Abstract

Traditional approaches to optical resolution enhancement have involved either the design of appropriate image-formation systems or some type of postprocessing of an image that has already been formed. Results presented in this paper suggest that improved images can be obtained if the image-gathering system is designed specifically to enhance the performance of the image-restoration algorithm to be used. We consider a class of problems in which the total available spatial bandwidth is fixed but the location of this bandwidth along the spatial-frequency axis is to some extent under our control. For example, we might consider either a low-pass system or a bandpass system of the same total bandwidth. We show that system performance can be substantially improved by proper allocation of the available bandwidth in the spatial-frequency domain. The optimum allocation is shown to be a function of the signal-to-noise ratio. We also describe coherent and incoherent optical image-gathering systems that can achieve the desired spatial-frequency passbands.

© 1984 Optical Society of America

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References

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  1. R. W. Gerchberg, “Super-resolution through error-energy reduction,” Opt. Acta 21, 709–720 (1974).
    [CrossRef]
  2. G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento Suppl. 3 IX, 426–438 (1952).
    [CrossRef]
  3. R. C. Hansen, “Fundamental limits in antennas,” Proc. IEEE 69, 170–182 (1981).
    [CrossRef]
  4. B. R. Frieden, “On arbitrarily perfect imagery with a finite aperture,” Opt. Acta 16, 795–807 (1969).
    [CrossRef]
  5. B. R. Frieden, “The extrapolating pupil, image synthesis, and some thought applications,” Appl. Opt. 9, 2489–2496 (1970).
    [CrossRef] [PubMed]
  6. G. R. Boyer, “Pupil filters for moderate superresolution,” Appl. Opt. 15, 3089–3093 (1976).
    [CrossRef] [PubMed]
  7. R. Biovin, A. Biovin, “Optimized amplitude filtering for superresolution over a restricted field,” Opt. Acta 27, 587–610 (1980).
    [CrossRef]
  8. M. A. Grimm, A. W. Lohmann, “Super-resolution image for one-dimensional objects,” J. Opt. Soc. Am. 6, 1151–1156 (1966).
    [CrossRef]
  9. A. W. Lohmann, D. P. Paris, “Super-resolution for nonbirefringent objects,” Appl. Opt. 3, 1037–1043 (1964).
    [CrossRef]
  10. W. Lukosz, “Optical systems with resolving powers exceeding the classical limit,” J. Opt. Soc. Am. 56, 1463–1472 (1966).
    [CrossRef]
  11. W. Lukosz, “Optical systems with resolving powers exceeding the classical limit. II,” J. Opt. Soc. Am. 57, 932–941 (1967).
    [CrossRef]
  12. M. Ueda, T. Sata, M. Kondo, “Superresolution by multiple superposition of image holograms having different carrier frequencies,” Opt. Acta 20, 403–410 (1973).
    [CrossRef]
  13. G. Toraldo di Francia, “Resolving power and information,” J. Opt. Soc. Am. 45, 497–501 (1955).
    [CrossRef]
  14. R. C. Jones, “Information capacity of a beam of light,” J. Opt. Soc. Am. 52, 493–501 (1962).
    [CrossRef]
  15. N. J. Bershad, “Resolution, optical-channel capacity and information theory,” J. Opt. Soc. Am. 59, 157–163 (1969).
    [CrossRef]
  16. M. Bendinelli, A. Consortini, L. Ronchi, B. R. Frieden, “Degrees of freedom, and eigenfunctions, for the noisy image,” J. Opt. Soc. Am. 64, 1498–1502 (1974).
    [CrossRef]
  17. B. Saleh, “A priori information and the degrees of freedom of noisy images,” J. Opt. Soc. Am. 67, 71–76 (1977).
    [CrossRef]
  18. C. K. Rushforth, R. W. Harris, “Restoration, resolution, and noise,” J. Opt. Soc. Am. 58, 539–545 (1968).
    [CrossRef]
  19. B. R. Frieden, “Restoring with maximum likelihood and maximum entropy,” J. Opt. Soc. Am. 62, 511–518 (1972).
    [CrossRef] [PubMed]
  20. R. Kikuchi, B. H. Soffer, “Maximum entropy image restoration. I. The entropy expression,” J. Opt. Soc. Am. 67, 1656–1665 (1977).
    [CrossRef]
  21. W. H. Richardson, “Bayesian-based interative method of image restoration,” J. Opt. Soc. Am. 62, 55–59 (1972).
    [CrossRef]
  22. G. Cesini, G. Guattari, G. Lucarini, C. Palma, “An iterative method of restoring noisy images,” Opt. Acta 25, 501–508 (1978).
    [CrossRef]
  23. D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
    [CrossRef]
  24. C. K. Rushforth, R. L. Frost, “Comparison of some algorithms for reconstructing space-limited images,” J. Opt. Soc. Am. 70, 1539–1544 (1980).
    [CrossRef]
  25. R. J. Marks, “Coherent optical extrapolation of 2-D band-limited signals: processor theory,” Appl. Opt. 19, 1670–1672 (1980).
    [CrossRef]
  26. E. J. Kelly, I. S. Reed, W. L. Root, “The detection of radar echoes in noise. I,” J. Soc. Indust. Appl. Math. 8, 309–341 (1960).
    [CrossRef]
  27. E. J. Kelly, I. S. Reed, W. L. Root, “The detection of radar echoes in noise. II,” J. Soc. Indust. Appl. Math. 8, 481–507 (1960).
    [CrossRef]
  28. C. K. Rushforth, A. E. Crawford, Y. Zhou, “Least-squares reconstruction of objects with missing high-frequency components,” J. Opt. Soc. Am. 72, 204–211, 1982.
    [CrossRef]
  29. Y. S. Shim, Z. H. Cho, “SVD pseudoinversion image reconstruction,” IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 904–909 (1981).
    [CrossRef]
  30. J. W. Goodman, ed., Synthetic Aperture Optics (National Academy of Sciences, Washington, D.C., 1967).
  31. B. R. Frieden, “Picture processing and digital filtering,” in Topics in Applied Physics, T. S. Huang, ed. (Springer-Verlag, New York, 1975), Vol. 6, pp. 177–248.
    [CrossRef]
  32. C. W. Helstrom, “Image restoration by the method of least squares,” J. Opt. Soc. Am. 57, 297–303 (1967).
    [CrossRef]
  33. B. H. Soffer, R. Kikuchi, “Maximum entropy image estimation: analysis of image formation with thinned random arrays,” (U.S. Air Force Office of Scientific Research, Washington, D.C., 1981).
  34. A. W. Rihaczek, Principles of High-Resolution Radar (McGraw-Hill, New York, 1969).
  35. Y. Zhou, C. K. Rushforth, R. L. Frost, “Signal restoration following the suppression of arbitrary frequencies,” to be presented at the International Conference on Acoustics, Speech, and Signal Processing, San Diego, California, March 1984.
  36. We thank Lai Chang Ling for computer programming the algorithms leading to the solutions shown in Figs. 8–10.
  37. W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974), Chap. 8.
  38. L. C. Martin, The Theory of the Microscope (Blackie, London, 1966), Sec. 6.06.
  39. W. T. Cathey, “Single side band imaging,” J. Opt. Soc. Am. 61, 1555 (A) (1971).
  40. P. Chavel, S. Lowenthal, “A method of incoherent optical-image processing using synthetic holograms,” J. Opt. Soc. Am 66, 14–23 (1976).
    [CrossRef]

1982 (1)

1981 (2)

Y. S. Shim, Z. H. Cho, “SVD pseudoinversion image reconstruction,” IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 904–909 (1981).
[CrossRef]

R. C. Hansen, “Fundamental limits in antennas,” Proc. IEEE 69, 170–182 (1981).
[CrossRef]

1980 (3)

1978 (2)

G. Cesini, G. Guattari, G. Lucarini, C. Palma, “An iterative method of restoring noisy images,” Opt. Acta 25, 501–508 (1978).
[CrossRef]

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
[CrossRef]

1977 (2)

1976 (2)

G. R. Boyer, “Pupil filters for moderate superresolution,” Appl. Opt. 15, 3089–3093 (1976).
[CrossRef] [PubMed]

P. Chavel, S. Lowenthal, “A method of incoherent optical-image processing using synthetic holograms,” J. Opt. Soc. Am 66, 14–23 (1976).
[CrossRef]

1974 (2)

1973 (1)

M. Ueda, T. Sata, M. Kondo, “Superresolution by multiple superposition of image holograms having different carrier frequencies,” Opt. Acta 20, 403–410 (1973).
[CrossRef]

1972 (2)

1971 (1)

W. T. Cathey, “Single side band imaging,” J. Opt. Soc. Am. 61, 1555 (A) (1971).

1970 (1)

1969 (2)

B. R. Frieden, “On arbitrarily perfect imagery with a finite aperture,” Opt. Acta 16, 795–807 (1969).
[CrossRef]

N. J. Bershad, “Resolution, optical-channel capacity and information theory,” J. Opt. Soc. Am. 59, 157–163 (1969).
[CrossRef]

1968 (1)

1967 (2)

1966 (2)

W. Lukosz, “Optical systems with resolving powers exceeding the classical limit,” J. Opt. Soc. Am. 56, 1463–1472 (1966).
[CrossRef]

M. A. Grimm, A. W. Lohmann, “Super-resolution image for one-dimensional objects,” J. Opt. Soc. Am. 6, 1151–1156 (1966).
[CrossRef]

1964 (1)

1962 (1)

1960 (2)

E. J. Kelly, I. S. Reed, W. L. Root, “The detection of radar echoes in noise. I,” J. Soc. Indust. Appl. Math. 8, 309–341 (1960).
[CrossRef]

E. J. Kelly, I. S. Reed, W. L. Root, “The detection of radar echoes in noise. II,” J. Soc. Indust. Appl. Math. 8, 481–507 (1960).
[CrossRef]

1955 (1)

1952 (1)

G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento Suppl. 3 IX, 426–438 (1952).
[CrossRef]

Bendinelli, M.

Bershad, N. J.

Biovin, A.

R. Biovin, A. Biovin, “Optimized amplitude filtering for superresolution over a restricted field,” Opt. Acta 27, 587–610 (1980).
[CrossRef]

Biovin, R.

R. Biovin, A. Biovin, “Optimized amplitude filtering for superresolution over a restricted field,” Opt. Acta 27, 587–610 (1980).
[CrossRef]

Boyer, G. R.

Cathey, W. T.

W. T. Cathey, “Single side band imaging,” J. Opt. Soc. Am. 61, 1555 (A) (1971).

W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974), Chap. 8.

Cesini, G.

G. Cesini, G. Guattari, G. Lucarini, C. Palma, “An iterative method of restoring noisy images,” Opt. Acta 25, 501–508 (1978).
[CrossRef]

Chavel, P.

P. Chavel, S. Lowenthal, “A method of incoherent optical-image processing using synthetic holograms,” J. Opt. Soc. Am 66, 14–23 (1976).
[CrossRef]

Cho, Z. H.

Y. S. Shim, Z. H. Cho, “SVD pseudoinversion image reconstruction,” IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 904–909 (1981).
[CrossRef]

Consortini, A.

Crawford, A. E.

Frieden, B. R.

Frost, R. L.

C. K. Rushforth, R. L. Frost, “Comparison of some algorithms for reconstructing space-limited images,” J. Opt. Soc. Am. 70, 1539–1544 (1980).
[CrossRef]

Y. Zhou, C. K. Rushforth, R. L. Frost, “Signal restoration following the suppression of arbitrary frequencies,” to be presented at the International Conference on Acoustics, Speech, and Signal Processing, San Diego, California, March 1984.

Gerchberg, R. W.

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

Grimm, M. A.

M. A. Grimm, A. W. Lohmann, “Super-resolution image for one-dimensional objects,” J. Opt. Soc. Am. 6, 1151–1156 (1966).
[CrossRef]

Guattari, G.

G. Cesini, G. Guattari, G. Lucarini, C. Palma, “An iterative method of restoring noisy images,” Opt. Acta 25, 501–508 (1978).
[CrossRef]

Hansen, R. C.

R. C. Hansen, “Fundamental limits in antennas,” Proc. IEEE 69, 170–182 (1981).
[CrossRef]

Harris, R. W.

Helstrom, C. W.

Jones, R. C.

Kelly, E. J.

E. J. Kelly, I. S. Reed, W. L. Root, “The detection of radar echoes in noise. I,” J. Soc. Indust. Appl. Math. 8, 309–341 (1960).
[CrossRef]

E. J. Kelly, I. S. Reed, W. L. Root, “The detection of radar echoes in noise. II,” J. Soc. Indust. Appl. Math. 8, 481–507 (1960).
[CrossRef]

Kikuchi, R.

R. Kikuchi, B. H. Soffer, “Maximum entropy image restoration. I. The entropy expression,” J. Opt. Soc. Am. 67, 1656–1665 (1977).
[CrossRef]

B. H. Soffer, R. Kikuchi, “Maximum entropy image estimation: analysis of image formation with thinned random arrays,” (U.S. Air Force Office of Scientific Research, Washington, D.C., 1981).

Kondo, M.

M. Ueda, T. Sata, M. Kondo, “Superresolution by multiple superposition of image holograms having different carrier frequencies,” Opt. Acta 20, 403–410 (1973).
[CrossRef]

Lohmann, A. W.

M. A. Grimm, A. W. Lohmann, “Super-resolution image for one-dimensional objects,” J. Opt. Soc. Am. 6, 1151–1156 (1966).
[CrossRef]

A. W. Lohmann, D. P. Paris, “Super-resolution for nonbirefringent objects,” Appl. Opt. 3, 1037–1043 (1964).
[CrossRef]

Lowenthal, S.

P. Chavel, S. Lowenthal, “A method of incoherent optical-image processing using synthetic holograms,” J. Opt. Soc. Am 66, 14–23 (1976).
[CrossRef]

Lucarini, G.

G. Cesini, G. Guattari, G. Lucarini, C. Palma, “An iterative method of restoring noisy images,” Opt. Acta 25, 501–508 (1978).
[CrossRef]

Lukosz, W.

Marks, R. J.

Martin, L. C.

L. C. Martin, The Theory of the Microscope (Blackie, London, 1966), Sec. 6.06.

Palma, C.

G. Cesini, G. Guattari, G. Lucarini, C. Palma, “An iterative method of restoring noisy images,” Opt. Acta 25, 501–508 (1978).
[CrossRef]

Paris, D. P.

Reed, I. S.

E. J. Kelly, I. S. Reed, W. L. Root, “The detection of radar echoes in noise. I,” J. Soc. Indust. Appl. Math. 8, 309–341 (1960).
[CrossRef]

E. J. Kelly, I. S. Reed, W. L. Root, “The detection of radar echoes in noise. II,” J. Soc. Indust. Appl. Math. 8, 481–507 (1960).
[CrossRef]

Richardson, W. H.

Rihaczek, A. W.

A. W. Rihaczek, Principles of High-Resolution Radar (McGraw-Hill, New York, 1969).

Ronchi, L.

Root, W. L.

E. J. Kelly, I. S. Reed, W. L. Root, “The detection of radar echoes in noise. II,” J. Soc. Indust. Appl. Math. 8, 481–507 (1960).
[CrossRef]

E. J. Kelly, I. S. Reed, W. L. Root, “The detection of radar echoes in noise. I,” J. Soc. Indust. Appl. Math. 8, 309–341 (1960).
[CrossRef]

Rushforth, C. K.

Saleh, B.

Sata, T.

M. Ueda, T. Sata, M. Kondo, “Superresolution by multiple superposition of image holograms having different carrier frequencies,” Opt. Acta 20, 403–410 (1973).
[CrossRef]

Shim, Y. S.

Y. S. Shim, Z. H. Cho, “SVD pseudoinversion image reconstruction,” IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 904–909 (1981).
[CrossRef]

Soffer, B. H.

R. Kikuchi, B. H. Soffer, “Maximum entropy image restoration. I. The entropy expression,” J. Opt. Soc. Am. 67, 1656–1665 (1977).
[CrossRef]

B. H. Soffer, R. Kikuchi, “Maximum entropy image estimation: analysis of image formation with thinned random arrays,” (U.S. Air Force Office of Scientific Research, Washington, D.C., 1981).

Toraldo di Francia, G.

G. Toraldo di Francia, “Resolving power and information,” J. Opt. Soc. Am. 45, 497–501 (1955).
[CrossRef]

G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento Suppl. 3 IX, 426–438 (1952).
[CrossRef]

Ueda, M.

M. Ueda, T. Sata, M. Kondo, “Superresolution by multiple superposition of image holograms having different carrier frequencies,” Opt. Acta 20, 403–410 (1973).
[CrossRef]

Youla, D. C.

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
[CrossRef]

Zhou, Y.

C. K. Rushforth, A. E. Crawford, Y. Zhou, “Least-squares reconstruction of objects with missing high-frequency components,” J. Opt. Soc. Am. 72, 204–211, 1982.
[CrossRef]

Y. Zhou, C. K. Rushforth, R. L. Frost, “Signal restoration following the suppression of arbitrary frequencies,” to be presented at the International Conference on Acoustics, Speech, and Signal Processing, San Diego, California, March 1984.

Appl. Opt. (4)

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

Y. S. Shim, Z. H. Cho, “SVD pseudoinversion image reconstruction,” IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 904–909 (1981).
[CrossRef]

IEEE Trans. Circuits Syst. (1)

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
[CrossRef]

J. Opt. Soc. Am (1)

P. Chavel, S. Lowenthal, “A method of incoherent optical-image processing using synthetic holograms,” J. Opt. Soc. Am 66, 14–23 (1976).
[CrossRef]

J. Opt. Soc. Am. (16)

W. T. Cathey, “Single side band imaging,” J. Opt. Soc. Am. 61, 1555 (A) (1971).

C. W. Helstrom, “Image restoration by the method of least squares,” J. Opt. Soc. Am. 57, 297–303 (1967).
[CrossRef]

C. K. Rushforth, R. L. Frost, “Comparison of some algorithms for reconstructing space-limited images,” J. Opt. Soc. Am. 70, 1539–1544 (1980).
[CrossRef]

M. A. Grimm, A. W. Lohmann, “Super-resolution image for one-dimensional objects,” J. Opt. Soc. Am. 6, 1151–1156 (1966).
[CrossRef]

C. K. Rushforth, A. E. Crawford, Y. Zhou, “Least-squares reconstruction of objects with missing high-frequency components,” J. Opt. Soc. Am. 72, 204–211, 1982.
[CrossRef]

W. Lukosz, “Optical systems with resolving powers exceeding the classical limit,” J. Opt. Soc. Am. 56, 1463–1472 (1966).
[CrossRef]

W. Lukosz, “Optical systems with resolving powers exceeding the classical limit. II,” J. Opt. Soc. Am. 57, 932–941 (1967).
[CrossRef]

G. Toraldo di Francia, “Resolving power and information,” J. Opt. Soc. Am. 45, 497–501 (1955).
[CrossRef]

R. C. Jones, “Information capacity of a beam of light,” J. Opt. Soc. Am. 52, 493–501 (1962).
[CrossRef]

N. J. Bershad, “Resolution, optical-channel capacity and information theory,” J. Opt. Soc. Am. 59, 157–163 (1969).
[CrossRef]

M. Bendinelli, A. Consortini, L. Ronchi, B. R. Frieden, “Degrees of freedom, and eigenfunctions, for the noisy image,” J. Opt. Soc. Am. 64, 1498–1502 (1974).
[CrossRef]

B. Saleh, “A priori information and the degrees of freedom of noisy images,” J. Opt. Soc. Am. 67, 71–76 (1977).
[CrossRef]

C. K. Rushforth, R. W. Harris, “Restoration, resolution, and noise,” J. Opt. Soc. Am. 58, 539–545 (1968).
[CrossRef]

B. R. Frieden, “Restoring with maximum likelihood and maximum entropy,” J. Opt. Soc. Am. 62, 511–518 (1972).
[CrossRef] [PubMed]

R. Kikuchi, B. H. Soffer, “Maximum entropy image restoration. I. The entropy expression,” J. Opt. Soc. Am. 67, 1656–1665 (1977).
[CrossRef]

W. H. Richardson, “Bayesian-based interative method of image restoration,” J. Opt. Soc. Am. 62, 55–59 (1972).
[CrossRef]

J. Soc. Indust. Appl. Math. (2)

E. J. Kelly, I. S. Reed, W. L. Root, “The detection of radar echoes in noise. I,” J. Soc. Indust. Appl. Math. 8, 309–341 (1960).
[CrossRef]

E. J. Kelly, I. S. Reed, W. L. Root, “The detection of radar echoes in noise. II,” J. Soc. Indust. Appl. Math. 8, 481–507 (1960).
[CrossRef]

Nuovo Cimento Suppl. 3 (1)

G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento Suppl. 3 IX, 426–438 (1952).
[CrossRef]

Opt. Acta (5)

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

B. R. Frieden, “On arbitrarily perfect imagery with a finite aperture,” Opt. Acta 16, 795–807 (1969).
[CrossRef]

M. Ueda, T. Sata, M. Kondo, “Superresolution by multiple superposition of image holograms having different carrier frequencies,” Opt. Acta 20, 403–410 (1973).
[CrossRef]

G. Cesini, G. Guattari, G. Lucarini, C. Palma, “An iterative method of restoring noisy images,” Opt. Acta 25, 501–508 (1978).
[CrossRef]

R. Biovin, A. Biovin, “Optimized amplitude filtering for superresolution over a restricted field,” Opt. Acta 27, 587–610 (1980).
[CrossRef]

Proc. IEEE (1)

R. C. Hansen, “Fundamental limits in antennas,” Proc. IEEE 69, 170–182 (1981).
[CrossRef]

Other (8)

B. H. Soffer, R. Kikuchi, “Maximum entropy image estimation: analysis of image formation with thinned random arrays,” (U.S. Air Force Office of Scientific Research, Washington, D.C., 1981).

A. W. Rihaczek, Principles of High-Resolution Radar (McGraw-Hill, New York, 1969).

Y. Zhou, C. K. Rushforth, R. L. Frost, “Signal restoration following the suppression of arbitrary frequencies,” to be presented at the International Conference on Acoustics, Speech, and Signal Processing, San Diego, California, March 1984.

We thank Lai Chang Ling for computer programming the algorithms leading to the solutions shown in Figs. 8–10.

W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974), Chap. 8.

L. C. Martin, The Theory of the Microscope (Blackie, London, 1966), Sec. 6.06.

J. W. Goodman, ed., Synthetic Aperture Optics (National Academy of Sciences, Washington, D.C., 1967).

B. R. Frieden, “Picture processing and digital filtering,” in Topics in Applied Physics, T. S. Huang, ed. (Springer-Verlag, New York, 1975), Vol. 6, pp. 177–248.
[CrossRef]

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Figures (16)

Fig. 1
Fig. 1

Coherent transfer function of a perfect bandpass filter.

Fig. 2
Fig. 2

Plots of point-spread function for b = a and for b = a/2.

Fig. 3
Fig. 3

Reconstruction of a low-pass-filtered image.

Fig. 4
Fig. 4

Reconstruction of a bandpass-filtered image.

Fig. 5
Fig. 5

Improved reconstruction when the system passes higher frequencies.

Fig. 6
Fig. 6

Another reconstruction of a bandpass-filtered image.

Fig. 7
Fig. 7

Signal spectrum of the form ϕ0(u) = cos2(2πau): (a) a = 1/4 uc, (b) a = 1/2uc.

Fig. 8
Fig. 8

Optimum pupils (symmetric) and corresponding MTF’s for various power spectra ϕ0/ϕn. Fill factor is 0.80.

Fig. 9
Fig. 9

Optimum pupils (symmetric) and corresponding MTF’s for various power spectra ϕ0/ϕn. Fill factor is 0.60.

Fig. 10
Fig. 10

Optimum pupils (symmetric) and corresponding MTF’s for various power spectra ϕ0/ϕn. Fill factor is 0.40.

Fig. 11
Fig. 11

(a) Transfer function with spatial frequency u and cutoff spatial frequencies −uc and uc. (b) S, source; O, object; F, Fourier plane; I, image plane.

Fig. 12
Fig. 12

(a) Transfer function shifted by amount us because of change in source position. (b) Imaging system with off-axis illumination.

Fig. 13
Fig. 13

(a) Angular spectrum of 1 + (1/2)cos 2πσx + (1/2)cos 4πσx. (b) Passband with oblique illumination to achieve shift of us.

Fig. 14
Fig. 14

Illustration of use of three sources and three references to record three different bands in the spatial spectrum.

Fig. 15
Fig. 15

Shifted object spectrum and optical transfer function.

Fig. 16
Fig. 16

(a) Two-point-source illumination to produce sinusoidal fringe pattern. (b) Maximum displacement (in geometrical-optics limit) required to ensure linear-in-intensity imaging.

Tables (1)

Tables Icon

Table 1 Error As a Function of Signal-to-Noise Ratioa

Equations (27)

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

T ( u ) = rect ( u / a ) [ δ ( u - b ) + δ ( u + b ) ] ,
rect ( v ) = { 1 - ( 1 / 2 ) v ( 1 / 2 ) 0 elsewhere .
u = x / λ f ,
t ( x ) = 2 a sinc ( a x ) cos ( 2 π b x ) ,
t ( x ) = 2 a sinc ( 2 a x ) ,
σ 2 = - 1 ρ Q ( 0 ) ,
Q ( x ) = sinc ( a x ) cos ( 2 π b x ) ,
σ 2 = 1 ρ [ 3 π 2 ( a 2 + 12 b 2 ) ] .
Y ( u ) = τ * ( u ) ϕ 0 ( u ) τ ( u ) 2 ϕ 0 ( u ) + ϕ n ( u ) ,
2 = - 2 u c 2 u c ϕ 0 ( u ) ϕ n ( u ) τ ( u ) 2 ϕ 0 ( u ) + ϕ n ( u ) d u ,
τ ( u ) = P ( u ) P ( u ) ,
f = n / N
11100 00111 11010 01011 11001 10011 10110 01101 10101 10101 10011 11001 01110 01110 01101 10110 01011 11010 00111 11100
M = ( N / 2 ) ! ( N f / 2 ) ! [ N ( 1 - f ) / 2 ] ! .
ϕ n ( u ) ϕ n .
ϕ 0 ( u ) ϕ 0
ϕ 0 ( u ) = cos 2 ( 2 π a u ) .
2 = - 2 u c 2 u c ϕ n ( u ) τ ( u ) 2 + ϕ n ( u ) / ϕ 0 ( u ) du .
0 ( x ) = 1 + ( 1 / 2 ) cos ( 2 π σ x ) + ( 1 / 2 ) cos ( 4 π σ x ) .
I ( x ) ~ ( 1 / 4 ) exp ( i 2 π σ x ) + ( 1 / 4 ) exp ( i 4 π σ x ) 2 = 1 / 8 + ( 1 / 8 ) cos 2 π σ x .
I ( x ) ~ R + ( 1 / 4 ) exp ( i 2 π σ x ) + ( 1 / 4 ) exp ( i 4 π σ x ) 2 = ( R 2 + 1 / 8 ) + ( R 2 + 1 / 8 ) cos ( 2 π σ x ) + R 2 cos ( 4 π σ ) ,
g c ( x ) = [ f ( x ) ( 1 + cos 2 π u s x ) ] h ( x ) ,
g s ( x ) = [ f ( x ) ( 1 + sin 2 π u s x ) ] h ( x ) ,
g ( x ) = f ( x ) h ( x ) ,
g ( x ) = g c ( x ) - i g s ( x ) - ( 1 - i ) g ( x )
g ( x ) = [ f ( x ) ( cos 2 π u s x - i sin 2 π u s x ) ] h ( x ) = [ f ( x ) exp ( - i 2 π u s x ) ] h ( x ) .
G ( u ) = F ( u + u s ) H ( u ) ,

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