It is shown that circular nonmoving targets, detected at threshold, which appear as a “point source” for low values of the subtended visual angle (target diameter), pass smoothly into a subjective annular shape for larger diameters. The annulus is the locale of the luminance gradient and therefore provides the significant visual information. The annulus width is 0.9 min of arc when the adapting luminance B is 102 ft-L and rises to 1.3 min as B falls to 10−3 ft-L. Because of this continuous transition from a point source to an annulus, the “critical visual angle” is at best only an approximation.
Over a range of B from 10−3 to 102 ft-L and a duration of stimulus from 10−2 to 1 sec the threshold energy for a point source rises 140 fold. The corresponding rise in the threshold energy per sq min of annulus area is about 1600 fold.
The trends in threshold energy and in threshold contrast with increasing target size are believed to measure some of the effects of retinal (or neural) interaction and inhibition in the human fovea.
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By definition, one troland is the retinal illuminance E corresponding to an adapting luminance B of 1 candle/meter2 when viewed through an artificial pupil of 1 mm2 area.
Table II
Ricco constants R (troland-sec), for various durations of stimulus t (sec), and adapting luminances B (ft-L).
1/t
B = 0.001
B = 0.01
B = 0.1
B = 1
B = 10
B = 100
1000
1.68
1.69
2.15
3.35
7.63
28.6
300
1.73
1.79
2.22
3.45
7.85
29.5
100
1.68
1.69
2.15
3.35
7.98
28.6
30
1.88
1.78
2.12
3.71
9.55
33.7
10
2.29
2.11
2.42
4.49
13.3
51.8
3
3.91
3.54
4.07
7.88
24.1
114.5
1
10.5
9.36
9.92
18.06
53.7
238.
Table III
Threshold energies ΔQ as a function of target diameter at t = 0.01 sec, B = 0.1 ft − L, for values of d from 4′ to 60′. E = 5.14 trolands. The value of w is calculated to be 1.26′.
Visual angle d′
ΔQ = tEC(πdw)
ΔQ = 2.15 + 0.041 (d − 5.04)
% Difference
4
2.94
2.15 (Ricco)
(27)
6
2.46
2.19
(11)
10
2.36
2.35
0.4
14
2.48
2.52
1.6
20
2.74
2.76
0.7
30
3.17
3.17
0
40
3.58
3.58
0
50
3.99
3.99
0
60
4.02
4.41
10
Table IV
Ring width, w, for 42 combinations of B and t. The unit for w is minutes of visual angle.
1/t
B = 0.001
B = 0.01
B = 0.1
B = 1
B = 10
B = 100
1000
1.29
1.30
1.26
1.14
1.05
0.87
300
1.28
1.32
1.28
1.18
1.05
0.89
100
1.29
1.29
1.26
1.14
1.04
0.87
30
1.28
(1.05)
1.27
1.13
1.06
0.91
10
1.29
1.28
1.24
1.16
1.04
(0.78)
3
(1.63)
1.16
1.15
1.10
0.86
0.94
1
1.18
1.13
1.15
1.19
0.98
0.96
Av
1.27
1.25
1.23
1.15
1.03
0.90
Table V
Critical visual angle dc = 4w for 42 combinations of B and t. The unit for dc is minutes of visual angle.
1/t
B = 0.001
B = 0.01
B = 0.1
B = 1
B = 10
B = 100
1000
5.17
5.18
5.04
4.57
4.19
3.49
300
5.13
5.28
5.10
4.74
4.20
3.54
100
5.15
5.14
5.04
4.57
4.18
3.49
30
5.12
(4.21)
5.09
4.52
4.23
3.64
10
5.15
5.13
4.98
4.65
4.14
(3.12)
3
(6.53)
4.62
4.59
4.42
3.43
3.77
1
4.72
4.52
4.59
4.74
3.91
3.84
Av
5.1
5.0
4.9
4.6
4.1
3.6
Table VI
Critical visual angle, Blackwell 1946.
log B (ft − L)
log C.V.A.
C.V.A.
dc (this paper)
−3
0.8
6.3′
5.1′
−2
0.42
2.6
5.0
−1
0.14
1.4
4.9
0
0.9
4.6
1
0.75
4.1
2
0.7
3.6
3
0.7
⋯
Table VII
Dependence of the critical visual angle dc on the pupil diameter
and the Stiles–Crawford factor S.
log B (ft − L)
dc (min)
S
2
3.6
2.72
0.96
1.38
1
4.1
3.55
0.86
1.34
0
4.6
4.38
0.80
1.31
−1
4.9
5.13
0.73
1.31
−2
5.0
5.80
0.70
1.23
−3
5.1
6.30
0.65
1.25
Table VIII
Coefficient b for various exposure times and adapting luminances, b is expressed in troland-seconds per πw square minutes of annulus subtend of the target.
1/t
B = 0.001
B = 0.01
B = 0.1
B = 1
B = 10
B = 100
1000
0.035
0.035
0.042
0.079
0.310
1.76
300
0.035
0.037
0.043
0.083
0.323
1.91
100
0.034
0.034
0.041
0.079
0.320
1.76
30
0.038
0.029
0.042
0.101
0.431
2.25
10
0.049
0.060
0.046
0.187
0.936
2.57
3
0.234
0.150
0.166
0.445
1.92
13.03
1
0.564
0.449
0.505
1.491
7.46
41.6
Table IX
Values of β = b/πw. The units are troland-seconds per square minute of annulus subtend of the target. B is in ft − L, and E is in trolands.
B = 0.001
B = 0.01
B = 0.1
B = 1
B = 10
B = 100
1/t
E = 0.0706
E = 0.632
E = 50.14
E = 41.5
E = 292
E = 1906
1000
0.0086
0.0085
0.0105
0.0220
0.094
0.642
300
0.0087
0.0088
0.0108
0.0223
0.098
0.687
100
0.0084
0.0084
0.0104
0.0220
0.097
0.642
30
0.0095
0.0088
0.0105
0.0283
0.130
0.785
10
0.0121
0.0149
0.0117
0.0512
0.287
1.62
3
0.0455
0.0413
0.0461
0.128
0.713
4.40
1
0.152
0.126
0.140
0.400
2.43
13.8
Table X
Threshold limen ΔB vs target diameter d, when B = 10−1ft-L, and t = 0.01 sec.
d (min of arc)
ΔB (ft-L)
d (min of arc)
ΔB (ft-L)
1.0
5.15
10
0.112
1.33
2.93
14
0.084
2.0
1.29
20
0.065
2.8
0.66
30
0.050
4.
0.35
40
0.043
6
0.20
50
0.038
Table IV A
Smoothed threshold data, 1-sec duration, values of log threshold contrast.
log target diameter (min)
log background luminance (ft-L)
2.00
1.00
0.00
−1.00
−2.00
−3.00
0.000
−0.798
−0.631
−0.256
0.376
1.276
2.276
0.125
−1.044
−0.866
−0.501
0.131
1.031
2.031
0.301
−1.400
−1.233
−0.858
−0.226
0.674
1.674
0.447
−1.692
−1.525
−1.150
−0.518
0.382
1.382
0.602
−1.892
−1.743
−1.400
−0.777
0.123
1.123
0.778
−2.002
−1.874
−1.603
−0.993
−0.093
0.907
1.000
−2.064
−1.958
−1.746
−1.179
−0.279
0.721
1.146
−2.086
−1.992
−1.831
−1.265
−0.365
0.635
1.301
−2.102
−2.017
−1.880
−1.339
−0.439
0.561
1.477
−2.114
−2.036
−1.921
−1.410
−0.510
0.490
1.602
−2.122
−2.045
−1.941
−1.451
−0.551
0.449
1.699
−2.126
−2.053
−1.952
−1.482
−0.582
0.418
1.778
−2.128
−2.057
−1.960
−1.506
−0.606
0.394
Tables (11)
Table I
Retinal illuminance E for several levels of background luminance B.
By definition, one troland is the retinal illuminance E corresponding to an adapting luminance B of 1 candle/meter2 when viewed through an artificial pupil of 1 mm2 area.
Table II
Ricco constants R (troland-sec), for various durations of stimulus t (sec), and adapting luminances B (ft-L).
1/t
B = 0.001
B = 0.01
B = 0.1
B = 1
B = 10
B = 100
1000
1.68
1.69
2.15
3.35
7.63
28.6
300
1.73
1.79
2.22
3.45
7.85
29.5
100
1.68
1.69
2.15
3.35
7.98
28.6
30
1.88
1.78
2.12
3.71
9.55
33.7
10
2.29
2.11
2.42
4.49
13.3
51.8
3
3.91
3.54
4.07
7.88
24.1
114.5
1
10.5
9.36
9.92
18.06
53.7
238.
Table III
Threshold energies ΔQ as a function of target diameter at t = 0.01 sec, B = 0.1 ft − L, for values of d from 4′ to 60′. E = 5.14 trolands. The value of w is calculated to be 1.26′.
Visual angle d′
ΔQ = tEC(πdw)
ΔQ = 2.15 + 0.041 (d − 5.04)
% Difference
4
2.94
2.15 (Ricco)
(27)
6
2.46
2.19
(11)
10
2.36
2.35
0.4
14
2.48
2.52
1.6
20
2.74
2.76
0.7
30
3.17
3.17
0
40
3.58
3.58
0
50
3.99
3.99
0
60
4.02
4.41
10
Table IV
Ring width, w, for 42 combinations of B and t. The unit for w is minutes of visual angle.
1/t
B = 0.001
B = 0.01
B = 0.1
B = 1
B = 10
B = 100
1000
1.29
1.30
1.26
1.14
1.05
0.87
300
1.28
1.32
1.28
1.18
1.05
0.89
100
1.29
1.29
1.26
1.14
1.04
0.87
30
1.28
(1.05)
1.27
1.13
1.06
0.91
10
1.29
1.28
1.24
1.16
1.04
(0.78)
3
(1.63)
1.16
1.15
1.10
0.86
0.94
1
1.18
1.13
1.15
1.19
0.98
0.96
Av
1.27
1.25
1.23
1.15
1.03
0.90
Table V
Critical visual angle dc = 4w for 42 combinations of B and t. The unit for dc is minutes of visual angle.
1/t
B = 0.001
B = 0.01
B = 0.1
B = 1
B = 10
B = 100
1000
5.17
5.18
5.04
4.57
4.19
3.49
300
5.13
5.28
5.10
4.74
4.20
3.54
100
5.15
5.14
5.04
4.57
4.18
3.49
30
5.12
(4.21)
5.09
4.52
4.23
3.64
10
5.15
5.13
4.98
4.65
4.14
(3.12)
3
(6.53)
4.62
4.59
4.42
3.43
3.77
1
4.72
4.52
4.59
4.74
3.91
3.84
Av
5.1
5.0
4.9
4.6
4.1
3.6
Table VI
Critical visual angle, Blackwell 1946.
log B (ft − L)
log C.V.A.
C.V.A.
dc (this paper)
−3
0.8
6.3′
5.1′
−2
0.42
2.6
5.0
−1
0.14
1.4
4.9
0
0.9
4.6
1
0.75
4.1
2
0.7
3.6
3
0.7
⋯
Table VII
Dependence of the critical visual angle dc on the pupil diameter
and the Stiles–Crawford factor S.
log B (ft − L)
dc (min)
S
2
3.6
2.72
0.96
1.38
1
4.1
3.55
0.86
1.34
0
4.6
4.38
0.80
1.31
−1
4.9
5.13
0.73
1.31
−2
5.0
5.80
0.70
1.23
−3
5.1
6.30
0.65
1.25
Table VIII
Coefficient b for various exposure times and adapting luminances, b is expressed in troland-seconds per πw square minutes of annulus subtend of the target.
1/t
B = 0.001
B = 0.01
B = 0.1
B = 1
B = 10
B = 100
1000
0.035
0.035
0.042
0.079
0.310
1.76
300
0.035
0.037
0.043
0.083
0.323
1.91
100
0.034
0.034
0.041
0.079
0.320
1.76
30
0.038
0.029
0.042
0.101
0.431
2.25
10
0.049
0.060
0.046
0.187
0.936
2.57
3
0.234
0.150
0.166
0.445
1.92
13.03
1
0.564
0.449
0.505
1.491
7.46
41.6
Table IX
Values of β = b/πw. The units are troland-seconds per square minute of annulus subtend of the target. B is in ft − L, and E is in trolands.
B = 0.001
B = 0.01
B = 0.1
B = 1
B = 10
B = 100
1/t
E = 0.0706
E = 0.632
E = 50.14
E = 41.5
E = 292
E = 1906
1000
0.0086
0.0085
0.0105
0.0220
0.094
0.642
300
0.0087
0.0088
0.0108
0.0223
0.098
0.687
100
0.0084
0.0084
0.0104
0.0220
0.097
0.642
30
0.0095
0.0088
0.0105
0.0283
0.130
0.785
10
0.0121
0.0149
0.0117
0.0512
0.287
1.62
3
0.0455
0.0413
0.0461
0.128
0.713
4.40
1
0.152
0.126
0.140
0.400
2.43
13.8
Table X
Threshold limen ΔB vs target diameter d, when B = 10−1ft-L, and t = 0.01 sec.
d (min of arc)
ΔB (ft-L)
d (min of arc)
ΔB (ft-L)
1.0
5.15
10
0.112
1.33
2.93
14
0.084
2.0
1.29
20
0.065
2.8
0.66
30
0.050
4.
0.35
40
0.043
6
0.20
50
0.038
Table IV A
Smoothed threshold data, 1-sec duration, values of log threshold contrast.