Abstract

A method for continuing Fourier spectra applicable to spectra given by the discrete Fourier transform is presented. From the principle of minimizing the sum of the square of the negative values of the restored function, a set of simultaneous nonlinear equations is obtained. The only means of solution at present is iterative, but computational time is comparable with that for noniterative methods. Excellent restoration is obtained for the sharply attenuated spectrum of deconvolved infared peaks. The numerical procedure developed here lends itself easily to the inclusion of additional constraints to enhance resolution further. The constraints of minimum negativity and finite extent may both be enforced together on pertinent data with only slight modification of the procedure.

© 1981 Optical Society of America

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References

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  1. S. J. Howard, “Method for continuing Fourier spectra given by the fast Fourier transform,” J. Opt. Soc. Am. 71, 95–98 (1981).
    [CrossRef]
  2. B. R. Frieden, in Picture Processing and Digital Filtering, T. S. Huang, ed. (Springer-Verlag, New York, 1975), Chap. 5.
  3. A. C. Schell, “Enhancing the angular resolution of incoherent sources,” Radio Electron. Eng. 29, 21–26 (1965).
    [CrossRef]
  4. Y. Biraud, “A new approach for increasing the resolving power by data processing,” Astron. Astrophys. 1, 124–127 (1969).
  5. P. A. Jansson, R. H. Hunt, and E. K. Plyer, “Resolution enhancement of spectra,” J. Opt. Soc. Am. 60, 596–599 (1970).
    [CrossRef]
  6. J. P. Burg, “Maximum entropy spectral analysis,” presented at the 37th Annual Society of Exploration Geophysicists Meeting, Oklahoma City, Okla., 1967.
  7. F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill, New York, 1965), pp. 450–451.

1981 (1)

1970 (1)

1969 (1)

Y. Biraud, “A new approach for increasing the resolving power by data processing,” Astron. Astrophys. 1, 124–127 (1969).

1965 (1)

A. C. Schell, “Enhancing the angular resolution of incoherent sources,” Radio Electron. Eng. 29, 21–26 (1965).
[CrossRef]

Biraud, Y.

Y. Biraud, “A new approach for increasing the resolving power by data processing,” Astron. Astrophys. 1, 124–127 (1969).

Burg, J. P.

J. P. Burg, “Maximum entropy spectral analysis,” presented at the 37th Annual Society of Exploration Geophysicists Meeting, Oklahoma City, Okla., 1967.

Frieden, B. R.

B. R. Frieden, in Picture Processing and Digital Filtering, T. S. Huang, ed. (Springer-Verlag, New York, 1975), Chap. 5.

Hildebrand, F. B.

F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill, New York, 1965), pp. 450–451.

Howard, S. J.

Hunt, R. H.

Jansson, P. A.

Plyer, E. K.

Schell, A. C.

A. C. Schell, “Enhancing the angular resolution of incoherent sources,” Radio Electron. Eng. 29, 21–26 (1965).
[CrossRef]

Astron. Astrophys. (1)

Y. Biraud, “A new approach for increasing the resolving power by data processing,” Astron. Astrophys. 1, 124–127 (1969).

J. Opt. Soc. Am. (2)

Radio Electron. Eng. (1)

A. C. Schell, “Enhancing the angular resolution of incoherent sources,” Radio Electron. Eng. 29, 21–26 (1965).
[CrossRef]

Other (3)

B. R. Frieden, in Picture Processing and Digital Filtering, T. S. Huang, ed. (Springer-Verlag, New York, 1975), Chap. 5.

J. P. Burg, “Maximum entropy spectral analysis,” presented at the 37th Annual Society of Exploration Geophysicists Meeting, Oklahoma City, Okla., 1967.

F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill, New York, 1965), pp. 450–451.

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Figures (2)

Fig. 1
Fig. 1

Restoration of Fourier spectrum of deconvolution of two noisy infrared peaks using constraint of minimum negativity. (a) Two merged infrared peaks. (b) Deconvolution of infrared peaks with spectrum truncated after 10th coefficient. (c) Spectrum restored by minimizing the sum of the square of the negative regions of the deconvolution. Sixteen (unique complex) coefficients were restored.

Fig. 2
Fig. 2

Restoration of Fourier spectrum of deconvolution of strongly merged infrared peaks using constraint of minimum negativity. (a) Noisy infrared data. (b) Deconvolution of infrared data with spectrum truncated after 15th coefficient. (c) Spectrum restored by minimizing the sum of the square of the negative regions of the deconvolution. Sixteen (complex) coefficients were restored.

Equations (10)

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f ( χ i ) = A k cos [ 2 π N ( k - 1 ) χ i ] + B k sin [ 2 π N ( k - 1 ) χ i ] + A k + 1 cos [ 2 π N ( k ) χ i ] + B k + 1 sin [ 2 π N ( k ) χ i ] + A k + 2 cos [ 2 π N ( k + 1 ) χ i ] + ,
i = 1 N ( H { - [ F ( χ i ) + f ( χ i ) ] } [ F ( χ i ) + f ( χ i ) ] ) 2 .
H ( y ) = 0 y < 0 , H ( y ) = ½ y = 0 , H ( y ) = 1 y > 0.
i = 1 N ( 1 1 + exp { K [ F ( χ i ) + f ( χ i ) ] } [ F ( χ i ) + f ( χ i ) ] ) 2 ,
A k i = 1 N { G ( χ i ) [ F ( χ i ) + f ( χ i ) ] } 2 = 0 , B k i = 1 N { G ( χ i ) [ F ( χ i ) + f ( χ i ) ] } 2 = 0 , A k + 1 i = 1 N { G ( χ i ) [ F ( χ i ) + f ( χ i ) ] } 2 = 0 , B k + n i = 1 N { G ( χ i ) [ F ( χ i ) + f ( χ i ) ] } 2 = 0.
i = 1 N A k ( 1 1 + exp { K [ F ( χ i ) + f ( χ i ) ] } [ F ( χ i ) + f ( χ i ) ] ) 2 = 2 i = 1 N cos [ 2 π N ( k - 1 ) χ i ] ( [ F ( χ i ) + f ( χ i ) ] 1 + exp { K [ F ( χ i ) + f ( χ i ) } ) × { 1 1 + exp { K [ F ( χ i ) + f ( χ i ) ] } - K exp { K [ F ( χ i ) + f ( χ i ) ] } × [ F ( χ i ) + f ( χ i ) ] ( 1 + exp { K [ F ( χ i ) + f ( χ i ) ] } ) 2 } = 0.
i = 1 N [ F ( χ i ) + f ( χ i ) ] ( 1 + exp { K [ F ( χ i ) + f ( χ i ) ] } ) 2 cos [ 2 π N ( k - 1 ) χ i ] = 0.
i = 1 N G 2 ( χ i ) [ F ( χ i ) + f ( χ i ) ] cos [ 2 π N ( k - 1 ) χ i ] = 0 , i = 1 N G 2 ( χ i ) [ F ( χ i ) + f ( χ i ) ] sin [ 2 π N ( k - 1 ) χ i ] = 0 , i = 1 N G 2 ( χ i ) [ F ( χ i ) + f ( χ i ) ] cos [ 2 π N ( k ) χ i ] = 0 , i = 1 N G 2 ( χ i ) [ F ( χ i ) + f ( χ i ) ] sin [ 2 π N ( k + n - 1 ) χ i ] = 0.
A k = i { F ( χ i ) + f ( χ i ) - A k cos [ 2 π N ( N - 1 ) χ i ] } ( 1 - exp { K [ F ( χ i ) + ( χ i ) ] } ) 2 cos [ 2 π N ( k - 1 ) χ i ] i cos 2 [ 2 π N ( k - 1 ) χ i ] ( 1 - exp { K [ F ( χ i ) + f ( χ i ) ] } ) 2 , B k = - i { F ( χ i ) + f ( χ i ) - B k sin [ 2 π N ( k - 1 ) χ i ] } ( 1 + exp { K [ F ( χ i ) + f ( χ i ) ] } ) 2 sin ( 2 π N ( k - 1 ) χ i ) i sin 2 [ 2 π N ( k - 1 ) χ i ] ( 1 + exp { K [ F ( χ i ) + f ( χ i ) ] } ) 2 , A k + 1 = - i { F ( χ i ) + f ( χ i ) - A k + 1 cos [ 2 π N ( k ) χ i ] } ( 1 + exp { K [ F ( χ i ) + f ( χ i ) ] } ) 2 cos ( 2 π N ( k ) χ i ) i cos 2 ( 2 π N ( k ) χ i ) ( 1 + exp { K [ F ( χ i ) + f ( χ i ) ] } ) 2 , B k + n = - i { F ( χ i ) + f ( χ i ) - B k + n sin [ 2 π N ( k + n - 1 ) χ i ] } ( 1 + exp { K [ F ( χ i ) + f ( χ i ) ] } ) 2 sin ( 2 π N ( k + n - 1 ) χ i ) i sin 2 [ 2 π N ( k + n - 1 ) χ i ] ( 1 + exp { K [ F ( χ i ) + f ( χ i ) ] } ) 2 .
A k = - { F ( χ i ) + f ( χ i ) - A k cos [ 2 π N ( k - 1 ) χ i ] } cos [ 2 π N ( k - 1 ) χ i ] cos 2 [ 2 π N ( k - 1 ) χ i ]             for F ( χ i ) + f ( χ i ) < 0.