Abstract

A new hybrid scattering series is derived that incorporates as special cases both the Born and Rytov scattering series, and includes a parameter so that the behavior can be continuously varied between the two series. The parameter enables the error to be shifted between the Born and Rytov error terms to improve accuracy. The linearized hybrid approximation is derived as well as its condition of validity. Higher order terms of the hybrid series are also found. Also included is the integral equation that defines the exact solution to the forward scattering problem as well as its Fréchet derivative, which is used for the solution of inverse multiple scattering problems. Finally, the linearized hybrid approximation is demonstrated by simulations of inverse scattering off of uniform circular cylinders, where it is shown that the hybrid approximation achieves smaller error than either the Born or Rytov approximations alone.

© 2006 Optical Society of America

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References

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  1. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, Piscataway, NJ, 1995).
  2. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).
  3. G. Beylkin and M. L. Oristaglio, "Distorted-wave Born and distorted-wave Rytov approximations," Opt. Commun. 53, 213-216 (1985).
    [CrossRef]
  4. S. D. Rajan and G. V. Frisk, "A comparison between the Born and Rytov approximations for the inverse backscattering problem," Geophysics 54, 864-871 (1989).
    [CrossRef]
  5. M. J. Woodward, "Wave-equation tomography," Geophysics 57, 15-26 (1992).
    [CrossRef]
  6. M. I. Sancer and A. D. Varvatsis, "A comparison of the Born and Rytov methods," Proc. IEEE 58, 140-141 (1970).
    [CrossRef]
  7. M. Slaney, A. C. Kak, and L. E. Larsen, "Limitations of imaging with first-order diffraction tomography," IEEE Trans. Microwave Theory Tech. MTT-32, 860-874 (1984).
    [CrossRef]
  8. F. C. Lin and M. A. Fiddy, "The Born-Rytov controversy: I. Comparing the analytical and approximate expressions for the one-dimensional deterministic case," J. Opt. Soc. Am. A 9, 1102-1110 (1992).
    [CrossRef]
  9. F. C. Lin and M. A. Fiddy, "Born-Rytov controversy: II Applications to nonlinear and stochastic scattering problems in one-dimensional half-space media," J. Opt. Soc. Am. A 10, 1971-1983 (1993).
    [CrossRef]
  10. G. Gbur and E. Wolf, "Relation between computed tomography and diffraction tomography," J. Opt. Soc. Am. A 18, 2132-2137 (2001).
    [CrossRef]
  11. Z. Q. Lu, "Multidimensional structure diffraction tomography for varying object orientation through generalized scattered waves," Inv. Prob. 1, 339-356 (1985).
    [CrossRef]
  12. Z.-Q. Lu, "JKM Perturbation Theory, Relaxation Perturbation Theory, and their Applications to Inverse Scattering: Theory and Reconstruction Algorithms," IEEE Trans. Ultra. Ferroelectr. Freq. Control UFFC-32, 722-730 (1986).
  13. G. A. Tsihrintzis and A. J. Devaney, "Higher order (nonlinear) diffraction tomography: inversion of the Rytov series," IEEE Trans. Inf. Theory 46, 1748-1761 (2000).
    [CrossRef]
  14. W. C. Chew and Y. M. Wang, "Reconstruction of two-dimensional permittivity distribution using distorted Born iterative method," IEEE Trans. Med. Imaging 9, 218-225 (1990).
    [CrossRef] [PubMed]
  15. R. E. Kleinman and P. M. van der Berg, "A modified gradient method for two-dimensional problems in tomography," J. Comput. Appl. Math 42, 17-35 (1992).
    [CrossRef]
  16. R. E. Kleinman and P. M. van der Berg, "An extended-range modified gradient technique for profile inversion," Radio Sci. 28, 877-884 (1993).
    [CrossRef]
  17. K. Belkebir and A. G. Tijhuis, "Modified gradient method and modified Born method for solving a two dimensional inverse scattering problem," Inv. Prob. 17, 1671-1688 (2001).
    [CrossRef]
  18. G. A. Tsihrintzis and A. J. Devaney, "Higher-order (Nonlinear) Diffraction Tomography: Reconstruction Algorithms and Computer Simulation," IEEE Trans. Image Process. 9, 1560-1572 (2000).
    [CrossRef]
  19. G. A. Tsihrintzis and A. J. Devaney, "A Volterra series approach to nonlinear traveltime tomography," IEEE Trans. Geosco. Remote Sens. 38, 1733-1742 (2000).
    [CrossRef]
  20. A. J. Devaney, "A filtered back propagation algorithm for diffraction tomography," Ultrason. Imaging 4, 336-350 (1982).
    [CrossRef] [PubMed]
  21. V. A. Markel, J. A. O’Sullivan, and J. C. Schotland, "Inverse problem in optical diffusion tomography. IV. Nonlinear inversion formulas," J. Opt. Soc. Am. A 20, 903-912 (2003).
    [CrossRef]
  22. M. Slaney, "Diffraction Tomography Algorithms from Malcolm Slaney PhD Dissertation," obtained from http://rvl4.ecn.purdue.edu/˜malcolm/purdue/diffract.tar.Z.

2003 (1)

2001 (2)

G. Gbur and E. Wolf, "Relation between computed tomography and diffraction tomography," J. Opt. Soc. Am. A 18, 2132-2137 (2001).
[CrossRef]

K. Belkebir and A. G. Tijhuis, "Modified gradient method and modified Born method for solving a two dimensional inverse scattering problem," Inv. Prob. 17, 1671-1688 (2001).
[CrossRef]

2000 (3)

G. A. Tsihrintzis and A. J. Devaney, "Higher-order (Nonlinear) Diffraction Tomography: Reconstruction Algorithms and Computer Simulation," IEEE Trans. Image Process. 9, 1560-1572 (2000).
[CrossRef]

G. A. Tsihrintzis and A. J. Devaney, "A Volterra series approach to nonlinear traveltime tomography," IEEE Trans. Geosco. Remote Sens. 38, 1733-1742 (2000).
[CrossRef]

G. A. Tsihrintzis and A. J. Devaney, "Higher order (nonlinear) diffraction tomography: inversion of the Rytov series," IEEE Trans. Inf. Theory 46, 1748-1761 (2000).
[CrossRef]

1993 (2)

1992 (3)

F. C. Lin and M. A. Fiddy, "The Born-Rytov controversy: I. Comparing the analytical and approximate expressions for the one-dimensional deterministic case," J. Opt. Soc. Am. A 9, 1102-1110 (1992).
[CrossRef]

M. J. Woodward, "Wave-equation tomography," Geophysics 57, 15-26 (1992).
[CrossRef]

R. E. Kleinman and P. M. van der Berg, "A modified gradient method for two-dimensional problems in tomography," J. Comput. Appl. Math 42, 17-35 (1992).
[CrossRef]

1990 (1)

W. C. Chew and Y. M. Wang, "Reconstruction of two-dimensional permittivity distribution using distorted Born iterative method," IEEE Trans. Med. Imaging 9, 218-225 (1990).
[CrossRef] [PubMed]

1989 (1)

S. D. Rajan and G. V. Frisk, "A comparison between the Born and Rytov approximations for the inverse backscattering problem," Geophysics 54, 864-871 (1989).
[CrossRef]

1986 (1)

Z.-Q. Lu, "JKM Perturbation Theory, Relaxation Perturbation Theory, and their Applications to Inverse Scattering: Theory and Reconstruction Algorithms," IEEE Trans. Ultra. Ferroelectr. Freq. Control UFFC-32, 722-730 (1986).

1985 (2)

Z. Q. Lu, "Multidimensional structure diffraction tomography for varying object orientation through generalized scattered waves," Inv. Prob. 1, 339-356 (1985).
[CrossRef]

G. Beylkin and M. L. Oristaglio, "Distorted-wave Born and distorted-wave Rytov approximations," Opt. Commun. 53, 213-216 (1985).
[CrossRef]

1984 (1)

M. Slaney, A. C. Kak, and L. E. Larsen, "Limitations of imaging with first-order diffraction tomography," IEEE Trans. Microwave Theory Tech. MTT-32, 860-874 (1984).
[CrossRef]

1982 (1)

A. J. Devaney, "A filtered back propagation algorithm for diffraction tomography," Ultrason. Imaging 4, 336-350 (1982).
[CrossRef] [PubMed]

1970 (1)

M. I. Sancer and A. D. Varvatsis, "A comparison of the Born and Rytov methods," Proc. IEEE 58, 140-141 (1970).
[CrossRef]

Belkebir, K.

K. Belkebir and A. G. Tijhuis, "Modified gradient method and modified Born method for solving a two dimensional inverse scattering problem," Inv. Prob. 17, 1671-1688 (2001).
[CrossRef]

Beylkin, G.

G. Beylkin and M. L. Oristaglio, "Distorted-wave Born and distorted-wave Rytov approximations," Opt. Commun. 53, 213-216 (1985).
[CrossRef]

Chew, W. C.

W. C. Chew and Y. M. Wang, "Reconstruction of two-dimensional permittivity distribution using distorted Born iterative method," IEEE Trans. Med. Imaging 9, 218-225 (1990).
[CrossRef] [PubMed]

Devaney, A. J.

G. A. Tsihrintzis and A. J. Devaney, "A Volterra series approach to nonlinear traveltime tomography," IEEE Trans. Geosco. Remote Sens. 38, 1733-1742 (2000).
[CrossRef]

G. A. Tsihrintzis and A. J. Devaney, "Higher-order (Nonlinear) Diffraction Tomography: Reconstruction Algorithms and Computer Simulation," IEEE Trans. Image Process. 9, 1560-1572 (2000).
[CrossRef]

G. A. Tsihrintzis and A. J. Devaney, "Higher order (nonlinear) diffraction tomography: inversion of the Rytov series," IEEE Trans. Inf. Theory 46, 1748-1761 (2000).
[CrossRef]

A. J. Devaney, "A filtered back propagation algorithm for diffraction tomography," Ultrason. Imaging 4, 336-350 (1982).
[CrossRef] [PubMed]

Fiddy, M. A.

Frisk, G. V.

S. D. Rajan and G. V. Frisk, "A comparison between the Born and Rytov approximations for the inverse backscattering problem," Geophysics 54, 864-871 (1989).
[CrossRef]

Gbur, G.

Kak, A. C.

M. Slaney, A. C. Kak, and L. E. Larsen, "Limitations of imaging with first-order diffraction tomography," IEEE Trans. Microwave Theory Tech. MTT-32, 860-874 (1984).
[CrossRef]

Kleinman, R. E.

R. E. Kleinman and P. M. van der Berg, "An extended-range modified gradient technique for profile inversion," Radio Sci. 28, 877-884 (1993).
[CrossRef]

R. E. Kleinman and P. M. van der Berg, "A modified gradient method for two-dimensional problems in tomography," J. Comput. Appl. Math 42, 17-35 (1992).
[CrossRef]

Larsen, L. E.

M. Slaney, A. C. Kak, and L. E. Larsen, "Limitations of imaging with first-order diffraction tomography," IEEE Trans. Microwave Theory Tech. MTT-32, 860-874 (1984).
[CrossRef]

Lin, F. C.

Lu, Z. Q.

Z. Q. Lu, "Multidimensional structure diffraction tomography for varying object orientation through generalized scattered waves," Inv. Prob. 1, 339-356 (1985).
[CrossRef]

Lu, Z.-Q.

Z.-Q. Lu, "JKM Perturbation Theory, Relaxation Perturbation Theory, and their Applications to Inverse Scattering: Theory and Reconstruction Algorithms," IEEE Trans. Ultra. Ferroelectr. Freq. Control UFFC-32, 722-730 (1986).

Markel, V. A.

O’Sullivan, J. A.

Oristaglio, M. L.

G. Beylkin and M. L. Oristaglio, "Distorted-wave Born and distorted-wave Rytov approximations," Opt. Commun. 53, 213-216 (1985).
[CrossRef]

Rajan, S. D.

S. D. Rajan and G. V. Frisk, "A comparison between the Born and Rytov approximations for the inverse backscattering problem," Geophysics 54, 864-871 (1989).
[CrossRef]

Sancer, M. I.

M. I. Sancer and A. D. Varvatsis, "A comparison of the Born and Rytov methods," Proc. IEEE 58, 140-141 (1970).
[CrossRef]

Schotland, J. C.

Slaney, M.

M. Slaney, A. C. Kak, and L. E. Larsen, "Limitations of imaging with first-order diffraction tomography," IEEE Trans. Microwave Theory Tech. MTT-32, 860-874 (1984).
[CrossRef]

Tijhuis, A. G.

K. Belkebir and A. G. Tijhuis, "Modified gradient method and modified Born method for solving a two dimensional inverse scattering problem," Inv. Prob. 17, 1671-1688 (2001).
[CrossRef]

Tsihrintzis, G. A.

G. A. Tsihrintzis and A. J. Devaney, "Higher-order (Nonlinear) Diffraction Tomography: Reconstruction Algorithms and Computer Simulation," IEEE Trans. Image Process. 9, 1560-1572 (2000).
[CrossRef]

G. A. Tsihrintzis and A. J. Devaney, "A Volterra series approach to nonlinear traveltime tomography," IEEE Trans. Geosco. Remote Sens. 38, 1733-1742 (2000).
[CrossRef]

G. A. Tsihrintzis and A. J. Devaney, "Higher order (nonlinear) diffraction tomography: inversion of the Rytov series," IEEE Trans. Inf. Theory 46, 1748-1761 (2000).
[CrossRef]

van der Berg, P. M.

R. E. Kleinman and P. M. van der Berg, "An extended-range modified gradient technique for profile inversion," Radio Sci. 28, 877-884 (1993).
[CrossRef]

R. E. Kleinman and P. M. van der Berg, "A modified gradient method for two-dimensional problems in tomography," J. Comput. Appl. Math 42, 17-35 (1992).
[CrossRef]

Varvatsis, A. D.

M. I. Sancer and A. D. Varvatsis, "A comparison of the Born and Rytov methods," Proc. IEEE 58, 140-141 (1970).
[CrossRef]

Wang, Y. M.

W. C. Chew and Y. M. Wang, "Reconstruction of two-dimensional permittivity distribution using distorted Born iterative method," IEEE Trans. Med. Imaging 9, 218-225 (1990).
[CrossRef] [PubMed]

Wolf, E.

Woodward, M. J.

M. J. Woodward, "Wave-equation tomography," Geophysics 57, 15-26 (1992).
[CrossRef]

Geophysics (2)

S. D. Rajan and G. V. Frisk, "A comparison between the Born and Rytov approximations for the inverse backscattering problem," Geophysics 54, 864-871 (1989).
[CrossRef]

M. J. Woodward, "Wave-equation tomography," Geophysics 57, 15-26 (1992).
[CrossRef]

IEEE Trans. Geosco. Remote Sens. (1)

G. A. Tsihrintzis and A. J. Devaney, "A Volterra series approach to nonlinear traveltime tomography," IEEE Trans. Geosco. Remote Sens. 38, 1733-1742 (2000).
[CrossRef]

IEEE Trans. Image Process. (1)

G. A. Tsihrintzis and A. J. Devaney, "Higher-order (Nonlinear) Diffraction Tomography: Reconstruction Algorithms and Computer Simulation," IEEE Trans. Image Process. 9, 1560-1572 (2000).
[CrossRef]

IEEE Trans. Inf. Theory (1)

G. A. Tsihrintzis and A. J. Devaney, "Higher order (nonlinear) diffraction tomography: inversion of the Rytov series," IEEE Trans. Inf. Theory 46, 1748-1761 (2000).
[CrossRef]

IEEE Trans. Med. Imaging (1)

W. C. Chew and Y. M. Wang, "Reconstruction of two-dimensional permittivity distribution using distorted Born iterative method," IEEE Trans. Med. Imaging 9, 218-225 (1990).
[CrossRef] [PubMed]

IEEE Trans. Microwave Theory Tech. (1)

M. Slaney, A. C. Kak, and L. E. Larsen, "Limitations of imaging with first-order diffraction tomography," IEEE Trans. Microwave Theory Tech. MTT-32, 860-874 (1984).
[CrossRef]

IEEE Trans. Ultra. Ferroelectr. Freq. Control (1)

Z.-Q. Lu, "JKM Perturbation Theory, Relaxation Perturbation Theory, and their Applications to Inverse Scattering: Theory and Reconstruction Algorithms," IEEE Trans. Ultra. Ferroelectr. Freq. Control UFFC-32, 722-730 (1986).

J. Comput. Appl. Math (1)

R. E. Kleinman and P. M. van der Berg, "A modified gradient method for two-dimensional problems in tomography," J. Comput. Appl. Math 42, 17-35 (1992).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Commun. (1)

G. Beylkin and M. L. Oristaglio, "Distorted-wave Born and distorted-wave Rytov approximations," Opt. Commun. 53, 213-216 (1985).
[CrossRef]

Prob. (2)

K. Belkebir and A. G. Tijhuis, "Modified gradient method and modified Born method for solving a two dimensional inverse scattering problem," Inv. Prob. 17, 1671-1688 (2001).
[CrossRef]

Z. Q. Lu, "Multidimensional structure diffraction tomography for varying object orientation through generalized scattered waves," Inv. Prob. 1, 339-356 (1985).
[CrossRef]

Proc. IEEE (1)

M. I. Sancer and A. D. Varvatsis, "A comparison of the Born and Rytov methods," Proc. IEEE 58, 140-141 (1970).
[CrossRef]

Radio Sci. (1)

R. E. Kleinman and P. M. van der Berg, "An extended-range modified gradient technique for profile inversion," Radio Sci. 28, 877-884 (1993).
[CrossRef]

Ultrason. Imaging (1)

A. J. Devaney, "A filtered back propagation algorithm for diffraction tomography," Ultrason. Imaging 4, 336-350 (1982).
[CrossRef] [PubMed]

Other (3)

M. Slaney, "Diffraction Tomography Algorithms from Malcolm Slaney PhD Dissertation," obtained from http://rvl4.ecn.purdue.edu/˜malcolm/purdue/diffract.tar.Z.

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, Piscataway, NJ, 1995).

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).

Supplementary Material (3)

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Figures (2)

Fig. 1.
Fig. 1.

Profile plots of computed reconstructions of cylinders with various refractive index and radii using the Born, Rytov, and hybrid approximations. The blue curve is the Born reconstruction of the refractive index contrast, the magenta curve the Rytov reconstruction, and the black curve the hybrid reconstruction. The text in the upper left corners of each subfigure (a) to (j) specifies the radius and index contrast of each cylinder, and the optimal exponent for which the error was minimized. The text in the upper right corners is the RMS error for the Born, Rytov, and hybrid reconstructions.

Fig. 2.
Fig. 2.

More profile plots of computed reconstructions of cylinders with various refractive index and radii using the Born, Rytov, and hybrid approximations. The blue curve is the Born reconstruction of the refractive index contrast, the magenta curve the Rytov reconstruction, and the black curve the hybrid reconstruction. The text in the upper left corners of each subfigure (a) to (h) specifies the radius and index contrast of each cylinder, and the optimal exponent for which the error was minimized. The text in the upper right corners is the RMS error for the Born, Rytov, and hybrid reconstructions.

Equations (46)

Equations on this page are rendered with MathJax. Learn more.

lim n ( 1 + x n ) n = exp x
u = u 0 lim n ( 1 + ϕ n ) n
2 u + k 2 ( r ) u = 0
u ( r ) = u 0 ( r ) ( 1 + ϕ ( r ) n ) n
· [ u 0 ( 1 + ϕ n ) n ] + k 2 [ u 0 ( 1 + ϕ n ) n ] = 0
( 1 + ϕ n ) n 2 u 0 + 2 ( 1 + ϕ n ) n 1 u 0 · ϕ + u 0 ( 1 + ϕ n ) n 1 2 ϕ +
u 0 n 1 n ( 1 + ϕ n ) n 2 ( ϕ · ϕ ) + k 2 u 0 ( 1 + ϕ n ) n = 0
2 u 0 · ϕ + u 0 2 ϕ + u 0 n 1 n ( 1 + ϕ n ) 1 ( ϕ · ϕ ) + ε κ u 0 ( 1 + ϕ n ) = 0
2 ( u 0 ϕ ) + k 0 2 ( u 0 ϕ ) = u 0 n 1 n ( 1 + ϕ n ) 1 ( ϕ · ϕ ) ε κ u 0 ( 1 + ϕ n )
u 0 ( r ' ) ϕ ( r ' ) = V d 3 r G ( r ' , r ) u 0 ( r ) [ n 1 n ( 1 + ϕ n ) 1 ( ϕ · ϕ ) + ε κ ( r ) ( 1 + ϕ n ) ]
G ( r ' , r ) u 0 ( r ) [ n 1 n ( 1 + ϕ n ) 1 ( ϕ · ϕ ) ] = 𝒪 ( ε 2 )
G ( r ' , r ) u 0 ( r ) ε κ ( r ) = ( ε 1 )
G ( r ' , r ) u 0 ( r ) ε κ ( r ) ϕ n = 𝒪 ( ε 2 )
u 0 ( r ' ) ϕ ( r ' ) = ε V d 3 r G ( r ' , r ) κ ( r ) u 0 ( r )
u 0 ( r ) 1 V d 3 r G ( r ' , r ) u 0 ( r ) [ n 1 n ( 1 + ϕ n ) 1 ( ϕ · ϕ ) + ε κ ( r ) ϕ n ] 1
1 R n 1 n V d 3 r ϕ 2 + 2 k 0 Δ k R 1 n V d 3 r ϕ 1
2 u 0 u 0 · ϕ ( 1 + ϕ n ) + 2 ϕ ( 1 + ϕ n ) + n 1 n ( ϕ · ϕ ) + ε κ ( 1 + ϕ n ) 2 = 0
2 u 0 u 0 . ( ϕ p + 1 n m = 0 p 1 ϕ p m ϕ m ) + ( 2 ϕ p + 1 n m = 0 p 1 ϕ p m 2 ϕ m ) +
n 1 n ( m = 1 p 1 ϕ p m · ϕ m ) + κ ( 2 ϕ p 1 n + 1 n 2 m = 0 p 1 ϕ p m 1 ϕ m ) = 0
( 2 + k 0 2 ) ( u 0 ϕ p ) = [ 1 n m = 0 p 1 ϕ p m ( ( 2 + k 0 2 ) ( u 0 ϕ m ) ) +
n 1 n u 0 m = 1 p 1 ϕ p m · ϕ m + κ n 2 u 0 m = 0 p 1 ϕ p m 1 ϕ m + 2 κ n u 0 ϕ p 1 ]
( 2 + k 0 2 ) ( u 0 ϕ 1 ) = u 0 κ
( 2 + k 0 2 ) ( u 0 ϕ 2 ) = u 0 [ n 1 n ( ϕ 1 · ϕ 1 ) + 1 n ϕ 1 κ ]
( 2 + k 0 2 ) ( u 0 ϕ 3 ) = u 0 [ n 1 n ( 2 ϕ 1 · ϕ 2 1 n ϕ 1 ( ϕ 1 · ϕ 1 ) ) + 1 n ϕ 2 κ ]
u 1 u 0 = ϕ 1
u 2 u 0 = ϕ 2 + n 1 2 n ϕ 1 2
u 3 u 0 = ϕ 3 + n 1 n ϕ 1 ϕ 2 + ( n 1 ) ( n 2 ) 6 n 2 ϕ 1 3
L = V d 3 r ' [ M γ + q ( κ , κ ) ]
where M = u 0 ( r ' ) ϕ ( r ' ) + V d 3 r u 0 [ n 1 n M R + 1 n M B + G ( r ' , r ) ϕ ( r ) ]
M R = G ( r ' , r ) ( 1 + ϕ n ) 1 ( ϕ · ϕ )
M B = G ( r ' , r ) κ ( r ) ϕ ( r )
δ ( M γ ) δ ϕ = γ M γ 2 M * δ M δ ϕ
δ M δ ϕ = u 0 + V d 3 r u 0 [ n 1 n δ M R δ ϕ + 1 n δ M B δ ϕ ]
δ M R δ ϕ = 2 [ ( 1 + ϕ n ) 1 ( r ' G ( r ' , r ) · r ϕ ( r ) ) ] [ 1 n G ( r ' , r ) ( 1 + ϕ n ) 2 ( ϕ · ϕ ) ]
δ M B δ ϕ = κ ( r ) G ( r ' , r )
δ ( M γ + q ) δ κ = γ M γ 2 M * [ V d 3 r G ( r ' , r ) u 0 ( 1 + ϕ n ) ] + q κ r ' · q κ
ϕ = n [ ( u u 0 ) 1 n 1 ] = n [ exp ( 1 n log u u 0 ) 1 ]
[ u 0 ( r ' ) G i ( κ ) ] = V d 3 r G ( r ' , r ) κ ( r ) [ u 0 ( r ) G i 1 ( κ ) ]
and G 0 ( κ ) = 1
Φ 1 ( κ ) = G 1 ( κ )
Φ 2 ( κ ) = G 2 ( κ ) n 1 2 n Φ 1 ( κ ) 2
Φ 3 ( κ ) = G 3 ( κ ) n 1 n Φ 1 ( κ ) Φ 2 ( κ ) ( n 1 ) ( n 2 ) 6 n 2 Φ 1 ( κ ) 3
κ 1 = B ( { ϕ } )
κ 2 = B ( { Φ 2 ( κ 1 ) } )
n opt = 1 + 0.205 ( N 1 ) 1.08 R 1.93
RMS opt = 2.34 ( N 1 ) 0.93 R 0.46

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