The problem of detecting symmetry has been studied by using digitally generated images with random pixel values. The statistical efficiency of humans and a computerized observer, the cross correlator of the image halves, has been evaluated. The efficiency of humans is approximately 100% when the image comprises only a few pixels and is notably better than that of the cross correlator. When the number of pixels in the image is increased, the detectability of symmetry gets better. For human observers detectability saturates, however, on a level corresponding to a modest number of pixels. Human efficiency in detecting symmetry is thus low when the image matrix size is large.
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Detectability Indices of the Ideal Observer (dI′), Cross-Correlating Observer (dc′), and Human Observer A (dh′) for Various Image Parameter Valuesa
s
n
dI′
dc′
dh′
Fh
0.03
0.035
2.5 ± 0.2 (2.71)
2.1 ± 0.2 (1.93)
1.7 ± 0.3
0.38
0.07
2.3 ± 0.2 (2.18)
1.9 ± 0.1 (1.63)
1.3 ± 0.2
0.33
0.14
1.4 ± 0.1 (1.52)
1.1 ± 0.1 (1.17)
0.7 ± 0.2
0.21
0.15
0.175
— (4.32)
2.4 ± 0.2 (2.45)
1.7 ± 0.2
0.15
0.35
3.1 ± 0.3 (2.88)
1.6 ± 0.1 (1.70)
0.8 ± 0.2
0.08
0.70
1.5 ± 0.1 (1.73)
1.0 ± 0.1 (0.97)
0.3 ± 0.2
0.03
0.30
0.35
— (4.68)
2.5 ± 0.2 (2.31)
1.5 ± 0.2
0.10
0.70
3.0 ± 0.2 (2.99)
1.6 ± 0.1 (1.48)
0.7 ± 0.2
0.05
0.60
0.70
— (4.81)
1.9 ± 0.1 (1.98)
1.7 ± 0.2
0.12
1.40
3.1 ± 0.3 (3.03)
1.2 ± 0.1 (1.19)
0.8 ± 0.2
0.07
2.80
1.9 ± 0.1 (1.77)
0.6 ± 0.1 (0.60)
0.6 ± 0.2
0.12
The number of pixel pairs in the image k = 100. s and n are the mean values of the underlying Poisson distributions for the signal and noise contributions, respectively. The detectability indices have been estimated by using Eq. (2). Values in parentheses have been obtained by using Eqs. (12). The last column shows human statistical efficiency Fh. Dashes indicate 100% correct test results; d′ could not be estimated.
Table 2
Detectability Indices of the Ideal Observer (dI′), Cross-Correlating Observer (dc′), and Human Observer A (dh′) for Various Image Parameter Valuesa
k
s
n
dI′
dc′
dh′
Fh
1
0.15
0.35
0.4 ± 0.1
0.3 ± 0.1
0.0 ± 0.1
0.0
4
0.6 ± 0.1
0.4 ± 0.1
0.6 ± 0.1
1.00
16
1.2 ± 0.1 (1.15)
0.6 ± 0.1 (0.68)
0.9 ± 0.2
0.55
100
3.1 ± 0.3 (2.88)
1.6 ± 0.1 (1.70)
0.8 ± 0.2
0.08
625
— (7.21)
— (4.26)
1.1 ± 0.2
0.02
2500
— (14.4)
— (8.51)
0.9 ± 0.2
0.004
1
0.30
0.70
0.3 ± 0.1
0.1 ± 0.1
0.2 ± 0.1
0.65
4
0.6 ± 0.1
0.2 ± 0.1
0.4 ± 0.2
0.40
25
1.00
0.30
— (4.68)
1.6 ± 0.1 (1.48)
2.1 ± 0.3
0.20
0.60
3.6 ± 0.5 (3.48)
1.3 ± 0.1 (1.20)
1.4 ±0.2
0.16
1.00
2.6 ± 0.2 (2.65)
0.8 ± 0.1 (0.93)
1.4 ± 0.2
0.26
k is the number of pixel pairs in the image; s and n are the mean values of the underlying Poisson distributions for the signal and noise contributions, respectively. The detectability indices have been estimated by using Eq. (2). Values in parentheses have been obtained by using Eqs. (12). The last column shows human statistical efficiency Fh. Dashes indicate 100% correct test results; d′ could not be estimated.
Tables (2)
Table 1
Detectability Indices of the Ideal Observer (dI′), Cross-Correlating Observer (dc′), and Human Observer A (dh′) for Various Image Parameter Valuesa
s
n
dI′
dc′
dh′
Fh
0.03
0.035
2.5 ± 0.2 (2.71)
2.1 ± 0.2 (1.93)
1.7 ± 0.3
0.38
0.07
2.3 ± 0.2 (2.18)
1.9 ± 0.1 (1.63)
1.3 ± 0.2
0.33
0.14
1.4 ± 0.1 (1.52)
1.1 ± 0.1 (1.17)
0.7 ± 0.2
0.21
0.15
0.175
— (4.32)
2.4 ± 0.2 (2.45)
1.7 ± 0.2
0.15
0.35
3.1 ± 0.3 (2.88)
1.6 ± 0.1 (1.70)
0.8 ± 0.2
0.08
0.70
1.5 ± 0.1 (1.73)
1.0 ± 0.1 (0.97)
0.3 ± 0.2
0.03
0.30
0.35
— (4.68)
2.5 ± 0.2 (2.31)
1.5 ± 0.2
0.10
0.70
3.0 ± 0.2 (2.99)
1.6 ± 0.1 (1.48)
0.7 ± 0.2
0.05
0.60
0.70
— (4.81)
1.9 ± 0.1 (1.98)
1.7 ± 0.2
0.12
1.40
3.1 ± 0.3 (3.03)
1.2 ± 0.1 (1.19)
0.8 ± 0.2
0.07
2.80
1.9 ± 0.1 (1.77)
0.6 ± 0.1 (0.60)
0.6 ± 0.2
0.12
The number of pixel pairs in the image k = 100. s and n are the mean values of the underlying Poisson distributions for the signal and noise contributions, respectively. The detectability indices have been estimated by using Eq. (2). Values in parentheses have been obtained by using Eqs. (12). The last column shows human statistical efficiency Fh. Dashes indicate 100% correct test results; d′ could not be estimated.
Table 2
Detectability Indices of the Ideal Observer (dI′), Cross-Correlating Observer (dc′), and Human Observer A (dh′) for Various Image Parameter Valuesa
k
s
n
dI′
dc′
dh′
Fh
1
0.15
0.35
0.4 ± 0.1
0.3 ± 0.1
0.0 ± 0.1
0.0
4
0.6 ± 0.1
0.4 ± 0.1
0.6 ± 0.1
1.00
16
1.2 ± 0.1 (1.15)
0.6 ± 0.1 (0.68)
0.9 ± 0.2
0.55
100
3.1 ± 0.3 (2.88)
1.6 ± 0.1 (1.70)
0.8 ± 0.2
0.08
625
— (7.21)
— (4.26)
1.1 ± 0.2
0.02
2500
— (14.4)
— (8.51)
0.9 ± 0.2
0.004
1
0.30
0.70
0.3 ± 0.1
0.1 ± 0.1
0.2 ± 0.1
0.65
4
0.6 ± 0.1
0.2 ± 0.1
0.4 ± 0.2
0.40
25
1.00
0.30
— (4.68)
1.6 ± 0.1 (1.48)
2.1 ± 0.3
0.20
0.60
3.6 ± 0.5 (3.48)
1.3 ± 0.1 (1.20)
1.4 ±0.2
0.16
1.00
2.6 ± 0.2 (2.65)
0.8 ± 0.1 (0.93)
1.4 ± 0.2
0.26
k is the number of pixel pairs in the image; s and n are the mean values of the underlying Poisson distributions for the signal and noise contributions, respectively. The detectability indices have been estimated by using Eq. (2). Values in parentheses have been obtained by using Eqs. (12). The last column shows human statistical efficiency Fh. Dashes indicate 100% correct test results; d′ could not be estimated.
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