Abstract

Superresolution has been an active field during the last 50 years. Many approaches for achieving superresolution have been suggested, all based on having a priori information on the signal. We discuss the space–bandwidth product adaptation process, which is a generalization of all previous approaches. This new point of view is based on handling the Wigner chart of the input signal as well as the Wigner chart of the signal that can be accepted by the system. We also take into account the number of degrees of freedom of the signal and the system. Examples that demonstrate the suggested approach are illustrated in a companion paper [J. Opt. Soc. Am. A 14, 563 (1997)].

© 1997 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), Chap. 7.
  2. Ref. 1, Chap. 2.
  3. A. W. Lohmann, “Utilization of a priori information for getting better image,” in Proceedings of the Conference on Synthetic Aperture Optics (National Academy of Sciences–National Research Council, Washington, D.C., 1967), App. 33.
  4. A. W. Lohmann, “Some thoughts on super resolution,” in Proceedings of Workshop on Limits of Passive Imaging, C. Aleksoff, ed. [R. Guenther (ARO), Raleigh, N. Car., 1983], p. 37.
  5. J. W. Goodman, “Synthetic aperture optics,” Prog. Opt. 8, 1–50 (1970).
    [CrossRef]
  6. G. Toraldo Di Francia, “Resolving power and information,” J. Opt. Soc. Am. 45, 497–501 (1955).
    [CrossRef]
  7. G. Toraldo Di Francia, “Degrees of freedom of an image,” J. Opt. Soc. Am. 59, 799–804 (1969).
    [CrossRef] [PubMed]
  8. M. Francon, “Amélioration de résolution d’optique,” Nuovo Cimento Suppl. 9, 283–290 (1952).
  9. W. Lukosz, “Optical systems with resolving powers exceeding the classical limit,” J. Opt. Soc. Am. 56, 1463–1472 (1966).
    [CrossRef]
  10. A. I. Kartashev, “Optical systems with enhanced resolving power,” Opt. Spectrosc. (USSR) 9, 204–206 (1960).
  11. H. Bartelt, A. W. Lohmann, “Optical processing of 1-D signals,” Opt. Commun. 42, 87–91 (1982).
    [CrossRef]
  12. W. Gartner, A. W. Lohmann, “An experiment going beyond Abbe’s limit of diffraction,” Z. Phys. 174, 18–24 (1963).
  13. B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
  14. A. W. Lohmann, “Image rotation, Wigner rotation and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  15. A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky, C. Ferreira, “Space–bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470–473 (1996).
    [CrossRef]
  16. D. Mendlovic, A. W. Lohmann, Z. Zalevsky, “Space–bandwidth product adaptation and its application for superresolution: examples,” J. Opt. Soc. Am. A 14, 563–567 (1997).
    [CrossRef]
  17. H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domain and their relation to chirp and wavelet transform,” J. Opt. Soc. Am. A 11, 547–559 (1994).
    [CrossRef]

1997

1996

1994

1993

1982

H. Bartelt, A. W. Lohmann, “Optical processing of 1-D signals,” Opt. Commun. 42, 87–91 (1982).
[CrossRef]

1970

J. W. Goodman, “Synthetic aperture optics,” Prog. Opt. 8, 1–50 (1970).
[CrossRef]

1969

1966

1963

W. Gartner, A. W. Lohmann, “An experiment going beyond Abbe’s limit of diffraction,” Z. Phys. 174, 18–24 (1963).

1960

A. I. Kartashev, “Optical systems with enhanced resolving power,” Opt. Spectrosc. (USSR) 9, 204–206 (1960).

1955

1952

M. Francon, “Amélioration de résolution d’optique,” Nuovo Cimento Suppl. 9, 283–290 (1952).

Barshan, B.

Bartelt, H.

H. Bartelt, A. W. Lohmann, “Optical processing of 1-D signals,” Opt. Commun. 42, 87–91 (1982).
[CrossRef]

Dorsch, R. G.

Ferreira, C.

Francon, M.

M. Francon, “Amélioration de résolution d’optique,” Nuovo Cimento Suppl. 9, 283–290 (1952).

Gartner, W.

W. Gartner, A. W. Lohmann, “An experiment going beyond Abbe’s limit of diffraction,” Z. Phys. 174, 18–24 (1963).

Goodman, J. W.

J. W. Goodman, “Synthetic aperture optics,” Prog. Opt. 8, 1–50 (1970).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), Chap. 7.

Kartashev, A. I.

A. I. Kartashev, “Optical systems with enhanced resolving power,” Opt. Spectrosc. (USSR) 9, 204–206 (1960).

Lohmann, A. W.

D. Mendlovic, A. W. Lohmann, Z. Zalevsky, “Space–bandwidth product adaptation and its application for superresolution: examples,” J. Opt. Soc. Am. A 14, 563–567 (1997).
[CrossRef]

A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky, C. Ferreira, “Space–bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470–473 (1996).
[CrossRef]

A. W. Lohmann, “Image rotation, Wigner rotation and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
[CrossRef]

H. Bartelt, A. W. Lohmann, “Optical processing of 1-D signals,” Opt. Commun. 42, 87–91 (1982).
[CrossRef]

W. Gartner, A. W. Lohmann, “An experiment going beyond Abbe’s limit of diffraction,” Z. Phys. 174, 18–24 (1963).

A. W. Lohmann, “Utilization of a priori information for getting better image,” in Proceedings of the Conference on Synthetic Aperture Optics (National Academy of Sciences–National Research Council, Washington, D.C., 1967), App. 33.

A. W. Lohmann, “Some thoughts on super resolution,” in Proceedings of Workshop on Limits of Passive Imaging, C. Aleksoff, ed. [R. Guenther (ARO), Raleigh, N. Car., 1983], p. 37.

Lukosz, W.

Mendlovic, D.

Onural, L.

Ozaktas, H. M.

Saleh, B. E. A.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

Teich, M. C.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

Toraldo Di Francia, G.

Zalevsky, Z.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Nuovo Cimento Suppl.

M. Francon, “Amélioration de résolution d’optique,” Nuovo Cimento Suppl. 9, 283–290 (1952).

Opt. Commun.

H. Bartelt, A. W. Lohmann, “Optical processing of 1-D signals,” Opt. Commun. 42, 87–91 (1982).
[CrossRef]

Opt. Spectrosc. (USSR)

A. I. Kartashev, “Optical systems with enhanced resolving power,” Opt. Spectrosc. (USSR) 9, 204–206 (1960).

Prog. Opt.

J. W. Goodman, “Synthetic aperture optics,” Prog. Opt. 8, 1–50 (1970).
[CrossRef]

Z. Phys.

W. Gartner, A. W. Lohmann, “An experiment going beyond Abbe’s limit of diffraction,” Z. Phys. 174, 18–24 (1963).

Other

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), Chap. 7.

Ref. 1, Chap. 2.

A. W. Lohmann, “Utilization of a priori information for getting better image,” in Proceedings of the Conference on Synthetic Aperture Optics (National Academy of Sciences–National Research Council, Washington, D.C., 1967), App. 33.

A. W. Lohmann, “Some thoughts on super resolution,” in Proceedings of Workshop on Limits of Passive Imaging, C. Aleksoff, ed. [R. Guenther (ARO), Raleigh, N. Car., 1983], p. 37.

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Figures (5)

Fig. 1
Fig. 1

Influence of some basic optical systems on the Wigner chart: (a) original object, (b) lens, (c) free-space propagation (FSP), (d) Fourier transform (FT), (e) fractional Fourier transform (FRT), (f) magnification.

Fig. 2
Fig. 2

Wigner chart and its projections for SW estimation.

Fig. 3
Fig. 3

Demonstration of the definition of the SWY chart for a spatial/spectral limited system.

Fig. 4
Fig. 4

Number of degrees of freedom and the SW chart.

Fig. 5
Fig. 5

Block diagram of the SW adaptation process with an example for demonstration.

Tables (1)

Tables Icon

Table 1 SW Behavior with Respect to the Several Common Optical Operations

Equations (19)

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δxRESλf#=λfB,
SW=ΔxΔν.
W(x, ν)=- fx+x2f*x-x2exp(-i2πνx)dx.
SWB(x, ν)=1W(x, ν)>Wthresh0otherwise.
SW(x, ν)=STSWB(x, ν),
Total energy=SWB(x, ν)W(x, ν)dxdν=SW(x, ν)dxdν.
ST=SWB(x, ν)W(x, ν)dxdνSWB(x, ν)dxdν.
NOUTNIN.
NOUT=NIN.
SWI(x, ν)SWY(x, ν).
[area(SWI)=] NsignalNsystem [=area(SWY)].
SWI(x, ν)SWY(x, ν).
NsignalNsystem,SWI(x, ν)SWY(x, ν).
SW(x, νx, y, νy, t, λ, POL, ).
ϕ=π2P.
f(x)=1f*(0)W(x/2, ν)exp(i2πνx)dν.
W(x, ν)dxdν=SW(x, ν)dxdν.
W(x, ν)dν=|f(x)|2,
SWIN(x, ν)dxdν=SWOUT(x, ν)dxdν.

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