Abstract

Within the framework of inverse scattering problems, the quantifying of the degree of nonlinearity of the problem at hand provides an interesting possibility for evaluating the validity range of the Born series and for quantifying the difficulty of both forward and inverse problems. With reference to the two-dimensional scalar problem, new tools are proposed that allow the determination of the degree of nonlinearity in scattering problems when the maximum value, dimensions, and spatial-frequency content of the unknown permittivity are changed at the same time. As such, the proposed tools make it possible to identify useful guidelines for the solution of both forward and inverse problems and suggest an effective solution procedure for the latter. Numerical examples are reported to confirm the usefulness of the tools introduced and of the procedure proposed.

© 2001 Optical Society of America

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References

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  1. J. B. Keller, “Accuracy and validity of the Born and Rytov approximations,” J. Opt. Soc. Am. 59, 1003–1004 (1969).
  2. M. Slaney, A. Kak, L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
    [CrossRef]
  3. D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Theory, (Springer-Verlag, Berlin, 1992).
  4. R. E. Kleinman, P. M. Van den Berg, “An extended modified gradient technique for profile inversion,” Radio Sci. 28, 877–884 (1993).
    [CrossRef]
  5. T. Isernia, V. Pascazio, R. Pierri, “A non-linear estimation method in tomographic imaging,” IEEE Trans. Geosci. Remote Sens. 35, 910–923 (1997).
    [CrossRef]
  6. R. Pierri, A. Brancaccio, “Imaging of a rotationally symmetric dielectric cylinder by a quadratic approach,” J. Opt. Soc. Am. A 14, 2777–2785 (1997).
    [CrossRef]
  7. W. C. Chew, Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Med. Imaging 9, 218–225 (1990).
    [CrossRef] [PubMed]
  8. T. M. Habashy, R. W. Groom, B. P. Spies, “Beyond the Born and Rytov approximations: a non-linear approach to electromagnetic scattering,” J. Geophys. Res. 98 (B2), 1759–1776 (1993).
    [CrossRef]
  9. W. C. Chew, Waves and Fields in Inhomogenous Media, (IEEE Computer Society, Los Alamitos, Calif., 1995).
  10. A. Kolmogorov, S. V. Fomine, Eléments de la théorie des fonctions et de l’analyse fonctionelle (MIR Editions, Moscow, 1973).
  11. R. E. Kleinman, G. F. Roach, P. M. Van den Berg, “Convergent Born series for large refractive indices,” J. Opt. Soc. Am. A 7, 890–897 (1990).
    [CrossRef]
  12. O. M. Bucci, T. Isernia, “Electromagnetic inverse scattering: retrievable information and measurement strategies,” Radio Sci. 32, 2123–2138 (1997).
    [CrossRef]
  13. A. Friedman, Partial Different Equations (Krieger, Malaba, Fla., 1976).
  14. O. M. Bucci, L. Crocco, T. Isernia, “Improving the reconstruction capabilities in inverse scattering problems by exploiting ‘near proximity’ set-ups,” J. Opt. Soc. Am. A 16, 1788–1798 (1999).
    [CrossRef]
  15. W. C. Chew, J. H. Lin, “A frequency-hopping approach for microwave imaging of large inhomogeneous bodies,” IEEE Microwave Guided Wave Lett. 5, 439–441 (1995).
    [CrossRef]
  16. O. M. Bucci, L. Crocco, T. Isernia, V. Pascazio, “Inverse scattering problems with multi-frequency data: reconstruction capabilities and solution strategies,” IEEE Trans. Geosci. Remote Sens. 38, 1749–1756 (2000).
    [CrossRef]
  17. M. Bertero, “Linear inverse and ill-posed problems,” in Advances in Electronics and Electronic Physics (Academic, New York, 1989), pp. 1–120.
  18. T. C. Wedberg, J. J. Stamnes, “Experimental examination of the quantitative imaging properties of optical diffraction tomography,” J. Opt. Soc. Am. A 12, 493–500 (1995).
    [CrossRef]
  19. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products, A. Jeffrey, ed. (Academic, London, 1997).
  20. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

2000 (1)

O. M. Bucci, L. Crocco, T. Isernia, V. Pascazio, “Inverse scattering problems with multi-frequency data: reconstruction capabilities and solution strategies,” IEEE Trans. Geosci. Remote Sens. 38, 1749–1756 (2000).
[CrossRef]

1999 (1)

1997 (3)

O. M. Bucci, T. Isernia, “Electromagnetic inverse scattering: retrievable information and measurement strategies,” Radio Sci. 32, 2123–2138 (1997).
[CrossRef]

T. Isernia, V. Pascazio, R. Pierri, “A non-linear estimation method in tomographic imaging,” IEEE Trans. Geosci. Remote Sens. 35, 910–923 (1997).
[CrossRef]

R. Pierri, A. Brancaccio, “Imaging of a rotationally symmetric dielectric cylinder by a quadratic approach,” J. Opt. Soc. Am. A 14, 2777–2785 (1997).
[CrossRef]

1995 (2)

W. C. Chew, J. H. Lin, “A frequency-hopping approach for microwave imaging of large inhomogeneous bodies,” IEEE Microwave Guided Wave Lett. 5, 439–441 (1995).
[CrossRef]

T. C. Wedberg, J. J. Stamnes, “Experimental examination of the quantitative imaging properties of optical diffraction tomography,” J. Opt. Soc. Am. A 12, 493–500 (1995).
[CrossRef]

1993 (2)

T. M. Habashy, R. W. Groom, B. P. Spies, “Beyond the Born and Rytov approximations: a non-linear approach to electromagnetic scattering,” J. Geophys. Res. 98 (B2), 1759–1776 (1993).
[CrossRef]

R. E. Kleinman, P. M. Van den Berg, “An extended modified gradient technique for profile inversion,” Radio Sci. 28, 877–884 (1993).
[CrossRef]

1990 (2)

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

R. E. Kleinman, G. F. Roach, P. M. Van den Berg, “Convergent Born series for large refractive indices,” J. Opt. Soc. Am. A 7, 890–897 (1990).
[CrossRef]

1984 (1)

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

1969 (1)

Abramowitz, M.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Bertero, M.

M. Bertero, “Linear inverse and ill-posed problems,” in Advances in Electronics and Electronic Physics (Academic, New York, 1989), pp. 1–120.

Brancaccio, A.

R. Pierri, A. Brancaccio, “Imaging of a rotationally symmetric dielectric cylinder by a quadratic approach,” J. Opt. Soc. Am. A 14, 2777–2785 (1997).
[CrossRef]

Bucci, O. M.

O. M. Bucci, L. Crocco, T. Isernia, V. Pascazio, “Inverse scattering problems with multi-frequency data: reconstruction capabilities and solution strategies,” IEEE Trans. Geosci. Remote Sens. 38, 1749–1756 (2000).
[CrossRef]

O. M. Bucci, L. Crocco, T. Isernia, “Improving the reconstruction capabilities in inverse scattering problems by exploiting ‘near proximity’ set-ups,” J. Opt. Soc. Am. A 16, 1788–1798 (1999).
[CrossRef]

O. M. Bucci, T. Isernia, “Electromagnetic inverse scattering: retrievable information and measurement strategies,” Radio Sci. 32, 2123–2138 (1997).
[CrossRef]

Chew, W. C.

W. C. Chew, J. H. Lin, “A frequency-hopping approach for microwave imaging of large inhomogeneous bodies,” IEEE Microwave Guided Wave Lett. 5, 439–441 (1995).
[CrossRef]

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

W. C. Chew, Waves and Fields in Inhomogenous Media, (IEEE Computer Society, Los Alamitos, Calif., 1995).

Colton, D.

D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Theory, (Springer-Verlag, Berlin, 1992).

Crocco, L.

O. M. Bucci, L. Crocco, T. Isernia, V. Pascazio, “Inverse scattering problems with multi-frequency data: reconstruction capabilities and solution strategies,” IEEE Trans. Geosci. Remote Sens. 38, 1749–1756 (2000).
[CrossRef]

O. M. Bucci, L. Crocco, T. Isernia, “Improving the reconstruction capabilities in inverse scattering problems by exploiting ‘near proximity’ set-ups,” J. Opt. Soc. Am. A 16, 1788–1798 (1999).
[CrossRef]

Fomine, S. V.

A. Kolmogorov, S. V. Fomine, Eléments de la théorie des fonctions et de l’analyse fonctionelle (MIR Editions, Moscow, 1973).

Friedman, A.

A. Friedman, Partial Different Equations (Krieger, Malaba, Fla., 1976).

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products, A. Jeffrey, ed. (Academic, London, 1997).

Groom, R. W.

T. M. Habashy, R. W. Groom, B. P. Spies, “Beyond the Born and Rytov approximations: a non-linear approach to electromagnetic scattering,” J. Geophys. Res. 98 (B2), 1759–1776 (1993).
[CrossRef]

Habashy, T. M.

T. M. Habashy, R. W. Groom, B. P. Spies, “Beyond the Born and Rytov approximations: a non-linear approach to electromagnetic scattering,” J. Geophys. Res. 98 (B2), 1759–1776 (1993).
[CrossRef]

Isernia, T.

O. M. Bucci, L. Crocco, T. Isernia, V. Pascazio, “Inverse scattering problems with multi-frequency data: reconstruction capabilities and solution strategies,” IEEE Trans. Geosci. Remote Sens. 38, 1749–1756 (2000).
[CrossRef]

O. M. Bucci, L. Crocco, T. Isernia, “Improving the reconstruction capabilities in inverse scattering problems by exploiting ‘near proximity’ set-ups,” J. Opt. Soc. Am. A 16, 1788–1798 (1999).
[CrossRef]

O. M. Bucci, T. Isernia, “Electromagnetic inverse scattering: retrievable information and measurement strategies,” Radio Sci. 32, 2123–2138 (1997).
[CrossRef]

T. Isernia, V. Pascazio, R. Pierri, “A non-linear estimation method in tomographic imaging,” IEEE Trans. Geosci. Remote Sens. 35, 910–923 (1997).
[CrossRef]

Kak, A.

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

Keller, J. B.

Kleinman, R. E.

R. E. Kleinman, P. M. Van den Berg, “An extended modified gradient technique for profile inversion,” Radio Sci. 28, 877–884 (1993).
[CrossRef]

R. E. Kleinman, G. F. Roach, P. M. Van den Berg, “Convergent Born series for large refractive indices,” J. Opt. Soc. Am. A 7, 890–897 (1990).
[CrossRef]

Kolmogorov, A.

A. Kolmogorov, S. V. Fomine, Eléments de la théorie des fonctions et de l’analyse fonctionelle (MIR Editions, Moscow, 1973).

Kress, R.

D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Theory, (Springer-Verlag, Berlin, 1992).

Larsen, L. E.

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

Lin, J. H.

W. C. Chew, J. H. Lin, “A frequency-hopping approach for microwave imaging of large inhomogeneous bodies,” IEEE Microwave Guided Wave Lett. 5, 439–441 (1995).
[CrossRef]

Pascazio, V.

O. M. Bucci, L. Crocco, T. Isernia, V. Pascazio, “Inverse scattering problems with multi-frequency data: reconstruction capabilities and solution strategies,” IEEE Trans. Geosci. Remote Sens. 38, 1749–1756 (2000).
[CrossRef]

T. Isernia, V. Pascazio, R. Pierri, “A non-linear estimation method in tomographic imaging,” IEEE Trans. Geosci. Remote Sens. 35, 910–923 (1997).
[CrossRef]

Pierri, R.

T. Isernia, V. Pascazio, R. Pierri, “A non-linear estimation method in tomographic imaging,” IEEE Trans. Geosci. Remote Sens. 35, 910–923 (1997).
[CrossRef]

R. Pierri, A. Brancaccio, “Imaging of a rotationally symmetric dielectric cylinder by a quadratic approach,” J. Opt. Soc. Am. A 14, 2777–2785 (1997).
[CrossRef]

Roach, G. F.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products, A. Jeffrey, ed. (Academic, London, 1997).

Slaney, M.

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

Spies, B. P.

T. M. Habashy, R. W. Groom, B. P. Spies, “Beyond the Born and Rytov approximations: a non-linear approach to electromagnetic scattering,” J. Geophys. Res. 98 (B2), 1759–1776 (1993).
[CrossRef]

Stamnes, J. J.

Stegun, I.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Van den Berg, P. M.

R. E. Kleinman, P. M. Van den Berg, “An extended modified gradient technique for profile inversion,” Radio Sci. 28, 877–884 (1993).
[CrossRef]

R. E. Kleinman, G. F. Roach, P. M. Van den Berg, “Convergent Born series for large refractive indices,” J. Opt. Soc. Am. A 7, 890–897 (1990).
[CrossRef]

Wang, Y. M.

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

Wedberg, T. C.

IEEE Microwave Guided Wave Lett. (1)

W. C. Chew, J. H. Lin, “A frequency-hopping approach for microwave imaging of large inhomogeneous bodies,” IEEE Microwave Guided Wave Lett. 5, 439–441 (1995).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (2)

O. M. Bucci, L. Crocco, T. Isernia, V. Pascazio, “Inverse scattering problems with multi-frequency data: reconstruction capabilities and solution strategies,” IEEE Trans. Geosci. Remote Sens. 38, 1749–1756 (2000).
[CrossRef]

T. Isernia, V. Pascazio, R. Pierri, “A non-linear estimation method in tomographic imaging,” IEEE Trans. Geosci. Remote Sens. 35, 910–923 (1997).
[CrossRef]

IEEE Trans. Med. Imaging (1)

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

IEEE Trans. Microwave Theory Tech. (1)

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

J. Opt. Soc. Am. A (1)

R. Pierri, A. Brancaccio, “Imaging of a rotationally symmetric dielectric cylinder by a quadratic approach,” J. Opt. Soc. Am. A 14, 2777–2785 (1997).
[CrossRef]

J. Geophys. Res. (1)

T. M. Habashy, R. W. Groom, B. P. Spies, “Beyond the Born and Rytov approximations: a non-linear approach to electromagnetic scattering,” J. Geophys. Res. 98 (B2), 1759–1776 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Radio Sci. (2)

O. M. Bucci, T. Isernia, “Electromagnetic inverse scattering: retrievable information and measurement strategies,” Radio Sci. 32, 2123–2138 (1997).
[CrossRef]

R. E. Kleinman, P. M. Van den Berg, “An extended modified gradient technique for profile inversion,” Radio Sci. 28, 877–884 (1993).
[CrossRef]

Other (7)

A. Friedman, Partial Different Equations (Krieger, Malaba, Fla., 1976).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products, A. Jeffrey, ed. (Academic, London, 1997).

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

M. Bertero, “Linear inverse and ill-posed problems,” in Advances in Electronics and Electronic Physics (Academic, New York, 1989), pp. 1–120.

W. C. Chew, Waves and Fields in Inhomogenous Media, (IEEE Computer Society, Los Alamitos, Calif., 1995).

A. Kolmogorov, S. V. Fomine, Eléments de la théorie des fonctions et de l’analyse fonctionelle (MIR Editions, Moscow, 1973).

D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Theory, (Springer-Verlag, Berlin, 1992).

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Figures (10)

Fig. 1
Fig. 1

(Radially symmetric) spectrum G ^ 2 ( u ,   v ) along the v = 0 axis, for u 0 .

Fig. 2
Fig. 2

Comparison between the spectra of G ^ 2 ( u ,   0 ) for different values of the losses in the background medium.

Fig. 3
Fig. 3

Comparison between the different singular functions u 0 ( β r ,   β R MAX ) for different values of β R MAX . Real and imaginary parts are respectively shown in parts (a) and (b).

Fig. 4
Fig. 4

Universal curves u 0 r ( β r ) and u 0 i ( β r ) , as defined in Eq. (20).

Fig. 5
Fig. 5

Sketch of the (numerically computed) U ^ 2 r ( u ,   0 ) and U ^ 2 i ( u ,   0 ) , as defined in Eq. (22).

Fig. 6
Fig. 6

Comparison between the X ^ 2 ( u ,   v ) spectra for the three different profiles: (a) constant profile (solid curve), (b) cos   ( x )   cos   ( y ) profile (dashed curve), (c) cos   ( 2 x )   cos   ( y ) (dotted curve). The cut along the v = 0 axis of the three spectra is reported.

Fig. 7
Fig. 7

Comparison between the (numerically computed) actual norm of oscillating contrasts having different periods [see Eq. (30)] and the estimated bounds, for different electrical dimensions of the scatterer. (a), β R MAX = π ; (b) β R MAX = 2 π . Note that in (a) the norm is almost coincident with the estimated lower bound.

Fig. 8
Fig. 8

Real part of the actual and reconstructed cosinusoidally oscillating profiles, as defined in Eq. (30). (a) and (b) n = 1 , m = 0 ( χ MAX = 2.4 ) ; (c) and (d) n = 1 , m = 1 ( χ MAX = 3.2 ) ; (e) and (f) n = 2 , m = 0 ( χ MAX = 4.0 ) ; (g) and (h) n = 2 , m = 2 ( χ MAX = 4.5 ) .

Fig. 9
Fig. 9

(a) Case 1, real part of the actual profile; (b), result obtained with inversion approach in Ref. 5; (c), reconstructed profile exploiting the proposed procedure.

Fig. 10
Fig. 10

(a) Case 2, real part of the actual profile; (b) result obtained with inversion approach in Ref. 5; (c) reconstructed profile exploiting the proposed procedure.

Tables (1)

Tables Icon

Table 1 Comparison between Maximum Contrasts with Reference to the Harmonic Content of the Oscillating Contrast Profile

Equations (51)

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2 E v + β 2 ( 1 + χ ) E v = 0 , r Ω
2 E S v + β 2 E S v = 0 , r Ω
E v = E inc v + E S v ,
E S v ( r )
= Ω g ( r ,   r ) χ ( r ) E v ( r ) d r A e ( χ E v ) , r Γ ,
E v ( r ) - E inc v ( r )
= Ω g ( r ,   r ) χ ( r ) E v ( r ) d r A i ( χ E v ) , r Ω ,
E v = ( I - A i X ) - 1 E inc v ,
A i X < 1 ,
( I - A i X ) - 1 = I + A i X + ( A i X ) 2 + + ( A i X ) n +   .
A i X 1 .
K ( e ( r ) ) Ω k ( r ,   r ) e ( r ) d r   r ,   r     Ω ,
A i X ( e ) 2 < Ω   | k ( r ,   r ) | 2 d r d r Ω   | e ( r ) | 2 d r = k L 2 ( Ω × Ω ) 2 e L 2 ( Ω ) 2 .
A i X 2 < k L 2 ( Ω × Ω ) 2 = Ω d r Ω d r | g ( r - r ) | 2 | χ ( r ) | 2 .
A i X 2 < Ω d r - + G ^ 2 ( u ) X ^ 2 ( u ) exp ( j u r ) d u ,
A i X 2 < - + G ^ 2 ( u ) X ^ 2 ( u ) F s ( u ) d u ,
F s ( u ) = Ω exp ( j u r ) d r
G ^ 2 ( u ,   v ) = 0 2 π 0 + | H 0 ( 2 ) ( β ρ ) | 2 × exp [ - j ρ ( u   cos   θ + v   sin   θ ) ] ρ d ρ d θ .
G ^ 2 ( u ,   v ) = 8 π β 2 1 ( 2 ξ - ξ 2 ) 1 / 2 arctan ( 2 ξ - ξ 2 ) 1 / 2 ξ ,
G ^ 2 ( u ,   v ) = 4 π β 2 1 ( ξ 2 - 2 ξ ) 1 / 2 ln ξ + ( ξ 2 - 2 ξ ) 1 / 2 ξ - ( ξ 2 - 2 ξ ) 1 / 2
G ^ 2 ( 0 ,   0 ) = 2 π   log ( β * ) - log ( β ) β 2 - ( β * ) 2 .
( A i X ) + = X + A i + = X * A i + ,
A i X = X * A i + = sup e = 1 X * A i + e e X * A i + ( v 0 ) v 0 ,
A i X X * σ 0 u 0 = σ 0 X * u 0 ,
u 0 ( β r ,   β R MAX ) = u 0 r ( β r ) ( β R MAX ) - 3 / 2 - ju 0 i ( β r ) ( β R MAX ) - 1 / 2 ,
σ 0 ( β R MAX ) β R MAX = σ 0 ( β R MAX ) β R MAX ,
A i X 2 > A 2 β 2 R MAX 2 - + U ^ 2 ( u ,   β R MAX ) X ^ 2 ( u ) d u ,
σ 0 2 U ^ 2 ( u ,   β R MAX ) = A 2 U ^ 2 r ( u ) β R MAX + U ^ 2 i ( u ) β R MAX ,
 
A i X 2 > A 2 β 2 R MAX - + × [ U ^ 2 ( u ,   β R MAX ) * X ^ 2 ( u ) ] F S ( u ) d u .
X ^ 2 ( u ) = F [ χ χ * ] = X ˆ ( u ) * [ X ˆ ( - u ) ] * ,
χ = χ 0 + Δ χ .
E = E i + A i χ 0 ( E i ) + A i Δ χ ( E i ) + A i χ 0 [ A i χ 0 ( E i ) ] + A i χ 0 [ A i Δ χ ( E i ) ] + A i Δ χ [ A i Δ χ ( E i ) ] + ,
Φ ( χ ¯ t ) = v E M v - E trial v ( χ ¯ t ) 2 ,
χ MIN χ ¯ t χ MAX ,
χ ( x ,   y ) = χ MAX ( 1 - 0 ,   1 j ) cos 2 π   x λ ,
χ ( x ,   y ) = χ MAX ( 1 - 0 ,   1 j ) cos 2 π   x λ cos 2 π   y λ ,
χ ( x ,   y ) = χ MAX ( 1 - 0 ,   1 j ) cos 2 π   nx λ cos 2 π   my λ
G ^ 2 ( u ,   v ) = 0 + 0 2 π | H 0 ( 2 ) ( β ρ ) | 2 exp [ - j ρ ( u 2 + v 2 ) 1 / 2 × cos ( θ - ϕ ) ] ρ d ρ d θ ,
cos   ϕ = u ( u 2 + v 2 ) 1 / 2 , sin   ϕ = v ( u 2 + v 2 ) 1 / 2 .
2 π J 0 [ ρ ( u 2 + v 2 ) 1 / 2 ] = 0 2 π exp [ - j ρ ( u 2 + v 2 ) 1 / 2 × cos ( θ - ϕ ) ] d θ ,
| H 0 ( 2 ) ( β ρ ) | 2 = 8 π 2 0 + K 0 [ 2 β ρ sinh ( t ) ] d t ,
G ^ 2 ( u ,   v ) = 16 π 0 + 0 + K 0 [ 2 β ρ sinh ( t ) ] × J 0 [ ρ ( u 2 + v 2 ) 1 / 2 ] ρ d ρ d t .
0 + K 0 [ 2 β ρ sinh ( t ) ] J 0 [ ρ ( u 2 + v 2 ) 1 / 2 ] ρ d ρ
= 1 [ 2 β sinh ( t ) ] 2 + u 2 + v 2 ,
G ^ 2 ( u ,   v ) = 16 π 0 + 1 [ 2 β sinh ( t ) ] 2 + u 2 + v 2 d t = 8 β 2 π 0 + [ cosh ( 2 t ) + ( ξ - 1 ) ] - 1 d t ,
G ^ 2 ( u ,   v ) = 8 β 2 π   [ 1 - ( ξ - 1 ) 2 ] - 1 / 2 × arctan [ 1 - ( ξ - 1 ) 2 ] 1 / 2 ξ ,
G ^ 2 ( u ,   v ) = 4 β 2 π   [ ( ξ - 1 ) 2 - 1 ] - 1 / 2 × ln ξ + [ ( ξ - 1 ) 2 - 1 ] 1 / 2 ξ - [ ( ξ - 1 ) 2 - 1 ] 1 / 2 .
G ^ 2 ( u ,   v ) = G ˆ ( u ,   v )   *   [ G ˆ ( - u ,   - v ) ] * ,
G ˆ ( u ,   v ) = 1 β 2 - ( u 2 + v 2 ) ,
G ^ 2 ( 0 ,   0 ) = 2 π   log ( β * ) - log ( β ) β 2 - ( β * ) 2 .

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