Abstract

In cases in which there are zeros in the optical transfer function, display of the absolute value of the Fourier transform of a blurred image may allow these zeros to be seen, if the noise level is low enough. This can be used to identify the optical transfer function, provided that it is one of the suitable common simple forms, or to determine certain parameters of the optical transfer function if its form is known. Examples of the use of this technique in the generation of restoring filters for image enhancement are presented.

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References

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  1. P. Elias, D. S. Grey, and D. Z. Robinson, J. Opt. Soc. Am. 42, 127 (1952).
    [CrossRef]
  2. J. L. Harris, J. Opt. Soc. Am. 56, 569 (1966).
    [CrossRef]
  3. B. R. Frieden, J. Opt. Soc. Am. 62, 511 (1972).
    [CrossRef] [PubMed]
  4. D. B. Gennery, in Seventh Space Congress Proceedings (Canaveral Council of Technical Societies, Cocoa Beach, Fla.,1970), p. 12–19.
  5. R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra (Dover, New York, 1958).
  6. B. L. McGlamery, J. Opt. Soc. Am. 57, 293 (1967).
    [CrossRef]

1972 (1)

B. R. Frieden, J. Opt. Soc. Am. 62, 511 (1972).
[CrossRef] [PubMed]

1967 (1)

B. L. McGlamery, J. Opt. Soc. Am. 57, 293 (1967).
[CrossRef]

1966 (1)

J. L. Harris, J. Opt. Soc. Am. 56, 569 (1966).
[CrossRef]

1952 (1)

P. Elias, D. S. Grey, and D. Z. Robinson, J. Opt. Soc. Am. 42, 127 (1952).
[CrossRef]

Blackman, R. B.

R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra (Dover, New York, 1958).

Elias, P.

P. Elias, D. S. Grey, and D. Z. Robinson, J. Opt. Soc. Am. 42, 127 (1952).
[CrossRef]

Frieden, B. R.

B. R. Frieden, J. Opt. Soc. Am. 62, 511 (1972).
[CrossRef] [PubMed]

Gennery, D. B.

D. B. Gennery, in Seventh Space Congress Proceedings (Canaveral Council of Technical Societies, Cocoa Beach, Fla.,1970), p. 12–19.

Grey, D. S.

P. Elias, D. S. Grey, and D. Z. Robinson, J. Opt. Soc. Am. 42, 127 (1952).
[CrossRef]

Harris, J. L.

J. L. Harris, J. Opt. Soc. Am. 56, 569 (1966).
[CrossRef]

McGlamery, B. L.

B. L. McGlamery, J. Opt. Soc. Am. 57, 293 (1967).
[CrossRef]

Robinson, D. Z.

P. Elias, D. S. Grey, and D. Z. Robinson, J. Opt. Soc. Am. 42, 127 (1952).
[CrossRef]

Tukey, J. W.

R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra (Dover, New York, 1958).

Other (6)

P. Elias, D. S. Grey, and D. Z. Robinson, J. Opt. Soc. Am. 42, 127 (1952).
[CrossRef]

J. L. Harris, J. Opt. Soc. Am. 56, 569 (1966).
[CrossRef]

B. R. Frieden, J. Opt. Soc. Am. 62, 511 (1972).
[CrossRef] [PubMed]

D. B. Gennery, in Seventh Space Congress Proceedings (Canaveral Council of Technical Societies, Cocoa Beach, Fla.,1970), p. 12–19.

R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra (Dover, New York, 1958).

B. L. McGlamery, J. Opt. Soc. Am. 57, 293 (1967).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Blurred photograph of numerals (enlarged from 20-mm by 16-mm region on negative). Rectangle denotes portion used to obtain OTF.

Fig. 2
Fig. 2

Fourier transform of image in Fig. 1. Frequency scales are in cycles/mm.

Fig. 3
Fig. 3

Fourier transform of image in Fig. 1, with increased contrast compared to Fig. 2.

Fig. 4
Fig. 4

Result of processing image in Fig. 1.

Fig. 5
Fig. 5

Blurred photograph of foliage (enlarged from 8.5-mm by 8.5-mm region on negative).

Fig. 6
Fig. 6

Fourier transform of image in Fig. 5. Frequency scales are in cycles/mm.

Fig. 7
Fig. 7

Result of processing image in Fig. 5.

Fig. 8
Fig. 8

Unblurred photograph of scene in Fig. 5.

Tables (2)

Tables Icon

Table I Some common types of blurring.

Tables Icon

Table II Zeros in OTF due to out-of-focus camera with annular aperture, with r and ρr as outer and inner radii of annulus of confusion.

Equations (15)

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H i ( f x , f y ) = S i ( f x , f y ) H t ( f x , f y ) ,
H ( f x ) = a b h ( x ) exp ( 2 π i f x x ) d x ,
exp [ 2 π i n x / ( b a ) ] = exp { 2 π i n [ x + m ( b a ) ] / ( b a ) } ,
w 1 , 0 = 1 , w 0 , 1 = 1 , w 0 , 0 = 4 w 0 , 1 = 1 , w 1 , 0 = 1
W ( f x , f y ) = j , k w j , k exp [ 2 π i ( f x j Δ x + f y k Δ y ) ] ,
W ( f x , f y ) = 4 [ sin 2 ( π f x Δ x ) + sin 2 ( π f y Δ y ) ]
S i ( f x , f y ) = J 1 ( 2 π r f ) π r f ,
S p ( f x , f y ) = S ¯ i ( f x , f y ) | S i ( f x , f y ) | 2 + β 2 ,
S i ( f x , f y ) = sin ( π a f x ) π a f x .
x = r cos ( 2 π t p ) ,
S ( f ) = + s ( x ) exp ( 2 π i f x ) d x .
S ( f ) = 0 p exp [ 2 π i f r cos ( 2 π t p ) ] d t .
S ( f ) = 0 2 π exp ( 2 π i f r cos θ ) d θ = 2 π J 0 ( 2 π f r ) .
S ( f ) = J 0 ( 2 π f r ) .
J 1 ( 2 π r f ) π r f ρ 2 J 1 ( 2 π ρ r f ) π ρ r f .