Abstract

Analytical expressions of average transmittance and the transmittance autocorrelation function of a random-dot model that has a size distribution of grain aggregations are presented. The results are expressed by use of an extended random-dot model that has been developed by the authors. Numerical illustrations of the Wiener spectrum are shown in order to illustrate the effects of uniform distributions of the radius of grain aggregations.

© 1985 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. Tanaka, S. Uchida, “Extended random-dot model,” J. Opt. Soc. Am. 73, 1312–1319 (1983).
    [CrossRef]

1983 (1)

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Extended random-dot model in which grain aggregations (mottles) are allowed to have a size distribution and approximation by a sandwich of N independent sublayers.

Fig. 2
Fig. 2

Continuous distribution of grain aggregations (mottles) size and discrete approximation.

Fig. 3
Fig. 3

Wiener spectrum for grain aggregations (mottles) that have a uniform distribution of radius.

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

t ¯ i = exp [ Q p ( λ i ) d λ × ( 1 exp { [ 1 f ( | x x | ) ] ¯ × λ i ( x x | ) d x } ) d x ] ,
T ¯ = t ¯ 1 t ¯ 2 t ¯ i t ¯ N ,
T ¯ = exp { i = 1 N [ Q p ( λ i ) d λ × ( 1 exp { [ 1 f ( | x x | ) ] ¯ × λ i ( | x x | ) d x } ) d x ] } .
T ¯ = exp [ Q p ( λ ) d λ × ( 1 exp { [ 1 f ( | x x | ¯ ) ] × λ ( | x x | ) d x } ) d x ] .
Φ ( l ) = T ( x ) T ( x + l i x ) ¯ = exp [ Q p ( λ ) d λ × ( 1 exp { [ 1 f ( x x ) f ( x x + l i x ) ] ¯ × λ ( | x x | ) d x } ) d x ] ,
p ( R ) = { 1 / ( R 2 R 1 ) , R 1 < R < R 2 0 , R < R 1 , R > R 2 ,
T ¯ = exp { Q [ 1 exp ( q a ) ] Ā } ,
Φ ( l ) = T ¯ 2 exp [ 2 Q [ 1 exp ( q a ) ] Q ( 1 exp { 2 q a + q a α [ l / ( 2 r ) ] } ) A α [ l / ( 2 R ) ] ¯ ] ,
α ( x ) = { 2 ( cos 1 x x 1 x 2 ) / π , x 1 0 , x > 1 ,
T ¯ = exp { q a [ 1 exp ( q a ) ] × ( M S 1 2 + M S 1 M S 2 + M S 2 2 ) / ( 3 M c ) } ,
Φ ( l ) = T ¯ 2 exp { [ 2 q a [ 1 exp ( q a ) ] / M c q a ( 1 exp { 2 q a + q a α [ l / ( 2 r ) ] } ) / M c ] × { M S 1 2 α [ l / ( 2 M S 1 r ) ] + [ 1 + M S 1 / ( M S 2 M S 1 ) ] × { M S 2 2 α [ l / ( 2 M S 2 r ) ] M S 1 2 α [ l / ( 2 M S 1 r ) ] } + 1 / ( M S 2 M S 1 ) { M S 2 3 β [ l / ( 2 M S 2 r ) ] M S 1 3 β [ l / ( 2 M S 1 r ) ] } ) } ,
β ( x ) = { 2 / ( 3 π ) [ 2 cos 1 x + x 1 x 2 x 3 × ln | x / ( 1 + 1 x 2 ) | ] x 1 0 x > 1 ,
M c = q / Q , M S 1 = R 1 / r , M S 2 = R 2 / r ,
W ( ω ) = 2 π [ Φ ( l ) T ¯ 2 ] J 0 ( 2 π ω l ) l d l ,

Metrics