Abstract

We designed an optical–digital system that delivers near-diffraction-limited imaging performance with a large depth of field. This system is the standard incoherent optical system modified by a phase mask with digital processing of the resulting intermediate image. The phase mask alters or codes the received incoherent wave front in such a way that the point-spread function and the optical transfer function do not change appreciably as a function of misfocus. Focus-independent digital filtering of the intermediate image is used to produce a combined optical–digital system that has a nearly diffraction limited point-spread function. This high-resolution extended depth of field is obtained through the expense of an increased dynamic range of the incoherent system. We use both the ambiguity function and the stationary-phase method to design these phase masks.

© 1995 Optical Society of America

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References

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  1. M. Mino, Y. Okano, “Improvement in the optical transfer function of a defocused optical system through the use of shaded apertures,” Appl. Opt. 10, 2219–2225 (1971).
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  3. J. Ojeda-Castaneda, R. Ramos, A. Noyola-Isgleas, “High focal depth by apodization and digital restoration,” Appl. Opt. 27, 2583–2586 (1988).
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  4. J. Ojeda-Castaneda, E. Tepichin, A. Diaz, “Arbitrary high focal depth with a quasioptimum real and positive transmittance apodizer,” Appl. Opt. 28, 2666–2670 (1989).
    [CrossRef] [PubMed]
  5. J. Ojeda-Castaneda, L. R. Berriel-Valdos, “Zone plate for arbitrarily high focal depth,” Appl. Opt. 29, 994–997 (1990).
    [CrossRef] [PubMed]
  6. G. Häusler, “A method to increase the depth of focus by two step image processing,” Opt. Commun. 6, 38–42 (1972).
    [CrossRef]
  7. W. T. Cathey, B. R. Frieden, W. T. Rhodes, C. K. Rushforth, “Image gathering and processing for enhanced resolution,” Opt. Soc. Am. A 1, 241–250 (1984).
    [CrossRef]
  8. C. E. Cook, M. Bernfeld, Radar Signals (Academic, New York, 1967), Chap. 4, pp. 59–108.
  9. A. W. Rihaczek, Principles of High Resolution Radar (McGraw-Hill, New York, 1969), Chaps. 4 and 5, pp. 87–157.
  10. K. Brenner, A. Lohmann, J. O. Casteneda, “The ambiguity function as a polar display of the OFT,” Opt. Commun. 44, 323–326 (1983).
    [CrossRef]
  11. E. L. Key, E. N. Fowle, R. D. Haggarty, “A method of designing signals of large time–bandwidth product,” IRE Int. Conv. Rec. 4, 146–155 (1961).
  12. E. N. Fowle, “The design of FM pulse compression signals,” IEEE Trans. Inf. Theory IT-10, 61–67 (1964).
    [CrossRef]
  13. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1989), App. III, pp. 747–754.
  14. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 6, pp. 101–140.
  15. W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974), Chap. 8, pp. 219–255.
  16. H. Bartelt, J. O. Casteneda, E. S. Enrique, “Misfocus tolerance seen by simple inspection of the ambiguity function,” Appl. Opt. 23, 2693–2696 (1984).
    [CrossRef] [PubMed]
  17. T. M. Cover, J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991), Chap. 12, 279–335.
    [CrossRef]
  18. L. L. Scharf, Statistical Signal Processing (Addison-Wesley, Reading, Mass.1991), Chap. 6, pp. 209–276.

1990 (1)

1989 (1)

1988 (1)

1986 (1)

1984 (2)

H. Bartelt, J. O. Casteneda, E. S. Enrique, “Misfocus tolerance seen by simple inspection of the ambiguity function,” Appl. Opt. 23, 2693–2696 (1984).
[CrossRef] [PubMed]

W. T. Cathey, B. R. Frieden, W. T. Rhodes, C. K. Rushforth, “Image gathering and processing for enhanced resolution,” Opt. Soc. Am. A 1, 241–250 (1984).
[CrossRef]

1983 (1)

K. Brenner, A. Lohmann, J. O. Casteneda, “The ambiguity function as a polar display of the OFT,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

1972 (1)

G. Häusler, “A method to increase the depth of focus by two step image processing,” Opt. Commun. 6, 38–42 (1972).
[CrossRef]

1971 (1)

1964 (1)

E. N. Fowle, “The design of FM pulse compression signals,” IEEE Trans. Inf. Theory IT-10, 61–67 (1964).
[CrossRef]

1961 (1)

E. L. Key, E. N. Fowle, R. D. Haggarty, “A method of designing signals of large time–bandwidth product,” IRE Int. Conv. Rec. 4, 146–155 (1961).

Andres, P.

Bartelt, H.

Bernfeld, M.

C. E. Cook, M. Bernfeld, Radar Signals (Academic, New York, 1967), Chap. 4, pp. 59–108.

Berriel-Valdos, L. R.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1989), App. III, pp. 747–754.

Brenner, K.

K. Brenner, A. Lohmann, J. O. Casteneda, “The ambiguity function as a polar display of the OFT,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Casteneda, J. O.

H. Bartelt, J. O. Casteneda, E. S. Enrique, “Misfocus tolerance seen by simple inspection of the ambiguity function,” Appl. Opt. 23, 2693–2696 (1984).
[CrossRef] [PubMed]

K. Brenner, A. Lohmann, J. O. Casteneda, “The ambiguity function as a polar display of the OFT,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Cathey, W. T.

W. T. Cathey, B. R. Frieden, W. T. Rhodes, C. K. Rushforth, “Image gathering and processing for enhanced resolution,” Opt. Soc. Am. A 1, 241–250 (1984).
[CrossRef]

W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974), Chap. 8, pp. 219–255.

Cook, C. E.

C. E. Cook, M. Bernfeld, Radar Signals (Academic, New York, 1967), Chap. 4, pp. 59–108.

Cover, T. M.

T. M. Cover, J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991), Chap. 12, 279–335.
[CrossRef]

Diaz, A.

Enrique, E. S.

Fowle, E. N.

E. N. Fowle, “The design of FM pulse compression signals,” IEEE Trans. Inf. Theory IT-10, 61–67 (1964).
[CrossRef]

E. L. Key, E. N. Fowle, R. D. Haggarty, “A method of designing signals of large time–bandwidth product,” IRE Int. Conv. Rec. 4, 146–155 (1961).

Frieden, B. R.

W. T. Cathey, B. R. Frieden, W. T. Rhodes, C. K. Rushforth, “Image gathering and processing for enhanced resolution,” Opt. Soc. Am. A 1, 241–250 (1984).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 6, pp. 101–140.

Haggarty, R. D.

E. L. Key, E. N. Fowle, R. D. Haggarty, “A method of designing signals of large time–bandwidth product,” IRE Int. Conv. Rec. 4, 146–155 (1961).

Häusler, G.

G. Häusler, “A method to increase the depth of focus by two step image processing,” Opt. Commun. 6, 38–42 (1972).
[CrossRef]

Key, E. L.

E. L. Key, E. N. Fowle, R. D. Haggarty, “A method of designing signals of large time–bandwidth product,” IRE Int. Conv. Rec. 4, 146–155 (1961).

Lohmann, A.

K. Brenner, A. Lohmann, J. O. Casteneda, “The ambiguity function as a polar display of the OFT,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Mino, M.

Noyola-Isgleas, A.

Ojeda-Castaneda, J.

Okano, Y.

Ramos, R.

Rhodes, W. T.

W. T. Cathey, B. R. Frieden, W. T. Rhodes, C. K. Rushforth, “Image gathering and processing for enhanced resolution,” Opt. Soc. Am. A 1, 241–250 (1984).
[CrossRef]

Rihaczek, A. W.

A. W. Rihaczek, Principles of High Resolution Radar (McGraw-Hill, New York, 1969), Chaps. 4 and 5, pp. 87–157.

Rushforth, C. K.

W. T. Cathey, B. R. Frieden, W. T. Rhodes, C. K. Rushforth, “Image gathering and processing for enhanced resolution,” Opt. Soc. Am. A 1, 241–250 (1984).
[CrossRef]

Scharf, L. L.

L. L. Scharf, Statistical Signal Processing (Addison-Wesley, Reading, Mass.1991), Chap. 6, pp. 209–276.

Tepichin, E.

Thomas, J. A.

T. M. Cover, J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991), Chap. 12, 279–335.
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1989), App. III, pp. 747–754.

Appl. Opt. (5)

IEEE Trans. Inf. Theory (1)

E. N. Fowle, “The design of FM pulse compression signals,” IEEE Trans. Inf. Theory IT-10, 61–67 (1964).
[CrossRef]

IRE Int. Conv. Rec. (1)

E. L. Key, E. N. Fowle, R. D. Haggarty, “A method of designing signals of large time–bandwidth product,” IRE Int. Conv. Rec. 4, 146–155 (1961).

Opt. Commun. (2)

K. Brenner, A. Lohmann, J. O. Casteneda, “The ambiguity function as a polar display of the OFT,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

G. Häusler, “A method to increase the depth of focus by two step image processing,” Opt. Commun. 6, 38–42 (1972).
[CrossRef]

Opt. Lett. (1)

Opt. Soc. Am. A (1)

W. T. Cathey, B. R. Frieden, W. T. Rhodes, C. K. Rushforth, “Image gathering and processing for enhanced resolution,” Opt. Soc. Am. A 1, 241–250 (1984).
[CrossRef]

Other (7)

C. E. Cook, M. Bernfeld, Radar Signals (Academic, New York, 1967), Chap. 4, pp. 59–108.

A. W. Rihaczek, Principles of High Resolution Radar (McGraw-Hill, New York, 1969), Chaps. 4 and 5, pp. 87–157.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1989), App. III, pp. 747–754.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 6, pp. 101–140.

W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974), Chap. 8, pp. 219–255.

T. M. Cover, J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991), Chap. 12, 279–335.
[CrossRef]

L. L. Scharf, Statistical Signal Processing (Addison-Wesley, Reading, Mass.1991), Chap. 6, pp. 209–276.

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Figures (10)

Fig. 1
Fig. 1

Ambiguity function of a rectangular aperture. The radial line has a slope of π/2, corresponding to an OTF with misfocus ψ = π2/2.

Fig. 2
Fig. 2

Misfocus OTF of the standard optical system with misfocus parameter ψ = π2/2.

Fig. 3
Fig. 3

Magnitude of the ambiguity function of the cubic-pm function with α = 90. Radial lines through this function are insensitive to angle for a broad range of angles.

Fig. 4
Fig. 4

Magnitude of the OTF of the cubic-pm system with α = 90 and misfocus ψ = 15. The smooth curve is the stationary-phase approximation of the OTF. The other curve is the calculated OTF.

Fig. 5
Fig. 5

Magnitude of the OTF’s from the cubic-pm system with α = 90 and misfocus ψ of 0, 15, and 30.

Fig. 6
Fig. 6

Magnitude of OTF’s from the standard optical system. The solid curve denotes the OTF with misfocus ψ of 0, the dashed curve is for ψ of 15, and the dashed–dotted curve is for ψ of 30. The vertical scale is different than that of Fig. 5.

Fig. 7
Fig. 7

Normalized full width at half-maximum amplitude(FWHM) of the PSF of the cubic-pm optical–digital system with comparison to that of the standard optical system.

Fig. 8
Fig. 8

Simulated images of a spoke target from a standard optical system (first column) and a cubic-pm optical–digital system (second column). (a), (b) (Geometrically in focus; (c), (d) mild misfocus; (e), (f) extreme misfocus.

Fig. 9
Fig. 9

Magnitude of the digital filter transfer function used in simulations of the cubic-pm optical–digital system.

Fig. 10
Fig. 10

Ratio of the Fisher information of misfocus, assuming a point object, of the standard optical system over the Fisher information of misfocus for the cubic-pm optical–digital system.

Equations (24)

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P ( x ) = { 1 2 exp [ j θ ( x ) ] for x 1 0 otherwise ,
H ( u , ψ ) = { P ( x + u / 2 ) exp [ j ( x + u / 2 ) 2 ψ ] } × { P * ( x - u / 2 ) exp [ - j ( x - u / 2 ) 2 ψ ] } d x
ψ = π L 2 4 λ ( 1 f - 1 d o - 1 d i ) = 2 π λ W 20 = k W 20 ,
A ( u , v ) = P ( x + u / 2 ) P * ( x - u / 2 ) exp ( j 2 π v x ) d x .
H ( u , ψ ) = A ( u , u ψ / π ) .
P ( x ) = { 1 2 exp ( j α x 3 ) for x 1 0 otherwise             α 20 ,
H ( u , ψ ) { ( π 12 α u ) 1 / 2 exp ( j α u 3 4 ) α 20 u 0 1 u = 0 .
J ( ψ ) = | ψ H ( u , ψ ) | 2 d u ,
A ( u , v ) = 1 2 - ( 1 - u / 2 ) ( 1 - u / 2 ) exp [ j θ ( x + u / 2 ) ] × exp [ - j θ ( x - u / 2 ) ] exp ( j 2 π v x ) d x ,             u 2 .
θ ( x ) = α x γ ,             γ { 0 , 1 } ,             α 0.
A ( u , v ) = 1 2 - ( 1 - u / 2 ) ( 1 - u / 2 ) exp [ j α ( x + u / 2 ) γ ] × exp [ - j α ( x - u / 2 ) γ ] exp ( j 2 π v x ) d x ,             u 2 = 1 2 - ( 1 ( u / 2 ) ( 1 - u / 2 ) exp [ j ϑ ( x ) ] exp ( j 2 π v x ) d x ,             u 2 ;
ϑ ( x ) = α [ ( x + u / 2 ) γ - ( x - u / 2 ) γ ] .
A ( u , v ) 1 2 [ 2 π ϑ ( x i ) ] 1 / 2 exp [ j ϕ ( v ) ] = 1 2 ( | x i v | ) 1 / 2 exp [ j ϕ ( v ) ] ,
ϕ ( v ) = 2 π v x i + ϑ ( x i ) .
( / x i ) [ 2 π v x i + ϑ ( x i ) ] = 0 , 2 π v + γ α ( x i + u / 2 ) γ - 1 - γ α ( x i - u / 2 ) γ - 1 = 0.
x i = - π v 3 α u ,             u 0.
A ( u , v ) 1 2 ( | x i v | ) 1 / 2 = ( π 12 α u ) 1 / 2 ,             u 0.
ϕ ( v ) α u 3 4 - π 2 v 2 3 α u ,             u 0.
A ( u , v ) ( π 12 α u ) 1 / 2 exp ( j α u 3 4 ) exp ( - j π 2 v 2 3 α u ) ,             u 0.
H ( u , ψ ) ( π 12 α u ) 1 / 2 exp ( j α u 3 4 ) exp ( - j ψ 2 u 3 α ) ,             u 0.
H ( u , ψ ) ( π 12 α u ) 1 / 2 exp ( j α u 3 4 ) for large α ,             u 0.
BW = max x x θ ( x ) = max x x α x 3 = 3 α .
SBP = 2 ( 3 α ) = 6 α 100
α 20.

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