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References

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  1. E. Linfoot, Fourier Methods in Optical Image Evaluation (Focal Press, London, 1964).
  2. H. H. Hopkins, “The Frequency Response of a Defocused Optical System,” Proc. R. Soc. London Ser. A 231, 91 (1955).
    [CrossRef]
  3. A. W. Lohmann, J. Ojeda-Castaneda, N. Streibl, “Symmetries in Coherent and in Partially Coherent Fields,” Opt. Acta 30, 399 (1983).
    [CrossRef]
  4. G. Häusler, E. Körner, “Expansion of Depth of Focus by Image De-puzzling” in Proceedings, Sixth International Conference on Pattern Recognition, Munich (1982), p. 1201.

1983 (1)

A. W. Lohmann, J. Ojeda-Castaneda, N. Streibl, “Symmetries in Coherent and in Partially Coherent Fields,” Opt. Acta 30, 399 (1983).
[CrossRef]

1955 (1)

H. H. Hopkins, “The Frequency Response of a Defocused Optical System,” Proc. R. Soc. London Ser. A 231, 91 (1955).
[CrossRef]

Häusler, G.

G. Häusler, E. Körner, “Expansion of Depth of Focus by Image De-puzzling” in Proceedings, Sixth International Conference on Pattern Recognition, Munich (1982), p. 1201.

Hopkins, H. H.

H. H. Hopkins, “The Frequency Response of a Defocused Optical System,” Proc. R. Soc. London Ser. A 231, 91 (1955).
[CrossRef]

Körner, E.

G. Häusler, E. Körner, “Expansion of Depth of Focus by Image De-puzzling” in Proceedings, Sixth International Conference on Pattern Recognition, Munich (1982), p. 1201.

Linfoot, E.

E. Linfoot, Fourier Methods in Optical Image Evaluation (Focal Press, London, 1964).

Lohmann, A. W.

A. W. Lohmann, J. Ojeda-Castaneda, N. Streibl, “Symmetries in Coherent and in Partially Coherent Fields,” Opt. Acta 30, 399 (1983).
[CrossRef]

Ojeda-Castaneda, J.

A. W. Lohmann, J. Ojeda-Castaneda, N. Streibl, “Symmetries in Coherent and in Partially Coherent Fields,” Opt. Acta 30, 399 (1983).
[CrossRef]

Streibl, N.

A. W. Lohmann, J. Ojeda-Castaneda, N. Streibl, “Symmetries in Coherent and in Partially Coherent Fields,” Opt. Acta 30, 399 (1983).
[CrossRef]

Opt. Acta (1)

A. W. Lohmann, J. Ojeda-Castaneda, N. Streibl, “Symmetries in Coherent and in Partially Coherent Fields,” Opt. Acta 30, 399 (1983).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

H. H. Hopkins, “The Frequency Response of a Defocused Optical System,” Proc. R. Soc. London Ser. A 231, 91 (1955).
[CrossRef]

Other (2)

E. Linfoot, Fourier Methods in Optical Image Evaluation (Focal Press, London, 1964).

G. Häusler, E. Körner, “Expansion of Depth of Focus by Image De-puzzling” in Proceedings, Sixth International Conference on Pattern Recognition, Munich (1982), p. 1201.

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Figures (2)

Fig. 1
Fig. 1

(a) Measured image intensity I(xn;z) vs defocusing z. As an example, three arbitrary pixels xn were selected from the image of a radial test pattern. (b) the focus function R(z) is calculated from I(x;z): R ( z ) = n = 1 20 I ( x n ; z ) / z ) . R(z) shows a distinct minimum [R(z0) ≈ 0] at the best focus z0 = 0.

Fig. 2
Fig. 2

By local application of the criterion to a 3-D object it is possible to find its shape z0(x). In the experiment, the object was a planar test pattern, oblique to the optical axis.

Equations (8)

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I ( x ; z ) = O ( x - x ) F ( x ; z ) d x ,
I ( x ; z ) z O ( x - x ) · F ( x ; z ) z d x .
F ( x ; z ) z z 0 = 0 for all x .
F ˜ ( ν ; z ) z z 0 = 0 for all ν ,
F ˜ ( ν ; z ) z f ˜ ( ν 1 - ν 2 ) f ˜ * ( ν 1 + ν 2 ) · ν ν 1 × exp ( - 2 i π λ z sin 2 u · ν ν 1 ) d ν 1             ( u = aperture angle ) .
I ( x ; z ) = o ( x - x 1 ) o * ( x - x 2 ) × Γ ( x 1 - x 2 ) f ( x 1 ; z ) f * ( x 2 ; z ) d x 1 d x 2 .
I ( x ; z ) z = o ( x - x 1 ) o * ( x - x 2 ) Γ ( x 1 - x 2 ) [ B ] d x 1 d x 2 ,
B = f ( x 1 ; z ) z f * ( x 2 ; z ) + f * ( x 2 ; z ) z f ( x 1 ; z ) .

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