Abstract

The mean-squared error between an object and image formed by scanning was used by Beall as a measure of the degree of fidelity of the image. A similar approach to the image transfer by fiber optics is presented. In this paper, the one-dimensional line-scan analysis proposed by Beall is extended to encompass a two-dimensional sampling process for evaluating the image transfer in a fiber bundle. Interesting results occur when the evaluation technique is extended to the comparison of the imaging properties of a static bundle and a bundle that is dynamically scanned. This analysis indicates that there is a significant difference of resolution between the static case and the dynamically scanned case. The difference is found to be largely determined by the fiber configuration within the bundle. Experimental results in the case of the static fiber will be presented.

© 1971 Optical Society of America

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References

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  1. N. S. Kapany, J. A. Eyer, and R. F. Keim, J. Opt. Soc. Am. 47, 413 (1957).
    [CrossRef]
  2. P. G. Roetling and W. P. Ganley, J. Opt. Soc. Am. 52, 99 (1962).
    [CrossRef]
  3. R. Drougard, J. Opt. Soc. Am. 54, 907 (1964).
    [CrossRef]
  4. H. Ohzu, T. Sawatari, and K. Sayanagi, J. Appl. Phys. (Japan) Suppl. 1, 4, 323 (1965);T. Sawatari and K. Sayanagi, Oyo Butsuri 34, 3 (1965).
  5. W. H. Beall, J. Opt. Soc. Am. 54, 492 (1964).
    [CrossRef]
  6. For example, Y. W. Lee, Statistical Theory of Communication (Wiley, New York, 1960), p. 219.
  7. J. R. Ragazzini and G. R. Franklin, Sampled Data Control Systems (McGraw–Hill, New York, 1958), p. 250.
  8. E. R. Kretzman, Bell System Tech. J. 31, 751 (1952).
    [CrossRef]
  9. H. Ohzu and H. Kubota, Oyo Butsuri 26, 96 (1957).

1965 (1)

H. Ohzu, T. Sawatari, and K. Sayanagi, J. Appl. Phys. (Japan) Suppl. 1, 4, 323 (1965);T. Sawatari and K. Sayanagi, Oyo Butsuri 34, 3 (1965).

1964 (2)

1962 (1)

1957 (2)

1952 (1)

E. R. Kretzman, Bell System Tech. J. 31, 751 (1952).
[CrossRef]

Beall, W. H.

Drougard, R.

Eyer, J. A.

Franklin, G. R.

J. R. Ragazzini and G. R. Franklin, Sampled Data Control Systems (McGraw–Hill, New York, 1958), p. 250.

Ganley, W. P.

Kapany, N. S.

Keim, R. F.

Kretzman, E. R.

E. R. Kretzman, Bell System Tech. J. 31, 751 (1952).
[CrossRef]

Kubota, H.

H. Ohzu and H. Kubota, Oyo Butsuri 26, 96 (1957).

Lee, Y. W.

For example, Y. W. Lee, Statistical Theory of Communication (Wiley, New York, 1960), p. 219.

Ohzu, H.

H. Ohzu, T. Sawatari, and K. Sayanagi, J. Appl. Phys. (Japan) Suppl. 1, 4, 323 (1965);T. Sawatari and K. Sayanagi, Oyo Butsuri 34, 3 (1965).

H. Ohzu and H. Kubota, Oyo Butsuri 26, 96 (1957).

Ragazzini, J. R.

J. R. Ragazzini and G. R. Franklin, Sampled Data Control Systems (McGraw–Hill, New York, 1958), p. 250.

Roetling, P. G.

Sawatari, T.

H. Ohzu, T. Sawatari, and K. Sayanagi, J. Appl. Phys. (Japan) Suppl. 1, 4, 323 (1965);T. Sawatari and K. Sayanagi, Oyo Butsuri 34, 3 (1965).

Sayanagi, K.

H. Ohzu, T. Sawatari, and K. Sayanagi, J. Appl. Phys. (Japan) Suppl. 1, 4, 323 (1965);T. Sawatari and K. Sayanagi, Oyo Butsuri 34, 3 (1965).

Bell System Tech. J. (1)

E. R. Kretzman, Bell System Tech. J. 31, 751 (1952).
[CrossRef]

J. Appl. Phys. (Japan) Suppl. (1)

H. Ohzu, T. Sawatari, and K. Sayanagi, J. Appl. Phys. (Japan) Suppl. 1, 4, 323 (1965);T. Sawatari and K. Sayanagi, Oyo Butsuri 34, 3 (1965).

J. Opt. Soc. Am. (4)

Oyo Butsuri (1)

H. Ohzu and H. Kubota, Oyo Butsuri 26, 96 (1957).

Other (2)

For example, Y. W. Lee, Statistical Theory of Communication (Wiley, New York, 1960), p. 219.

J. R. Ragazzini and G. R. Franklin, Sampled Data Control Systems (McGraw–Hill, New York, 1958), p. 250.

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Figures (6)

F. 1
F. 1

(a) Relative spectral densities (gaussian shape) of random objects, where the parameter is the standard deviation of the distribution; and (b) the optical transfer function of the dynamic image obtained for a square fiber assembly, where the spatial frequency is normalized by the center-to-center spacing between the adjacent fibers.

F. 2
F. 2

Relative image degradations computed for the static image and the dynamic image, where the subscripts s and d indicate the static and dynamic images, respectively, and 1 and 1 2 show the ratio of the fiber diameter to the spacing of the two adjacent fibers.

F. 3
F. 3

Optical correlator used for the measurement of the image degradation. S, Light source (mercury lamp); L1, condenser lens; PN, pinhole; L2, collimator lens; P1 and P2, transparencies tested; M, microscope stage; L3, condenser lens; Pd, pinhole with diffuser; D, photodetector; and R, recorder.

F. 4
F. 4

(a) Photomicrographs of a photographic emulsion; (b) photomicrographs of images obtained through a fiber bundle; the normalized standard deviations of the object spectral densities (ρ) are indicated.

F. 5
F. 5

(a) Diffraction patterns of the objects (b) those of the image by a fiber bundle (7-μ fiber diameter).

F. 6
F. 6

The relative image degradations of the static images for random objects whose spectral densities are gaussian in shape: experimental (dotted line) and theoretical (solid line).

Equations (26)

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f ( x , y ) = o ( x , y ) t 1 ( x x , y y ) dxdy .
f ( x , y ) = f ( x , y ) δ ( x n α , y m β ) , ( n and m = 0 , ± 1 , ± 2 ) ,
i s ( x , y ) = f ( x , y ) t 2 ( x x , y y ) d x d y .
i d ( x , y ) = o ( x , y ) t ( x x , y y ) dxdy ,
t ( x , y ) = t 1 ( ξ , η ) t 2 ( x ξ , y η ) d ξ d η .
ō ( x , y ) = o ( x , y ) K o
ī ( x , y ) = ( K o / K i ) [ i ( x , y ) K i ] ,
K o = 1 A A o ( x , y ) dxdy
K i = 1 A A i ( x , y ) dxdy .
e ( x , y ) = ō ( x , y ) ī ( x , y ) .
= e 2 ( x , y ) A o 2 A + ( K o 2 / K i 2 ) i 2 A 2 ( K o / K i ) o i A ,
o 2 A = φ o o ( 0 , 0 ) = Φ o o ( ν , τ ) d ν d τ i 2 A = ϕ i i ( 0 , 0 ) = Φ i i ( ν , τ ) d ν d τ
i o A = φ i o ( 0 , 0 ) = Φ i o ( ν , τ ) d ν d τ ,
φ j k ( x , y ) = j ( x x 2 , y x 2 ) k ( x + x 2 , y + y 2 ) d x d y Φ j k ( ν , τ ) = φ j k ( x , y ) e i 2 π ( x ν + y τ ) dxdy , ( j , k = 0 or i ) ,
¯ = 1 K n { φ o o ( 0 , 0 ) + ( K o K i ) 2 φ i i ( 0 , 0 ) 2 K o K i φ i o ( 0 , 0 ) } ,
= 1 K n { [ Φ o o ( ν , τ ) + ( K o K i ) 2 Φ i i ( ν , τ ) 2 K o K i Φ i o ( ν , τ ) ] d ν d τ } ,
K n = ō 2 ( x , y ) = o 2 ( x , y ) K o 2 .
Φ f f ( ν , τ ) = Φ o o ( ν , τ ) | T 1 ( ν , τ ) | 2 ,
Φ f f ( ν , τ ) = 1 α β Φ f f ( ν + n α , τ + m β ) .
Φ i i ( ν , τ ) = | T 2 ( ν , τ ) | 2 1 ( α β ) 2 Φ o o ( ν + n α , τ + m β ) × | T 1 ( ν + n α , τ + m β ) | 2 .
Φ i o ( ν , τ ) = 1 α β Φ o o ( ν , τ ) T 1 ( ν , τ ) T 2 ( ν , τ ) .
¯ s = 1 K n [ Φ o o ( ν , τ ) + 1 ( α β ) 2 K o 2 K i 2 | T 2 ( ν , τ ) | 2 × Φ o o ( ν + n α , τ + m β ) | T 1 ( ν + n α , τ + m β ) | 2 2 α β K o K i Φ o o ( ν , τ ) T 1 ( ν , τ ) T 2 ( ν , τ ) ] d ν d τ .
¯ d = 1 K n Φ o o ( ν , τ ) × [ 1 1 α β K o K i T 1 ( ν , τ ) T 2 ( ν , τ ) ] 2 d ν d τ ,
T 1 ( ν , τ ) = T 2 ( ν , τ ) = sin π a ν π a ν sin π a τ π a τ
Φ o o ( ν , τ ) = C 1 e ρ 2 ( ν 2 + τ 2 ) / 2 + C 2 δ ( ν , τ ) ,
C 1 + C 2 = K o 2 = φ o o ( , ) ,