Abstract

The thesis of this paper is that the only fully adequate way to describe the granularity of photographic materials is by means of a film noise spectrum. The film noise spectrum bears the same relation to granularity that the power spectrum (of electrical circuit theory) bears to electrical noise. The film noise spectrum includes all of the information in the previous granularity measures and readily interrelates them. It goes beyond the previous measures in that it leads to the solution of many of the signal-to-noise problems that arise in connection with the detection of target images on photographic materials.

The film noise spectrum is defined and discussed in Parts 2 and 3. Its relation to the older methods of description involving granularity coefficients, both ordinary and syzygetic, is the subject of Part 4. The superiority of the new method is explained in detail in Part 5. Part 6 employs the film noise spectrum to derive the relation between ordinary granularity and syzygetic granularity. Part 7 discusses photoelectric scanning of the film and derives general expressions for the signal and for the noise. These expressions are used in Parts 8 and 9 to derive a particularly convenient method for the measurement of the film noise spectrum, and to solve the general problem of maximizing the signal-to-noise ratio in the detection of target images. Part 10 discusses the film noise spectrum of Super-XX film.

© 1955 Optical Society of America

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References

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  1. Norbert Wiener, Acta Math. (Stockholm) 55, 117–258 (1930).
  2. S. O. Rice, Bell System Tech. J. 23, 282–333 (1944), pp. 285, 311.
    [Crossref]
  3. J. L. Lawson and G. E. Uhlenbeck, Threshold Signals (McGraw-Hill Book Company, Inc., New York, 1950), pp. 33–42, esp. pp. 39–40.
  4. A. Van der Ziel, Noise (Prentice Hall, Inc., New York, 1954), pp. 310–318, esp. pp. 314–318.
  5. Norbert Wiener, Stationary Time Series (John Wiley and Sons, Inc., New York, 1949), pp. 42–43.
  6. Elias, Grey, and Robinson, J. Opt. Soc. Am. 42, 127–134 (1952).
    [Crossref]
  7. Peter B. Fellgett, J. Opt. Soc. Am. 43, 271–282 (1953).
    [Crossref]
  8. L. A. Jones and G. C. Higgins, J. Opt. Soc. Am. 36, 203–227 (1946).
    [Crossref]
  9. E. T. Whittaker and G. N. Watson, Modern Analysis (Cambridge University Press, London, 1935), fourth edition, pp. 188–189.
  10. G. N. Watson, Bessel Functions (Cambridge University Press, London, 1944), second edition, p. 47.
  11. See reference 10, p. 45.
  12. L. A. Jones and G. C. Higgins, J. Opt. Soc. Am. 37, 217–263 (1947).
    [Crossref] [PubMed]
  13. See reference 10, pp. 24, 100, 393, and 394.
  14. H. J. Zweig, J. Opt. Soc. Am. 45, 410 (1955).

1955 (1)

H. J. Zweig, J. Opt. Soc. Am. 45, 410 (1955).

1953 (1)

1952 (1)

1947 (1)

1946 (1)

1944 (1)

S. O. Rice, Bell System Tech. J. 23, 282–333 (1944), pp. 285, 311.
[Crossref]

1930 (1)

Norbert Wiener, Acta Math. (Stockholm) 55, 117–258 (1930).

Elias,

Fellgett, Peter B.

Grey,

Higgins, G. C.

Jones, L. A.

Lawson, J. L.

J. L. Lawson and G. E. Uhlenbeck, Threshold Signals (McGraw-Hill Book Company, Inc., New York, 1950), pp. 33–42, esp. pp. 39–40.

Rice, S. O.

S. O. Rice, Bell System Tech. J. 23, 282–333 (1944), pp. 285, 311.
[Crossref]

Robinson,

Uhlenbeck, G. E.

J. L. Lawson and G. E. Uhlenbeck, Threshold Signals (McGraw-Hill Book Company, Inc., New York, 1950), pp. 33–42, esp. pp. 39–40.

Van der Ziel, A.

A. Van der Ziel, Noise (Prentice Hall, Inc., New York, 1954), pp. 310–318, esp. pp. 314–318.

Watson, G. N.

E. T. Whittaker and G. N. Watson, Modern Analysis (Cambridge University Press, London, 1935), fourth edition, pp. 188–189.

G. N. Watson, Bessel Functions (Cambridge University Press, London, 1944), second edition, p. 47.

Whittaker, E. T.

E. T. Whittaker and G. N. Watson, Modern Analysis (Cambridge University Press, London, 1935), fourth edition, pp. 188–189.

Wiener, Norbert

Norbert Wiener, Acta Math. (Stockholm) 55, 117–258 (1930).

Norbert Wiener, Stationary Time Series (John Wiley and Sons, Inc., New York, 1949), pp. 42–43.

Zweig, H. J.

H. J. Zweig, J. Opt. Soc. Am. 45, 410 (1955).

Acta Math. (Stockholm) (1)

Norbert Wiener, Acta Math. (Stockholm) 55, 117–258 (1930).

Bell System Tech. J. (1)

S. O. Rice, Bell System Tech. J. 23, 282–333 (1944), pp. 285, 311.
[Crossref]

J. Opt. Soc. Am. (5)

Other (7)

See reference 10, pp. 24, 100, 393, and 394.

E. T. Whittaker and G. N. Watson, Modern Analysis (Cambridge University Press, London, 1935), fourth edition, pp. 188–189.

G. N. Watson, Bessel Functions (Cambridge University Press, London, 1944), second edition, p. 47.

See reference 10, p. 45.

J. L. Lawson and G. E. Uhlenbeck, Threshold Signals (McGraw-Hill Book Company, Inc., New York, 1950), pp. 33–42, esp. pp. 39–40.

A. Van der Ziel, Noise (Prentice Hall, Inc., New York, 1954), pp. 310–318, esp. pp. 314–318.

Norbert Wiener, Stationary Time Series (John Wiley and Sons, Inc., New York, 1949), pp. 42–43.

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Figures (7)

Fig. 1
Fig. 1

These two-dimensional sinusoidal densitance distributions play the same role in the present theory as the sinusoidal functions of time play in electrical circuit theory. The distribution in (a) has the wave number K=1.59, 1.08 cpc, and the distribution in (b) corresponds to K=0.49, −0.41 cpc, where cpc means cycles per centimeter.

Fig. 2
Fig. 2

The four functions here plotted show how each of the four granularity coefficients weights the film noise spectrum. The four functions are defined by Eqs. (4.6), (4.15), (6.3), and (6.6). The abscissa z is a dimensionless wave number defined by z=2πka.

Fig. 3
Fig. 3

The aperture employed in measuring syzygetic granularity. The mean densitance in the right half is reckoned positive, and the mean densitance in the left half is reckoned negative.

Fig. 4
Fig. 4

The upper curve shows the shape of the target given by Eq. (9.4), and the lower curve shows the shape of the optimum aperture as given by Eq. (9.6).

Fig. 5
Fig. 5

The (blurred) film noise spectrum of Super-XX film as computed from the Selwyn granularity data of Jones and Higgins by means of the approximate Eq. (10.1). The ordinate is the film noise spectrum in square microns, and the abscissa is the wave number in cycles per centimeter. The three curves correspond to three different mean densities. The dashed curve is Eqs. (10.2) and (10.7).

Fig. 6
Fig. 6

This plot considers in further detail the film noise spectrum of Super-XX film for a mean density D ¯=0.28. The abscissa is linear from 0 to 400, and logarithmic beyond. The curve through the circles is the curve plotted from the data in Table I and is the same as the lowest curve in Fig. 5. The lower solid curve is the film noise spectrum of the ideal film with randomly positioned grains as given by Eq. (10.4). The dashed curve corresponds to Eqs. (10.2) and (10.6).

Fig. 7
Fig. 7

This plot considers in further detail the film noise spectrum of Super-XX film for a mean density D ¯=1.76. The abscissa is linear from 0 to 80, and logarithmic beyond. The curve through the circles is the curve plotted from the data in Table I and is the same as the top curve in Fig. 5. The lower solid curve is the curve predicted by the ideal film in accordance with Eq. (10.5). The dashed curve is given by Eqs. (10.2) and (10.7).

Equations (75)

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u ( Y ) = 1 y 1 2 + y 2 2 < a 2 , u ( Y ) = 0 y 1 2 + y 2 2 > a 2 .
D ( X ) = A - 1 D ( X + Y ) u ( Y ) d y 1 d y 2 ,
D A - 1 A D ( X ) d x 1 d x 2
G ( X ) D ( X ) - D .
G ( X ) = A - 1 G ( X + Y ) u ( Y ) d y 1 d y 2 .
F ( K ) A G ( X ) e - 2 π i K X d x 1 d x 2 ,
K X k 1 x 1 + k 2 x 2 .
v ( K ) A - 1 u ( Y ) e - 2 π i K Y d y 1 d y 2 .
G ( X ) = F ( K ) e 2 π i K X d k 1 d k 2 ,
u ( Y ) = A v ( K ) e 2 π i K Y d k 1 d k 2 .
k 1 = 1.59 cycles per centimeter k 2 = 1.08 cycles per centimeter
k 1 =             0.49 cycles per centimeter k 2 = - 0.41 cycles per centimeter .
F ( K ) = v ( K ) F ( K ) .
ϕ ( Z ) a - 1 A G ( X + Z ) G ( X ) d x 1 d x 2
n ( K ) A - 1 F ( K ) 2
n ( K ) = ϕ ( Z ) cos 2 π K Z d z 1 d z 2
ϕ ( Z ) = n ( K ) cos 2 π K Z d z 1 d z 2 .
n ( K ) = v ( K ) 2 n ( K ) .
G 2 Av = ϕ ( 0 ) = n ( K ) d k 1 d k 2
z Z = ( z 1 2 + z 2 2 ) 1 2 .
k K = ( k 1 2 + k 2 2 ) 1 2 .
ϕ ( Z ) ϕ ( z )
n ( K ) n ( k ) .
n ( k ) = 2 π 0 J 0 ( 2 π k z ) ϕ ( z ) z d z ,
ϕ ( z ) = 2 π 0 J 0 ( 2 π k z ) n ( k ) k d k ,
G 2 Av = 2 π 0 n ( k ) k d k ,
v ( k ) = 2 π A - 1 0 J 0 ( 2 π k y ) u ( y ) y d y .
v ( k ) = ( 2 / a 2 ) 0 a J 0 ( 2 π k y ) y d y
v ( k ) = J 1 ( 2 π k a ) / ( π k a ) .
D = k - 1 n D n
G 2 Av = ( D - D ) 2 Av = k - 1 n ( D n - D ) 2 .
S 2 ( a ) 2 π a 2 G 2 Av = 2 A G 2 Av
S 2 ( a ) = 4 0 J 1 2 ( 2 π k a ) n ( k ) d k / k ,
S 2 ( a ) = 2 0 H ( z ) n ( z / 2 π a ) d z ,
H ( z ) 2 J 1 2 ( z ) / z .
S 2 = 2 n .
G 2 Av = k - 1 n Δ n 22 .
S s 2 ( a ) 2 A G 2 Av ,
A = 2 ρ π a 2
ρ 2 3 + 3 1 2 / 2 π = 0.942.
v ( k 1 , k 2 ) = sin ( 3 1 2 π k 1 a ) J 1 ( 2 π k a ) / π ρ k a ,
S s 2 ( a ) = ( 4 π / ρ ) sin 2 ( 3 1 2 π k 1 a ) × J 1 2 ( 2 π k a ) n ( k ) k - 2 d k 1 d k 2 .
S s 2 ( a ) = 2 0 M ( z ) n ( z / 2 π a ) d z
M ( z ) 2 J 1 2 ( z ) [ 1 - J 0 ( 3 1 2 z ) ] / ρ z .
S s 2 = S 2 = 2 n
S ¯ 2 ( a ) [ 64 S 2 ( a ) - 20 S 2 ( 2 a ) + S 2 ( 4 a ) ] / 45.
S ¯ 2 ( a ) = 2 0 H ¯ ( z ) n ( z / 2 π a ) d z
H ¯ ( z ) [ 128 J 1 2 ( z ) - 40 J 1 2 ( 2 z ) + 2 J 1 2 ( 4 z ) ] / 45 z .
S s 2 ( a ) = [ 64 S 2 ( a ) - 20 S 2 ( 2 a ) + S 2 ( 4 a ) ] / 45.
S ¯ s 2 ( a ) [ 1024 S s 2 ( a ) - 80 S s 2 ( 2 a ) + S s 2 ( 4 a ) ] / 945.
S ¯ s 2 ( a ) = 2 0 M ( z ) n ( z / 2 π a ) d z
M ¯ ( z ) [ 1024 M ( z ) - 160 M ( 2 z ) + 4 M ( 4 z ) ] / 945.
G ( f ) d E / d D .
N ( f ) = 2 s - 1 G ( f ) - v ( f / s , k 2 ) 2 n ( f / s , k 2 ) d k 2
( Δ E ) 2 = 0 N ( f ) d f
( Δ E ) 2 = G ( s k 1 ) v ( K ) 2 n ( K ) d k 1 d k 2 .
E ( t , x 2 ) = G ( s k 1 ) v ( K ) × θ ( K ) e - 2 π i ( k 1 s t + k 2 x 2 ) d k 1 d k 2 .
v ( K ) = sin π k 1 w π k 1 w · sin π k 2 l π k 2 l .
n ( k ) = ( π k w ) 2 sin 2 π k w · s l N ( f ) 2 G ( f ) 2 ,             k 1 / 2 l
k > 1 / [ ( 8 R ) 1 2 l ] ,             R Δ k / k
n ( k ) = 1 2 s l N ( s k ) / G ( f ) 2 ,             1 / 2 l k 1 / 2 w .
E 2 ( t , x 2 ) / ( Δ E ) 2 .
E 2 ( t , x 2 ) ( Δ E ) 2 θ ( K ) 2 n ( K ) d k 1 d k 2
v ( K ) = constant · θ * ( K ) / n ( K ) G ( s k 1 )
Q ( x ) = D 0 exp ( - x 2 / 2 b 2 )
n ( k ) = B / k 2 .
u ( y ) = ( 1 - y 2 / 2 b 2 ) exp ( - y 2 / 2 b 2 )
n ( k ) 1 2 S ¯ 2 ( 3.8 / k ) .
n ( k ) = 1.36 α 2 D ¯ Q [ J 1 ( 2 π k α ) / ( π k α ) ] 2
α = 0.75 micron Q = 0.8
n ( k ) = 0.172 [ J 1 ( 4.7 k ) / 2.35 k ] 2             for D ¯ = 0.28 ,
n ( k ) = 1.08 [ J 1 ( 4.7 k ) / 2.35 k ] 2             for D ¯ = 1.76 ,
α = 4.0 micron Q = 0.164             D ¯ = 0.28 ,
α = 16.0 micron Q = 0.0176             D ¯ = 1.76.
l = k p - 1 ,