Abstract

The ill-posed problem of restoring object information from finitely many measurements of its spectrum can be solved by using the best approximation in Hilbert spaces appropriately designed to include a priori information about object extent and shape and noise statistics. The procedures that are derived are noniterative, the linear ones extending the minimum-energy band-limited extrapolation methods (and thus related to Gerchberg–Papoulis iteration) and the nonlinear ones generalizing Burg’s maximum-entropy reconstruction of nonnegative objects.

© 1983 Optical Society of America

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  1. G. Toraldo di Francia, "Resolving power and information," J. Opt. Soc. Am. 45,497–501 (1955).
    [CrossRef]
  2. T. K. Sarkar, D. D. Weiner, and V. K. Jain, "Some mathematical considerations in dealing with the inverse problem," IEEE Trans. Antennas Propag. AP-29, 373–379 (1981).
    [CrossRef]
  3. J. B. Abbiss, C. de Mol, and H. S. Dhadwal, "Regularized iterative and noniterative procedures for object restoration from experimental data," Opt. Acta, 30, 107–124 (1983).
    [CrossRef]
  4. M. Bertero and C. de Mol, "Ill-posedness, regularization and number of degrees of freedom," Atti Fond. "Giorgio Ronchi" 36, 619–632 (1981).
  5. B. R. Frieden, in Progress in Optics, Vol. 9, E. Wolf, ed. (North-Holland, Amsterdam, 1971), Clap. VIII, pp. 311–407.
    [CrossRef]
  6. D. Slepian and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty—I," Bell Sys. Tech. J. 40,43–63, (1961); H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty—II," Bell Syst. Tech. J. 40, 65–84 (1961); "Prolate spheroidal wave functions, Fourier analysis and uncertainty—III; the dimension of the space of essentially time- and band-limited signals," Bell Syst. Tech. J. 41, 1295–1336 (1962).
    [CrossRef]
  7. J. L. Harris, "Diffraction and resolving power," J. Opt. Soc. Am. 54, 931–936 (1964).
    [CrossRef]
  8. R. W. Gerchberg, "Super-resolution through error energy reduction," Opt. Acta 21, 709–720 (1974).
    [CrossRef]
  9. A. Papoulis, "A new algorithm in spectral analysis and bandlimited extrapolation," IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
    [CrossRef]
  10. A. K. Jain and S. Ranganath, "Extrapolation algorithms for discrete signals, with application in spectral estimation," IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 830–845 (1981).
    [CrossRef]
  11. K. Miller, "Least squares methods for ill-posed problems with prescribed bound," SIAM J. Math Anal. 1, 52–74 (1970).
    [CrossRef]
  12. A. M. Darling, T. J. Hall, and M. A. Fiddy, "Stable, noniterative, object reconstruction from incomplete data using prior knowledge," in Digest of the Topical Meeting on Signal Recovery and Synthesis with Incomplete Information and Partial Constraints (Optical-Society of America, Washington, D.C., 1983).
  13. C. Byrne and R. Fitzgerald, "Reconstruction from partial information, with applications to tomography," SIAM J. Appl. Math. 42, 933–940 (1982).
    [CrossRef]
  14. L. M. Cheng, A. S. Ho, and R. E. Burge, "The use of prior knowlege in image reconstruction," submitted to J. Opt. Soc. Am.
  15. B. R. Frieden, "Image enhancement and restoration," in Picture Processing and Digital Filtering, Vol. 6 of Topics in Applied Physics, T. S. Huang, ed. (Springer-Verlag, New York, 1978).
  16. J. P. Burg, "Maximum entropy spectrum analysis," Ph.D. Thesis (Stanford University, Palo Alto, Calif., 1975).
  17. C. van Schooneveld, "Resolution enhancement: the 'maximum entropy method' and the 'high resolution method'," in Image Formation from Coherence Functions in Astronomy (Reidel, Dordrecht, The Netherlands, 1979), pp. 197–218.
    [CrossRef]
  18. R. T. Lacoss, "Data adaptive spectral analysis methods," Geophysics 36, 661–675 (1971).
    [CrossRef]
  19. A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977), p. 146.
  20. R. E. Burge, M. A. Fiddy, A. H. Greenaway, and G. Ross, "The phase problem," Proc. R. Soc. London, Ser. A 350, 191–212 (1976).
    [CrossRef]
  21. H. A. Ferwerda, "Optics in four dimensions—1980," AIP Conf. Proc. 65, 402–411 (1981).
    [CrossRef]
  22. J. R. Fienup, "Reconstruction of an object from the modulus of its Fourier transform," Opt. Lett. 3, 27–29 (1978).
    [CrossRef] [PubMed]
  23. T. J. Hall, A. M. Darling, and M. A. Fiddy, "Image compression and restoration incorporating prior knowledge," Opt. Lett. 7, 467–468 (1982).
    [CrossRef] [PubMed]

1983 (1)

J. B. Abbiss, C. de Mol, and H. S. Dhadwal, "Regularized iterative and noniterative procedures for object restoration from experimental data," Opt. Acta, 30, 107–124 (1983).
[CrossRef]

1982 (2)

T. J. Hall, A. M. Darling, and M. A. Fiddy, "Image compression and restoration incorporating prior knowledge," Opt. Lett. 7, 467–468 (1982).
[CrossRef] [PubMed]

C. Byrne and R. Fitzgerald, "Reconstruction from partial information, with applications to tomography," SIAM J. Appl. Math. 42, 933–940 (1982).
[CrossRef]

1981 (4)

H. A. Ferwerda, "Optics in four dimensions—1980," AIP Conf. Proc. 65, 402–411 (1981).
[CrossRef]

T. K. Sarkar, D. D. Weiner, and V. K. Jain, "Some mathematical considerations in dealing with the inverse problem," IEEE Trans. Antennas Propag. AP-29, 373–379 (1981).
[CrossRef]

A. K. Jain and S. Ranganath, "Extrapolation algorithms for discrete signals, with application in spectral estimation," IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 830–845 (1981).
[CrossRef]

M. Bertero and C. de Mol, "Ill-posedness, regularization and number of degrees of freedom," Atti Fond. "Giorgio Ronchi" 36, 619–632 (1981).

1978 (1)

1976 (1)

R. E. Burge, M. A. Fiddy, A. H. Greenaway, and G. Ross, "The phase problem," Proc. R. Soc. London, Ser. A 350, 191–212 (1976).
[CrossRef]

1975 (1)

A. Papoulis, "A new algorithm in spectral analysis and bandlimited extrapolation," IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

1974 (1)

R. W. Gerchberg, "Super-resolution through error energy reduction," Opt. Acta 21, 709–720 (1974).
[CrossRef]

1971 (1)

R. T. Lacoss, "Data adaptive spectral analysis methods," Geophysics 36, 661–675 (1971).
[CrossRef]

1970 (1)

K. Miller, "Least squares methods for ill-posed problems with prescribed bound," SIAM J. Math Anal. 1, 52–74 (1970).
[CrossRef]

1964 (1)

1955 (1)

Abbiss, J. B.

J. B. Abbiss, C. de Mol, and H. S. Dhadwal, "Regularized iterative and noniterative procedures for object restoration from experimental data," Opt. Acta, 30, 107–124 (1983).
[CrossRef]

Bertero, M.

M. Bertero and C. de Mol, "Ill-posedness, regularization and number of degrees of freedom," Atti Fond. "Giorgio Ronchi" 36, 619–632 (1981).

Burg, J. P.

J. P. Burg, "Maximum entropy spectrum analysis," Ph.D. Thesis (Stanford University, Palo Alto, Calif., 1975).

Burge, R. E.

R. E. Burge, M. A. Fiddy, A. H. Greenaway, and G. Ross, "The phase problem," Proc. R. Soc. London, Ser. A 350, 191–212 (1976).
[CrossRef]

L. M. Cheng, A. S. Ho, and R. E. Burge, "The use of prior knowlege in image reconstruction," submitted to J. Opt. Soc. Am.

Byrne, C.

C. Byrne and R. Fitzgerald, "Reconstruction from partial information, with applications to tomography," SIAM J. Appl. Math. 42, 933–940 (1982).
[CrossRef]

Cheng, L. M.

L. M. Cheng, A. S. Ho, and R. E. Burge, "The use of prior knowlege in image reconstruction," submitted to J. Opt. Soc. Am.

Darling, A. M.

T. J. Hall, A. M. Darling, and M. A. Fiddy, "Image compression and restoration incorporating prior knowledge," Opt. Lett. 7, 467–468 (1982).
[CrossRef] [PubMed]

A. M. Darling, T. J. Hall, and M. A. Fiddy, "Stable, noniterative, object reconstruction from incomplete data using prior knowledge," in Digest of the Topical Meeting on Signal Recovery and Synthesis with Incomplete Information and Partial Constraints (Optical-Society of America, Washington, D.C., 1983).

de Mol, C.

J. B. Abbiss, C. de Mol, and H. S. Dhadwal, "Regularized iterative and noniterative procedures for object restoration from experimental data," Opt. Acta, 30, 107–124 (1983).
[CrossRef]

M. Bertero and C. de Mol, "Ill-posedness, regularization and number of degrees of freedom," Atti Fond. "Giorgio Ronchi" 36, 619–632 (1981).

Dhadwal, H. S.

J. B. Abbiss, C. de Mol, and H. S. Dhadwal, "Regularized iterative and noniterative procedures for object restoration from experimental data," Opt. Acta, 30, 107–124 (1983).
[CrossRef]

Ferwerda, H. A.

H. A. Ferwerda, "Optics in four dimensions—1980," AIP Conf. Proc. 65, 402–411 (1981).
[CrossRef]

Fiddy, M. A.

T. J. Hall, A. M. Darling, and M. A. Fiddy, "Image compression and restoration incorporating prior knowledge," Opt. Lett. 7, 467–468 (1982).
[CrossRef] [PubMed]

R. E. Burge, M. A. Fiddy, A. H. Greenaway, and G. Ross, "The phase problem," Proc. R. Soc. London, Ser. A 350, 191–212 (1976).
[CrossRef]

A. M. Darling, T. J. Hall, and M. A. Fiddy, "Stable, noniterative, object reconstruction from incomplete data using prior knowledge," in Digest of the Topical Meeting on Signal Recovery and Synthesis with Incomplete Information and Partial Constraints (Optical-Society of America, Washington, D.C., 1983).

Fienup, J. R.

Fitzgerald, R.

C. Byrne and R. Fitzgerald, "Reconstruction from partial information, with applications to tomography," SIAM J. Appl. Math. 42, 933–940 (1982).
[CrossRef]

Frieden, B. R.

B. R. Frieden, "Image enhancement and restoration," in Picture Processing and Digital Filtering, Vol. 6 of Topics in Applied Physics, T. S. Huang, ed. (Springer-Verlag, New York, 1978).

B. R. Frieden, in Progress in Optics, Vol. 9, E. Wolf, ed. (North-Holland, Amsterdam, 1971), Clap. VIII, pp. 311–407.
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg, "Super-resolution through error energy reduction," Opt. Acta 21, 709–720 (1974).
[CrossRef]

Greenaway, A. H.

R. E. Burge, M. A. Fiddy, A. H. Greenaway, and G. Ross, "The phase problem," Proc. R. Soc. London, Ser. A 350, 191–212 (1976).
[CrossRef]

Hall, T. J.

T. J. Hall, A. M. Darling, and M. A. Fiddy, "Image compression and restoration incorporating prior knowledge," Opt. Lett. 7, 467–468 (1982).
[CrossRef] [PubMed]

A. M. Darling, T. J. Hall, and M. A. Fiddy, "Stable, noniterative, object reconstruction from incomplete data using prior knowledge," in Digest of the Topical Meeting on Signal Recovery and Synthesis with Incomplete Information and Partial Constraints (Optical-Society of America, Washington, D.C., 1983).

Harris, J. L.

Ho, A. S.

L. M. Cheng, A. S. Ho, and R. E. Burge, "The use of prior knowlege in image reconstruction," submitted to J. Opt. Soc. Am.

Jain, A. K.

A. K. Jain and S. Ranganath, "Extrapolation algorithms for discrete signals, with application in spectral estimation," IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 830–845 (1981).
[CrossRef]

Jain, V. K.

T. K. Sarkar, D. D. Weiner, and V. K. Jain, "Some mathematical considerations in dealing with the inverse problem," IEEE Trans. Antennas Propag. AP-29, 373–379 (1981).
[CrossRef]

Lacoss, R. T.

R. T. Lacoss, "Data adaptive spectral analysis methods," Geophysics 36, 661–675 (1971).
[CrossRef]

Miller, K.

K. Miller, "Least squares methods for ill-posed problems with prescribed bound," SIAM J. Math Anal. 1, 52–74 (1970).
[CrossRef]

Papoulis, A.

A. Papoulis, "A new algorithm in spectral analysis and bandlimited extrapolation," IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977), p. 146.

Pollak, H. O.

D. Slepian and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty—I," Bell Sys. Tech. J. 40,43–63, (1961); H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty—II," Bell Syst. Tech. J. 40, 65–84 (1961); "Prolate spheroidal wave functions, Fourier analysis and uncertainty—III; the dimension of the space of essentially time- and band-limited signals," Bell Syst. Tech. J. 41, 1295–1336 (1962).
[CrossRef]

Ranganath, S.

A. K. Jain and S. Ranganath, "Extrapolation algorithms for discrete signals, with application in spectral estimation," IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 830–845 (1981).
[CrossRef]

Ross, G.

R. E. Burge, M. A. Fiddy, A. H. Greenaway, and G. Ross, "The phase problem," Proc. R. Soc. London, Ser. A 350, 191–212 (1976).
[CrossRef]

Sarkar, T. K.

T. K. Sarkar, D. D. Weiner, and V. K. Jain, "Some mathematical considerations in dealing with the inverse problem," IEEE Trans. Antennas Propag. AP-29, 373–379 (1981).
[CrossRef]

Schooneveld, C. van

C. van Schooneveld, "Resolution enhancement: the 'maximum entropy method' and the 'high resolution method'," in Image Formation from Coherence Functions in Astronomy (Reidel, Dordrecht, The Netherlands, 1979), pp. 197–218.
[CrossRef]

Slepian, D.

D. Slepian and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty—I," Bell Sys. Tech. J. 40,43–63, (1961); H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty—II," Bell Syst. Tech. J. 40, 65–84 (1961); "Prolate spheroidal wave functions, Fourier analysis and uncertainty—III; the dimension of the space of essentially time- and band-limited signals," Bell Syst. Tech. J. 41, 1295–1336 (1962).
[CrossRef]

Toraldo di Francia, G.

Weiner, D. D.

T. K. Sarkar, D. D. Weiner, and V. K. Jain, "Some mathematical considerations in dealing with the inverse problem," IEEE Trans. Antennas Propag. AP-29, 373–379 (1981).
[CrossRef]

AIP Conf. Proc. (1)

H. A. Ferwerda, "Optics in four dimensions—1980," AIP Conf. Proc. 65, 402–411 (1981).
[CrossRef]

Geophysics (1)

R. T. Lacoss, "Data adaptive spectral analysis methods," Geophysics 36, 661–675 (1971).
[CrossRef]

Giorgio Ronchi (1)

M. Bertero and C. de Mol, "Ill-posedness, regularization and number of degrees of freedom," Atti Fond. "Giorgio Ronchi" 36, 619–632 (1981).

IEEE Trans. Acoust. Speech Signal Process. (1)

A. K. Jain and S. Ranganath, "Extrapolation algorithms for discrete signals, with application in spectral estimation," IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 830–845 (1981).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

T. K. Sarkar, D. D. Weiner, and V. K. Jain, "Some mathematical considerations in dealing with the inverse problem," IEEE Trans. Antennas Propag. AP-29, 373–379 (1981).
[CrossRef]

IEEE Trans. Circuits Syst. (1)

A. Papoulis, "A new algorithm in spectral analysis and bandlimited extrapolation," IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Acta (2)

R. W. Gerchberg, "Super-resolution through error energy reduction," Opt. Acta 21, 709–720 (1974).
[CrossRef]

J. B. Abbiss, C. de Mol, and H. S. Dhadwal, "Regularized iterative and noniterative procedures for object restoration from experimental data," Opt. Acta, 30, 107–124 (1983).
[CrossRef]

Opt. Lett. (2)

Proc. R. Soc. London,Ser. (1)

R. E. Burge, M. A. Fiddy, A. H. Greenaway, and G. Ross, "The phase problem," Proc. R. Soc. London, Ser. A 350, 191–212 (1976).
[CrossRef]

SIAM J. Appl. Math. (1)

C. Byrne and R. Fitzgerald, "Reconstruction from partial information, with applications to tomography," SIAM J. Appl. Math. 42, 933–940 (1982).
[CrossRef]

SIAM J. Math Anal. (1)

K. Miller, "Least squares methods for ill-posed problems with prescribed bound," SIAM J. Math Anal. 1, 52–74 (1970).
[CrossRef]

Other (8)

A. M. Darling, T. J. Hall, and M. A. Fiddy, "Stable, noniterative, object reconstruction from incomplete data using prior knowledge," in Digest of the Topical Meeting on Signal Recovery and Synthesis with Incomplete Information and Partial Constraints (Optical-Society of America, Washington, D.C., 1983).

L. M. Cheng, A. S. Ho, and R. E. Burge, "The use of prior knowlege in image reconstruction," submitted to J. Opt. Soc. Am.

B. R. Frieden, "Image enhancement and restoration," in Picture Processing and Digital Filtering, Vol. 6 of Topics in Applied Physics, T. S. Huang, ed. (Springer-Verlag, New York, 1978).

J. P. Burg, "Maximum entropy spectrum analysis," Ph.D. Thesis (Stanford University, Palo Alto, Calif., 1975).

C. van Schooneveld, "Resolution enhancement: the 'maximum entropy method' and the 'high resolution method'," in Image Formation from Coherence Functions in Astronomy (Reidel, Dordrecht, The Netherlands, 1979), pp. 197–218.
[CrossRef]

B. R. Frieden, in Progress in Optics, Vol. 9, E. Wolf, ed. (North-Holland, Amsterdam, 1971), Clap. VIII, pp. 311–407.
[CrossRef]

D. Slepian and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty—I," Bell Sys. Tech. J. 40,43–63, (1961); H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty—II," Bell Syst. Tech. J. 40, 65–84 (1961); "Prolate spheroidal wave functions, Fourier analysis and uncertainty—III; the dimension of the space of essentially time- and band-limited signals," Bell Syst. Tech. J. 41, 1295–1336 (1962).
[CrossRef]

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977), p. 146.

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Figures (9)

Fig. 1
Fig. 1

DFT of O(n), |n| ≤ 6 for the object shown in background. Object also has rectangular component of amplitude 0.0001 on [−π, π].

Fig. 2
Fig. 2

Linear restoration from O(n), |n| ≤ 6 using p(x) = r(−1.8, 1.8, x) in Eq. (2.9).

Fig. 3
Fig. 3

Linear restoration from O(n), |n| ≤ 6 using p(x) = r(−1.8 1.8, x) in Eq. (2.9) after changing O(0) from 0.3459333 to 0.3458333.

Fig. 4
Fig. 4

Linear restoration from O(n), |n| ≤ 6 using p(x) = r(−1.8, 1.8, x) + (0.0001) r(−π, π, x) in Eq. (2.9).

Fig. 5
Fig. 5

Linear restoration from O(n), |n| ≤ 6 using p(x) = 0.5 r(−π/2, π/2, x) + 0.0001 r(−π, π, x) in Eq. (2.9).

Fig. 6
Fig. 6

Linear restoration from O(n), |n| ≤ 6 using p(x) = 0.5 (r(−π/6, π/6, x) + 0.5 r(−π/2, π/2, x) + 0.0001 r(−π, π, x) in Eq. (2.9).

Fig. 7
Fig. 7

MEM restoration from O(n), |n| ≤ 6.

Fig. 8
Fig. 8

Nonlinear restoration from O(n), |n| ≤ 6 using p(x) = 0.5 r(−π/2, π/2, x) + 0.0001 r(−π, π, x) in Eq. (3.18).

Fig. 9
Fig. 9

Nonlinear restoration from O(n), |n| ≤ 6 using P(x) = 0.5 r(−π/6, π/6, x) + 0.5 r(−π/2, π/2, x) + 0.0001 r(−π, π, x) in Eq. (3.18).

Equations (32)

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

i ( y ) = - o ( x ) s ( y , x ) d x ,
i ( y ) = - X X o ( x ) sin [ Ω ( y - x ) ] π ( y - x ) d x ,
o ( x ) = n = 0 λ n - 1 [ - X X i ( y ) ϕ n ( y ) d y ] ϕ n ( x ) ,
O ( ω ) = - X X o ( x ) exp ( - i x ω ) d x / 2 π ,
o ( x ) = d n = - O ( n d ) exp ( i n d x ) ,
O ( ω ) = d n = - O ( n d ) sin [ X ( ω - n d ) ] π ( ω - n d ) .
O ( m Δ ) = d n = - O ( n d ) sin [ X ( m Δ - n d ) ] π ( m Δ - n d ) .
O ( m Δ ) = n = 1 M a n sin [ X ( m - n ) Δ ] π ( m - n ) Δ ,
o ^ ( x ) = p ( x ) n = 1 M a n exp ( i n Δ x ) ,
p ( x ) = { 1 , x X 0 , x > X .
p ( x ) = { 1 + , x X , X < x < π / Δ
O ( m Δ ) = n = 1 M a n p ( x ) exp [ i ( m - n ) Δ x ] d x .
error = o ( x ) - o ^ ( x ) 2 p - 1 ( x ) d x ,
H = - π / Δ π / Δ log o ^ ( x ) d x
O ( m Δ ) = - π / Δ π / Δ o ^ ( x ) exp ( - i m Δ x ) d x / 2 π ,
o ^ ( x ) = c 0 / | m = 0 M c m exp ( i m Δ x ) | 2 ,
m = 0 M c m O ( n Δ - m Δ ) = { 1 , n = 0 0 , n = 1 , , M .
o - 1 ( x ) = h ( x ) 2 ,
h ( x ) = n = 0 H n exp ( i n Δ x )
0 = - π / Δ π / Δ [ o - 1 ( x ) - c h ( x ) ] o ( x ) exp ( - i n Δ x ) d x .
f , g = - π / Δ π / Δ f ( x ) g ( x ) * o ( x ) d x ,
q M ( x ) = m = 0 M c m exp ( i m Δ x )
error = - π / Δ π / Δ o - 1 ( x ) - q M ( x ) 2 o ( x ) d x
error = - π / Δ π / Δ p ( x ) o - 1 ( x ) - q M ( x ) 2 o ( x ) d x .
m = 0 M c m O ( n Δ - m Δ ) = - π / Δ π / Δ p ( x ) exp ( - i n Δ x ) d x = P ( n Δ ) .
0 = - π / Δ π / Δ [ p ( x ) o - 1 ( x ) - q ( x ) ] exp ( - i n Δ x ) o ( x ) d x = - π / Δ π / Δ [ p ( x ) - q ( x ) o ( x ) ] exp ( - i n Δ x ) d x
p ( x ) + = Δ n = 0 P ( n Δ ) exp ( i n Δ x ) .
[ q M ( x ) o ( x ) ] + = q m ( x ) o ( x ) + + j ( x ) ,
j ( x ) = c 1 O ( - Δ ) + c 2 [ O ( - 2 Δ ) + O ( - Δ ) exp ( i Δ x ) ] + + c M { O ( - M Δ ) + + O ( - Δ ) exp [ i ( M - 1 ) Δ x ] } .
( x ^ ) + = [ p ( x ) + - j ( x ) ] / q M ( x ) .
o ^ ( x ) = 2 Re [ o ^ ( x ) + ] - O ( 0 ) .
f ( z ) = m = 0 M c m z m ,