Abstract

A procedure for designing pupil functions for four-channel, three-channel, and two-channel syntheses of complex incoherent point-spread functions is presented. This procedure is then generalized for synthesis of multiple bipolar point-spread functions. Examples of pupil-function design for complex syntheses are also presented.

© 1987 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. W. Lohmann, W. T. Rhodes, “Two-pupil synthesis of optical transfer functions,” Appl. Opt. 17, 1141–1151 (1978).
    [CrossRef] [PubMed]
  2. W. T. Rhodes, A. A. Sawchuk, “Incoherent optical processing,” in Optical Information Processing, S. H. Lee, ed., Vol. 48 of Topics in Applied Physics (Springer-Verlag, New York, 1981), pp. 69–110.
    [CrossRef]
  3. A. W. Lohmann, “Incoherent optical processing of complex data,” Appl. Opt. 16, 261–263 (1977).
    [CrossRef] [PubMed]
  4. I. Glaser, “Representing bipolar and complex imagery in noncoherent optical image processing systems: comparison of approaches,” Opt. Eng. 20, 568–573 (1981).
    [CrossRef]
  5. A. W. Lohmann, Ch. Thum, “Two-way translation by computer-generated holographic filters,” Opt. Commun. 46, 74–78 (1983).
    [CrossRef]
  6. J. N. Mait, “Pupil-function design for bipolar incoherent spatial filtering,” J. Opt. Soc. Am. A 3, 1826–1832 (1986).
    [CrossRef]
  7. J. M. Mait, W. T. Rhodes, “Two-pupil synthesis of optical transfer functions: 2. Pupil function relationships,” Appl. Opt. 25, 2003–2007 (1986).
    [CrossRef] [PubMed]
  8. J. N. Mait, “Existence conditions for two-pupil synthesis of bipolar incoherent point-spread functions,” J. Opt. Soc. Am. A 3, 437–445 (1986).
    [CrossRef]
  9. J. N. Mait, “Pupil function optimization for bipolar incoherent spatial filtering,” Ph.D. dissertation (Georgia Institute of Technology, Atlanta, Ga., 1985).
  10. A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
    [CrossRef]
  11. G. Indebetouw, T.-C. Poon, “Parallel synthesis of bipolar point spread functions in a scanning heterodyne optical system,” Opt. Acta 33, 827–834 (1986).
    [CrossRef]
  12. D. Gorlitz, F. Lanzl, “Colour encoded aperture masks used for incoherent filtering of images,” Opt. Commun. 28, 283–286 (1979).
    [CrossRef]

1986 (4)

1983 (1)

A. W. Lohmann, Ch. Thum, “Two-way translation by computer-generated holographic filters,” Opt. Commun. 46, 74–78 (1983).
[CrossRef]

1981 (1)

I. Glaser, “Representing bipolar and complex imagery in noncoherent optical image processing systems: comparison of approaches,” Opt. Eng. 20, 568–573 (1981).
[CrossRef]

1979 (1)

D. Gorlitz, F. Lanzl, “Colour encoded aperture masks used for incoherent filtering of images,” Opt. Commun. 28, 283–286 (1979).
[CrossRef]

1978 (1)

1977 (1)

1963 (1)

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
[CrossRef]

Glaser, I.

I. Glaser, “Representing bipolar and complex imagery in noncoherent optical image processing systems: comparison of approaches,” Opt. Eng. 20, 568–573 (1981).
[CrossRef]

Gorlitz, D.

D. Gorlitz, F. Lanzl, “Colour encoded aperture masks used for incoherent filtering of images,” Opt. Commun. 28, 283–286 (1979).
[CrossRef]

Indebetouw, G.

G. Indebetouw, T.-C. Poon, “Parallel synthesis of bipolar point spread functions in a scanning heterodyne optical system,” Opt. Acta 33, 827–834 (1986).
[CrossRef]

Lanzl, F.

D. Gorlitz, F. Lanzl, “Colour encoded aperture masks used for incoherent filtering of images,” Opt. Commun. 28, 283–286 (1979).
[CrossRef]

Lohmann, A. W.

Mait, J. M.

Mait, J. N.

Poon, T.-C.

G. Indebetouw, T.-C. Poon, “Parallel synthesis of bipolar point spread functions in a scanning heterodyne optical system,” Opt. Acta 33, 827–834 (1986).
[CrossRef]

Rhodes, W. T.

Sawchuk, A. A.

W. T. Rhodes, A. A. Sawchuk, “Incoherent optical processing,” in Optical Information Processing, S. H. Lee, ed., Vol. 48 of Topics in Applied Physics (Springer-Verlag, New York, 1981), pp. 69–110.
[CrossRef]

Thum, Ch.

A. W. Lohmann, Ch. Thum, “Two-way translation by computer-generated holographic filters,” Opt. Commun. 46, 74–78 (1983).
[CrossRef]

Walther, A.

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
[CrossRef]

Appl. Opt. (3)

J. Opt. Soc. Am. A (2)

Opt. Acta (2)

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
[CrossRef]

G. Indebetouw, T.-C. Poon, “Parallel synthesis of bipolar point spread functions in a scanning heterodyne optical system,” Opt. Acta 33, 827–834 (1986).
[CrossRef]

Opt. Commun. (2)

D. Gorlitz, F. Lanzl, “Colour encoded aperture masks used for incoherent filtering of images,” Opt. Commun. 28, 283–286 (1979).
[CrossRef]

A. W. Lohmann, Ch. Thum, “Two-way translation by computer-generated holographic filters,” Opt. Commun. 46, 74–78 (1983).
[CrossRef]

Opt. Eng. (1)

I. Glaser, “Representing bipolar and complex imagery in noncoherent optical image processing systems: comparison of approaches,” Opt. Eng. 20, 568–573 (1981).
[CrossRef]

Other (2)

W. T. Rhodes, A. A. Sawchuk, “Incoherent optical processing,” in Optical Information Processing, S. H. Lee, ed., Vol. 48 of Topics in Applied Physics (Springer-Verlag, New York, 1981), pp. 69–110.
[CrossRef]

J. N. Mait, “Pupil function optimization for bipolar incoherent spatial filtering,” Ph.D. dissertation (Georgia Institute of Technology, Atlanta, Ga., 1985).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

A two-channel incoherent spatial filtering system (from Ref. 1). Abbreviations P1 and P2, pupil functions; A1 and A2, attenuations; Φ, a phase shift; BS, a beam splitter; M, a mirror.

Fig. 2
Fig. 2

(a) Magnitude of the complex PSF g(x) = sinc2(x) exp(2πu0x);(b) the corresponding OTF.

Fig. 3
Fig. 3

The system pupil functions (a) P1(u) and (b) P2(u) for the synthesis of g(x) = sinc2(x)exp(2πu0x) as described by Eqs. (30), using the two-channel system of Fig. 1.

Fig. 4
Fig. 4

Comparison of the bias ψ(x) as given by Eq. (42) (dashed curve) with the minimum bias ψMB(x) given by Eq. (43) (solid curve).

Fig. 5
Fig. 5

(a) Magnitude of the complex PSF g(x) = [sinc2(x)/(1 − x2)]exp(2πu0x); (b) the corresponding OTF.

Fig. 6
Fig. 6

The magnitudes of pupil functions (a) P1(u) and (b) P2(u) for the synthesis of g(x) = [sinc2(x)/(1 − x2)exp(2πu0x) as described by Eqs. (30), using the two-channel system of Fig. 1.

Equations (103)

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

i ( x ) = - o ( ζ ) z ( x - ζ ) d ζ = o ( x ) * z ( x ) ,
z ( x ) = p ( x ) 2 .
P eff ( 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 eff ( 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 ( x ) = f + ( x ) - f - ( x ) ,
f + ( x ) = [ f ( x ) + ψ ( x ) ] ,
f - ( x ) = [ - f ( x ) + ψ ( x ) ] ,
ψ MB ( x ) = f ( x )
f + ( x ) f - ( x ) = 0.
f + ( x ) = p + ( x ) 2 ,
f - ( x ) = p - ( x ) 2 ,
p + ( x ) = A ˜ 1 + p 1 ( x ) + A ˜ 2 + p 2 ( x ) ,
p - ( x ) = A ˜ 1 - p 1 ( x ) + A ˜ 2 - p 2 ( x ) .
p 1 ( x ) = A ˜ 2 - p + ( x ) - A ˜ 2 + p - ( x ) A ˜ 1 + A ˜ 2 - - A ˜ 2 + A ˜ 1 - ,
p 2 ( x ) = - A ˜ 1 - p + ( x ) - A ˜ 1 + p - ( x ) A ˜ 1 + A ˜ 2 - - A ˜ 2 + A ˜ 1 - .
g ( x ) = f 1 ( x ) + j f 2 ( x ) = [ z 1 ( x ) - z 2 ( x ) ] + j [ z 3 ( x ) - z 4 ( x ) ] ,
z 1 ( x ) = ( 1 / 2 ) [ f 1 ( x ) + ψ 1 ( x ) ] ,
z 2 ( x ) = ( 1 / 2 ) [ - f 1 ( x ) + ψ 1 ( x ) ] ,
z 3 ( x ) = ( 1 / 2 ) [ f 2 ( x ) + ψ 2 ( x ) ] ,
z 4 ( x ) = ( 1 / 2 ) [ - f 2 ( x ) + ψ 2 ( x ) ] ,
ψ ( x ) = ψ 1 ( x ) + j ψ 2 ( x ) .
ψ MB 1 ( x ) = f 1 ( x ) ,
ψ MB 2 ( x ) = f 2 ( x ) .
z 1 ( x ) = ( 1 / 2 ) [ f 1 ( x ) + ψ ( x ) ] ,
z 2 ( x ) = ( 1 / 2 ) [ - f 1 ( x ) + ψ ( x ) ] ,
z 3 ( x ) = ( 1 / 2 ) [ f 2 ( x ) + ψ ( x ) ] ,
z 4 ( x ) = ( 1 / 2 ) [ - f 2 ( x ) + ψ ( x ) ] ,
ψ MB ( x ) = - min [ - f 1 ( x ) , - f 2 ( x ) , f 1 ( x ) , f 2 ( x ) ] = max [ f 1 ( x ) , f 2 ( x ) ] .
j = 1 4 z j ( x ) = 0.
ψ ( x ) = ψ 1 ( x ) + ψ 2 ( x ) .
ψ MB ( x ) = - min [ f ( x ) , - f ( x ) ] .
z j ( x ) = p eff ( j ) ( x ) 2 , j = [ 1 , 4 ] .
[ p eff ( 1 ) ( x ) p eff ( 2 ) ( x ) p eff ( 3 ) ( x ) p eff ( 4 ) ( x ) ] = [ A ˜ 11 A ˜ 12 A ˜ 13 A ˜ 14 A ˜ 21 A ˜ 22 A ˜ 23 A ˜ 24 A ˜ 31 A ˜ 32 A ˜ 33 A ˜ 34 A ˜ 41 A ˜ 42 A ˜ 43 A ˜ 44 ] [ p 1 ( x ) p 2 ( x ) p 3 ( x ) p 4 ( x ) ] .
g ( x ) = h 1 ( x ) + h 2 ( x ) exp ( j β ) + h 3 ( x ) exp ( - j β ) ,
h 1 ( x ) = f 1 ( x ) ,
h 2 ( x ) = f 2 ( x ) 2 sin β ,
h 3 ( x ) = - f 2 ( x ) 2 sin β .
z 1 ( x ) = h 1 ( x ) + ψ ( x ) ,
z 2 ( x ) = h 2 ( x ) + ψ ( x ) ,
z 3 ( x ) = h 3 ( x ) + ψ ( x ) .
ψ MB ( x ) = - min [ h 1 ( x ) , h 2 ( x ) , h 3 ( x ) ] ,
j = 1 3 z j ( x ) = 0.
g ( x ) = z 1 ( x ) + ρ z 2 ( x ) exp ( j β ) + ρ z 3 ( x ) exp ( - j β ) ,
ρ exp ( j β ) = - ½ + j sin β .
g ( x ) = z 1 ( x ) - ( 1 / 2 ) [ z 1 ( x ) + z 3 ( x ) ] - j ( sin β ) [ z 2 ( x ) - z 3 ( x ) ] .
g ( x ) = z 1 ( x ) + z 2 ( x ) exp ( j β ) + z 3 ( x ) exp ( - j β ) .
f ( x ; ½ , ½ , 0 ) = ( 1 / 4 ) p 1 ( x ) 2 + ( 1 / 4 ) p 2 ( x ) 2 + ( 1 / 2 ) p 1 ( x ) p 2 ( x ) cos [ θ 1 ( x ) - θ 2 ( x ) ] .
p 1 ( x ) p 2 * ( x ) = g ( x ) ,
ψ ( x ) = ( ½ ) [ p 1 ( x ) 2 + p 2 ( x ) 2 ] .
z 1 ( x ) = f ( x ; ½ , ½ , 0 ) ,
z 2 ( x ) = f ( x ; ½ , ½ , π ) ,
z 3 ( x ) = f ( x ; ½ , ½ , - π / 2 ) ,
z 4 ( x ) = f ( x ; ½ , ½ , π / 2 ) .
g ( x ) = [ z 1 ( x ) - z 2 ( x ) ] + j [ z 3 ( x ) - z 4 ( x ) ] .
[ p eff ( 1 ) ( x ) p eff ( 2 ) ( x ) p eff ( 3 ) ( x ) p eff ( 4 ) ( x ) ] = [ ½ ½ ½ - ½ ½ - j / 2 ½ j / 2 ] [ p 1 ( x ) p 2 ( x ) ] .
z 1 ( x ) = f ( x ; ½ , ½ , 0 ) = ( ½ ) [ Re { g ( x ) } + ψ ( x ) ] ,
z 2 ( x ) = f ( x ; ½ , ½ , - α ) = ( 1 / 2 ) [ Re { g ( x ) exp ( - j α ) } + ψ ( x ) ] ,
z 3 ( x ) = f ( x ; ½ , ½ , α ) = ( 1 / 2 ) [ Re { g ( x ) } exp ( - j α ) } + ψ ( x ) ] .
[ p eff ( 1 ) ( x ) p eff ( 2 ) ( x ) p eff ( 3 ) ( x ) ] = [ ½ ½ ½ exp ( - j α ) / 2 ½ exp ( j α ) / 2 ] [ p 1 ( x ) p 2 ( x ) ] .
g ( x ) = 2 z 1 ( x ) - [ z 2 ( x ) + z 3 ( x ) ] 1 - cos α + j z 2 ( x ) - z 3 ( x ) sin α .
[ z 1 ( x ) · · · z N ( x ) z N + 1 ( x ) ] = [ a 11 a N 1 b · · · · · · · · · · · · a N 1 a N N b a ( N + 1 ) 1 a ( N + 1 ) N b ] × [ f 1 ( x ) · · · f N ( x ) ψ ( x ) ] .
a = [ 1 0 1 0 ( 2 sin β ) - 1 1 0 - ( 2 sin β ) - 1 1 ] ,
b = [ 1 / 2 0 1 / 2 ( cos α ) / 2 ( sin α ) / 2 1 / 2 ( cos α ) / 2 - ( sin α ) / 2 1 / 2 ] .
c = [ 1 0 1 0 1 1 0 0 1 ] .
ψ M B ( x ) = - min j [ 1 , N + 1 ] k = 1 N a j k b f k ( x )
j = 1 N + 1 z j ( x ) = 0.
1 b k = 1 N j = 1 N + 1 a j k f k ( x ) 1 b j = 1 N + 1 | k = 1 N a j k f k ( x ) | ψ M B ( x ) .
ψ ( x ) = 1 b k = 1 N j = 1 N + 1 a j k ψ k ( x ) ,
z j ( x ) = p eff ( j ) ( x ) 2 ,             j = [ 1 , N + 1 ] ,
p eff = Ap .
p 1 ( x ) = p 2 * ( x ) = [ g ( x ) ] 1 / 2 exp [ j θ g ( x ) / 2 ] ,
θ g ( x ) = tan - 1 [ f 2 ( x ) f 1 ( x ) ] .
p 1 ( x ) = sinc ( x ) exp ( j 2 π u 0 2 x ) ,
p 2 ( x ) = sinc ( x ) exp ( - j 2 π u 0 2 x ) .
f 1 ( x ) = sinc 2 ( x ) cos ( 2 π u 0 x ) ,
f 2 ( x ) = sinc 2 ( x ) sin ( 2 π u 0 x ) .
ψ ( x ) = sinc 2 ( x ) .
ψ M B ( x ) = sinc 2 ( x ) { [ cos ( 2 π u 0 x ) rect ( 4 u 0 x ) ] * 4 u 0 comb ( 4 u 0 x ) } .
z 1 ( x ) = ( 1 / 2 ) sinc 2 ( x ) [ 1 + cos ( 2 π u 0 x ) ] ,
z 2 ( x ) = ( 1 / 2 ) sinc 2 ( x ) [ 1 + sin ( 2 π u 0 x ) ] ,
z 3 ( x ) = ( 1 / 2 ) sinc 2 ( x ) [ 1 - cos ( 2 π u 0 x ) ] .
p eff ( 1 ) ( x ) = sinc ( x ) cos ( 2 π u 0 2 x ) ,
p eff ( 2 ) ( x ) = sinc ( x ) cos ( 2 π u 0 2 x - π 4 ) ,
p eff ( 3 ) ( x ) = sinc ( x ) cos ( 2 π u 0 2 x + π 4 ) .
[ p 1 ( x ) p 2 ( x ) ] = [ - 2 1 + exp ( j π / 4 ) 1 + exp ( - j π / 4 ) 2 2 - 1 j - exp ( j π / 4 ) 2 - 1 - j - exp ( j π / 4 ) 2 - 1 ] × [ p eff ( 1 ) ( x ) p eff ( 2 ) ( x ) p eff ( 3 ) ( x ) ] .
f 1 ( x ) = sinc 2 ( x ) 1 - x 2 cos ( 2 π u 0 x ) ,
f 2 ( x ) = sinc 2 ( x ) 1 - x 2 sin ( 2 π u 0 x ) .
ψ ( x ) = [ sinc ( x ) 1 - x 2 ] 2 ( 1 + x 2 ) .
z 1 ( x ) = ( 1 / 2 ) sinc 2 ( x ) 1 - x 2 [ 1 + x 2 1 - x 2 + cos ( 2 π u 0 x ) ] ,
z 2 ( x ) = ( 1 / 2 ) sinc 2 ( x ) 1 - x 2 [ 1 + x 2 1 - x 2 + sin ( 2 π u 0 x ) ] ,
z 3 ( x ) = ( 1 / 2 ) sinc 2 ( x ) 1 - x 2 [ 1 + x 2 1 - x 2 - sin ( 2 π u 0 x ) ] .
p eff ( 1 ) ( x ) = sinc ( x ) 1 - x 2 { cos ( 2 π u 0 2 x ) + j [ x sin ( 2 π u 0 2 x ) ] } ,
p eff ( 2 ) ( x ) = sinc ( x ) 1 - x 2 { cos ( 2 π u 0 2 x - π 4 ) + j [ x sin ( 2 π u 0 2 x - π 4 ) ] } ,
p eff ( 3 ) ( x ) = sinc ( x ) 1 - x 2 { cos ( 2 π u 0 2 x + π 4 ) + j [ x sin ( 2 π u 0 2 x + π 4 ) ] } .
p 1 ( x ) = sinc ( x ) 1 - x 2 ( 1 + x ) exp ( j 2 π u 0 2 x ) = sinc ( x ) 1 - x exp ( j 2 π u 0 2 x ) ,
p 2 ( x ) = sinc ( x ) 1 - x 2 ( 1 + x ) exp ( - j 2 π u 0 2 x ) = sinc ( x ) 1 + x exp ( - j 2 π u 0 2 x ) = p 1 ( - x ) .
z m ( x ) = f 1 ( x ) cos ( 2 π u c x ) - f 2 ( x ) sin ( 2 π u c x ) + ψ ( x ) = g ( x ) 2 exp ( j 2 π u c x ) + g * ( x ) 2 exp ( - j 2 π u c x ) + ψ ( x ) .
z m ( x ) = sinc 2 ( x ) { 1 + cos [ 2 π ( u 0 + u c ) x ] } .
p ( x ) = sinc ( x ) cos [ 2 π ( u 0 + u c ) 2 x ] = sinc ( x ) exp ( j 2 π u 0 2 x ) exp ( j 2 π u c 2 x ) + sinc ( x ) exp ( - j 2 π u 0 2 x ) exp ( - j 2 π u c 2 x ) ,
z m ( x ) = sinc 2 ( x ) 1 - x 2 { 1 + x 2 1 - x 2 + cos [ 2 π ( u 0 + u c ) x ] }
p ( x ) = sinc ( x ) 1 - x exp ( j 2 π u 0 2 x ) exp ( j 2 π u c 2 x ) + sinc ( x ) 1 + x exp ( - j 2 π u 0 2 ) exp ( - j 2 π u c 2 x ) .
ψ ( x ) = g ( x ) .
p ( x ) = [ g ( x ) ] 1 / 2 exp [ j θ g ( x ) / 2 ] exp ( j 2 π u c 2 x ) + [ g ( x ) ] 1 / 2 exp [ - j θ g ( x ) / 2 ] exp ( - j 2 π u c 2 x )

Metrics