Figure 8 represents the contrast thresholds in the absence of noise as a function of the photopic luminance intensity for different combinations of spatial and temporal frequencies with red and blue displays. As in the first experiment, contrast thresholds were lower with the blue display compared to the red display at low luminance intensities (<∼5 photopic Td), which suggests a considerable contribution of rods at low luminance intensities with a blue display.
The contrast thresholds for noise as a function of the temporal and spatial frequencies with red and blue displays are shown in
Figure 9. For each spatiotemporal frequency (8 or 16 cpc, and 1 or 4 Hz), similar contrast thresholds were observed with the blue and red displays, suggesting that the calculation efficiency was relatively independent of the display color, as in the first experiment. The calculation efficiency (
k) for each spatial and temporal frequency was therefore assumed to be the same for the two colors and was derived from the geometric mean contrast threshold in noise (
Equation 2).
For each spatiotemporal frequency and each display color, the EIN as a function of luminance intensity was derived using
Equation 3 based on the calculation efficiency (
k) and the contrast thresholds in the absence of noise as a function of luminance intensity (
Figure 8). The EIN levels of each participant were found to roughly match the de Vries–Rose law (i.e., a log–log slope of –1) over an intermediate range of luminance intensities for each color displays (see
Figure 10 in which the fits closely match the data).
For each testing condition (two colors, two temporal frequencies, and two spatial frequencies), the photon noise was derived from the EIN by fitting a curve with a slope of –1 (i.e., de Vries–Rose law; see light blue and red lines in
Figure 10) using the same fitting procedure as in
Experiment 1.
Figure 11 shows the photon noise (geometric mean across participants) as a function of temporal and spatial frequencies for the red and blue displays. A repeated-measures 2 × 2 × 2 ANOVA on the photon noise in log units showed a significant large effect of color,
F(1,3) = 808.514,
p < 0.001, ω
2 = 0.989, but no significant effect of spatial frequency,
F(1,3) = 0.666,
p = 0.474, ω
2 = 0, or temporal frequency,
F(1,3) = 1.099,
p = 0.372, ω
2 = 0.011, and no significant interaction was observed. Thus, the photon noise was found to be roughly independent of the spatial and temporal frequencies. Furthermore, in
Figure 10, note that, when approaching the de Vries–Rose law, the EIN is roughly independent of the spatial and temporal frequencies, as can be seen with the overlapping curves for each display. In contrast, when the EIN deviated from the de Vries–Rose law at higher luminance intensities, variability was observed between the spatial and temporal frequencies, as can be seen by the curves being very distinct from each other.
The results were drastically different at high luminance intensities as a repeated-measures 2 × 2 × 2 ANOVA of the EIN in log units at the highest luminance intensity (without any filter) showed a significant large effect of temporal frequency,
F(1,3) = 11.13,
p = 0.045, ω
2 = 0.185, and spatial frequency,
F(1,3) = 70.24,
p = 0.004, ω
2 = 0.580 (
Figure 12). Although no significant effect of color was found,
F(1,3) = 2.47,
p = 0.214, ω
2 = 0.056, a significant interaction between color and temporal frequency was observed,
F(1,3) = 10.11,
p = 0.05, ω
2 = 0.065. These results show that, at the highest luminance intensity, the EIN considerably varied with color, spatial, and temporal frequency. Furthermore, at high luminance intensities, the slope of the EIN was much flatter than the de Vries–Rose law, as can clearly be seen in
Figure 10. Conversely, the slope was much steeper at the very low luminance intensities.
The fitting method used was simple and effective, but it implicitly assumed that a portion of the function followed the de Vries–Rose law. To objectively test if this was the case, an objective fitting procedure was conducted in which a wide variety of models were fitted (see
Supplementary Material). A model with photon noise was found to provide a significantly better fit than other models without photon noise. This analysis was also consistent with the current findings that photon noise significantly varied with the display color, but not the spatial and temporal frequency, and late noise varied with temporal frequency.
In the second experiment, less variability was observed within participants, as can be seen in
Figure 10, compared to the variability noted in the first experiment (
Figure 5). Moreover, participants did not report any difficulty in maintaining fixation at the center of the display and mentioned that the black annulus was helpful to keep gazing at the central fixation when they could no longer perceive the central black dot under low luminance intensities conditions.