In recent years several attempts have been made to approximate experimentally determined sine-wave responses of photographic emulsions by a mathematical function. In this paper, curve-fitting techniques, using four types of functions with a single parameter, were applied to those sine-wave response curves which did not show appreciable adjacency effects. The Bravais-Pearson correlation coeffcients indicated that the 1/(1+N2)-type function, suggested by several authors, fits the sine-wave response curves of the tested emulsions best.
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Line-spread functions and sine-wave responses as used by various authors to fit experimental data, and corresponding point-spread functions. (K0 and K1 are Bessel functions of the third kind with complex argument.a)
See, for instance, G. N. Watson, Theory of Bessel Functions (The Macmillan Company, New York, 1944), 2nd ed., pp. 78–80.
Table II
Bravais-Pearson correlation coefficients indicating the precision of fit of four approximation functions to experimental sine-wave responses of photographic emulsions. The highest coefficient for each emulsion is in boldface type. The emulsions marked “B” among Hendeberg’s curves were exposed 27 months before date of expiration, those marked “A” 6 to 12 months before this date. (The value one means that the correlation coefficient is greater than or equal to 0.995).
The parameters “a” for the four approximation functions (for convenience measured in units of 100 mm−1). The emulsions are arranged in order of decreasing “a” so that the total number of deviations from this order is least.
exp(−2πN/a)
exp[−(2πN/a)2]
Microfile
7.1
4.9
6.8
6.8
Lamberts
AGFA-IFF B
7.4
4.6
6.1
5.9
Hendeberg
AGFA-IFF A (different devel.)
6.9
4.5
5.9
5.6
AGFA-IFF A
5.7
3.7
4.9
4.7
Microfile
5.1
3.5
4.9
4.9
Schade
AGFA-IF
5.1
3.3
4.4
4.3
Hendeberg
Fine Grain Aero Exp.
4.4
3.2
4.2
4.5
Eastman Kodak
Panatomic-X (No. 5240)
5.3
3.2
4.2
3.9
Kelly
AGFA-IU B
4.7
3.0
3.9
3.8
Hendeberg
Plus-X Aerecon (thin base)
4.0
2.8
3.9
3.9
Tarkington
AGFA-ISS
4.4
2.8
3.7
3.7
Hendeberg
Plus-X (No. 4231)
3.9
2.5
3.8
3.1
Kelly
Panatomic-X
3.5
2.3
3.0
2.8
Ingelstam et al.
AGFA-IU A
3.5
2.2
2.9
2.9
Hendeberg
Tri-X (No. 5233)
3.6
2.2
2.9
2.7
Kelly
Kodachrome
2.7
2.0
2.7
2.7
Schade
Kodak Fine Grain Pos. Film
2.6
1.9
2.6
2.7
Higgins et al.
Super-XX Aerial Reconn.
2.6
1.9
2.6
2.6
Eastman Kodak
Kodak Background-X Neg. Film
2.6
1.7
2.2
2.1
Kelly
Tri-X Aerecon
2.2
1.5
2.1
2.1
Tarkington
Super-XX Aerographic
2.0
1.4
1.9
1.9
Super-XX
1.9
1.4
1.8
1.8
Schade
Tri-X Plate
1.7
1.0
1.4
1.3
Eyer
103 G Plate
1.6
1.0
1.3
1.2
Note added in proof.—After submission of this paper we became aware of a very recent publication, in which the sine-wave response of photographic emulsions was approximated by the expression (1+2πN/a) exp(−2πN/a) [N. W. Lewis, Nature 189, 909 (1961)]. Using this function and applying our curve-fitting techniques to the available data we got a mean of 0.98 for the correlation coefficient.
Tables (3)
Table I
Line-spread functions and sine-wave responses as used by various authors to fit experimental data, and corresponding point-spread functions. (K0 and K1 are Bessel functions of the third kind with complex argument.a)
See, for instance, G. N. Watson, Theory of Bessel Functions (The Macmillan Company, New York, 1944), 2nd ed., pp. 78–80.
Table II
Bravais-Pearson correlation coefficients indicating the precision of fit of four approximation functions to experimental sine-wave responses of photographic emulsions. The highest coefficient for each emulsion is in boldface type. The emulsions marked “B” among Hendeberg’s curves were exposed 27 months before date of expiration, those marked “A” 6 to 12 months before this date. (The value one means that the correlation coefficient is greater than or equal to 0.995).
The parameters “a” for the four approximation functions (for convenience measured in units of 100 mm−1). The emulsions are arranged in order of decreasing “a” so that the total number of deviations from this order is least.
exp(−2πN/a)
exp[−(2πN/a)2]
Microfile
7.1
4.9
6.8
6.8
Lamberts
AGFA-IFF B
7.4
4.6
6.1
5.9
Hendeberg
AGFA-IFF A (different devel.)
6.9
4.5
5.9
5.6
AGFA-IFF A
5.7
3.7
4.9
4.7
Microfile
5.1
3.5
4.9
4.9
Schade
AGFA-IF
5.1
3.3
4.4
4.3
Hendeberg
Fine Grain Aero Exp.
4.4
3.2
4.2
4.5
Eastman Kodak
Panatomic-X (No. 5240)
5.3
3.2
4.2
3.9
Kelly
AGFA-IU B
4.7
3.0
3.9
3.8
Hendeberg
Plus-X Aerecon (thin base)
4.0
2.8
3.9
3.9
Tarkington
AGFA-ISS
4.4
2.8
3.7
3.7
Hendeberg
Plus-X (No. 4231)
3.9
2.5
3.8
3.1
Kelly
Panatomic-X
3.5
2.3
3.0
2.8
Ingelstam et al.
AGFA-IU A
3.5
2.2
2.9
2.9
Hendeberg
Tri-X (No. 5233)
3.6
2.2
2.9
2.7
Kelly
Kodachrome
2.7
2.0
2.7
2.7
Schade
Kodak Fine Grain Pos. Film
2.6
1.9
2.6
2.7
Higgins et al.
Super-XX Aerial Reconn.
2.6
1.9
2.6
2.6
Eastman Kodak
Kodak Background-X Neg. Film
2.6
1.7
2.2
2.1
Kelly
Tri-X Aerecon
2.2
1.5
2.1
2.1
Tarkington
Super-XX Aerographic
2.0
1.4
1.9
1.9
Super-XX
1.9
1.4
1.8
1.8
Schade
Tri-X Plate
1.7
1.0
1.4
1.3
Eyer
103 G Plate
1.6
1.0
1.3
1.2
Note added in proof.—After submission of this paper we became aware of a very recent publication, in which the sine-wave response of photographic emulsions was approximated by the expression (1+2πN/a) exp(−2πN/a) [N. W. Lewis, Nature 189, 909 (1961)]. Using this function and applying our curve-fitting techniques to the available data we got a mean of 0.98 for the correlation coefficient.
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