An improved time domain linear sampling method for Robin and Neumann obstacles, Appl. Anal., 93 (2014), 369-390.  doi: 10.1080/00036811.2013.772583.

[26]

M. Ikehata, The enclosure method for inverse obstacle scattering problems with dynamical data over a finite time interval: III. Sound-soft obstacle and bistatic data, Inverse Problems, 29 (2013), 085013, 35 pp. doi: 10.1088/0266-5611/26/5/055010.

[27]

O. Ivanyshyn and T. Johansson, Nonlinear integral equation methods for the reconstruction of an acoustically sound-soft obstacle, J. Integral Equations Appl., 19 (2007), 289-308.  doi: 10.1216/jiea/1190905488.

[28]

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.

[29]

A. Kirsch and S. Ritter, The Nyström method for solving a class of singular integral equations and applications in 3D late elasticity, Math. Method. Appl. Sci., 22 (1999), 177-197.  doi: 10.1002/(SICI)1099-1476(19990125)22:2<177::AID-MMA36>3.0.CO;2-F.

[30]

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.

[31]

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.

[32]

J. Li, H. Liu, Y. Sun and Q. Wang, Ground detection by a single electromagnetic measurement, J. Comput. Phys., 257, (2014), 554–571.

[33]

J. LiH. LiuH. Sun and J. Zou, Imaging obstacles by hypersingular point sources, Inverse Probl. Imaging, 7 (2013), 545-563.  doi: 10.3934/ipi.2013.7.545.

[34]

P. Li and L. Zhang, Analysis of transient acoustic scattering by an elastic obstacle, Commun. Math. Sci., 17 (2019), 1671-1698.  doi: 10.4310/CMS.2019.v17.n6.a8.

[35]

Y. Liu, Y. Guo and J. Sun, A deterministic-statistical approach to reconstruct moving sources using sparse partial data, 2021, arXiv: 2101.01290v2.

[36]

H. Liu and G. Uhlmann, Determining both sound speed and internal source in thermo- and photo-acoustic tomography, Inverse Problems, 31 (2015), 105005, 10 pp. doi: 10.1088/0266-5611/31/10/105005.

[37]

D. R. Luke and R. Potthast, The point source method for inverse scattering in the time domain, Math. Methods Appl. Sci., 29 (2006), 1501-1521.  doi: 10.1002/mma.738.

[38]

F.-J. Sayas, Retarded Potentials and Time Domain Boundary Integral Equations, Springer, Switzerland, 2016. doi: 10.1007/978-3-319-26645-9.

[39]

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, 19 pp. doi: 10.1088/1361-6420/aa873f.

[40]

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.

[41]

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, 18 pp. doi: 10.1088/1361-6420/aa573c.

[42]

D. Zhang, Y. Guo, J. Li and H. Liu, Locating multiple multipolar acoustic sources using the direct sampling method, Commun. Comput. Phys., 25 (2019), 1328–1356. arXiv: 1801.05584v1. doi: 10.4208/cicp.oa-2018-0020.

[43]

L. Zhao, H. Dong and F. Ma, Time-domain analysis of forward obstacle scattering for elastic wave, Discrete Contin. Dyn. Syst. Ser. B, preprint. doi: 10.3934/dcdsb.2020276.

show all references

References:
[1]

L. Banjai and S. Sauter, Rapid solution of the wave equation in unbounded domains, SIAM J. Numer. Anal., 47 (2008), 227-249.  doi: 10.1137/070690754.

[2]

G. BaoY. Gao and P. Li, Time domain analysis of an acoustic-elastic interaction problem, Arch. Ration. Mech. Anal., 229 (2018), 835-884.  doi: 10.1007/s00205-018-1228-2.

[3]

G. BaoB. HuP. Li and J. Wang, Analysis of time-domain Maxwell's equations in biperiodic structures, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 259-286.  doi: 10.3934/dcdsb.2019181.

[4]

C. Burkard and R. Potthast, A time-domain probe method for three-dimensional rough surface reconstructions, Inverse Probl. Imaging, 3 (2009), 259-274.  doi: 10.3934/ipi.2009.3.259.

[5]

F. CakoniH. Haddar and A. Lechleiter, On the factorization method for a far field inverse scattering problem in the time domain, SIAM J. Math. Anal., 51 (2019), 854-872.  doi: 10.1137/18M1214809.

[6]

Z. Chen, Convergence of the time-domain perfectly matched layer method for acoustic scattering problems, Int. J. Numer. Anal. Model., 6 (2009), 124-146. 

[7]

Q. Chen, H. Haddar, A. Lechleiter and P. Monk, A sampling method for inverse scattering in the time domain, Inverse Problems, 26 (2010), 085001, 17 pp. doi: 10.1088/0266-5611/26/8/085001.

[8]

B. Chen, Y. Guo, F. Ma and Y. Sun, Numerical schemes to reconstruct three dimensional time-dependent point sources of acoustic waves, Inverse Problems, 36 (2020), 075009, 21 pp. doi: 10.1088/1361-6420/ab8f85.

[9]

B. ChenF. Ma and Y. Guo, Time domain scattering and inverse scattering problems in a locally perturbed half-plane, Appl. Anal., 96 (2017), 1303-1325.  doi: 10.1080/00036811.2016.1188288.

[10]

Z. Chen and J.-C. N$\acute{e}$d$\acute{e}$lec, On Maxwell equations with the transparent boundary condition, J. Comput. Math., 26 (2008), 284-296. 

[11]

Z. Chen and X. Wu, Long-time stability and convergence of the uniaxial perfectly matched layer method for time-domain acoustic scattering problems, SIAM J. Numer. Anal., 50 (2012), 2632-2655.  doi: 10.1137/110835268.

[12]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3$^{nd}$ edition, Springer, New York, 2013. doi: 10.1007/978-3-030-30351-8.

[13]

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.

[14]

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.

[15]

H. DongJ. Lai and P. Li, Inverse obstacle scattering for elastic waves with phased orphaseless far-field data, SIAM J. Imaging Sci., 12 (2019), 809-838.  doi: 10.1137/18M1227263.

[16]

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.

[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.

[18]

Y. Gao and P. Li, Analysis of time-domain scattering by periodic structures, J. Differential Equations, 261 (2016), 5094-5118.  doi: 10.1016/j.jde.2016.07.020.

[19]

Y. Gao and P. Li, Electromagnetic scattering for time-domain Maxwell's equations in an unbounded structure, Math. Models Methods Appl. Sci., 27 (2017), 1843-1870.  doi: 10.1142/S0218202517500336.

[20]

Y. GaoP. Li and Y. Li, Analysis of time-domain elastic scattering by an unbounded structure, Math. Methods Appl. Sci., 41 (2018), 7032-7054.  doi: 10.1002/mma.5214.

[21]

Y. GaoP. Li and B. Zhang, Analysis of transient acoustic-elastic interaction in an unbounded structure, SIAM J. Math. Anal., 49 (2017), 3951-3972.  doi: 10.1137/16M1090326.

[22]

Y. GuoD. HömbergG. HuJ. Li and H. Liu, A time domain sampling method for inverse acoustic scattering problems, J. Comput. Phys., 314 (2016), 647-660.  doi: 10.1016/j.jcp.2016.03.046.

[23]

Y. Guo, P. Monk and D. Colton, Toward a time domain approach to the linear sampling method, Inverse Problems, 29 (2013), 095016, 17 pp. doi: 10.1088/0266-5611/29/9/095016.

[24]

Y. GuoP. Monk and D. Colton, The linear sampling method for sparse small aperture data, Appl. Anal., 95 (2016), 1599-1615.  doi: 10.1080/00036811.2015.1065317.

[25]

H. HaddarA. Lechleiter and S. Marmorat, An improved time domain linear sampling method for Robin and Neumann obstacles, Appl. Anal., 93 (2014), 369-390.  doi: 10.1080/00036811.2013.772583.

[26]

M. Ikehata, The enclosure method for inverse obstacle scattering problems with dynamical data over a finite time interval: III. Sound-soft obstacle and bistatic data, Inverse Problems, 29 (2013), 085013, 35 pp. doi: 10.1088/0266-5611/26/5/055010.

[27]

O. Ivanyshyn and T. Johansson, Nonlinear integral equation methods for the reconstruction of an acoustically sound-soft obstacle, J. Integral Equations Appl., 19 (2007), 289-308.  doi: 10.1216/jiea/1190905488.

[28]

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.

[29]

A. Kirsch and S. Ritter, The Nyström method for solving a class of singular integral equations and applications in 3D late elasticity, Math. Method. Appl. Sci., 22 (1999), 177-197.  doi: 10.1002/(SICI)1099-1476(19990125)22:2<177::AID-MMA36>3.0.CO;2-F.

[30]

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.

[31]

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.

[32]

J. Li, H. Liu, Y. Sun and Q. Wang, Ground detection by a single electromagnetic measurement, J. Comput. Phys., 257, (2014), 554–571.

[33]

J. LiH. LiuH. Sun and J. Zou, Imaging obstacles by hypersingular point sources, Inverse Probl. Imaging, 7 (2013), 545-563.  doi: 10.3934/ipi.2013.7.545.

[34]

P. Li and L. Zhang, Analysis of transient acoustic scattering by an elastic obstacle, Commun. Math. Sci., 17 (2019), 1671-1698.  doi: 10.4310/CMS.2019.v17.n6.a8.

[35]

Y. Liu, Y. Guo and J. Sun, A deterministic-statistical approach to reconstruct moving sources using sparse partial data, 2021, arXiv: 2101.01290v2.

[36]

H. Liu and G. Uhlmann, Determining both sound speed and internal source in thermo- and photo-acoustic tomography, Inverse Problems, 31 (2015), 105005, 10 pp. doi: 10.1088/0266-5611/31/10/105005.

[37]

D. R. Luke and R. Potthast, The point source method for inverse scattering in the time domain, Math. Methods Appl. Sci., 29 (2006), 1501-1521.  doi: 10.1002/mma.738.

[38]

F.-J. Sayas, Retarded Potentials and Time Domain Boundary Integral Equations, Springer, Switzerland, 2016. doi: 10.1007/978-3-319-26645-9.

[39]

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, 19 pp. doi: 10.1088/1361-6420/aa873f.

[40]

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.

[41]

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, 18 pp. doi: 10.1088/1361-6420/aa573c.

[42]

D. Zhang, Y. Guo, J. Li and H. Liu, Locating multiple multipolar acoustic sources using the direct sampling method, Commun. Comput. Phys., 25 (2019), 1328–1356. arXiv: 1801.05584v1. doi: 10.4208/cicp.oa-2018-0020.

[43]

L. Zhao, H. Dong and F. Ma, Time-domain analysis of forward obstacle scattering for elastic wave, Discrete Contin. Dyn. Syst. Ser. B, preprint. doi: 10.3934/dcdsb.2020276.

Figure 1.  Reconstructions of an apple-shaped obstacle at different levels of noise, the radius of the initial guess is $ r_0 = 0.3 $, the position of the initial guess is $ (c_1^{(0)},c_2^{(0)}) = (-0.7,0.3) $ and the launch position of the incident wave is $ (1.5\cos\theta,1.5\sin\theta ) $, $ \theta = 7\pi/10 $
Figure 2.  Reconstructions of a peanut-shaped obstacle at different levels of noise, the radius of the initial guess is $ r_0 = 0.3 $, the position of the initial guess is $ (c_1^{(0)},c_2^{(0)}) = (0.75,-0.55) $ and the launch position of the incident wave is $ (1.5\cos\theta,1.5\sin\theta ) $, $ \theta = 7\pi/20 $
Figure 3.  Reconstructions of an apple-shaped obstacle at different positions of the initial guess $ (c_1^{(0)},c_2^{(0)}) = (-0.75.-0.4) $ and $ (c_1^{(0)},c_2^{(0)}) = (1,0.2) $, the radius of the initial guess is $ r_0 = 0.3 $, the noise level is 1$ \% $ and the launch position of the incident wave is $ (1.5\cos\theta,1.5\sin\theta ) $, $ \theta = \pi/2 $
Figure 4.  Reconstructions of a peanut-shaped obstacle at different positions of the initial guess $ (c_1^{(0)},c_2^{(0)}) = (-0.5,-0.65) $ and $ (c_1^{(0)},c_2^{(0)}) = (0.46,0.7) $, the radius of the initial guess is $ r_0 = 0.3 $, the noise level is 1$ \% $ and the launch position of the incident wave is $ (1.5\cos\theta,1.5\sin\theta ) $, $ \theta = 13\pi/20 $
Figure 5.  Reconstructions of an apple-shaped obstacle at different launch positions of the incident wave $ (1.5\cos\theta,1.5\sin\theta) $, the values of $ \theta $ are given in figure. The radius of the initial guess is $ r_0 = 0.3 $, the position of the initial guess is $ (c_1^{(0)},c_2^{(0)}) = (-0.9,-0.2) $ and the noise level is 1$ \% $
Figure 6.  Reconstructions of a peanut-shaped obstacle at different launch positions of the incident wave $ (1.5\cos\theta,1.5\sin\theta) $, the values of $ \theta $ are given in figure. The radius of the initial guess is $ r_0 = 0.3 $, the position of the initial guess is $ (c_1^{(0)},c_2^{(0)}) = (0.7,0.3) $ and the noise level is 1$ \% $
Figure 7.  Reconstructions of an apple-shaped obstacle at different numbers of launch position or different levels of noise, the radius of the initial guess is $ r_0 = 0.3 $, the position of the initial guess is $ (c_1^{(0)},c_2^{(0)}) = (0.9,-0.2) $ and the launch position of the incident wave is $ (1.5\cos\theta,1.5\sin\theta ) $, $ \theta_1 = 17\pi/20 $ and $ \theta_2 = 37\pi/20 $
Figure 8.  Reconstructions of a peanut-shaped obstacle at different numbers of launch position or different levles of noise, the radius of the initial guess is $ r_0 = 0.3 $, the position of the initial guess is $ (c_1^{(0)},c_2^{(0)}) = (-0.7,-0.4) $ and the launch position of the incident wave is $ (1.5\cos\theta,1.5\sin\theta ) $, $ \theta_1 = 11\pi/20 $ and $ \theta_2 = 31\pi/20 $
Table 1.  Parametrization of the exact boundary curves
Type Parametrization
apple-shaped $ p_D(\theta)= \frac{0.55(1+0.9\cos{\theta}+0.1\sin{2\theta})}{1+0.75\cos{\theta}}(\cos{\theta},\sin{\theta}), \quad \theta\in [0,2\pi] $
peanut-shaped $ p_D(t)=0.5\sqrt{0.25\cos^2{\theta}+\sin^2{\theta}}(\cos{\theta},\sin{\theta}), \quad \theta\in[0,2\pi] $
Type Parametrization
apple-shaped $ p_D(\theta)= \frac{0.55(1+0.9\cos{\theta}+0.1\sin{2\theta})}{1+0.75\cos{\theta}}(\cos{\theta},\sin{\theta}), \quad \theta\in [0,2\pi] $
peanut-shaped $ p_D(t)=0.5\sqrt{0.25\cos^2{\theta}+\sin^2{\theta}}(\cos{\theta},\sin{\theta}), \quad \theta\in[0,2\pi] $
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