Recent work by Jones and Higgins has led to an objective method for evaluating the granular structure of the developed photographic image by which materials of widely differing granularity can be ordered in correspondence with subjective graininess measurements. The present paper gives the theoretical calculations of statistical fluctuations of density and syzygetic density differences (SΔD) for simplified density patterns when scanned with varied-size apertures. Average syzygetic density values, 〈SΔD〉Av, versus density curves are derived first for square, checkerboard density patterns and second for circular patterns containing uniform-size grains with random arrangement. By combining the experimental, threshold, visual, gradient sensitivity curve (〈SΔD〉Avversus density for critical aperture), obtained by Jones and Higgins, with theoretical curves of 〈SΔD〉Avversus density calculated here, it is shown how the graininess-density curve can be obtained.
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The fs values listed in column 4 are “standardized” values of P(m,n) obtained by dividing all the P(m,n) values by the maximum value of P(m,n) in the third column. Values of the transmission T are given in column 5 and the corresponding values of density in column 6.
Table II
Transformed density-frequency data calculated for checkerboard density pattern of Fig. 3.
fs
T
fs×T
(fs×T)s
D=Log1/T
0.019
1.00
0.019
0.033
0.000
0.120
0.90
0.108
0.185
0.046
0.365
0.80
0.292
0.499
0.097
0.715
0.70
0.501
0.856
0.155
0.876
0.65
0.569
0.973
0.187
0.975
0.60
0.585
1.000
0.221
1.000
0.55
0.550
0.940
0.260
0.955
0.50
0.478
0.816
0.301
0.625
0.40
0.250
0.427
0.398
0.310
0.30
0.093
0.159
0.523
0.105
0.20
0.021
0.036
0.699
0.020
0.10
0.002
0.003
1.000
0.000
0.00
0.000
0.000
∞
Table III
SΔD-frequency data calculated for checkerboard density pattern of Fig. 3 for average density 0.2558.
SΔD
f
fs
fs×SΔD
0.000
8.5186
1.000
0.00000
0.05
8.1296
0.954
0.04770
0.10
7.0892
0.932
0.08320
0.15
5.6956
0.669
0.10035
0.20
4.2702
0.501
0.10020
0.25
3.0362
0.356
0.08900
0.30
2.0804
0.244
0.07320
0.35
1.3894
0.163
0.05705
0.40
0.9082
0.107
0.04280
0.45
0.5850
0.069
0.03105
0.50
0.3756
0.044
0.04400
0.55
0.2416
0.028
0.01540
0.60
0.1532
0.018
0.01080
0.65
0.0916
0.011
0.00715
0.70
0.0474
0.006
0.00420
0.75
0.0190
0.002
0.00150
0.80
0.0050
0.001
0.00080
Table IV
Theoretical values of syzygetic density difference 〈SΔD〉Av for the density pattern of Fig. 3, calculated for different-sized scanning apertures φ and at different density levels.
φ
d
No. elem. areas
DAv
Statistical law
〈SΔD〉Av
2μ
1μ
4
3/4
0.1240
Binomial
0.1428
2/4
0.3010
Binomial
0.2816
3μ
1μ
9
7/9
0.1089
Binomial
0.0789
5/9
0.2558
Binomial
0.1371
6μ
1μ
36
30/36
0.0790
Binomial
0.0284
18/36
0.3010
Binomial
0.0669
20μ
1μ
400
300/400
0.1250
Binomial
0.0130
100/400
0.6020
Binomial
0.0365
Table V
Theoretical values of syzygetic density difference 〈SΔD〉Av for the density pattern of Fig. 8, calculated for different-sized scanning apertures φ and at different density levels.
φ
d
a
DAv
Statistical law
〈SΔD〉Av
2μ
1μ
1/4
1
0.1085
Poisson
0.0960
8
0.868
Poisson
0.3080
5μ
1μ
1/25
18
0.312
Normal
0.0680
52
0.900
Normal
0.1220
88
1.520
Normal
0.1520
10μ
1μ
1/100
100
0.434
Normal
0.0411
250
1.085
Normal
0.0622
500
2.170
Normal
0.0925
15μ
1μ
1/225
208
0.400
Normal
0.0274
518
1.000
Normal
0.0411
830
1.600
Normal
0.0547
20μ
1μ
1/400
400
0.434
Normal
0.0207
1200
1.302
Normal
0.0350
25μ
1μ
1/625
578
0.401
Normal
0.0160
1500
1.040
Normal
0.0243
Tables (5)
Table I
Transmission-frequency data calculated by binomial theorem for checkerboard density pattern of Fig. 3.
The fs values listed in column 4 are “standardized” values of P(m,n) obtained by dividing all the P(m,n) values by the maximum value of P(m,n) in the third column. Values of the transmission T are given in column 5 and the corresponding values of density in column 6.
Table II
Transformed density-frequency data calculated for checkerboard density pattern of Fig. 3.
fs
T
fs×T
(fs×T)s
D=Log1/T
0.019
1.00
0.019
0.033
0.000
0.120
0.90
0.108
0.185
0.046
0.365
0.80
0.292
0.499
0.097
0.715
0.70
0.501
0.856
0.155
0.876
0.65
0.569
0.973
0.187
0.975
0.60
0.585
1.000
0.221
1.000
0.55
0.550
0.940
0.260
0.955
0.50
0.478
0.816
0.301
0.625
0.40
0.250
0.427
0.398
0.310
0.30
0.093
0.159
0.523
0.105
0.20
0.021
0.036
0.699
0.020
0.10
0.002
0.003
1.000
0.000
0.00
0.000
0.000
∞
Table III
SΔD-frequency data calculated for checkerboard density pattern of Fig. 3 for average density 0.2558.
SΔD
f
fs
fs×SΔD
0.000
8.5186
1.000
0.00000
0.05
8.1296
0.954
0.04770
0.10
7.0892
0.932
0.08320
0.15
5.6956
0.669
0.10035
0.20
4.2702
0.501
0.10020
0.25
3.0362
0.356
0.08900
0.30
2.0804
0.244
0.07320
0.35
1.3894
0.163
0.05705
0.40
0.9082
0.107
0.04280
0.45
0.5850
0.069
0.03105
0.50
0.3756
0.044
0.04400
0.55
0.2416
0.028
0.01540
0.60
0.1532
0.018
0.01080
0.65
0.0916
0.011
0.00715
0.70
0.0474
0.006
0.00420
0.75
0.0190
0.002
0.00150
0.80
0.0050
0.001
0.00080
Table IV
Theoretical values of syzygetic density difference 〈SΔD〉Av for the density pattern of Fig. 3, calculated for different-sized scanning apertures φ and at different density levels.
φ
d
No. elem. areas
DAv
Statistical law
〈SΔD〉Av
2μ
1μ
4
3/4
0.1240
Binomial
0.1428
2/4
0.3010
Binomial
0.2816
3μ
1μ
9
7/9
0.1089
Binomial
0.0789
5/9
0.2558
Binomial
0.1371
6μ
1μ
36
30/36
0.0790
Binomial
0.0284
18/36
0.3010
Binomial
0.0669
20μ
1μ
400
300/400
0.1250
Binomial
0.0130
100/400
0.6020
Binomial
0.0365
Table V
Theoretical values of syzygetic density difference 〈SΔD〉Av for the density pattern of Fig. 8, calculated for different-sized scanning apertures φ and at different density levels.