Figure 1.
Spatial luminance contrast sensitivity for six of the seven subjects of this study.
Both eyes were separately evaluated. Each data point represents either the right or left eye monocular contrast sensitivity at 11 different spatial frequencies (circles). Dashed curves represent the upper and lower tolerance limits estimated from control subjects (n = 62, 16–30 years old).
Figure 2.
The psychophysical measurements used a modification of the two interval forced choice task, 2IFC. A reference stimulus was compared with a test stimulus having different spatial frequency or different spatial extent. Blank fields were interleaved with reference stimulus and test stimulus. A beep indicated to the subject the moment to respond. Two forced choices were available as subject responses, either “equal stimuli” or “different stimuli”. After each trial, the spatial frequency or the spatial extent of the test stimulus was changed following the method of constant stimuli. A total of 21 different envelope's standard deviations around the reference standard deviation and 19 different grating's spatial frequencies around the reference spatial frequency were tested.
Figure 3.
Psychometric functions obtained from Subject GSS for spatial frequency discrimination and spatialextent discrimination (left and right columns, respectively) at four different Michelson contrasts (100%, 10%, 5%, and 2% from top to bottom, respectively).
Reference stimulus: 0.4 cycles/degree and 1 degree. Data points represent percent of correct responses (filled circles) or incorrect responses (empty squares). Curves are Gaussian fits to the data. The standard deviations of these Gaussian functions were used as measurement of entropy in the spatial frequency domain or spatial domain, respectively.
Figure 4.
Psychometric functions obtained from Subject GSS for spatial frequency discrimination and spatial extent discrimination (left and right columns, respectively) at four different Michelson contrasts (100%, 10%, 5%, and 2% from top to bottom, respectively).
Reference stimulus: 2 cycles/degree and 1 degree. Data points represent percent of correct responses (filled circles) or incorrect responses (empty squares). Curves are Gaussian fits to the data. The standard deviations of these Gaussian functions were used as measurement of entropy in the spatial frequency domain or spatial domain, respectively.
Figure 5.
Psychometric functions obtained from Subject GSS for spatial frequency discrimination and spatial extent discrimination (left and right columns, respectively) at four different Michelson contrasts (100%, 10%, 5%, and 2% from top to bottom, respectively).
Reference stimulus: 10 cycles/degree and 1 degree. Data points represent percent of correct responses (filled circles) or incorrect responses (empty squares). Curves are Gaussian fits to the data. The standard deviations of these Gaussian functions were used as measurement of entropy in the spatial frequency domain or spatial domain, respectively.
Figure 6.
Mean psychometric functions for spatial frequency discrimination and spatialextent discrimination (left and right columns, respectively) at four different Michelson contrasts (100%, 10%, 5%, and 2% from top to bottom, respectively).
Reference stimulus: 0.4 cycles/degree and 1 degree. Filled circles and empty squares represent means for correct responses and incorrect responses, respectively, obtained from the seven subjects. Vertical bars represent the standard errors of the means. Curves are Gaussian fits to the data. The standard deviations of these Gaussian functions were used as measurement of entropy in the spatial frequency domain or spatial domain, respectively.
Figure 7.
Mean psychometric functions for spatial frequency discrimination and spatial extent discrimination (left and right columns, respectively) at four different Michelson contrasts (100%, 10%, 5%, and 2% from top to bottom, respectively).
Reference stimulus: 2 cycles/degree and 1 degree. Filled circles and empty squares represent means for correct responses and incorrect responses, respectively, obtained from the seven subjects. Vertical bars represent the standard errors of the means. Curves are Gaussian fits to the data. The standard deviations of these Gaussian functions were used as measurement of entropy in the spatial frequency domain or spatial domain, respectively.
Figure 8.
Mean psychometric functions for spatial frequency discrimination and spatial extent discrimination (left and right columns, respectively) at four different Michelson contrasts (100%, 10%, 5%, and 2% from top to bottom, respectively).
Reference stimulus: 10 cycles/degree and 1 degree. Filled circles and empty squares represent means for correct responses and incorrect responses, respectively, obtained from the seven subjects. Vertical bars represent the standard errors of the means. Curves are Gaussian fits to the data. The standard deviations of these Gaussian functions were used as measurement of entropy in the spatial frequency domain or spatial domain, respectively.
Table 1.
Results for all seven subjects tested, means and standard errors of the means for the sample, as well as estimate results for an “average subject” (ASU; see text for details).
Table 2.
Results for all seven subjects tested, means and standard errors of the means for the sample, as well as estimate results for an “average subject” (ASU; see text for details).
Table 3.
Results for all seven subjects tested, means and standard errors of the means for the sample, as well as estimate results for an “average subject” (ASU; see text for details).
Figure 9.
Statistical comparisons for the joint entropy measurements obtained from individual subjectspresented in Tables 1–3.
Individual values for each subject, as well as means and standard errors of the means for the sample are plotted for different contrasts at 0.4, 2, and 10 cycles/degree (left panels, top to bottom) and for the three spatial frequencies at 5%, 10%, and 100% Michelson contrasts (right panels, top to bottom). Results for conditions with two or less measurements were not plotted (2% contrast at 0.4 and 10 cycles/degree). In all frequencies there was a trend for joint entropy to decrease when contrast was increased, but this reached the level of statistical significance only for the comparison between 5% and 100% contrasts (p<0.05; One-Way ANOVA, Tukey Multiple Comparison Test). For all contrasts, joint entropy was significantly lower at 0.4 and 2 cycles/degree when compared with 10 cycles/degree (p<0.05).
Table 4.
Estimate results for an “average subject” (ASU; see text for details).
Figure 10.
ASU (“average subject”) spatial and spatial frequency joint entropy as a function of Michelson contrast for three different spatial frequencies: 0.4, 2, and 10 cycles/degree (diamonds, square, and triangles, respectively).
The dashed line represents the theoretical minimum for the 1D joint entropy of a system comprising only linear interactions between its subsystems. Using standard deviation as the entropy parameter and cycles/degree as spatial frequency metrics, the joint entropy minimum corresponds to 1/4π or 0.0796, and can only be attained by the product of the joint entropies for a Gábor function and its Fourier transform [14], [15], [22]. For all spatial frequencies, the joint entropy was higher at low contrasts and decreased when contrast was raised. At low and intermediate spatial frequencies and high contrasts, joint entropy reached levels below the minimum, an effect particularly pronounced at 0.4 cycles/degree and high contrast. This effect is suggestive that non-linear interactions between two or more visual mechanisms occur at these ranges of contrast and spatial frequency (see text for details).
Figure 11.
Spatial and spatial frequency joint entropy as a function of Michelson contrast for seven different subjects (circles) and for the “average subject” (crosses).
From top to bottom the results for 0.4, 2, and 10/degree are presented, respectively. The dashed line represents the theoretical minimum for the 1D joint entropy of a system comprising only linear interactions between its subsystems, 0.0796. For most of cases, 10 out 12 conditions, the joint entropy for the “average subject” fells in the range of results for individual subjects. The two exceptions were conditions where only one or two individual results were obtainable by data fitting. Nevertheless, all data were used to estimate ASU results. For all subjects and spatial frequencies, the joint entropy was higher at low contrasts and decreased when contrast was raised. At low and intermediate spatial frequencies and high contrasts, joint entropy reached levels below the theoretical minimum, suggesting non-linear interactions between two or more visual mechanisms (see text for details).