June  2018, 12(3): 635-665. doi: 10.3934/ipi.2018027

A fast direct imaging method for the inverse obstacle scattering problem with nonlinear point scatterers

1. 

School of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang 310027, China

2. 

College of Data Science, Taiyuan University of Technology, Taiyuan, Shanxi 030024, China

3. 

Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA

4. 

Institute for Mathematics and its Applications, Minneapolis, MN 55455, USA

* Corresponding author: Ming Li

Received  June 2017 Revised  September 2017 Published  March 2018

Fund Project: The first author was supported in part by the Funds for Creative Research Groups of NSFC (No. 11621101) and the Major Research Plan of NSFC (No. 91630309). The second author was supported partially by the National Natural Science Foundation of China (Grant no 11771321). The third author was supported in part by the NSF grant DMS-1151308.

Consider the scattering of a time-harmonic plane wave by heterogeneous media consisting of linear or nonlinear point scatterers and extended obstacles. A generalized Foldy–Lax formulation is developed to take fully into account of the multiple scattering by the complex media. A new imaging function is proposed and an FFT-based direct imaging method is developed for the inverse obstacle scattering problem, which is to reconstruct the shape of the extended obstacles. The novel idea is to utilize the nonlinear point scatterers to excite high harmonic generation so that enhanced imaging resolution can be achieved. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.

Citation: 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 and Imaging, 2018, 12 (3) : 635-665. doi: 10.3934/ipi.2018027
References:
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S. Albeverio, F. Gesztesy, R. Hoegh-Krohn and H. Holden, Solvable Models in Quantum Mechanics (with an appendix by Pavel Exner), 2nd Ed, AMS Chelsea Publishing, Providence, RI, 2005

[2]

B. Alpert, Hybrid Gauss-trapezoidal quadrature rules, SIAM J. Sci. Comput., 20 (1999), 1551-1584.  doi: 10.1137/S1064827597325141.

[3]

H. AmmariJ. GarnierH. KangM. Lim and K. Solna, Multistatic imaging of extended targets, SIAM J. Imaging Sci., 5 (2012), 564-600.  doi: 10.1137/10080631X.

[4]

H. AmmariJ. Carnier and P. Millien, Backprojection imaging in nonlinear harmonic holography in the presence of measurement and medium noises, SIAM J. Imaging Sci., 7 (2014), 239-276.  doi: 10.1137/130926717.

[5]

H. AmmariH. KangE. KimM. Lim and K. Louati, A direct algorithm for ultrasound imaging of internal corrosion, SIAM J. Numer. Anal., 49 (2011), 1177-1193.  doi: 10.1137/100784710.

[6]

G. BaoS. Hou and P. Li, Inverse scattering by a continuation method with initial guesses from a direct imaging algorithm, J. Comput. Phys., 227 (2007), 755-762.  doi: 10.1016/j.jcp.2007.08.020.

[7]

G. BaoK. HuangP. Li and H. Zhao, A direct imaging method for inverse scattering using the generalized Foldy–Lax formulation, Contemp. Math., 615 (2014), 49-70. 

[8]

G. BaoP. LiG. Lin and F. Triki, Invese scattering problems with multi-frequencies, Inverse Problems, 31 (2015), 093001, 21pp. 

[9]

P. BlomgrenG. Papanicolaou and H. Zhao, Super-resolution in time-reversal acoustics, J. Acoust. Soc. Am., 111 (2002), 230-248.  doi: 10.1121/1.1421342.

[10]

L. BorceaG. GarnierG. Papanicolaou and C. Tsogka, Enhanced statistical stability in coherent interferometric imaging, Inverse Problems, 27 (2011), 085004, 33pp. 

[11]

L. BorceaW. LiA.V. Mamonov and J. Schotland, Mamonov and J. Schotland, Second-harmonic imaging in random media, Inverse Problems, 33 (2017), 065004, 37pp. 

[12]

R. W. Boyd, Nonlinear Optics, 3rd Edition, Academic Press, New York, 2008.

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F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory, Springer-Verlag, Berlin, 2006.

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D. P. ChallaG. Hu and M. Sini, Multiple scattering of electromagnetic waves by a finite number of point-like obstacles, Math. Models Methods Appl. Sci., 24 (2014), 863-899.  doi: 10.1142/S021820251350070X.

[15]

D.P. Challa and M. Sini, Inverse scattering by point-like scatterers in the Foldy regime, Inverse Problems, 22 (2012), 125006, 39pp. 

[16]

D. P. Challa and M. Sini, On the justification of the Foldy–Lax approximation for the acoustic scattering by small rigid bodies of arbitrary shapes, Multiscale Model. Simul., 12 (2014), 55-108.  doi: 10.1137/130919313.

[17]

M. Cheney, The linear sampling method and the MUSIC algorithm, Inverse Problems, 17 (2001), 591-595.  doi: 10.1088/0266-5611/17/4/301.

[18]

J. ChengJ. Liu and G. Nakamura, The numerical realization of the probe method for the inverse scattering problems from the near-field data, Inverse Problems, 21 (2005), 839-855.  doi: 10.1088/0266-5611/21/3/004.

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D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383-393.  doi: 10.1088/0266-5611/12/4/003.

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D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, Pure Appl. Math., John Wiley, New York, 1983.

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M. Danckwerts and L. Novotny, Optical frequency mixing at coupled gold nanoparticles, Phys. Rev. Lett., 98 (2007), 026104.  doi: 10.1103/PhysRevLett.98.026104.

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P. de VriesD. van Coevorden and A. Lagendijk, Point scatterers for classical waves, Rev. Mod. Phys., 70 (1998), 447-466.  doi: 10.1103/RevModPhys.70.447.

[23]

A. Devaney, Super-resolution processing of multi-static data using time-reversal and MUSIC, preprint.

[24]

K. Erhard and R. Potthast, A numerical study of the probe method, SIAM J. Sci. Comput., 28 (2006), 1597-1612.  doi: 10.1137/040607149.

[25]

L. Foldy, The multiple scattering of waves I. General theory of isotropic scattering by randomly distributed scatterers, Phys. Rev., 67 (1945), 107-119.  doi: 10.1103/PhysRev.67.107.

[26]

L. Greengard and J.-Y. Lee, Accelerating the nonuniform fast Fourier transform, SIAM Rev., 46 (2004), 443-454.  doi: 10.1137/S003614450343200X.

[27]

L. Greengard and V. Rokhlin, A fast algorithm for particle simulations, J. Comput. Phys., 73 (1987), 325-348.  doi: 10.1016/0021-9991(87)90140-9.

[28]

B. Gremaud and T. Wellens, Nonlinear coherent transport of waves in disordered media, Phys. Rev. Lett., 100 (2008), 033902. 

[29]

B. Gremaud and T. Wellens, Speckle instability: Coherent effects in nonlinear disordered media, Phys. Rev. Lett., 104 (2010), 133901.  doi: 10.1103/PhysRevLett.104.133901.

[30]

F. GruberE. Marengo and A. Devaney, Time-reversal imaging with multiple signal classification considering multiple scattering between the targets, J. Acoust. Soc. Am., 115 (2004), 3042-3047.  doi: 10.1121/1.1738451.

[31]

S. HouK. Solna and H. Zhao, Imaging of location and geometry for extended targets using the response matrix, J. Comput. Phys., 199 (2004), 317-338.  doi: 10.1016/j.jcp.2004.02.010.

[32]

S. HouK. Solna and H. Zhao, A direct imaging algorithm for extended targets, Inverse Problems, 22 (2006), 1151-1178.  doi: 10.1088/0266-5611/22/4/003.

[33]

S. HouK. Solna and H. Zhao, A direct imaging method using far-field data, Inverse Problems, 23 (2007), 1533-1546.  doi: 10.1088/0266-5611/23/4/010.

[34]

S. HouK. HuangK. Solna and H. Zhao, A phase and space coherent direct imaging method, J. Acoust. Soc. Am., 125 (2009), 227-238.  doi: 10.1121/1.3035835.

[35]

G. HuA. Mantile and M. Sini, Direct and inverse acoustic scattering by a collection of extended and point-like scatterers, Multiscale Model. Simul., 12 (2014), 996-1027.  doi: 10.1137/130932107.

[36]

K. Huang and P. Li, A two-scale multiple scattering problem, Multiscale Model. Simul., 8 (2010), 1511-1534.  doi: 10.1137/090771090.

[37]

K. HuangP. Li and H. Zhao, An efficient algorithm for the generalized Foldy–Lax formulation, J. Comput. Phys., 234 (2013), 376-398.  doi: 10.1016/j.jcp.2012.09.027.

[38]

K. HuangK. Solna and H. Zhao, Generalized Foldy–Lax formulation, J. Comput. Phys., 229 (2010), 4544-4553.  doi: 10.1016/j.jcp.2010.02.021.

[39]

M. Ikehata, Reconstruction of an obstacle from the scattering amplitude at a fixed frequency, Inverse Problems, 14 (1998), 949-954.  doi: 10.1088/0266-5611/14/4/012.

[40]

E. KerbratC. Prada and M. Fink, Imaging in the presence of grain noise using the decomposition of the time reversal operator, J. Acoust. Soc. Am., 113 (2003), 1230-1240.  doi: 10.1121/1.1548156.

[41]

A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, Oxford, 2008.

[42]

J. LaiM. Kobayashi and L. Greengard, A fast solver for multi-particle scattering in a layered medium, Opt. Express, 22 (2014), 20481-20499.  doi: 10.1364/OE.22.020481.

[43]

M. Lax, Multiple scattering of waves, Rev. Modern Phys., 23 (1951), 287-310.  doi: 10.1103/RevModPhys.23.287.

[44]

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

[45]

W. Li and J. Schotland, Optical theorem for nonlinear media, Phys. Rev. A., 92 (2015), 043824.  doi: 10.1103/PhysRevA.92.043824.

[46]

X. Liu, A novel sampling method for multiple multiscale targets from scattering amplitude at a fixed frequency, Inverse Problems, 33 (2017), 085011, 20pp. 

[47]

P. Martin, Multiple Scattering: Interaction of Time-Harmonic Wave with $N$ Obstacles, Encyclopedia Math. Appl., 107, Cambridge University Press, Cambridge, 2006.

[48]

R. Potthast, Stability estimates and reconstructions in inverse acoustic scattering using singular sources, J. Comput. Appl. Math., 114 (2000), 247-274.  doi: 10.1016/S0377-0427(99)00201-0.

[49]

H. Zhao, Analysis of the response matrix for an extended target, SIAM J. Appl. Math., 64 (2004), 725-745.  doi: 10.1137/S0036139902415282.

show all references

References:
[1]

S. Albeverio, F. Gesztesy, R. Hoegh-Krohn and H. Holden, Solvable Models in Quantum Mechanics (with an appendix by Pavel Exner), 2nd Ed, AMS Chelsea Publishing, Providence, RI, 2005

[2]

B. Alpert, Hybrid Gauss-trapezoidal quadrature rules, SIAM J. Sci. Comput., 20 (1999), 1551-1584.  doi: 10.1137/S1064827597325141.

[3]

H. AmmariJ. GarnierH. KangM. Lim and K. Solna, Multistatic imaging of extended targets, SIAM J. Imaging Sci., 5 (2012), 564-600.  doi: 10.1137/10080631X.

[4]

H. AmmariJ. Carnier and P. Millien, Backprojection imaging in nonlinear harmonic holography in the presence of measurement and medium noises, SIAM J. Imaging Sci., 7 (2014), 239-276.  doi: 10.1137/130926717.

[5]

H. AmmariH. KangE. KimM. Lim and K. Louati, A direct algorithm for ultrasound imaging of internal corrosion, SIAM J. Numer. Anal., 49 (2011), 1177-1193.  doi: 10.1137/100784710.

[6]

G. BaoS. Hou and P. Li, Inverse scattering by a continuation method with initial guesses from a direct imaging algorithm, J. Comput. Phys., 227 (2007), 755-762.  doi: 10.1016/j.jcp.2007.08.020.

[7]

G. BaoK. HuangP. Li and H. Zhao, A direct imaging method for inverse scattering using the generalized Foldy–Lax formulation, Contemp. Math., 615 (2014), 49-70. 

[8]

G. BaoP. LiG. Lin and F. Triki, Invese scattering problems with multi-frequencies, Inverse Problems, 31 (2015), 093001, 21pp. 

[9]

P. BlomgrenG. Papanicolaou and H. Zhao, Super-resolution in time-reversal acoustics, J. Acoust. Soc. Am., 111 (2002), 230-248.  doi: 10.1121/1.1421342.

[10]

L. BorceaG. GarnierG. Papanicolaou and C. Tsogka, Enhanced statistical stability in coherent interferometric imaging, Inverse Problems, 27 (2011), 085004, 33pp. 

[11]

L. BorceaW. LiA.V. Mamonov and J. Schotland, Mamonov and J. Schotland, Second-harmonic imaging in random media, Inverse Problems, 33 (2017), 065004, 37pp. 

[12]

R. W. Boyd, Nonlinear Optics, 3rd Edition, Academic Press, New York, 2008.

[13]

F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory, Springer-Verlag, Berlin, 2006.

[14]

D. P. ChallaG. Hu and M. Sini, Multiple scattering of electromagnetic waves by a finite number of point-like obstacles, Math. Models Methods Appl. Sci., 24 (2014), 863-899.  doi: 10.1142/S021820251350070X.

[15]

D.P. Challa and M. Sini, Inverse scattering by point-like scatterers in the Foldy regime, Inverse Problems, 22 (2012), 125006, 39pp. 

[16]

D. P. Challa and M. Sini, On the justification of the Foldy–Lax approximation for the acoustic scattering by small rigid bodies of arbitrary shapes, Multiscale Model. Simul., 12 (2014), 55-108.  doi: 10.1137/130919313.

[17]

M. Cheney, The linear sampling method and the MUSIC algorithm, Inverse Problems, 17 (2001), 591-595.  doi: 10.1088/0266-5611/17/4/301.

[18]

J. ChengJ. Liu and G. Nakamura, The numerical realization of the probe method for the inverse scattering problems from the near-field data, Inverse Problems, 21 (2005), 839-855.  doi: 10.1088/0266-5611/21/3/004.

[19]

D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383-393.  doi: 10.1088/0266-5611/12/4/003.

[20]

D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, Pure Appl. Math., John Wiley, New York, 1983.

[21]

M. Danckwerts and L. Novotny, Optical frequency mixing at coupled gold nanoparticles, Phys. Rev. Lett., 98 (2007), 026104.  doi: 10.1103/PhysRevLett.98.026104.

[22]

P. de VriesD. van Coevorden and A. Lagendijk, Point scatterers for classical waves, Rev. Mod. Phys., 70 (1998), 447-466.  doi: 10.1103/RevModPhys.70.447.

[23]

A. Devaney, Super-resolution processing of multi-static data using time-reversal and MUSIC, preprint.

[24]

K. Erhard and R. Potthast, A numerical study of the probe method, SIAM J. Sci. Comput., 28 (2006), 1597-1612.  doi: 10.1137/040607149.

[25]

L. Foldy, The multiple scattering of waves I. General theory of isotropic scattering by randomly distributed scatterers, Phys. Rev., 67 (1945), 107-119.  doi: 10.1103/PhysRev.67.107.

[26]

L. Greengard and J.-Y. Lee, Accelerating the nonuniform fast Fourier transform, SIAM Rev., 46 (2004), 443-454.  doi: 10.1137/S003614450343200X.

[27]

L. Greengard and V. Rokhlin, A fast algorithm for particle simulations, J. Comput. Phys., 73 (1987), 325-348.  doi: 10.1016/0021-9991(87)90140-9.

[28]

B. Gremaud and T. Wellens, Nonlinear coherent transport of waves in disordered media, Phys. Rev. Lett., 100 (2008), 033902. 

[29]

B. Gremaud and T. Wellens, Speckle instability: Coherent effects in nonlinear disordered media, Phys. Rev. Lett., 104 (2010), 133901.  doi: 10.1103/PhysRevLett.104.133901.

[30]

F. GruberE. Marengo and A. Devaney, Time-reversal imaging with multiple signal classification considering multiple scattering between the targets, J. Acoust. Soc. Am., 115 (2004), 3042-3047.  doi: 10.1121/1.1738451.

[31]

S. HouK. Solna and H. Zhao, Imaging of location and geometry for extended targets using the response matrix, J. Comput. Phys., 199 (2004), 317-338.  doi: 10.1016/j.jcp.2004.02.010.

[32]

S. HouK. Solna and H. Zhao, A direct imaging algorithm for extended targets, Inverse Problems, 22 (2006), 1151-1178.  doi: 10.1088/0266-5611/22/4/003.

[33]

S. HouK. Solna and H. Zhao, A direct imaging method using far-field data, Inverse Problems, 23 (2007), 1533-1546.  doi: 10.1088/0266-5611/23/4/010.

[34]

S. HouK. HuangK. Solna and H. Zhao, A phase and space coherent direct imaging method, J. Acoust. Soc. Am., 125 (2009), 227-238.  doi: 10.1121/1.3035835.

[35]

G. HuA. Mantile and M. Sini, Direct and inverse acoustic scattering by a collection of extended and point-like scatterers, Multiscale Model. Simul., 12 (2014), 996-1027.  doi: 10.1137/130932107.

[36]

K. Huang and P. Li, A two-scale multiple scattering problem, Multiscale Model. Simul., 8 (2010), 1511-1534.  doi: 10.1137/090771090.

[37]

K. HuangP. Li and H. Zhao, An efficient algorithm for the generalized Foldy–Lax formulation, J. Comput. Phys., 234 (2013), 376-398.  doi: 10.1016/j.jcp.2012.09.027.

[38]

K. HuangK. Solna and H. Zhao, Generalized Foldy–Lax formulation, J. Comput. Phys., 229 (2010), 4544-4553.  doi: 10.1016/j.jcp.2010.02.021.

[39]

M. Ikehata, Reconstruction of an obstacle from the scattering amplitude at a fixed frequency, Inverse Problems, 14 (1998), 949-954.  doi: 10.1088/0266-5611/14/4/012.

[40]

E. KerbratC. Prada and M. Fink, Imaging in the presence of grain noise using the decomposition of the time reversal operator, J. Acoust. Soc. Am., 113 (2003), 1230-1240.  doi: 10.1121/1.1548156.

[41]

A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, Oxford, 2008.

[42]

J. LaiM. Kobayashi and L. Greengard, A fast solver for multi-particle scattering in a layered medium, Opt. Express, 22 (2014), 20481-20499.  doi: 10.1364/OE.22.020481.

[43]

M. Lax, Multiple scattering of waves, Rev. Modern Phys., 23 (1951), 287-310.  doi: 10.1103/RevModPhys.23.287.

[44]

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

[45]

W. Li and J. Schotland, Optical theorem for nonlinear media, Phys. Rev. A., 92 (2015), 043824.  doi: 10.1103/PhysRevA.92.043824.

[46]

X. Liu, A novel sampling method for multiple multiscale targets from scattering amplitude at a fixed frequency, Inverse Problems, 33 (2017), 085011, 20pp. 

[47]

P. Martin, Multiple Scattering: Interaction of Time-Harmonic Wave with $N$ Obstacles, Encyclopedia Math. Appl., 107, Cambridge University Press, Cambridge, 2006.

[48]

R. Potthast, Stability estimates and reconstructions in inverse acoustic scattering using singular sources, J. Comput. Appl. Math., 114 (2000), 247-274.  doi: 10.1016/S0377-0427(99)00201-0.

[49]

H. Zhao, Analysis of the response matrix for an extended target, SIAM J. Appl. Math., 64 (2004), 725-745.  doi: 10.1137/S0036139902415282.

Figure 1.  Schematic of the problem geometry
Figure 2.  Schematic of the imaging modality with nonlinear point scatterers
Figure 3.  Imaging of two extended scatterers surrounded by 1000 linear point scatterers. (a) Example 1: $\kappa = 10$; (b) Example 2: $\kappa = 50$
Figure 4.  Example 3: Imaging of one extended scatterer with two fixed quadratically nonlinear point scatterers. (a) Imaging with $\kappa_1 = 2$; (b) Imaging with $\kappa_2 = 4$
Figure 5.  Example 3: Imaging of one extended scatterer with two moving quadratically nonlinear point scatterers. (a) Imaging with $\kappa_1 = 2$; (b) Imaging with $\kappa_2 = 4$
Figure 6.  Example 4: Imaging of two extended scatterers with two quadratically nonlinear point scatterers close by. (a) Imaging with $\kappa_1 = 5$; (b) Imaging with $\kappa_2 = 10$
Figure 7.  Example 4: Imaging of two extended scatterers with two quadratically nonlinear point scatterers far away. (a) Imaging with $\kappa_1 = 5$; (b) Imaging with $\kappa_2 = 10$
Figure 8.  Example 5: Imaging of two extended scatterers with two cubically nonlinear point scatterers. (a) Imaging with $\kappa_1 = 2$; (b) Imaging with $\kappa_3 = 6$
Figure 9.  Example 6: Imaging of two extended scatterers with two cubically nonlinear point scatterers. (a) Imaging with $\kappa_1 = 5$; (b) Imaging with $\kappa_3 = 15$
Table 1.  Parameters used in the numerical experiments
$N_{\rm point}$number of point scatterers
$N_{\rm boundary}$number of points to discretize the boundary of extended scatterer(s)
$N_{\rm direction}$number of incident and observation directions
$N_{\rm sampling}$number of sampling points along the $x$-and $y$-direction
$T_{\rm invert}$time to invert (factorize) the scattering matrix
$T_{\rm solver}$time to solve the linear system for one incidence
$T_{\rm ffp}$time to evaluate the far-field patterns
$T_{\rm NUFFT}$time to apply the NUFFT to evaluate the imaging function
$N_{\rm point}$number of point scatterers
$N_{\rm boundary}$number of points to discretize the boundary of extended scatterer(s)
$N_{\rm direction}$number of incident and observation directions
$N_{\rm sampling}$number of sampling points along the $x$-and $y$-direction
$T_{\rm invert}$time to invert (factorize) the scattering matrix
$T_{\rm solver}$time to solve the linear system for one incidence
$T_{\rm ffp}$time to evaluate the far-field patterns
$T_{\rm NUFFT}$time to apply the NUFFT to evaluate the imaging function
Table 2.  Time (in seconds) to solve the linear system (68) on an HP workstation
$N_{\rm point}$ $N_{\rm boundary}$Method 1Method 2Method 3
$1000$ $600$ $0.16$ $1.42$ $0.89$
$10000$ $600$ $8.9$fail to convergefail to converge
$N_{\rm point}$ $N_{\rm boundary}$Method 1Method 2Method 3
$1000$ $600$ $0.16$ $1.42$ $0.89$
$10000$ $600$ $8.9$fail to convergefail to converge
Table 3.  Results for imaging two extended scatterers surrounded by linear point scatterers
$\kappa$$N_{\rm point}$ $N_{\rm boundary}$ $N_{\rm direction}$ $N_{\rm sampling}$
Example 1101000600360500
Example 250100048001800500
$T_{\rm invert}$ $T_{\rm sampling}$ $T_{\rm ffp}$ $T_{\rm NUFFT}$
Example 17.95e-22.56e-32.23e-22.46e-1
Example 21.612.17e-23.49e-13.70
$\kappa$$N_{\rm point}$ $N_{\rm boundary}$ $N_{\rm direction}$ $N_{\rm sampling}$
Example 1101000600360500
Example 250100048001800500
$T_{\rm invert}$ $T_{\rm sampling}$ $T_{\rm ffp}$ $T_{\rm NUFFT}$
Example 17.95e-22.56e-32.23e-22.46e-1
Example 21.612.17e-23.49e-13.70
Table 4.  Results for imaging the extended scatterers surrounded by quadratically nonlinear point scatterers
$\kappa$$N_{\rm point}$ $N_{\rm boundary}$ $N_{\rm direction}$ $N_{\rm sampling}$
Example 322600360500
Example 4521200360500
$T_{\rm invert}$ $T_{\rm solver}$ $T_{\rm ffp}$ $T_{\rm NUFFT}$
Example 31.22e-38.52e-31.39e-23.39e-1
Example 41.90e-13.99e-31.96e-23.84e-1
$\kappa$$N_{\rm point}$ $N_{\rm boundary}$ $N_{\rm direction}$ $N_{\rm sampling}$
Example 322600360500
Example 4521200360500
$T_{\rm invert}$ $T_{\rm solver}$ $T_{\rm ffp}$ $T_{\rm NUFFT}$
Example 31.22e-38.52e-31.39e-23.39e-1
Example 41.90e-13.99e-31.96e-23.84e-1
Table 5.  Results for imaging the extended scatterers surrounded by cubically nonlinear point scatterers
$\kappa$$N_{\rm point}$ $N_{\rm boundary}$ $N_{\rm direction}$ $N_{\rm sampling}$
Example 522600360500
Example 6521200360500
$T_{\rm invert}$ $T_{\rm solver}$ $T_{\rm ffp}$ $T_{\rm NUFFT}$
Example 51.24e-31.00e-21.22e-24.33e-1
Example 64.22e-31.91e-22.11e-24.41e-1
$\kappa$$N_{\rm point}$ $N_{\rm boundary}$ $N_{\rm direction}$ $N_{\rm sampling}$
Example 522600360500
Example 6521200360500
$T_{\rm invert}$ $T_{\rm solver}$ $T_{\rm ffp}$ $T_{\rm NUFFT}$
Example 51.24e-31.00e-21.22e-24.33e-1
Example 64.22e-31.91e-22.11e-24.41e-1
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