Abstract

When certain scenes are stored on photographic emulsions, the Fraunhofer diffraction pattern of these scenes exhibits an array of equally spaced bright spikes along the axes that are not accounted for by simple theory. Experimental evidence, obtained with random-checkerboard patterns, supports the proposed theory that these higher orders result from interaction of the stored density variations and the induced relief image in the emulsion.

© 1973 Optical Society of America

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References

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  1. E. Garcia, H. Stark, and R. C. Barker, Appl. Opt. 11, 1480 (1972).
    [Crossref] [PubMed]
  2. J. W. Goodman, J. Opt. Soc. Am. 57, 493 (1967).
    [Crossref] [PubMed]
  3. E. L. O’Neill, Introduction to Statistical Optics (Addison–Wesley, Reading, Mass., 1963), p. 114.
  4. C. B. Burckhardt, Appl. Opt. 9, 695 (1970).
    [Crossref] [PubMed]
  5. Y. Taheda, Y. Oshida, and Y. Miyamura, Appl. Opt. 11, 818 (1972).
    [Crossref]
  6. B. Julesz, Foundations of Cyclopean Perception (University of Chicago Press, Chicago, 1971), Ch. 8.
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill, New York, 1968), p. 14.
  8. H. M. Smith, J. Opt. Soc. Am. 58, 533 (1968).
    [Crossref]
  9. J. Minkoff, Ph.D. thesis, Electrical Eng. Dept., Columbia University (1967), p. 157.
  10. The sample was bleached in Kodak Bleach Bath R-9 (see Kodak Pamphlet No. P-230) and fixed in Kodak Rapid Fixer. The final sample was free from stains and any visually observable density variation.
  11. In the case of ergodic processes, the diffraction pattern of a single sample can be used as an estimate of the power spectrum, provided that the optical-system parameters are correctly chosen and sufficient smoothing is done in the frequency plane to reduce the variability. See, for example, H. Stark, W. R. Bennett, and M. Arm, Appl. Opt. 8, 2165 (1969).
    [Crossref] [PubMed]

1972 (2)

1970 (1)

1969 (1)

1968 (1)

1967 (1)

Arm, M.

Barker, R. C.

Bennett, W. R.

Burckhardt, C. B.

Garcia, E.

Goodman, J. W.

J. W. Goodman, J. Opt. Soc. Am. 57, 493 (1967).
[Crossref] [PubMed]

J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill, New York, 1968), p. 14.

Julesz, B.

B. Julesz, Foundations of Cyclopean Perception (University of Chicago Press, Chicago, 1971), Ch. 8.

Minkoff, J.

J. Minkoff, Ph.D. thesis, Electrical Eng. Dept., Columbia University (1967), p. 157.

Miyamura, Y.

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Addison–Wesley, Reading, Mass., 1963), p. 114.

Oshida, Y.

Smith, H. M.

Stark, H.

Taheda, Y.

Appl. Opt. (4)

J. Opt. Soc. Am. (2)

Other (5)

J. Minkoff, Ph.D. thesis, Electrical Eng. Dept., Columbia University (1967), p. 157.

The sample was bleached in Kodak Bleach Bath R-9 (see Kodak Pamphlet No. P-230) and fixed in Kodak Rapid Fixer. The final sample was free from stains and any visually observable density variation.

E. L. O’Neill, Introduction to Statistical Optics (Addison–Wesley, Reading, Mass., 1963), p. 114.

B. Julesz, Foundations of Cyclopean Perception (University of Chicago Press, Chicago, 1971), Ch. 8.

J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill, New York, 1968), p. 14.

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Figures (9)

Fig. 1
Fig. 1

An RC photograph used in the experiment.

Fig. 2
Fig. 2

The diffraction pattern of an RC photograph showing the axial array of bright spikes.

Fig. 3
Fig. 3

The diffraction pattern of an RC photograph in an index-matching liquid.

Fig. 4
Fig. 4

Relative powers in the diffracted orders: (a) without index-matching liquid; (b) with index-matching liquid.

Fig. 5
Fig. 5

Diffraction pattern of a bleached RC photograph.

Fig. 6
Fig. 6

Power variations in the first and second orders as functions of amplitude transmittance.

Fig. 7
Fig. 7

An example of a more-generalized RC model that displays higher orders in its diffraction pattern. (a) Amplitude transmittance of “on” cell; (b) typical sample function.

Fig. 8
Fig. 8

Schematic representation of edge effects in relief image: (a) cross section of emulsion showing silver image; (b) corresponding transmittance; (c) hypothetical emulsion thickness after drying; (d) portion of relief image that contributes to diffraction pattern.

Fig. 9
Fig. 9

Edge effects in relief image based on emulsion transfer characteristics: (a) characteristic transfer function obtained by Smith (Ref. 8), (b) the relief profile for a dry emulsion when Δ−1 < u0. Excess gelatin has accumulated at edges under pulling action of high-density emulsion. Net pulling forces at center of high-density region is zero; (c) relief profile when Δ−1 > u0. Edge effects are unresolved.

Equations (62)

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f N ( x ) = rect ( x Δ ) * k = 0 N 1 c k δ ( x k Δ ) + c 1 rect [ x Δ / 2 1 / 2 1 ] + c N rect [ x ( N 1 / 2 ) Δ 2 / 2 2 ] ,
rect ( x Δ ) = { 1 , | x | Δ / 2 0 , | x | > Δ / 2 .
F N ( u ) = F [ f N ( x ) ] = Δ sinc u Δ k = 0 N 1 c k e j 2 π u Δ k + 1 c 1 e j 2 π u ( Δ / 2 + 1 / 2 ) sinc u 1 + 2 c N e j 2 π u [ ( N 1 / 2 ) Δ + 2 / 2 ] sinc u 2 .
F N ( u ) = Δ sinc u Δ k = 0 N 1 c k e j 2 π u Δ k .
W ( u ) = lim N [ | F N ( u ) | 2 / N Δ ] .
| F N ( u ) | 2 = Δ 2 sin c 2 u Δ ( N σ 2 + μ 2 N 2 sin c 2 u N Δ sin c 2 u Δ ) ,
lim N | F N ( u ) | 2 N Δ = σ 2 Δ sin c 2 u Δ + μ 2 δ ( u ) ,
lim N N sin c 2 u N Δ sin c 2 u Δ = 1 Δ n δ ( u n Δ ) .
f N ( x ) = g ( x ) * k = 0 N 1 c k δ ( x k Δ ) ,
F N ( u ) = G ( u ) k = 0 N 1 c k e j 2 π u Δ k
| F N ( u ) | 2 = | G ( u ) | 2 N ( σ 2 + μ 2 N sin c 2 u N Δ sin c 2 u Δ ) .
f N ( x ) = [ rect ( x Δ ) * 0 N 1 c k δ ( x k Δ ) ] e j ϕ ( x ) ,
F N ( u ) = ( Δ sin π Δ π u Δ 0 N 1 c k e j 2 π k Δ u ) * [ δ ( u ) + j Φ ( u ) ] ,
| F N ( u ) | 2 = Δ 2 sin c 2 u Δ ( N σ 2 + μ 2 sin c 2 u N Δ sin c 2 u Δ ) + N μ C ϕ ( α ) Λ ( α Δ ) e j 2 π u α d α + k l N 1 l N 1 c k c l × C ϕ ( α ) Λ ( α ( l k ) Δ Δ ) e j 2 π u α d α ,
Λ ( x ) = { 1 | x | , | x | 1 0 , otherwise
| F N ( u ) | 2 = Δ 2 sin c 2 u Δ × ( N σ 2 + μ 2 N 2 sin c 2 u N Δ sin c 2 u Δ ) + N μ W ϕ ( u ) ,
f N ( x ) = k = 0 N 1 c k rect ( x k Δ Δ ) e j ϕ ( x ) ,
ϕ ( x ) = ϕ 0 + β n = 0 N 1 c n rect ( x n Δ Δ ) * h ( x ) ,
f N ( x ) = k = 0 N 1 c k rect ( x k Δ Δ ) × [ 1 + j β n = 0 N 1 c n rect ( x n Δ Δ ) * h ( x ) ] .
n = 0 N 1 c n rect ( x n Δ Δ ) * h ( x ) = l = 0 N b l g ( x l Δ + Δ 2 ) ,
b l = { 0 , c l 1 = c l ( adjacent cells have same density ) 1 , c l 1 c l ( adjacent cells have different density ) .
b l = Prob { b l = 1 | c l 1 = 1 } Prob { c l 1 = 1 } + Prob { b l = 1 | c l 1 = 0 } Prob { c l 1 = 0 } = 2 μ ( 1 μ ) .
b l 2 = 2 μ ( 1 μ ) , 0 l N b l b l 1 = μ ( 1 μ ) , N l 1 b l b l s = 4 μ 2 ( 1 μ ) 2 , 0 l s N , s 0 , 1 0 l N .
f N ( x ) = k = 0 N 1 c k rect ( x k Δ Δ ) + j β n = 0 N 1 l = 0 N 1 c k b l rect ( x k Δ Δ ) g ( x l Δ + Δ 2 ) ;
F N ( u ) = Δ sinc Δ u k = 0 N 1 c k e j 2 π u k Δ + j β k = 0 N 1 l = 0 N c k b l F ( u , k , l ) A N ( u ) + j B N ( u ) ,
F ( u , k , l ) = rect ( x k Δ Δ ) g ( x l Δ + Δ 2 ) e j 2 π u x d x , A N ( u ) = Δ sinc Δ u k = 0 N 1 c k e j 2 π u k Δ , B N ( u ) = β k = 0 N 1 l = 0 N c k b l F ( u , k , l ) .
| F N ( u ) | 2 = | A N ( u ) | 2 + | B N ( u ) | 2 + 2 | A N ( u ) B N ( u ) | sin [ arg A N ( u ) arg B N ( u ) ] .
| A N ( u ) | 2 = N σ 2 Δ 2 sin c 2 Δ u + μ 2 N 2 Δ 2 sin c 2 u N Δ
lim N | B N ( u ) | 2 N Δ = β 2 Δ 2 | Z ( u , Δ ) + Z ( u , Δ ) | 2 × [ μ ( 1 μ ) ] 2 δ ( u n Δ ) + R ( u ) ,
Z ( u , Δ ) = rect ( α Δ ) g ( α Δ 2 ) e j 2 π u α d α ,
f N ( b ) ( x ) = e i ϕ 0 e j [ ϕ ( x ) ϕ 0 ] e j ϕ 0 ( 1 + j [ ϕ ( x ) ϕ 0 ] ) = e j ϕ 0 [ 1 + j β n = 0 N 1 c n rect ( x n Δ Δ ) * [ δ ( x ) + h ( x ) ] ] .
F N ( b ) ( u ) = e j ϕ 0 [ N Δ sinc u N Δ + j β Δ sinc u Δ × ( n = 0 N 1 c n e j 2 π u n + H ( u ) e j π u Δ n = 0 N b n e j 2 π u n ) ] ,
Z ( u , Δ ) + Z ( u , Δ ) = rect ( x Δ ) [ h ( x ) * rect ( x Δ ) ] e j 2 π u x d x = Δ 2 sinc u Δ * H ( u ) sinc u Δ E ( u , Δ ) .
E ( n Δ , Δ ) = ( ) n + 1 π 2 sin 2 π u Δ u ( n / Δ u ) H ( u ) d u ,
B N ( u ) = β k = 0 N 1 l = 0 N c k b l F ( u , k , l ) ,
F ( u , k , l ) = rect ( x k Δ Δ ) g ( x l Δ + Δ 2 ) e j 2 π u x d x .
B N ( u ) = β l = 1 N b l c l 1 F ( u , l 1 , l ) + β l = 0 N 1 b l c l F ( u , l , l ) + l k l 1 k b l c k F ( u , k , l ) .
F ( u , l 1 , l ) = e j 2 π u ( l 1 ) Δ Z ( u , Δ ) ,
F ( u , l , l ) = e j 2 π u l Δ Z ( u , Δ ) ,
Z ( u , Δ ) = rect ( x Δ ) g ( x Δ 2 ) e j 2 π u x d x .
| B N ( u ) | 2 = β 2 | Z ( u , Δ ) | 2 l = 0 N 1 k = 0 N 1 b l + 1 c l b k + 1 c k e j 2 π u ( l k ) Δ + β 2 | Z ( u , Δ ) | 2 l = 0 N 1 k = 0 N 1 b l c l b k c k e j 2 π u ( l k ) Δ + β 2 Z * ( u , Δ ) Z ( u , Δ ) l = 0 N 1 k = 0 N 1 b l + 1 c l b k c k e + j 2 π u ( l k ) Δ + β 2 Z ( u , Δ ) Z * ( u , Δ ) l = 0 N 1 k = 0 N 1 b l + 1 c l b k c k e j 2 π u ( l k ) Δ .
β 2 | Z ( u , Δ ) | 2 l = 0 N 1 k = 0 N 1 b l + 1 c l b k + 1 c k e j 2 π u ( l k ) Δ = β 2 | Z ( u , Δ ) | 2 l = 0 ( k = 0 ) N 1 b l + 1 2 c l 2 [ N terms ] + l = 0 ( k = l + 1 ) N 2 b l + 1 b l + 2 c l c l + 1 e j 2 π u Δ [ ( N 1 ) terms ] + l = 1 ( k = l 1 ) N 1 b l + 1 c l b l c l 1 e j 2 π u Δ [ ( N 1 ) terms ] + l = 0 N 1 k = 0 N 1 ( k l , l ± 1 ) b l + 1 c l b k + 1 c k e j 2 π u ( l k ) u [ ( N 2 ) ( N 1 ) terms ] = β 2 | Z ( u , Δ ) | 2 [ N μ ( 1 μ ) + [ μ ( 1 μ ) ] 2 ( N 2 sin c 2 u N Δ sin c 2 u Δ N 2 ( N 1 ) cos 2 π u Δ ) ] ,
| B N ( u ) | 2 = β 2 | Z ( u , Δ ) + Z ( u , Δ ) | 2 × [ μ ( 1 μ ) ] 2 N 2 sin c 2 u N Δ sin c 2 u Δ + R N ( u ) ,
R N ( u ) = β 2 { | Z ( u , Δ ) | 2 + | Z ( u , Δ ) | 2 } × [ N μ ( 1 μ ) [ μ ( 1 μ ) ] 2 ( N + 2 ( N 1 ) cos 2 π u Δ ) ] + 2 β 2 Re { Z * ( u , Δ ) Z ( u , Δ ) [ ( 1 μ ) 2 μ N [ ( 1 μ ) μ ] 2 ( N + ( N 1 ) e j 2 π u Δ ) ] } .
| B N ( u ) | 2 N Δ | u ± n / Δ β 2 Δ | Z ( u , Δ ) + Z ( u , Δ ) | 2 × [ μ ( 1 μ ) ] 2 n = δ ( u n Δ ) .
lim N | B N ( u ) | 2 N Δ β 2 | sin c u Δ * H ( u ) sin c u Δ | 2 × [ μ ( 1 μ ) ] 2 δ ( u n Δ ) .
E ( n Δ ) = 2 ( ) n + 1 π 2 0 sin 2 π u Δ ( n / Δ ) 2 u 2 H ( u ) d u .
A N ( x , y ) = l = 0 Q 1 k = 0 Q 1 c k l rect ( x k Δ Δ ) rect ( y l Δ Δ ) ,
c k l = { 1 , with probability μ 0 , with probability 1 μ .
A N ( u , υ ) = Δ 2 sin c u Δ sinc υ Δ l = 0 Q 1 k = 0 Q 1 c k l e j 2 π ( u k + υ l ) Δ
| A N ( u , υ ) | 2 = ( Δ sin c u Δ ) 2 ( Δ sin c υ Δ ) 2 × 0 Q 1 0 Q 1 0 Q 1 0 Q 1 c k l c m n e j 2 π u ( k m ) Δ e j 2 π υ ( l n ) Δ ,
c k l c m n = { μ , k = m , l = n μ 2 , otherwise .
| A N ( u , υ ) | 2 = ( Δ sin c u Δ ) 2 ( Δ sin c υ Δ ) 2 × [ μ ( 1 μ ) N + μ 2 N 2 ( sin c 2 u Q Δ sin c 2 u Δ ) ( sin c 2 υ Q Δ sin c 2 υ Δ ) ] .
f N ( x , y ) = l = 0 Q 1 k = 0 Q 1 c k l rect ( x k Δ Δ ) rect ( y l Δ Δ ) × { 1 + j β m = 0 Q n = 0 Q [ b m n 0 g [ x ( m 1 2 ) Δ ] rect ( y n Δ Δ ) + b m n 1 g [ y ( n 1 2 ) Δ ] rect ( x m Δ Δ ) ] } ,
F N ( u , υ ) = A N ( u , υ ) = j B N ( u , υ ) ,
B N ( u , υ ) = β k = 0 Q 1 l = 0 Q 1 m = 0 Q 1 n = 0 Q 1 c k l { b m n 0 d x d y e j 2 π ( u x + υ y ) rect [ x k Δ Δ ] rect [ y l Δ Δ ] rect [ y n Δ Δ ] g [ x ( m 1 2 ) Δ ] + b m n 1 d x d y e j 2 π ( u x + υ y ) rect [ x k Δ Δ ] rect [ y l Δ Δ ] rect [ y m Δ Δ ] g [ y ( n 1 2 ) ] } .
b m n 0 = { 1 , if c m n c m 1 , n 0 , otherwise b m n 1 = { 1 , if c m n c m , n 1 0 , otherwise .
B N ( u , υ ) = β [ k = 0 Q 1 l = 0 Q 1 c k l b k l 0 F ( u , k , k ) Δ sinc Δ υ e j 2 π υ l Δ + k = 0 Q 1 l = 0 Q 1 c k l b k l 1 F ( υ , l , l ) Δ sinc Δ u e j 2 π u k Δ + k = 0 Q 1 l = 0 Q 1 c k l b k + 1 , l , 0 F ( u , k , k + 1 ) Δ sinc Δ υ e j 2 π υ l Δ + k = 0 Q 1 l = 0 Q 1 c k l b k , l + 1 , 1 F ( υ , l , l + 1 ) Δ sinc Δ u e j 2 π u l Δ ] .
B N ( u , υ ) = β Z ( u , Δ ) Δ sinc Δ υ 0 Q 1 0 Q 1 c k l b k l 0 e j 2 π Δ ( u k + υ l ) + β Z ( υ , Δ ) Δ sinc Δ u 0 Q 1 0 Q 1 c k l b k l 1 e j 2 π Δ ( u k + υ l ) + β Z ( υ , Δ ) Δ sinc Δ υ 0 Q 1 0 Q 1 c k l b k + 1 , l , 0 e j 2 π Δ ( u k + υ l ) + β Z ( υ , Δ ) Δ sinc Δ u 0 Q 1 0 Q 1 c k l b k , l + 1 , 1 e j 2 π Δ ( u k + υ l ) .
c k l b k l p c m n b m n q = c k l b k l p c m n b m n q = [ μ ( 1 μ ) ] 2 ,
| B N ( u , υ ) | 2 = β 2 Δ 2 N 2 [ μ ( 1 μ ) ] 2 ( sin c 2 Q u Δ sin c 2 u Δ ) ( sin c 2 Q υ Δ sin c 2 υ Δ ) × { sin c 2 u Δ | Z ( υ , Δ ) + Z ( υ , Δ ) | 2 + sin c 2 υ Δ | Z ( u , Δ ) + Z ( u , Δ ) | 2 + 2 sin c u Δ sinc υ Δ Re [ Z ( u , Δ ) + Z ( u , Δ ) ] [ Z ( υ , Δ ) + Z ( υ , Δ ) ] * } + R N ( u , υ ) ,
lim Q | B N ( u , υ ) | 2 [ Δ Q ] [ Δ Q ] = β 2 [ μ ( 1 μ ) ] 2 | Z ( υ , Δ ) + Z ( υ , Δ ) | 2 δ ( u , υ n Δ ) + β 2 [ μ ( 1 μ ) ] 2 × | Z ( u , Δ ) + Z ( u , Δ ) | 2 δ ( u n Δ , υ ) + β 2 [ μ ( 1 μ ) ] 2 δ ( u ) δ ( υ ) | Z ( 0 , Δ ) + Z ( 0 , Δ ) | 2 ,