Abstract

Streibl [ Optik 66, 341– 354 ( 1984)] has shown that afocal telecentric imaging systems are shift invariant in three dimensions. We show that afocal telecentric imaging systems are the only imaging systems that are shift invariant in three dimensions. In addition, we present a model that allows any imaging system to be modeled as an afocal telecentric imaging system preceded and succeeded by simple coordinate transformation operators. The model is derived for diffraction-limited imaging systems where the Fresnel approximation is valid. It is assumed that the object distribution is incoherently radiating and that multiple scattering and absorption within the object distribution are negligible. A physical analogy is given that provides insight into the mathematical model. Finally, a comparison with the work of Frieden is given.

© 1990 Optical Society of America

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References

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  1. P. A. Stokseth, “Properties of a Defocused Optical System,” J. Opt. Soc. Am. 59, 1314–1321 (1969).
    [Crossref]
  2. A. Erhardt, G. Zinser, D. Komitowski, J. R. Bille, “Reconstructing 3-D Light-Microscopic Images by Digital Image Processing,” Appl. Opt. 24, 194–200 (1985).
    [Crossref] [PubMed]
  3. K. R. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, NJ, 1979).
  4. D. A. Agard, Y. Hiraoka, P. Shaw, J. W. Sedat, “Fluorescence Microscopy in Three Dimensions,” Methods Cell Biol. 30, 353–377 (1989).
    [Crossref] [PubMed]
  5. N. Streibl, “Fundamental Restrictions for 3-D Light Distributions,” Optik 66, 341–354 (1984).
  6. N. Streibl, “Depth Transfer by an Imaging System,” Opt. Acta 31, 1233–1241 (1984).
    [Crossref]
  7. A. A. Sawchuk, “Space-Variant Image Motion Degradation and Restoration,” Proc. IEEE 60, 854–861 (1972).
    [Crossref]
  8. A. A. Sawchuk, “Space-Variant Image Restoration by Coordinate Transformations,” J. Opt. Soc. Am. 64, 138–144 (1974).
    [Crossref]
  9. G. M. Robbins, T. S. Huang “Inverse Filtering for Linear Shift-Variant Imaging Systems,” Proc. IEEE 60, 862–872 (1972).
    [Crossref]
  10. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  11. C. J. R. Sheppard, “Imaging in Optical Systems of Finite Fresnel Number,” J. Opt. Soc. Am. A 3, 1428–1432 (1986).
    [Crossref]
  12. S. F. Gibson, F. Lanni, “Diffraction by a Circular Aperture as a Model for Three-Dimensional Optical Microscopy,” J. Opt. Soc. Am. A 6, 1357–1367 (1989).
    [Crossref] [PubMed]
  13. Note that under the Fresnel approximation, spherical phase surfaces are approximated by quadratic phase surfaces.
  14. W. B. Wetherell, “Afocal Lenses,” in Applied Optics and Optical Engineering, Vol. X, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1987).
  15. B. R. Frieden, “Optical Transfer of the Three-Dimensional Object,” J. Opt. Soc. Am. 57, 56–66 (1967).
    [Crossref]
  16. M. Y. Chiu, H. H. Barrett, R. G. Simpson, C. Chou, J. W. Arendt, G. R. Gindi, “Three-Dimensional Radiographic Imaging with a Restricted View Angle,” J. Opt. Soc. Am. 69, no. 10, 1323–1333 (1979).
    [Crossref]

1989 (2)

D. A. Agard, Y. Hiraoka, P. Shaw, J. W. Sedat, “Fluorescence Microscopy in Three Dimensions,” Methods Cell Biol. 30, 353–377 (1989).
[Crossref] [PubMed]

S. F. Gibson, F. Lanni, “Diffraction by a Circular Aperture as a Model for Three-Dimensional Optical Microscopy,” J. Opt. Soc. Am. A 6, 1357–1367 (1989).
[Crossref] [PubMed]

1986 (1)

1985 (1)

1984 (2)

N. Streibl, “Fundamental Restrictions for 3-D Light Distributions,” Optik 66, 341–354 (1984).

N. Streibl, “Depth Transfer by an Imaging System,” Opt. Acta 31, 1233–1241 (1984).
[Crossref]

1979 (1)

1974 (1)

1972 (2)

G. M. Robbins, T. S. Huang “Inverse Filtering for Linear Shift-Variant Imaging Systems,” Proc. IEEE 60, 862–872 (1972).
[Crossref]

A. A. Sawchuk, “Space-Variant Image Motion Degradation and Restoration,” Proc. IEEE 60, 854–861 (1972).
[Crossref]

1969 (1)

1967 (1)

Agard, D. A.

D. A. Agard, Y. Hiraoka, P. Shaw, J. W. Sedat, “Fluorescence Microscopy in Three Dimensions,” Methods Cell Biol. 30, 353–377 (1989).
[Crossref] [PubMed]

Arendt, J. W.

Barrett, H. H.

Bille, J. R.

Castleman, K. R.

K. R. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, NJ, 1979).

Chiu, M. Y.

Chou, C.

Erhardt, A.

Frieden, B. R.

Gibson, S. F.

Gindi, G. R.

Goodman, J. W.

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

Hiraoka, Y.

D. A. Agard, Y. Hiraoka, P. Shaw, J. W. Sedat, “Fluorescence Microscopy in Three Dimensions,” Methods Cell Biol. 30, 353–377 (1989).
[Crossref] [PubMed]

Huang, T. S.

G. M. Robbins, T. S. Huang “Inverse Filtering for Linear Shift-Variant Imaging Systems,” Proc. IEEE 60, 862–872 (1972).
[Crossref]

Komitowski, D.

Lanni, F.

Robbins, G. M.

G. M. Robbins, T. S. Huang “Inverse Filtering for Linear Shift-Variant Imaging Systems,” Proc. IEEE 60, 862–872 (1972).
[Crossref]

Sawchuk, A. A.

A. A. Sawchuk, “Space-Variant Image Restoration by Coordinate Transformations,” J. Opt. Soc. Am. 64, 138–144 (1974).
[Crossref]

A. A. Sawchuk, “Space-Variant Image Motion Degradation and Restoration,” Proc. IEEE 60, 854–861 (1972).
[Crossref]

Sedat, J. W.

D. A. Agard, Y. Hiraoka, P. Shaw, J. W. Sedat, “Fluorescence Microscopy in Three Dimensions,” Methods Cell Biol. 30, 353–377 (1989).
[Crossref] [PubMed]

Shaw, P.

D. A. Agard, Y. Hiraoka, P. Shaw, J. W. Sedat, “Fluorescence Microscopy in Three Dimensions,” Methods Cell Biol. 30, 353–377 (1989).
[Crossref] [PubMed]

Sheppard, C. J. R.

Simpson, R. G.

Stokseth, P. A.

Streibl, N.

N. Streibl, “Fundamental Restrictions for 3-D Light Distributions,” Optik 66, 341–354 (1984).

N. Streibl, “Depth Transfer by an Imaging System,” Opt. Acta 31, 1233–1241 (1984).
[Crossref]

Wetherell, W. B.

W. B. Wetherell, “Afocal Lenses,” in Applied Optics and Optical Engineering, Vol. X, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1987).

Zinser, G.

Appl. Opt. (1)

J. Opt. Soc. Am. (4)

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

Methods Cell Biol. (1)

D. A. Agard, Y. Hiraoka, P. Shaw, J. W. Sedat, “Fluorescence Microscopy in Three Dimensions,” Methods Cell Biol. 30, 353–377 (1989).
[Crossref] [PubMed]

Opt. Acta (1)

N. Streibl, “Depth Transfer by an Imaging System,” Opt. Acta 31, 1233–1241 (1984).
[Crossref]

Optik (1)

N. Streibl, “Fundamental Restrictions for 3-D Light Distributions,” Optik 66, 341–354 (1984).

Proc. IEEE (2)

A. A. Sawchuk, “Space-Variant Image Motion Degradation and Restoration,” Proc. IEEE 60, 854–861 (1972).
[Crossref]

G. M. Robbins, T. S. Huang “Inverse Filtering for Linear Shift-Variant Imaging Systems,” Proc. IEEE 60, 862–872 (1972).
[Crossref]

Other (4)

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

K. R. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, NJ, 1979).

Note that under the Fresnel approximation, spherical phase surfaces are approximated by quadratic phase surfaces.

W. B. Wetherell, “Afocal Lenses,” in Applied Optics and Optical Engineering, Vol. X, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1987).

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Figures (5)

Fig. 1
Fig. 1

Model for nonafocal imaging systems.

Fig. 2
Fig. 2

Nonafocal imaging system. The Gaussian image is a distorted version of the object distribution, and the blurring is space variant.

Fig. 3
Fig. 3

Afocal telecentric imaging system. The Gaussian image is a replica of the object distribution, and the blurring is shift invariant.

Fig. 4
Fig. 4

Physical model for a nonafocal imaging system: (a) analysis geometry; (b) equivalent system (effects of equal-power positive and negative lenses cancel); (c) system of part (b) spread out to show the afocal–telecentric system in the middle and negative lenses that distort object and image space.

Fig. 5
Fig. 5

Considerable overlap occurs in image space when the imaged isotomic regions are blurred. A point in image space in general receives contributions from different isotomic regions, each of which has a different point spread function. The image space distribution is, therefore, related to the object distribution in a space variant manner.

Equations (18)

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I ( m x x , m y y , m z z ) = O ( α , β , γ ) × h ( x - α , y - β , z - γ ) d α d β d γ .
I ( x i , y i , z i ) = O ( x o , y o , z o ) h ( x i , y i , z i ; x o , y o , z o ) d x o d y o d z o ,
I ¯ ( x ¯ i , y ¯ i , z ¯ i ) = O ¯ ( x ¯ o , y ¯ o , z ¯ o ) × h ¯ ( x ¯ i - x ¯ o , y ¯ i - y ¯ o , z ¯ i - z ¯ o ) d x ¯ o d y ¯ o d z ¯ o ,
I ( x , y , z ) = { h F ( x - x o , y - y o ; d 1 - z o ) × p ( x / m 1 , y / m 1 ) } * * h F [ x , y ; - ( d 1 - z ) ] 2 ,
h F ( x , y ; z ) = 1 j λ z exp ( j k z ) exp [ j k 2 z ( x 2 + y 2 ) ] .
I ( x , y , z ) = | 1 λ 2 ( d 1 - z o ) ( d 1 - z ) - + - + p ( α / m 1 , β / m 1 ) × exp { j π ( α 2 + β 2 ) [ 1 λ ( d 1 - z o ) - 1 λ ( d 1 - z ) ] } × exp [ - j 2 π ( s α + t β ) ] d α d β | 2 ,
I ( x , y , z ) = | d 1 2 ( d 1 - z o ) ( d 1 - z ) [ m 1 λ d 1 ] 2 P ^ ( - m 1 u λ d 1 , - m 1 v λ d 1 ) × * * h F ( u , v ; d 1 2 d 1 - z - d 1 2 d 1 - z o ) | 2 ,
x = f l 2 - f + z i x i ,             y = f l 2 - f + z i y i ,             z = 1 M f l 2 - f + z i z i ,
h ( x i , y i , z i ; x o , y o , z o ) = { d 1 d 1 - z o } 2 { 1 M d 2 d 2 + z i } 2 × | [ m 1 λ d 1 ] 2 P ^ ( - m 1 u ¯ λ d 1 , - m 1 v ¯ λ d 1 ) * * h F ( u ¯ , v ¯ ; w ¯ ) | 2 ,
u ¯ = d 2 x i M ( d 2 + z i ) - d 1 x o ( d 1 - z o ) , v ¯ = d 2 y i M ( d 2 + z i ) - d 1 y o ( d 1 - z o ) , w ¯ = d 2 z i M 2 ( d 2 + z i ) - d 1 z 0 ( d 1 - z o ) .
h ¯ ( u ¯ , v ¯ , w ¯ ) = | [ m 1 λ d 1 ] 2 P ^ ( - m 1 u ¯ λ d 1 , - m 1 v ¯ λ d 1 ) * * h F ( u ¯ , v ¯ ; w ¯ ) | 2 ,
x ¯ o = d 1 x o ( d 1 - z o ) ,             y ¯ o = d 1 y o ( d 1 - z o ) ,             z ¯ o = d 1 z o ( d 1 - z o ) .
x ¯ i = d 2 x i M ( d 2 + z i ) ,             y ¯ i = d 2 y i M ( d 2 + z i ) ,             z ¯ i = d 2 z i M 2 ( d 2 + z i ) .
J ( x o , y o , z o ) = ( d 1 - z o d 1 ) 4 .
O ¯ ( x ¯ , y ¯ , z ¯ ) = O ( d 1 x ¯ d 1 + z ¯ , d 1 y ¯ d 1 + z ¯ , d 1 z ¯ d 1 + z ¯ ) ( d 1 d 1 + z ¯ ) 2 .
I ( x , y , z ) = I ¯ ( 1 M d 2 x d 2 + z , 1 M d 2 y d 2 + z , 1 M 2 d 2 z d 2 + z ) ( 1 M d 2 d 2 + z ) 2 .
h ¯ ( x ¯ , y ¯ , z ¯ ) = | [ m 1 λ d 1 ] 2 P ^ ( - m 1 x ¯ λ d 1 , - m 1 y ¯ λ d 1 ) * * h F ( x ¯ , y ¯ ; z ¯ ) | 2 ,
x = x i M ,             y = y i M ,             z = z i M 2 ,

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