Abstract

The effect of film platen vibration during exposure is considered from the standpoint of the degradation of the spectral density of the exposure random process. From statistical considerations, the correlation function of the exposure random process subjected to a vibrational spatial displacement is first derived. It is shown that the derived correlation function is the Fourier transform of the product of the square of the optical system transfer function, the spectral density of the input process, and a filter function whose attenuation characteristic is a function of exposure time, and spatial frequency. It is shown that the effect of vibration is equivalent to the insertion of an additional spatial filter in cascade with the optical system filter. The filter function attributable to vibration is shown to be the double time integral of the joint characteristic function of the vibration random process. The filter function is solved explicitly for the case where the vibration process is Gaussian or normal.

© 1963 Optical Society of America

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References

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  1. L. A. Zadeh, Proc. IRE 40, 564 (1952).
    [CrossRef]
  2. B. K. Wernike, “Collections of Graphs to be Used as Tables for Determining or Evaluating the Combined Effect of Image Motion and Camera-Lens-Film Resolving Power on the Capabilities of Aerial Camera,” WADC Tech. Note 58-321.
  3. Max R. Nagel, J. Opt. Soc. Am. 51, 780 (1961).
    [CrossRef]
  4. R. N. Wolfe and S. A. Tuccio, Phot. Sci. Eng. 4, 6 (1961).
  5. T. Trott, Photogrammetric Eng. XXXVI, 5 (1960).
  6. R. M. Scott, Phot. Sci. Rev. 3, 5 (1959).
  7. D. P. Paris, Phot. Sci. Eng. 6, 1 (1962).
  8. A. Lohmann, Opt. Acta 6, 4 (1959).

1962 (1)

D. P. Paris, Phot. Sci. Eng. 6, 1 (1962).

1961 (2)

Max R. Nagel, J. Opt. Soc. Am. 51, 780 (1961).
[CrossRef]

R. N. Wolfe and S. A. Tuccio, Phot. Sci. Eng. 4, 6 (1961).

1960 (1)

T. Trott, Photogrammetric Eng. XXXVI, 5 (1960).

1959 (2)

R. M. Scott, Phot. Sci. Rev. 3, 5 (1959).

A. Lohmann, Opt. Acta 6, 4 (1959).

1952 (1)

L. A. Zadeh, Proc. IRE 40, 564 (1952).
[CrossRef]

Lohmann, A.

A. Lohmann, Opt. Acta 6, 4 (1959).

Nagel, Max R.

Paris, D. P.

D. P. Paris, Phot. Sci. Eng. 6, 1 (1962).

Scott, R. M.

R. M. Scott, Phot. Sci. Rev. 3, 5 (1959).

Trott, T.

T. Trott, Photogrammetric Eng. XXXVI, 5 (1960).

Tuccio, S. A.

R. N. Wolfe and S. A. Tuccio, Phot. Sci. Eng. 4, 6 (1961).

Wernike, B. K.

B. K. Wernike, “Collections of Graphs to be Used as Tables for Determining or Evaluating the Combined Effect of Image Motion and Camera-Lens-Film Resolving Power on the Capabilities of Aerial Camera,” WADC Tech. Note 58-321.

Wolfe, R. N.

R. N. Wolfe and S. A. Tuccio, Phot. Sci. Eng. 4, 6 (1961).

Zadeh, L. A.

L. A. Zadeh, Proc. IRE 40, 564 (1952).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Acta (1)

A. Lohmann, Opt. Acta 6, 4 (1959).

Phot. Sci. Eng. (2)

D. P. Paris, Phot. Sci. Eng. 6, 1 (1962).

R. N. Wolfe and S. A. Tuccio, Phot. Sci. Eng. 4, 6 (1961).

Phot. Sci. Rev. (1)

R. M. Scott, Phot. Sci. Rev. 3, 5 (1959).

Photogrammetric Eng. (1)

T. Trott, Photogrammetric Eng. XXXVI, 5 (1960).

Proc. IRE (1)

L. A. Zadeh, Proc. IRE 40, 564 (1952).
[CrossRef]

Other (1)

B. K. Wernike, “Collections of Graphs to be Used as Tables for Determining or Evaluating the Combined Effect of Image Motion and Camera-Lens-Film Resolving Power on the Capabilities of Aerial Camera,” WADC Tech. Note 58-321.

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Figures (2)

Fig. 1
Fig. 1

Object illuminance sample function. Object luminance: u(x); object luminance spatially displaced by υ(t): u[x + υ(t)]; image illuminance: h ( α ) u [ x + υ ( t ) α ] d α ;

Fig. 2
Fig. 2

Effective filter function attributable to Gaussian vibration with autocorrelation of form σ2 exp(−a|τ|).

Equations (49)

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u ( x ) = h ( α ) u ( x α ) d α .
u ( x ) = h ( α ) u [ x + υ ( t ) α ] d α .
i ( x , T ) = 0 T h ( α ) u [ x + υ ( t ) α ] d α d t ,
i ( x 1 , T ) = 0 T h ( α ) u [ x 1 + υ ( t 1 ) α ] d α d t 1 ,
i ( x 2 , T ) = 0 T h ( β ) u [ x 2 + υ ( t 2 ) β ] d β d t 2 .
i ( x 1 , T ) i ( x 2 , T ) = 0 T 0 T h ( α ) h ( β ) u [ x 1 + υ ( t 1 ) α ] × u [ x 2 + υ ( t 2 ) β ] d α d β d t 1 d t 2 .
u ( x ) = 1 2 π g ( η ) exp ( j η x ) d η ,
u [ x 1 + υ ( t 1 ) α ] = 1 2 π g ( η 1 ) exp { j η 1 [ x 1 + υ ( t 1 ) α ] } d η 1 ,
u [ x 2 + υ ( t 2 ) β ] = 1 2 π g ( η 2 ) exp { j η 2 [ x 2 + υ ( t 2 ) β ] } d η 2 .
i ( x 1 , T ) i ( x 2 , T ) = 1 ( 2 π ) 2 0 T 0 T h ( α ) exp ( j η 1 α ) d α h ( β ) exp ( j η 2 β ) d β × g ( η 1 ) g ( η 2 ) exp [ j η 1 υ ( t 1 ) + j η 2 υ ( t 2 ) ] exp ( j η 1 x 1 + j η 2 x 2 ) d η 1 d η 2 d t 1 d t 2 .
g ( η ) = u ( x ) exp ( j η x ) d x ,
g ( η 1 ) = u ( ζ 1 ) exp ( j η 1 ζ 1 ) d ζ 1 ,
g ( η 2 ) = u ( ζ 2 ) exp ( j η 2 ζ 2 ) d ζ 2 .
i ( x 1 , T ) i ( x 2 , T ) = 1 ( 2 π ) 2 0 T 0 T h ( α ) exp ( j η 1 α ) d α h ( β ) exp ( j η 2 β ) d β × u ( ζ 1 ) u ( ζ 2 ) exp ( j η 1 ζ 1 j η 2 ζ 2 ) d ζ 1 d ζ 2 × exp [ j η 1 υ ( t 1 ) + j η 2 υ ( t 2 ) ] exp ( j η 1 x 1 + j η 2 x 2 ) d η 1 d η 2 d t 1 d t 2 .
E ( I 1 I 2 ) = 1 ( 2 π ) 2 0 T 0 T h ( α ) exp ( j η 1 α ) d α h ( β ) exp ( j η 2 β ) d β × E ( U 1 U 2 ) exp ( j η 1 ζ 1 j η 2 ζ 2 ) d ζ 1 d ζ 2 × E [ exp ( j η 1 V 1 + j η 2 V 2 ) ] exp ( j η 1 x 1 + j η 2 x 2 ) d η 1 d η 2 d t 1 d t 2 .
E ( U 1 U 2 ) = R ( ζ 2 ζ 1 ) ,
E ( U 1 U 2 ) exp ( j η 1 ζ 1 j η 2 ζ 2 ) d ζ 1 d ζ 2 = R ( ζ ) exp ( j η 1 ζ ) d ζ exp ( j η 1 ζ 1 j η 2 ζ 2 ) d ζ 2 .
E ( U 1 U 2 ) exp ( j η 1 ζ 1 j η 2 ζ 2 ) d ζ 1 d ζ 2 = R ( ζ ) exp ( j η 1 ζ ) d ζ exp [ j ζ 2 ( η 1 + η 2 ) ] d ζ 2 .
R ( ζ ) exp ( j η 1 ζ ) d ζ = S u ( η 1 ) .
E ( U 1 U 2 ) exp ( j η 1 ζ 1 j η 2 ζ 2 ) d ζ 1 d ζ 2 = S u ( η 1 ) exp [ j ζ 2 ( η 1 + η 2 ) ] d ζ 2 = S u ( η 1 ) 2 π δ ( η 1 + η 2 ) .
E ( I 1 I 2 ) = 1 2 π 0 T 0 T h ( α ) exp ( j η 1 α ) d α h ( β ) exp ( j η 2 β ) d β S u ( η 1 ) δ ( η 1 + η 2 ) × E [ exp ( j η 1 V 1 + j η 2 V 2 ) ] exp ( j η 1 x 1 + j η 2 x 2 ) d η 1 d η 2 d t 1 d t 2 .
E ( I 1 I 2 ) = 1 2 π 0 T 0 T h ( α ) exp ( j η 1 α ) d α h ( β ) exp ( j η 1 β ) d β S u ( η 1 ) × E [ exp ( j η 1 V 1 j η 1 V 2 ) ] exp [ j η 1 ( x 1 x 2 ) ] d η 1 d t 1 d t 2 .
h ( α ) exp ( j η 1 α ) d α h ( β ) exp ( j η 1 β ) d β = H ( j η 1 ) H * ( j η 1 ) = | H ( j η 1 ) | 2 .
E [ exp ( j η 1 V 1 j η 2 V 2 ) ] = M ( j η 1 , j η 1 ; t 2 t 1 ) .
E ( I 1 I 2 ) = 1 2 π 0 0 T 0 T | H ( j η ) | 2 S u ( η ) M ( j η , j η ; t 2 t 1 ) exp ( j ζ η ) d t 1 d t 2 d η .
R i ( ζ ) = 1 2 π 0 | H ( j η ) | 2 S u ( η ) 0 T 0 T M ( j η , j η ; t 2 t 1 ) d t 1 d t 2 exp ( j ζ η ) d η ,
G ( η , T ) = 0 T 0 T M ( j η , j η ; t 2 t 1 ) d t 1 d t 2 ,
R i ( ζ ) = 1 2 π | H ( j η ) | 2 S u ( η ) G ( η , T ) exp ( j ζ η ) d η ,
M ( j η , j η ; t 2 t 1 ) = exp { [ σ 2 R υ ( t 2 t 1 ) ] } η 2 ,
G ( η , T ) = 0 T 0 T exp { [ σ 2 R υ ( t 2 t 1 ) ] } η 2 d t 1 d t 2 .
G ( η , T ) = 2 exp ( σ 2 η 2 ) 0 T ( T τ ) exp [ R υ ( τ ) η 2 ] d τ ,
G ( η , T ) = exp ( σ 2 η 2 ) T T ( T | τ | ) exp [ R υ ( τ ) η 2 ] d τ ,
G ( η , T ) = exp ( σ 2 η 2 ) ( T | τ | ) exp [ R υ ( τ ) η 2 ] d τ | τ | T .
G ( η , T ) = exp ( σ 2 η 2 ) 2 π g 1 ( β ) g 2 ( β ) d β ,
g 1 ( β ) = ( T | τ | ) exp ( j β τ ) d τ = T 2 sin 2 ( β T / 2 ) ( β T / 2 ) 2 ,
g 2 ( β ) = exp [ R υ ( τ ) η 2 ] exp ( j β τ ) d τ
= k = 0 [ R υ ( τ ) η 2 ] k k ! exp ( j β τ ) d τ .
R υ ( τ ) = σ 2 exp ( a | τ | ) ,
g 2 ( β ) = k = 0 ( σ 2 η 2 ) k k ! [ 2 a k ( a k ) 2 + β 2 ] .
G ( η , T ) = T 2 exp ( σ 2 η 2 ) 2 π k = 0 { ( σ 2 η 2 ) k k ! × sin 2 ( β T / 2 ) ( β T / 2 ) 2 2 a k [ ( a k ) 2 + β 2 ] d β } .
G ( η , T ) = 2 T 2 exp ( σ 2 η 2 ) × k = 0 ( σ 2 η 2 ) k k ! [ k a T + exp ( k a T ) 1 ] ( k a T ) 2 ,
G ( η , T ) T 2 = exp ( σ 2 η 2 ) × { 1 + 2 k = 1 ( σ 2 η 2 ) k ! [ k a T + exp ( k a T ) 1 ] ( k a T ) 2 } .
lim σ η 0 [ G ( η , T ) / T 2 ] = 1 ,
lim σ η [ G ( η , T ) / T 2 ] = 0 ,
lim a T 0 [ G ( η , T ) / T 2 ] = 1
lim a T [ G ( η , T ) / T 2 ] = exp ( σ 2 η 2 ) .
i = 1 m exp ( a i | τ | cos β m τ ) ,
h ( α ) u [ x + υ ( t ) α ] d α ;
i ( x , T ) = 0 T h ( α ) u [ x + υ ( t ) α ] d α d t .