Abstract

Although different methods have been proposed for realizing bipolar point spread functions in hybrid two-channel incoherent spatial filtering systems, each method relies either on interference or noninterference between the two channels to achieve the desired synthesis. It is shown that the pupil functions appropriate for synthesis in these two regimes are related through a linear transformation. Furthermore, the pupil functions appropriate for a noninteractive synthesis can be used to specify pupil functions appropriate for any other synthesis.

© 1986 Optical Society of America

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References

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  1. P. Chavel, “A Method of Linear Image Processing with Synthetic Holograms using Spatially Incoherent Light,” in 1975 International Optical Conference (IEEE, New York, 1975), pp. 145–149.
  2. P. Chavel, S. Lowenthal, “Incoherent Optical-Image Processing Using Synthetic Holograms,” J. Opt. Soc. Am. 66, 14 (1976).
    [CrossRef]
  3. D. Gorlitz, F. Lanzl, “Methods of Zero-Order Non-Coherent Filtering,” Opt. Commun. 20, 68 (1977).
    [CrossRef]
  4. B. Braunecker, R. Hauck, “Grey Level on Axis Computer Holograms for Incoherent Image Processing,” Opt. Commun. 20, 234 (1977).
    [CrossRef]
  5. W. T. Rhodes, “Noncoherent Optical Processing with Two-Pupil Hybrid Systems,” Proc. Soc. Photo-Opt. Instrum. Eng. in Optical Signal and Image Processing, D. Casasent, ed., 118, 86 (1977).
  6. P. Chavel, S. Lowenthal, “Implementation of an Incoherent Optical Image Restoration Method: Limitations Related to Optical Subtraction,” Appl. Opt. 20, 1438 (1981).
    [CrossRef] [PubMed]
  7. A. Kozma, N. Massey, “Bias Level Reduction of Incoherent Holograms,” Appl. Opt. 8, 393 (1969).
    [CrossRef] [PubMed]
  8. W. T. Rhodes, “Temporal Frequency Carriers in Noncoherent Optical Processing,” in 1978 International Optical Computing Conference (IEEE, New York, 1978), pp. 163–168.
  9. A. W. Lohmann, “Incoherent Optical Processing of Complex Data,” Appl. Opt. 16, 261 (1977).
    [CrossRef] [PubMed]
  10. W. Stoner, “Edge Enhancement with Incoherent Optics,” Appl. Opt. 16, 1451 (1977).
    [CrossRef] [PubMed]
  11. W. Stoner, “Incoherent Optical Processing via Spatially Offset Pupil Masks,” Appl. Opt. 17, 2454 (1978).
    [CrossRef] [PubMed]
  12. W. T. Rhodes, “Phase-Switching Synthesis of Arbitrary Optical Transfer Functions,” in 1976 International Optical Computing Conference (IEEE, New York, 1976), pp. 62–64.
  13. W. T. Rhodes, “Bipolar Pointspread Function Synthesis by Phase Switching,” Appl. Opt. 16, 265 (1977).
    [CrossRef] [PubMed]
  14. A. W. Lohmann, W. T. Rhodes, “Two-Pupil Synthesis of Optical Transfer Functions,” Appl. Opt. 17, 1141 (1978).
    [CrossRef] [PubMed]
  15. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  16. B. Braunecker, R. Hauck, A. W. Lohmann, “Optical Character Recognition Based on Nonredundant Correlation Measurements,” Appl. Opt. 18, 2746 (1979).
    [CrossRef] [PubMed]
  17. D. Gorlitz, F. Lanzl, “Colour Encoded Aperture Masks used for Incoherent Filtering of Images,” Opt. Commun. 28, 283 (1979).
    [CrossRef]
  18. J. N. Mait, “Pupil Function Optimization for Bipolar Incoherent Spatial Filtering,” Ph.D. Dissertation Georgia Institute of Technology, Atlanta (June1985).

1981 (1)

1979 (2)

B. Braunecker, R. Hauck, A. W. Lohmann, “Optical Character Recognition Based on Nonredundant Correlation Measurements,” Appl. Opt. 18, 2746 (1979).
[CrossRef] [PubMed]

D. Gorlitz, F. Lanzl, “Colour Encoded Aperture Masks used for Incoherent Filtering of Images,” Opt. Commun. 28, 283 (1979).
[CrossRef]

1978 (2)

1977 (5)

1976 (1)

1969 (1)

Braunecker, B.

B. Braunecker, R. Hauck, A. W. Lohmann, “Optical Character Recognition Based on Nonredundant Correlation Measurements,” Appl. Opt. 18, 2746 (1979).
[CrossRef] [PubMed]

B. Braunecker, R. Hauck, “Grey Level on Axis Computer Holograms for Incoherent Image Processing,” Opt. Commun. 20, 234 (1977).
[CrossRef]

Chavel, P.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Gorlitz, D.

D. Gorlitz, F. Lanzl, “Colour Encoded Aperture Masks used for Incoherent Filtering of Images,” Opt. Commun. 28, 283 (1979).
[CrossRef]

D. Gorlitz, F. Lanzl, “Methods of Zero-Order Non-Coherent Filtering,” Opt. Commun. 20, 68 (1977).
[CrossRef]

Hauck, R.

B. Braunecker, R. Hauck, A. W. Lohmann, “Optical Character Recognition Based on Nonredundant Correlation Measurements,” Appl. Opt. 18, 2746 (1979).
[CrossRef] [PubMed]

B. Braunecker, R. Hauck, “Grey Level on Axis Computer Holograms for Incoherent Image Processing,” Opt. Commun. 20, 234 (1977).
[CrossRef]

Kozma, A.

Lanzl, F.

D. Gorlitz, F. Lanzl, “Colour Encoded Aperture Masks used for Incoherent Filtering of Images,” Opt. Commun. 28, 283 (1979).
[CrossRef]

D. Gorlitz, F. Lanzl, “Methods of Zero-Order Non-Coherent Filtering,” Opt. Commun. 20, 68 (1977).
[CrossRef]

Lohmann, A. W.

Lowenthal, S.

Mait, J. N.

J. N. Mait, “Pupil Function Optimization for Bipolar Incoherent Spatial Filtering,” Ph.D. Dissertation Georgia Institute of Technology, Atlanta (June1985).

Massey, N.

Rhodes, W. T.

A. W. Lohmann, W. T. Rhodes, “Two-Pupil Synthesis of Optical Transfer Functions,” Appl. Opt. 17, 1141 (1978).
[CrossRef] [PubMed]

W. T. Rhodes, “Bipolar Pointspread Function Synthesis by Phase Switching,” Appl. Opt. 16, 265 (1977).
[CrossRef] [PubMed]

W. T. Rhodes, “Phase-Switching Synthesis of Arbitrary Optical Transfer Functions,” in 1976 International Optical Computing Conference (IEEE, New York, 1976), pp. 62–64.

W. T. Rhodes, “Temporal Frequency Carriers in Noncoherent Optical Processing,” in 1978 International Optical Computing Conference (IEEE, New York, 1978), pp. 163–168.

W. T. Rhodes, “Noncoherent Optical Processing with Two-Pupil Hybrid Systems,” Proc. Soc. Photo-Opt. Instrum. Eng. in Optical Signal and Image Processing, D. Casasent, ed., 118, 86 (1977).

Stoner, W.

Appl. Opt. (8)

J. Opt. Soc. Am. (1)

Opt. Commun. (3)

D. Gorlitz, F. Lanzl, “Methods of Zero-Order Non-Coherent Filtering,” Opt. Commun. 20, 68 (1977).
[CrossRef]

B. Braunecker, R. Hauck, “Grey Level on Axis Computer Holograms for Incoherent Image Processing,” Opt. Commun. 20, 234 (1977).
[CrossRef]

D. Gorlitz, F. Lanzl, “Colour Encoded Aperture Masks used for Incoherent Filtering of Images,” Opt. Commun. 28, 283 (1979).
[CrossRef]

Other (6)

J. N. Mait, “Pupil Function Optimization for Bipolar Incoherent Spatial Filtering,” Ph.D. Dissertation Georgia Institute of Technology, Atlanta (June1985).

P. Chavel, “A Method of Linear Image Processing with Synthetic Holograms using Spatially Incoherent Light,” in 1975 International Optical Conference (IEEE, New York, 1975), pp. 145–149.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

W. T. Rhodes, “Temporal Frequency Carriers in Noncoherent Optical Processing,” in 1978 International Optical Computing Conference (IEEE, New York, 1978), pp. 163–168.

W. T. Rhodes, “Noncoherent Optical Processing with Two-Pupil Hybrid Systems,” Proc. Soc. Photo-Opt. Instrum. Eng. in Optical Signal and Image Processing, D. Casasent, ed., 118, 86 (1977).

W. T. Rhodes, “Phase-Switching Synthesis of Arbitrary Optical Transfer Functions,” in 1976 International Optical Computing Conference (IEEE, New York, 1976), pp. 62–64.

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Figures (5)

Fig. 1
Fig. 1

Single-channel incoherent spatial filtering system: O, object plane; P, pupil plane; I, image plane. The focal length of the lens is f, and the distance between the planes is 2f.

Fig. 2
Fig. 2

Two-channel incoherent spatial system: P1 and P2, pupil functions; A1 and A2, attentuations; ϕ, phase shift; BS, beam splitter; M, mirror (from Ref. 14).

Fig. 3
Fig. 3

Significance of the phase constant δ.

Fig. 4
Fig. 4

(a) Bipolar PSF f(x) and its associated OTF F(u).

Fig. 5
Fig. 5

Noninteractive and interactive pupil functions for the same PSF: (a) noninteractive pupil functions P+(u) and P(u); (b) interactive pupil functions obtained from Eq. (23) with δ = 0; (c) interactive pupil functions obtained from Eq. (23) with δ = π/2. Positive cross-hatching indicates a phase of π/4 rad; negative cross-hatching, −π/4 rad. (d) Interactive pupil functions obtained by adding π/4 rad to the phase of the pupil functions in (c). Striping indicates a phase of π/2 rad.

Equations (41)

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i ( x ) = c - o ( ζ ) p ( x - ζ ) 2 d ζ = c o ( x ) * p * ( x ) 2 ,
f ( x ) = p ( x ) 2 .
F ( u ) = F T { p ( x ) 2 } = - P ( ζ ) P * ( ζ - u ) d ζ , = P ( u ) P ( u ) ,
P ( u ; A 1 , A 2 , ϕ ) = A 1 P 1 ( u ) exp ( j ϕ ) + A 2 P 2 ( u ) ,
f ( x ; A 1 , A 2 , ϕ ) = p ( x ; A 1 , A 2 , ϕ ) 2 = A 1 2 p 1 ( x ) 2 + A 2 2 p 2 ( x ) 2 + 2 A 1 A 2 p 1 ( x ) p 2 ( x ) cos [ θ 1 ( x ) - θ 2 ( x ) + ϕ ] ,
f s ( x ) = f ( x ; A 1 + , A 2 + , ϕ + ) - f ( x ; A 1 - , A 2 - , ϕ - ) = ( A 1 + 2 - A 1 - 2 ) p 1 ( x ) 2 + ( A 2 + 2 - A 2 - 2 ) p 2 ( x ) 2 + 2 A 1 + A 2 + p 1 ( x ) p 2 ( x ) cos [ θ 1 ( x ) - θ 2 ( x ) + ϕ + ] - 2 A 1 - A 2 - p 1 ( x ) p 2 ( x ) cos [ θ 1 ( x ) - θ 2 ( x ) + ϕ - ] .
f s ( x ) = f ( x ; A 1 + , 0 , ϕ ) - f ( x ; 0 , A 2 - , ϕ ) , = A 1 + 2 p 1 ( x ) 2 - A 2 - 2 p 2 ( x ) 2 .
f s ( x ) = f ( x ; A 1 , A 2 , ϕ + ) - f ( x ; A 1 , A 2 , ϕ - ) , = 2 A 1 A 2 p 1 ( x ) p 2 ( x ) { cos [ ϕ 1 ( x ) - ϕ 2 ( x ) + ϕ + ] - cos [ ϕ 1 ( x ) - ϕ 2 ( x ) + ϕ - ] } .
f c ( x ; t ) = f [ x ; A 1 , A 2 , ϕ ( t ) ] .
f c ( x ; t ) = A 1 2 p 1 ( x ) 2 + A 2 2 p 2 ( x ) 2 + 2 A 1 A 2 p 1 ( x ) p 2 ( x ) cos [ 2 π ν c t + θ 1 ( x ) - θ 2 ( x ) ] .
f c ( x ; t ) = f [ x ; A 1 ( t ) , A 2 ( t ) , ϕ ] .
A 1 ( t ) A 2 ( t ) = 0.
A 1 ( t ) = ( 1 / 2 ) [ 1 + s w ( t ) ] ,
A 2 ( t ) = ( 1 / 2 ) [ 1 - s w ( t ) ] ,
f c ( x ; t ) = ( 1 / 2 ) [ p 1 ( x ) 2 + p 2 ( x ) 2 ] + ( 1 / 2 ) [ p 1 ( x ) 2 - p 2 ( x ) 2 ] s w ( t ) .
f c ( x ) = f [ x ; A 1 , A 2 , ϕ ( x ) ] = A 1 2 p 1 ( x ) 2 + A 2 2 p 2 ( x ) 2 + 2 A 1 A 2 p 1 ( x ) p 2 ( x ) cos [ 2 π u c x + θ 1 ( x ) - θ 2 ( x ) ] .
f s ( x ) = f + ( x ) - f - ( x ) ,
f + ( x ) = f ( x ; A + , A 2 + , ϕ + ) = p + ( x ) 2 ,
f - ( x ) = f ( x ; A 1 - , A 2 - , ϕ - ) = p - ( x ) 2 .
p + ( x ) = A 1 + p 1 ( x ) exp ( j ϕ + ) + A 2 + p 2 ( x ) ,
p - ( x ) = A 1 - p 1 ( x ) exp ( j ϕ - ) + A 2 - p 2 ( x ) .
p 1 n ( x ) = p + ( x ) ,
p 2 n ( x ) = p - ( x ) ,
p + ( x ) = A 1 + p 1 ( x ) exp ( j ϕ + ) + A 2 + p 2 ( x ) ,
p - ( x ) exp ( - j δ ) = A 1 - p 1 ( x ) exp ( j ϕ - ) + A 2 + p 2 ( x ) .
p 1 ( x ) = A 2 - p + ( x ) - A 2 - p - ( x ) exp ( - j δ ) A 1 + A 2 - exp ( j ϕ + ) - A 2 + A 1 - exp ( j ϕ - ) ,
p 2 ( x ) = - A 1 p + ( x ) exp ( j ϕ - ) + A 1 + p - ( x ) exp ( - j δ ) exp ( j ϕ + ) A 1 + A 2 - exp ( j ϕ + ) - A 2 + A 1 - exp ( j ϕ - ) .
f s ( x ) = γ 1 p 1 ( x ) 2 + γ 2 p 2 ( x ) 2 + α Re [ p 1 ( x ) p 2 * ( x ) ] + β Im [ p 1 ( x ) p 2 * ( x ) ] ,
γ 1 = A 1 + 2 - A 1 - 2 ,
γ 2 = A 2 + 2 - A 2 - 2 ,
α = 2 A 1 + A 2 + cos ϕ + - 2 A 1 - A 2 - cos ϕ - ,
β = - 2 A 1 + A 2 + sin ϕ + + 2 A 1 - A 2 - sin ϕ - .
p 1 i ( x ) = p + ( x ) - p - ( x ) exp ( - j δ ) 2 A 1 sin ( ϕ + - ϕ 1 2 ) × exp [ - j ( ϕ + + ϕ - - π 2 ) ] ;
p 2 i ( x ) = - p + ( x ) exp ( j ϕ - ) + p - ( x ) exp ( - j δ ) exp ( j ϕ + ) 2 A 2 sin ( ϕ + - ϕ - 2 ) × exp [ - j ( ϕ + + ϕ - - π 2 ) ] .
p 1 i ( x ) = [ p + ( x ) - p - ( x ) exp ( - j δ ) ] exp ( - j ϕ + ) ;
p 2 i ( x ) = p + ( x ) + p - ( x ) exp ( - j δ ) .
f s ( x ) = Re [ p 1 i ( x ) p 2 i * ( x ) exp ( j ϕ + ) ] .
f s ( x ) = cos ϕ + Re [ p 1 i ( x ) p 2 i * ( x ) ] - sin ϕ + Im [ p 1 i ( x ) p 2 i * ( x ) ] = α f ( x ) - 2 β Im [ p + ( x ) p - * ( x ) exp ( j δ ) ] .
δ = - - [ θ + ( x ) - θ - ( x ) ] d x = - θ ave ,
f + ( x ) = ( κ / 2 ) sinc 2 ( u 1 x ) { 1 + 2 cos ( 2 π u 0 x ) + ( 1 / 2 ) cos [ 2 π ( 2 u 0 ) x ] } , f - ( x ) = ( κ / 2 ) sinc 2 ( u 1 x ) { 1 - 2 cos ( 2 π u 0 x ) + ( 1 / 2 ) cos [ 2 π ( 2 u 0 ) x ] } ,
p + ( x ) = ( κ / 8 ) 1 / 2 sin c ( u 1 x ) [ 1 + 2 cos ( 2 π u 0 x ) ] , p - ( x ) = ( κ / 8 ) 1 / 2 sinc ( u 1 x ) [ 1 - 2 cos ( 2 π u 0 x ) ] .

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