Abstract

A modification of the equation of one-dimensional space-variant imaging is considered in which the output image is known both as a function of an image-plane coordinate and an object-position coordinate. Under this condition, the Fourier transform of the image intensity with respect to both variables equals the product of the object Fourier transform and a transfer function. A similar relation also occurs for observation of either the integrated image intensity or the intensity at a point versus an object-scan coordinate. Consequently, for the special situation considered, the transfer-function concept may be extended to the space-variant case. The implications of factorization of the transfer function are also considered.

© 1981 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  2. D. Korff, G. Dryden, and R. P. Leavitt, “Isoplanicity: the translation invariance of the atmospheric Green’s function,” J. Opt. Soc. Am. 65, 1321–1330 (1975).
    [Crossref]
  3. P. Nisenson and R. V. Stachnik, “Measurements of atmospheric isoplanatism using speckle interferometry,” J. Opt. Soc. Am. 68, 169–175 (1978).
    [Crossref]
  4. A. D. Code, “New generation optical telescope systems,” Ann. Rev. Astron. Astrophys. 11, 239–268 (1973).
    [Crossref]
  5. A. W. Lohmann and D. P. Paris, “Space-variant image formation,” J. Opt. Soc. Am. 55, 1007–1013 (1965).
  6. R. J. Marks, J. F. Walkup, and M. O. Hagler, “Line spread function notation,” Appl. Opt. 15, 2289–2290 (1976).
    [Crossref]
  7. R. J. Hanson and J. L. Phillips, “An adaptive numerical method for solving linear Fredholm integral equation of the first kind,” Numer. Math. 24, 291–307 (1975).
    [Crossref]
  8. H. C. Andrews, “Singular value decompositions and digital image processing,” IEEE Trans. Acoust. Speech Signal Process. ASSP-24, 26–53 (1976).
    [Crossref]
  9. J. W. Goodman, P. Kellman, and E. W. Hansen, “Linear space-variant optical processing of 1-D signals,” J. Opt. Soc. Am. 16, 733–738 (1977).
  10. Equivalently, the optical system, together with a detector fixed in the optical system, may move with respect to a stationary object.

1978 (1)

1977 (1)

J. W. Goodman, P. Kellman, and E. W. Hansen, “Linear space-variant optical processing of 1-D signals,” J. Opt. Soc. Am. 16, 733–738 (1977).

1976 (2)

H. C. Andrews, “Singular value decompositions and digital image processing,” IEEE Trans. Acoust. Speech Signal Process. ASSP-24, 26–53 (1976).
[Crossref]

R. J. Marks, J. F. Walkup, and M. O. Hagler, “Line spread function notation,” Appl. Opt. 15, 2289–2290 (1976).
[Crossref]

1975 (2)

R. J. Hanson and J. L. Phillips, “An adaptive numerical method for solving linear Fredholm integral equation of the first kind,” Numer. Math. 24, 291–307 (1975).
[Crossref]

D. Korff, G. Dryden, and R. P. Leavitt, “Isoplanicity: the translation invariance of the atmospheric Green’s function,” J. Opt. Soc. Am. 65, 1321–1330 (1975).
[Crossref]

1973 (1)

A. D. Code, “New generation optical telescope systems,” Ann. Rev. Astron. Astrophys. 11, 239–268 (1973).
[Crossref]

1965 (1)

Andrews, H. C.

H. C. Andrews, “Singular value decompositions and digital image processing,” IEEE Trans. Acoust. Speech Signal Process. ASSP-24, 26–53 (1976).
[Crossref]

Code, A. D.

A. D. Code, “New generation optical telescope systems,” Ann. Rev. Astron. Astrophys. 11, 239–268 (1973).
[Crossref]

Dryden, G.

Goodman, J. W.

J. W. Goodman, P. Kellman, and E. W. Hansen, “Linear space-variant optical processing of 1-D signals,” J. Opt. Soc. Am. 16, 733–738 (1977).

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

Hagler, M. O.

Hansen, E. W.

J. W. Goodman, P. Kellman, and E. W. Hansen, “Linear space-variant optical processing of 1-D signals,” J. Opt. Soc. Am. 16, 733–738 (1977).

Hanson, R. J.

R. J. Hanson and J. L. Phillips, “An adaptive numerical method for solving linear Fredholm integral equation of the first kind,” Numer. Math. 24, 291–307 (1975).
[Crossref]

Kellman, P.

J. W. Goodman, P. Kellman, and E. W. Hansen, “Linear space-variant optical processing of 1-D signals,” J. Opt. Soc. Am. 16, 733–738 (1977).

Korff, D.

Leavitt, R. P.

Lohmann, A. W.

Marks, R. J.

Nisenson, P.

Paris, D. P.

Phillips, J. L.

R. J. Hanson and J. L. Phillips, “An adaptive numerical method for solving linear Fredholm integral equation of the first kind,” Numer. Math. 24, 291–307 (1975).
[Crossref]

Stachnik, R. V.

Walkup, J. F.

Ann. Rev. Astron. Astrophys. (1)

A. D. Code, “New generation optical telescope systems,” Ann. Rev. Astron. Astrophys. 11, 239–268 (1973).
[Crossref]

Appl. Opt. (1)

IEEE Trans. Acoust. Speech Signal Process. (1)

H. C. Andrews, “Singular value decompositions and digital image processing,” IEEE Trans. Acoust. Speech Signal Process. ASSP-24, 26–53 (1976).
[Crossref]

J. Opt. Soc. Am. (4)

Numer. Math. (1)

R. J. Hanson and J. L. Phillips, “An adaptive numerical method for solving linear Fredholm integral equation of the first kind,” Numer. Math. 24, 291–307 (1975).
[Crossref]

Other (2)

Equivalently, the optical system, together with a detector fixed in the optical system, may move with respect to a stationary object.

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

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Equations (25)

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I ( x ) = - O ( x 1 ) S ( x - x 1 , x 1 ) d x 1 ,
I ( x , x 0 ) = - O ( x 1 - x 0 ) S ( x - x 1 , x 1 ) d x 1 ,
I ( x , x 0 ) = - Ô ( ν 2 ) exp [ 2 π i ν 2 ( x 1 - x 0 ) ] d ν 2 × - Ŝ ˆ ( ν 1 , ν 3 ) exp [ 2 π i ν 1 ( x - x 1 ) ] × exp ( 2 π i ν 3 x 1 ) d ν 1 d ν 3 d x 1 .
I ( x , x 0 ) = - Ô ( ν 2 ) Ŝ ˆ ( ν 1 , ν 1 - ν 2 ) × exp ( - 2 π i ν 2 x 0 ) exp ( 2 π i ν 1 x ) d ν 2 d ν 1 .
I ( x , x 0 ) = - Ô ( - ν 2 ) Ŝ ˆ ( ν 1 , ν 1 + ν 2 ) × exp [ 2 π i ( ν 2 x 0 + ν 1 x ) ] d ν 2 d ν 1 .
I ˆ ˆ ( ν 1 , ν 2 ) = Ô ( - ν 2 ) Ŝ ˆ ( ν 1 , ν 1 + ν 2 ) .
- I ( x , x 0 ) exp [ - 2 π i ( x ν 1 + x 0 ν 2 ) ] d x d x 0 = Ô ( - ν 2 ) Ŝ ˆ ( ν 1 , ν 1 + ν 2 ) ,
- [ - I ( x , x 0 ) d x ] exp ( - 2 π i x 0 ν 2 ) d x 0 = Ô ( - ν 2 ) Ŝ ˆ ( 0 , ν 2 ) .
E ( x 0 ) = - O ( x 1 - x 0 ) - S ( x - x 1 , x 1 ) d x d x 1 .
E ( x 0 ) = - O ( x 1 - x 0 ) × - { - Ŝ ( ν , x 1 ) exp [ 2 π i ν ( x - x 1 ) ] d ν } d x d x 1
E ( x 0 ) = - O ( x 1 - x 0 ) Ŝ ( ν = 0 , x 1 ) d x 1 .
- I ( x , x 0 ) d x 0 exp ( - 2 π i x ν 1 ) d x = Ô ( 0 ) Ŝ ˆ ( ν 1 , ν 1 ) .
I ( x , x 0 ) = - Ô ( - ν 2 ) [ Ŝ ˆ ( ν 1 , ν 1 + ν 2 ) × exp ( 2 π i ν 1 x ) d ν 1 ] exp ( 2 π i ν 2 x 0 ) d ν 2 .
I ˆ ( x , ν 2 ) = Ô ( - ν 2 ) - Ŝ ˆ ( ν 1 , ν 1 + ν 2 ) exp ( 2 π i ν 1 x ) d ν 1 .
I ˆ ( x = 0 , ν 2 ) = Ô ( - ν 2 ) - Ŝ ˆ ( ν 1 , ν 1 + ν 2 ) d ν 1 .
I ( x = 0 , x 0 ) = - O ( x 1 - x 0 ) S ( - x 1 , x 1 ) d x 1 .
- Ŝ ˆ ( ν 1 , ν 1 + ν 2 ) d ν 1
I ˆ ( x = 0 , ν 2 ) = Ô ( - ν 2 ) Ŝ ( - ν 2 ) .
I ( x , x 0 ) = - [ - Ô ( - ν 2 ) Ŝ ˆ ( ν 1 , ν 1 + ν 2 ) × exp ( 2 π i ν 2 x ) d ν 2 ] exp ( 2 π i ν 1 x ) d ν 1 ,
I ˆ ( ν 1 , x 0 ) = - Ô ( - ν 2 ) Ŝ ˆ ( ν 1 , ν 1 + ν 2 ) exp ( 2 π i ν 2 x 0 ) d ν 2 ,
I ˆ ( ν 1 , x 0 = 0 ) = - Ô ( - ν 2 ) Ŝ ˆ ( ν 1 , ν 1 + ν 2 ) d ν 2 .
I ˆ ( ν 1 , x 0 = 0 ) = Ô ( ν 1 ) Ŝ ( ν 1 ) ,
Ŝ ˆ ( ν 1 , ν 1 + ν 2 ) = Ŝ 1 ( ν 1 ) Ŝ 2 ( ν 1 + ν 2 ) .
I ( x , x 0 ) = - S 2 ( x 1 ) S 1 ( x - x 1 ) O ( x 1 - x 0 ) d x 1 .
I ˆ ( x = 0 , ν 2 ) = Ô ( - ν 2 ) - Ŝ 1 ( ν 1 ) Ŝ 2 ( ν 1 + ν 2 ) d ν 1 .