Abstract

Linear sampling method (LSM) is a qualitative method used to reconstruct the support of scatterers. This paper presents a modification of the LSM approach. The proposed method analyses the multipole expansion of the scattered field. Only monopole and dipole terms are used for the reconstruction of the scatterer support and all other higher order multipoles are truncated. It is shown that such modification performs better than the mathematical regularization typically used in LSM. The justification for truncation of higher order multipoles is presented. Various examples are presented to demonstrate the performance of the proposed method for dielectric as well as perfectly conducting scatterers in presence of significant amount of additive Gaussian noise.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  6. X. Chen, and Y. Zhong, "A robust noniterative method for obtaining scattering strengths of multiply scattering point targets," J. Acoust. Soc. Am. 122(3), 1325-1327 (2007).
    [CrossRef] [PubMed]
  7. H. Ammari, E. Iakovleva, D. Lesselier, and G. Perrusson, "Music-type electromagnetic imaging of a collection of small three-dimensional inclusions," SIAM J. Sci. Comput. 29(2), 674-709 (2007).
    [CrossRef]
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2009

X. Chen, and Y. Zhong, "MUSIC electromagnetic imaging with enhanced resolution for small inclusions," Inverse Probl. 25(1), 015008 (2009).
[CrossRef]

2008

K. Agarwal, and X. Chen, "Applicability of MUSIC-Type Imaging in Two-Dimensional Electromagnetic Inverse Problems," IEEE Trans. Antenn. Propag. 56(10), 3217-3223 (2008).
[CrossRef]

M. Brignone, G. Bozza, A. Randazzo, M. Piana, and M. Pastorino, "A Hybrid Approach to 3D Microwave Imaging by Using Linear Sampling and ACO," IEEE Trans. Antenn. Propag. 56(10), 3224-3232 (2008).
[CrossRef]

I. Catapano, L. Crocco, and T. Isernia, "Improved Sampling Methods for Shape Reconstruction of 3-D Buried Targets," IEEE Trans. Geosci. Rem. Sens. 46(10), 3265-3273 (2008).
[CrossRef]

2007

I. Catapano, L. Crocco, and T. Isernia, "On simple methods for shape reconstruction of unknown scatterers," IEEE Trans. Antenn. Propag. 55(5), 1431-1436 (2007).
[CrossRef]

I. Catapano, L. Crocco, M. D'Urso, and T. Isernia, "On the effect of support estimation and of a new model in 2-D inverse scattering problems," IEEE Trans. Antenn. Propag. 55(6), 1895-1899 (2007).
[CrossRef]

X. Chen, and Y. Zhong, "A robust noniterative method for obtaining scattering strengths of multiply scattering point targets," J. Acoust. Soc. Am. 122(3), 1325-1327 (2007).
[CrossRef] [PubMed]

H. Ammari, E. Iakovleva, D. Lesselier, and G. Perrusson, "Music-type electromagnetic imaging of a collection of small three-dimensional inclusions," SIAM J. Sci. Comput. 29(2), 674-709 (2007).
[CrossRef]

Y. Zhong, and X. Chen, "MUSIC imaging and electromagnetic inverse scattering of multiple-scattering small anisotropic spheres," IEEE Trans. Antenn. Propag. 55(12), 3542-3549 (2007).
[CrossRef]

2006

T. Rao, and X. Chen, "Analysis of the time-reversal operator for a single cylinder under two-dimensional settings," J. Electromagn. Waves Appl. 20(15), 2153-2165 (2006).
[CrossRef]

D. Colton, and R. Kress, "Using fundamental solutions in inverse scattering," Inverse Probl. 22(3), R49-R66 (2006).
[CrossRef]

2005

A. J. Devaney, "Time reversal imaging of obscured targets from multistatic data," IEEE Trans. Antenn. Propag. 53(5), 1600-1610 (2005).
[CrossRef]

2004

2003

N. Shelton, and K. F. Warnick, "Behavior of the regularized sampling inverse scattering method at internal resonance frequencies," J. Electromagn. Waves Appl. 17(3), 487-488 (2003).
[CrossRef]

D. Colton, H. Haddar, and M. Piana, "The linear sampling method in inverse electromagnetic scattering theory," Inverse Probl. 19(6), S105-S137 (2003).
[CrossRef]

T. Arens, and A. Kirsch, "The factorization method in inverse scattering from periodic structures," Inverse Probl. 19(5), 1195-1211 (2003).
[CrossRef]

2002

F. Cakoni, D. Colton, and H. Haddar, "The linear sampling method for anisotropic media," J. Comput. Appl. Math. 146(2), 285-299 (2002).
[CrossRef]

2001

M. Fink, and C. Prada, "Acoustic time-reversal mirrors," Inverse Probl. 17(1), 201 (2001).
[CrossRef]

2000

A. Kirsch, and S. Ritter, "A linear sampling method for inverse scattering from an open arc," Inverse Probl. 16(1), 89-105 (2000).
[CrossRef]

D. Colton, J. Coyle, and P. Monk, "Recent developments in inverse acoustic scattering theory," SIAM Rev. 42(3), 369-414 (2000).
[CrossRef]

1998

A. Kirsch, "Characterization of the shape of a scattering obstacle using the spectral data of the far field operator," Inverse Probl. 14(6), 1489-1512 (1998).
[CrossRef]

1990

K. Mayer, R. Marklein, K. J. Langenberg, and T. Kreutter, "Three-dimensional imaging system based on Fourier transform synthetic aperture focusing technique," Ultrasonics 28(4), 241-255 (1990).
[CrossRef]

Agarwal, K.

K. Agarwal, and X. Chen, "Applicability of MUSIC-Type Imaging in Two-Dimensional Electromagnetic Inverse Problems," IEEE Trans. Antenn. Propag. 56(10), 3217-3223 (2008).
[CrossRef]

Ammari, H.

H. Ammari, E. Iakovleva, D. Lesselier, and G. Perrusson, "Music-type electromagnetic imaging of a collection of small three-dimensional inclusions," SIAM J. Sci. Comput. 29(2), 674-709 (2007).
[CrossRef]

Arens, T.

T. Arens, and A. Kirsch, "The factorization method in inverse scattering from periodic structures," Inverse Probl. 19(5), 1195-1211 (2003).
[CrossRef]

Bozza, G.

M. Brignone, G. Bozza, A. Randazzo, M. Piana, and M. Pastorino, "A Hybrid Approach to 3D Microwave Imaging by Using Linear Sampling and ACO," IEEE Trans. Antenn. Propag. 56(10), 3224-3232 (2008).
[CrossRef]

Brignone, M.

M. Brignone, G. Bozza, A. Randazzo, M. Piana, and M. Pastorino, "A Hybrid Approach to 3D Microwave Imaging by Using Linear Sampling and ACO," IEEE Trans. Antenn. Propag. 56(10), 3224-3232 (2008).
[CrossRef]

Cakoni, F.

F. Cakoni, D. Colton, and H. Haddar, "The linear sampling method for anisotropic media," J. Comput. Appl. Math. 146(2), 285-299 (2002).
[CrossRef]

Catapano, I.

I. Catapano, L. Crocco, and T. Isernia, "Improved Sampling Methods for Shape Reconstruction of 3-D Buried Targets," IEEE Trans. Geosci. Rem. Sens. 46(10), 3265-3273 (2008).
[CrossRef]

I. Catapano, L. Crocco, M. D'Urso, and T. Isernia, "On the effect of support estimation and of a new model in 2-D inverse scattering problems," IEEE Trans. Antenn. Propag. 55(6), 1895-1899 (2007).
[CrossRef]

I. Catapano, L. Crocco, and T. Isernia, "On simple methods for shape reconstruction of unknown scatterers," IEEE Trans. Antenn. Propag. 55(5), 1431-1436 (2007).
[CrossRef]

Chen, X.

X. Chen, and Y. Zhong, "MUSIC electromagnetic imaging with enhanced resolution for small inclusions," Inverse Probl. 25(1), 015008 (2009).
[CrossRef]

K. Agarwal, and X. Chen, "Applicability of MUSIC-Type Imaging in Two-Dimensional Electromagnetic Inverse Problems," IEEE Trans. Antenn. Propag. 56(10), 3217-3223 (2008).
[CrossRef]

X. Chen, and Y. Zhong, "A robust noniterative method for obtaining scattering strengths of multiply scattering point targets," J. Acoust. Soc. Am. 122(3), 1325-1327 (2007).
[CrossRef] [PubMed]

Y. Zhong, and X. Chen, "MUSIC imaging and electromagnetic inverse scattering of multiple-scattering small anisotropic spheres," IEEE Trans. Antenn. Propag. 55(12), 3542-3549 (2007).
[CrossRef]

T. Rao, and X. Chen, "Analysis of the time-reversal operator for a single cylinder under two-dimensional settings," J. Electromagn. Waves Appl. 20(15), 2153-2165 (2006).
[CrossRef]

Colton, D.

D. Colton, and R. Kress, "Using fundamental solutions in inverse scattering," Inverse Probl. 22(3), R49-R66 (2006).
[CrossRef]

D. Colton, H. Haddar, and M. Piana, "The linear sampling method in inverse electromagnetic scattering theory," Inverse Probl. 19(6), S105-S137 (2003).
[CrossRef]

F. Cakoni, D. Colton, and H. Haddar, "The linear sampling method for anisotropic media," J. Comput. Appl. Math. 146(2), 285-299 (2002).
[CrossRef]

D. Colton, J. Coyle, and P. Monk, "Recent developments in inverse acoustic scattering theory," SIAM Rev. 42(3), 369-414 (2000).
[CrossRef]

Coyle, J.

D. Colton, J. Coyle, and P. Monk, "Recent developments in inverse acoustic scattering theory," SIAM Rev. 42(3), 369-414 (2000).
[CrossRef]

Crocco, L.

I. Catapano, L. Crocco, and T. Isernia, "Improved Sampling Methods for Shape Reconstruction of 3-D Buried Targets," IEEE Trans. Geosci. Rem. Sens. 46(10), 3265-3273 (2008).
[CrossRef]

I. Catapano, L. Crocco, M. D'Urso, and T. Isernia, "On the effect of support estimation and of a new model in 2-D inverse scattering problems," IEEE Trans. Antenn. Propag. 55(6), 1895-1899 (2007).
[CrossRef]

I. Catapano, L. Crocco, and T. Isernia, "On simple methods for shape reconstruction of unknown scatterers," IEEE Trans. Antenn. Propag. 55(5), 1431-1436 (2007).
[CrossRef]

Devaney, A. J.

A. J. Devaney, "Time reversal imaging of obscured targets from multistatic data," IEEE Trans. Antenn. Propag. 53(5), 1600-1610 (2005).
[CrossRef]

D'Urso, M.

I. Catapano, L. Crocco, M. D'Urso, and T. Isernia, "On the effect of support estimation and of a new model in 2-D inverse scattering problems," IEEE Trans. Antenn. Propag. 55(6), 1895-1899 (2007).
[CrossRef]

Fink, M.

M. Fink, and C. Prada, "Acoustic time-reversal mirrors," Inverse Probl. 17(1), 201 (2001).
[CrossRef]

Haddar, H.

D. Colton, H. Haddar, and M. Piana, "The linear sampling method in inverse electromagnetic scattering theory," Inverse Probl. 19(6), S105-S137 (2003).
[CrossRef]

F. Cakoni, D. Colton, and H. Haddar, "The linear sampling method for anisotropic media," J. Comput. Appl. Math. 146(2), 285-299 (2002).
[CrossRef]

Iakovleva, E.

H. Ammari, E. Iakovleva, D. Lesselier, and G. Perrusson, "Music-type electromagnetic imaging of a collection of small three-dimensional inclusions," SIAM J. Sci. Comput. 29(2), 674-709 (2007).
[CrossRef]

Isernia, T.

I. Catapano, L. Crocco, and T. Isernia, "Improved Sampling Methods for Shape Reconstruction of 3-D Buried Targets," IEEE Trans. Geosci. Rem. Sens. 46(10), 3265-3273 (2008).
[CrossRef]

I. Catapano, L. Crocco, M. D'Urso, and T. Isernia, "On the effect of support estimation and of a new model in 2-D inverse scattering problems," IEEE Trans. Antenn. Propag. 55(6), 1895-1899 (2007).
[CrossRef]

I. Catapano, L. Crocco, and T. Isernia, "On simple methods for shape reconstruction of unknown scatterers," IEEE Trans. Antenn. Propag. 55(5), 1431-1436 (2007).
[CrossRef]

Kirsch, A.

A. Kirsch, "The factorization method for Maxwell's equations," Inverse Probl. 20(6), S117-S134 (2004).
[CrossRef]

T. Arens, and A. Kirsch, "The factorization method in inverse scattering from periodic structures," Inverse Probl. 19(5), 1195-1211 (2003).
[CrossRef]

A. Kirsch, and S. Ritter, "A linear sampling method for inverse scattering from an open arc," Inverse Probl. 16(1), 89-105 (2000).
[CrossRef]

A. Kirsch, "Characterization of the shape of a scattering obstacle using the spectral data of the far field operator," Inverse Probl. 14(6), 1489-1512 (1998).
[CrossRef]

Kress, R.

D. Colton, and R. Kress, "Using fundamental solutions in inverse scattering," Inverse Probl. 22(3), R49-R66 (2006).
[CrossRef]

Kreutter, T.

K. Mayer, R. Marklein, K. J. Langenberg, and T. Kreutter, "Three-dimensional imaging system based on Fourier transform synthetic aperture focusing technique," Ultrasonics 28(4), 241-255 (1990).
[CrossRef]

Langenberg, K. J.

K. Mayer, R. Marklein, K. J. Langenberg, and T. Kreutter, "Three-dimensional imaging system based on Fourier transform synthetic aperture focusing technique," Ultrasonics 28(4), 241-255 (1990).
[CrossRef]

Lesselier, D.

H. Ammari, E. Iakovleva, D. Lesselier, and G. Perrusson, "Music-type electromagnetic imaging of a collection of small three-dimensional inclusions," SIAM J. Sci. Comput. 29(2), 674-709 (2007).
[CrossRef]

Li, M. L.

Li, P. C.

Liao, C. K.

Marklein, R.

K. Mayer, R. Marklein, K. J. Langenberg, and T. Kreutter, "Three-dimensional imaging system based on Fourier transform synthetic aperture focusing technique," Ultrasonics 28(4), 241-255 (1990).
[CrossRef]

Mayer, K.

K. Mayer, R. Marklein, K. J. Langenberg, and T. Kreutter, "Three-dimensional imaging system based on Fourier transform synthetic aperture focusing technique," Ultrasonics 28(4), 241-255 (1990).
[CrossRef]

Monk, P.

D. Colton, J. Coyle, and P. Monk, "Recent developments in inverse acoustic scattering theory," SIAM Rev. 42(3), 369-414 (2000).
[CrossRef]

Pastorino, M.

M. Brignone, G. Bozza, A. Randazzo, M. Piana, and M. Pastorino, "A Hybrid Approach to 3D Microwave Imaging by Using Linear Sampling and ACO," IEEE Trans. Antenn. Propag. 56(10), 3224-3232 (2008).
[CrossRef]

Perrusson, G.

H. Ammari, E. Iakovleva, D. Lesselier, and G. Perrusson, "Music-type electromagnetic imaging of a collection of small three-dimensional inclusions," SIAM J. Sci. Comput. 29(2), 674-709 (2007).
[CrossRef]

Piana, M.

M. Brignone, G. Bozza, A. Randazzo, M. Piana, and M. Pastorino, "A Hybrid Approach to 3D Microwave Imaging by Using Linear Sampling and ACO," IEEE Trans. Antenn. Propag. 56(10), 3224-3232 (2008).
[CrossRef]

D. Colton, H. Haddar, and M. Piana, "The linear sampling method in inverse electromagnetic scattering theory," Inverse Probl. 19(6), S105-S137 (2003).
[CrossRef]

Prada, C.

M. Fink, and C. Prada, "Acoustic time-reversal mirrors," Inverse Probl. 17(1), 201 (2001).
[CrossRef]

Randazzo, A.

M. Brignone, G. Bozza, A. Randazzo, M. Piana, and M. Pastorino, "A Hybrid Approach to 3D Microwave Imaging by Using Linear Sampling and ACO," IEEE Trans. Antenn. Propag. 56(10), 3224-3232 (2008).
[CrossRef]

Rao, T.

T. Rao, and X. Chen, "Analysis of the time-reversal operator for a single cylinder under two-dimensional settings," J. Electromagn. Waves Appl. 20(15), 2153-2165 (2006).
[CrossRef]

Ritter, S.

A. Kirsch, and S. Ritter, "A linear sampling method for inverse scattering from an open arc," Inverse Probl. 16(1), 89-105 (2000).
[CrossRef]

Shelton, N.

N. Shelton, and K. F. Warnick, "Behavior of the regularized sampling inverse scattering method at internal resonance frequencies," J. Electromagn. Waves Appl. 17(3), 487-488 (2003).
[CrossRef]

Warnick, K. F.

N. Shelton, and K. F. Warnick, "Behavior of the regularized sampling inverse scattering method at internal resonance frequencies," J. Electromagn. Waves Appl. 17(3), 487-488 (2003).
[CrossRef]

Zhong, Y.

X. Chen, and Y. Zhong, "MUSIC electromagnetic imaging with enhanced resolution for small inclusions," Inverse Probl. 25(1), 015008 (2009).
[CrossRef]

X. Chen, and Y. Zhong, "A robust noniterative method for obtaining scattering strengths of multiply scattering point targets," J. Acoust. Soc. Am. 122(3), 1325-1327 (2007).
[CrossRef] [PubMed]

Y. Zhong, and X. Chen, "MUSIC imaging and electromagnetic inverse scattering of multiple-scattering small anisotropic spheres," IEEE Trans. Antenn. Propag. 55(12), 3542-3549 (2007).
[CrossRef]

IEEE Trans. Antenn. Propag.

K. Agarwal, and X. Chen, "Applicability of MUSIC-Type Imaging in Two-Dimensional Electromagnetic Inverse Problems," IEEE Trans. Antenn. Propag. 56(10), 3217-3223 (2008).
[CrossRef]

Y. Zhong, and X. Chen, "MUSIC imaging and electromagnetic inverse scattering of multiple-scattering small anisotropic spheres," IEEE Trans. Antenn. Propag. 55(12), 3542-3549 (2007).
[CrossRef]

M. Brignone, G. Bozza, A. Randazzo, M. Piana, and M. Pastorino, "A Hybrid Approach to 3D Microwave Imaging by Using Linear Sampling and ACO," IEEE Trans. Antenn. Propag. 56(10), 3224-3232 (2008).
[CrossRef]

I. Catapano, L. Crocco, M. D'Urso, and T. Isernia, "On the effect of support estimation and of a new model in 2-D inverse scattering problems," IEEE Trans. Antenn. Propag. 55(6), 1895-1899 (2007).
[CrossRef]

I. Catapano, L. Crocco, and T. Isernia, "On simple methods for shape reconstruction of unknown scatterers," IEEE Trans. Antenn. Propag. 55(5), 1431-1436 (2007).
[CrossRef]

A. J. Devaney, "Time reversal imaging of obscured targets from multistatic data," IEEE Trans. Antenn. Propag. 53(5), 1600-1610 (2005).
[CrossRef]

IEEE Trans. Geosci. Rem. Sens.

I. Catapano, L. Crocco, and T. Isernia, "Improved Sampling Methods for Shape Reconstruction of 3-D Buried Targets," IEEE Trans. Geosci. Rem. Sens. 46(10), 3265-3273 (2008).
[CrossRef]

Inverse Probl.

T. Arens, and A. Kirsch, "The factorization method in inverse scattering from periodic structures," Inverse Probl. 19(5), 1195-1211 (2003).
[CrossRef]

A. Kirsch, "The factorization method for Maxwell's equations," Inverse Probl. 20(6), S117-S134 (2004).
[CrossRef]

A. Kirsch, "Characterization of the shape of a scattering obstacle using the spectral data of the far field operator," Inverse Probl. 14(6), 1489-1512 (1998).
[CrossRef]

A. Kirsch, and S. Ritter, "A linear sampling method for inverse scattering from an open arc," Inverse Probl. 16(1), 89-105 (2000).
[CrossRef]

D. Colton, H. Haddar, and M. Piana, "The linear sampling method in inverse electromagnetic scattering theory," Inverse Probl. 19(6), S105-S137 (2003).
[CrossRef]

D. Colton, and R. Kress, "Using fundamental solutions in inverse scattering," Inverse Probl. 22(3), R49-R66 (2006).
[CrossRef]

X. Chen, and Y. Zhong, "MUSIC electromagnetic imaging with enhanced resolution for small inclusions," Inverse Probl. 25(1), 015008 (2009).
[CrossRef]

M. Fink, and C. Prada, "Acoustic time-reversal mirrors," Inverse Probl. 17(1), 201 (2001).
[CrossRef]

J. Acoust. Soc. Am.

X. Chen, and Y. Zhong, "A robust noniterative method for obtaining scattering strengths of multiply scattering point targets," J. Acoust. Soc. Am. 122(3), 1325-1327 (2007).
[CrossRef] [PubMed]

J. Comput. Appl. Math.

F. Cakoni, D. Colton, and H. Haddar, "The linear sampling method for anisotropic media," J. Comput. Appl. Math. 146(2), 285-299 (2002).
[CrossRef]

J. Electromagn. Waves Appl.

T. Rao, and X. Chen, "Analysis of the time-reversal operator for a single cylinder under two-dimensional settings," J. Electromagn. Waves Appl. 20(15), 2153-2165 (2006).
[CrossRef]

N. Shelton, and K. F. Warnick, "Behavior of the regularized sampling inverse scattering method at internal resonance frequencies," J. Electromagn. Waves Appl. 17(3), 487-488 (2003).
[CrossRef]

Opt. Lett.

SIAM J. Sci. Comput.

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Figures (8)

Fig. 1.
Fig. 1.

Illustration of the current distribution for the sampling point r⃗p = (0,0) and the proposed multipole-based interpretation.

Fig. 2.
Fig. 2.

Comparison of MLSM (N = 20) and MLSM (N = 1) for noise-free and noisy (10% additive Gaussian noise) scenarios.

Fig. 3.
Fig. 3.

Effect of reduction of multipoles. After obtaining the values of υn by substituting the source distribution computed using conventional LSM and the matrix A̿ computed using Eq. (14) for N = 20 and N = 1 respectively, the absolute value of difference in υn corresponding to these cases for n = − 1 to 1 is computed. First, second, and third columns show this absolute value of difference for n equals to −1, 0, and 1 respectively.

Fig. 4.
Fig. 4.

Effect of noise on the multipoles. First column shows the values of υn obtained by the proposed method in the noise-free scenario. Second column is similar to the first column with the only difference being the noisy scenario (10% additive Gaussian noise). Third column displays the absolute value of the difference between the data plotted in first and second columns. Fourth column shows the difference relative to the magnitude of υn averaged over the noisy and noise-free cases.

Fig. 5.
Fig. 5.

Comparison of LSM and MLSM. In (b), ‘Error’ refers to the error measure defined in (21).The above results are in the presence of 10% noise.

Fig. 6.
Fig. 6.

Plot of error measure defined in (21) for the examples of dielectric scatterers presented in Fig. 7. The presented results are in the presence of 10% noise.

Fig. 7.
Fig. 7.

Examples of reconstruction of dielectric cylinders. The first column shows the scatterer profile (relative permittivity), the second column shows the reconstruction using conventional LSM and the third column shows the reconstruction using the proposed method. The presented results are in the presence of 10% noise.

Fig. 8.
Fig. 8.

Examples of reconstruction of perfectly conducting cylinders. The first column shows the scatterer profile (contours of cylinders), the second column shows the reconstruction using conventional LSM and the third column shows the reconstruction using the proposed method. The presented results are in the presence of 10% noise.

Equations (22)

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Γ E φ θ g θ r p = Φ φ r p .
u ( r ) = Φ r r p u ( r ) n = Φ r r p n } for r Ω ,
u ( r ) = Ω [ Φ r r u n ( r ) u ( r ) Φ r r n ( r ) ] ds ( r ) Ω [ 2 u ( r ) + k 2 u ( r ) ] Φ r r d r ,
u ( r ) = Ω [ Φ r r Φ r r p n ( r ) Φ r r p Φ r r n ( r ) ] ds ( r ) + Ω J ( r ) Φ r r d r .
Ω + Γ [ Φ r r Φ r r p n ( r ) Φ r r p Φ r r n ( r ) ] ds ( r ) ,
= 2 \ Ω - ( Φ r r 2 Φ r r p Φ r r p 2 Φ r r ) d r = 0
u ( r ) = Ω J ( r ) Φ r r d r , r Ω .
E r φ θ = J r θ Φ r φ r d r ,
E r φ θ = n = α ( n ) r p θ Φ ( n ) r φ r p ,
α ( n ) r p θ = J r θ J n ( k r p r ) e in arg ( r r p ) d r ,
Φ ( n ) r φ r p = i 4 H n ( 1 ) ( k r φ r p ) e in arg ( r φ r p ) .
Γ α ( n ) r p θ g θ r p = { 1 n = 0 0 otherwise .
υ n = Γ α ( n ) r p θ g θ r p d θ .
E φ θ n = N N α ( n ) r p θ Φ ( n ) φ r p .
E ¯ = Φ ̿ A ¯ ,
A ̿ g ¯ = D ¯ ,
g ¯ LSM = E ̿ + Φ ¯ ,
g ¯ LSM = s σ s σ s 2 + α 2 v ¯ s u ¯ s * Φ ¯ ,
A ¯ = Φ ̿ + E ¯ .
g ¯ MLSM = ( Φ ̿ + E ̿ ) + D ¯ .
= { r p : log 10 g θ r p > Min + β ( Max Min ) }
Error ( β ) = M err ( β ) M ,

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