Abstract

A closed-form expression is derived for the optical transfer function of a diffraction-limited system under polychromatic illumination. The expression is useful in a method for the accurate and efficient computation of the optical transfer function of an optical system. The expression is also useful in the theoretical analyses of the optical system. This method is illustrated with an example where the optical transfer function of the human eye is computed.

© 1990 Optical Society of America

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References

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  1. M. Subbarao, “Method and Apparatus for Determining the Distances Between Surface Patches of Three-Dimensional Spatial Scene and a Camera System,” U.S. Patent Application 126407 pending (Nov.1987).
  2. M. Subbarao, “Parallel Depth Recovery by Changing Camera Parameters,” in Proceedings, Second International Conference on Computer Vision, City, FL (Dec. 1988), pp. 149–155.
  3. B. K. P. Horn, Robot Vision (McGraw-Hill, New York, 1986), p. 126.
  4. W. F. Schreiber, Fundamentals of Electronic Imaging Systems (Springer-Verlag, Berlin, 1986), Sec. 2.5.2.
    [CrossRef]
  5. W. A. Martin, R. J. Fateman, “The macsyma System,” in Proceedings, ACM 2d Symposium on Symbolic and Algebraic Manipulation, Los Angeles (1971), pp. 23–25.
  6. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 118–120.
  7. R. W. Gubisch, “Optical Performance of the Human Eye,” J. Opt. Soc. Am. 57, 407–415 (1967).
    [CrossRef]

1967 (1)

Fateman, R. J.

W. A. Martin, R. J. Fateman, “The macsyma System,” in Proceedings, ACM 2d Symposium on Symbolic and Algebraic Manipulation, Los Angeles (1971), pp. 23–25.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 118–120.

Gubisch, R. W.

Horn, B. K. P.

B. K. P. Horn, Robot Vision (McGraw-Hill, New York, 1986), p. 126.

Martin, W. A.

W. A. Martin, R. J. Fateman, “The macsyma System,” in Proceedings, ACM 2d Symposium on Symbolic and Algebraic Manipulation, Los Angeles (1971), pp. 23–25.

Schreiber, W. F.

W. F. Schreiber, Fundamentals of Electronic Imaging Systems (Springer-Verlag, Berlin, 1986), Sec. 2.5.2.
[CrossRef]

Subbarao, M.

M. Subbarao, “Method and Apparatus for Determining the Distances Between Surface Patches of Three-Dimensional Spatial Scene and a Camera System,” U.S. Patent Application 126407 pending (Nov.1987).

M. Subbarao, “Parallel Depth Recovery by Changing Camera Parameters,” in Proceedings, Second International Conference on Computer Vision, City, FL (Dec. 1988), pp. 149–155.

J. Opt. Soc. Am. (1)

Other (6)

M. Subbarao, “Method and Apparatus for Determining the Distances Between Surface Patches of Three-Dimensional Spatial Scene and a Camera System,” U.S. Patent Application 126407 pending (Nov.1987).

M. Subbarao, “Parallel Depth Recovery by Changing Camera Parameters,” in Proceedings, Second International Conference on Computer Vision, City, FL (Dec. 1988), pp. 149–155.

B. K. P. Horn, Robot Vision (McGraw-Hill, New York, 1986), p. 126.

W. F. Schreiber, Fundamentals of Electronic Imaging Systems (Springer-Verlag, Berlin, 1986), Sec. 2.5.2.
[CrossRef]

W. A. Martin, R. J. Fateman, “The macsyma System,” in Proceedings, ACM 2d Symposium on Symbolic and Algebraic Manipulation, Los Angeles (1971), pp. 23–25.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 118–120.

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Figures (6)

Fig. 1
Fig. 1

Diffraction-limited system with circular exit pupil.

Fig. 2
Fig. 2

OTF of a diffraction-limited system with a circular exit pupil under incoherent quasimonochromatic illumination (after Goodman6).

Fig. 3
Fig. 3

Determining the limits for evaluation of the definite integrals.

Fig. 4
Fig. 4

Approximate visibility curve for a normal eye under white light illumination (see Schreiber4).

Fig. 5
Fig. 5

OTF of a diffraction-limited normal eye under white light illumination.

Fig. 6
Fig. 6

OTF of a diffraction-limited system with a square exit pupil under incoherent monochromatic illumination (after Goodman6).

Tables (1)

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Equations (25)

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H ( p , λ ) = { 2 π [ cos - 1 ( ρ 2 ρ 0 ) - ρ 2 ρ 0 1 - ( ρ 2 ρ 0 ) 2 ] for ρ 2 ρ 0 , 0 otherwise ,
ρ 0 = l 2 λ d i .
t = ρ d i l .
H ( ρ , λ ) = { 2 π [ cos - 1 ( t λ ) - ( t λ ) 1 - ( t λ ) 2 ] for t 1 λ , 0 otherwise .
G 0 ( ρ , λ ) = H ( ρ , λ ) d λ , = 2 π [ cos - 1 ( t λ ) d λ - t λ 1 - ( t λ ) 2 d λ ] = 2 π { λ cos - 1 ( t λ ) - 1 - ( t λ ) 2 t + [ 1 - ( t λ ) 2 ] 3 / 2 3 t } .
P ( λ ) = S ( λ ) M ( λ ) R ( λ ) ,
O ( ρ ) = 1 A - P ( λ ) H ( ρ , λ ) d λ ,
A = - P ( λ ) d λ .
P ( λ ) = { 1 for λ min λ < λ max , 0 otherwise . }
ρ min = l λ max d i , ρ max = l λ min d i ,
ρ min ρ < ρ max , λ = l ρ d i .
O ( ρ ) = { 1 λ max - λ min [ G 0 ( ρ , λ ) ] λ min λ max for 0 ρ < ρ min , 1 λ max - λ min [ G 0 ( ρ , λ ) ] λ min λ = l / ( ρ d i ) for ρ min ρ = ρ < ρ max , 0 for ρ ρ max .
G 1 ( ρ , λ ) = λ H ( ρ , λ ) d λ = 2 π [ λ cos - 1 ( t λ ) d λ - t λ 2 1 - ( t λ ) 2 d λ ] = 2 π { sin - 1 ( t λ ) 8 t 2 - 3 λ 1 - ( t λ ) 2 8 t + λ 2 cos - 1 ( t λ ) 2 + λ [ 1 - ( t λ ) 2 ] 3 / 2 4 t } .
G ( ρ , λ ) = P ( λ ) H ( ρ , λ ) d λ .
G ( ρ , λ ) = ( m i λ + c i ) H ( ρ , λ ) d λ = m i G i ( ρ , λ ) + c i S 0 ( ρ , λ ) ,
λ i λ i + 1 P ( λ ) H ( ρ , λ ) d λ = { [ G ( ρ , λ ) ] λ min λ max for 0 ρ < ρ min , [ G ( ρ , λ ) ] λ min λ = l / ( ρ d i ) for ρ min ρ = ρ < ρ max , 0 for ρ ρ max .
O ( ρ ) = 1 A i = 1 m λ i λ i + 1 P ( λ ) H ( ρ , λ ) d λ ,
A = i = 1 m λ i λ i + 1 P ( λ ) d λ
H ( f x , f y , λ ) = { ( 1 - λ d i f x l ) ( 1 - λ d i f y l ) for f x l λ d i , f y l λ d i 0 otherwise .
H ( f x , f y , λ ) = { ( 1 - λ d i f x l ) ( 1 - λ d i f y l ) for 0 f x l λ d i , 0 f y l λ d i , 0 otherwise .
G ( f x , f y , λ ) = P ( λ ) H ( f x , f y , λ ) d λ .
ρ = maximum ( f x , f y ) .
λ i λ i + 1 P ( λ ) H ( f x , f y , λ ) d λ = { [ G ( f x , f y , λ ) ] λ min λ max for 0 ρ < ρ min , [ G ( f x , f y , λ ) ] λ min λ = l / ( ρ d i ) for ρ min ρ = ρ < ρ max , 0 for ρ ρ max .
O ( f x , f y ) = 1 A i = 1 m λ i λ i + 1 P ( λ ) H ( f x , f y , λ ) d λ ,
λ 2 H ( ρ , λ ) = λ 3 cos - 1 ( λ t ) 3 + λ 2 [ 1 - ( λ t ) 2 ] 3 / 2 5 t + 2 [ 1 - ( λ t ) 2 ] 3 / 2 15 t 3 - λ 2 1 - ( λ t ) 2 9 t - 2 1 - ( λ t ) 2 9 t 3 , λ 3 H ( ρ , λ ) = - sin - 1 ( λ t ) 32 t 4 - λ 3 1 - ( λ t ) 2 16 t - 5 λ 1 - ( λ t ) 2 32 t 3 + λ 3 [ 1 - ( λ t ) 2 ] 3 / 2 6 t + λ [ 1 - ( λ t ) 2 ] 3 / 2 8 t 3 + λ 4 cos - 1 ( λ t ) 4 .

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