IV. Results of Experiments
IV.1. Experiment 1 : Color Perception for unrelated colors
IV.1.2. Brightness
Relationship between Luminance and Perceptual Brightness
In this section, stimulus 13 (Lv = 201.74cd/m2) was employed as an anchor stimulus, and the brightness of the modulus were assigned as 45. Figure 19 illustrates the relationship between the luminance levels of unrelated color stimuli and the perceived brightness. The horizontal axis of the graph represents the luminance values of the stimuli, which ranged between 7.31 and 331.64 cd/m2 under a photopic luminance condition. The vertical axis of the figure shows the magnitude of the perceived brightness. Overall, it appears that the magnitude of the perceptual brightness increased as the luminance level of the stimuli rose. However, it is clear that there is no linear relation between the luminance level of the stimuli and perceived brightness. The perceived brightness rose rapidly until it reached approximately 40 at a luminance level of 50 cd/m2. It then increased slowly at values greater than 50 cd/m2. This indicates that there is a compressive nonlinear relationship between the intensity of the luminance and the perceived magnitude of the brightness, which illustrates a decreasing sensitivity with increasing stimulus intensity.
Figure 19 Relationship between luminance and perceptual brightness
In terms of the relationship between the physical intensity and perceptual magnitude, many previous studies have derived a transformation of the physical stimulus intensity scale to a perceptual magnitude scale. Fechner’s law (Fechner, 1966) states that the perceived magnitude of a stimulus is proportional to the logarithm of the physical stimulus intensity. Stevens’ power law (Stevens, 1961) indicates that the relationship between the perceptual magnitude and stimulus intensity follows a
logarithmic function for the luminance, while the broken curve represents the power function of the luminance when the exponent is less than unity. The constants used for the functions were derived using the least square method. These two lines seem to have similar shapes and fit the perceptual brightness data well.
The following formula can be used for the logarithmic function explaining the relation between the luminance of the stimuli and perceived brightness.
Brightness ∙ ln
In this formula, a1 and a0 are constants with values of 8.25 and zero, respectively. The correlation coefficient (r) between the perceived brightness and logarithmic function was 0.90, indicating good correlation between these two scales. Furthermore, the coefficient of determination (r2), which provides a measure of how well outcomes are likely to be estimated by the model, was 0.81 and the adjusted r2 was 0.80. F-test was performed at significance level 0.05, and p-value was 0.00 (F = 199.83). As the p-value is much less than 0.05, the logarithmic function fit the perceived brightness data well in terms of linear regression. Figure 20 plots the relationship between the values estimated by the logarithmic function and the perceived brightness.
Figure 20 Comparison between brightness visual results and estimated data derived by logarithmic function depending on luminance
A power function could also be used to explain the relationship between the luminance and perceived brightness, as follows:
Brightness ∙
In this function, a0 and n are constants with values of about 12.45 and 0.26, respectively. n is the exponent of the function. The correlation coefficient between the magnitude of the perceptual brightness and the values estimated by the power function was 0.91, which implied a good correlation between these two scales. The coefficient of determination (r2) was 0.83 and the adjusted r2 was 0.82.
F-test was performed at significance level 0.05, and p-value was 0.00 (F = 225.84). As the p-value is much less than 0.05, the estimation function derived by the power function appropriately matched the perceptual data well in terms of linear regression. Figure 21 illustrates the relationship between the values estimated by the power function and the perceptual brightness data.
Figure 21 Comparison between brightness visual results and estimated data derived by power function depending on luminance
Performance of CAM97u for Brightness
Figure 22 illustrates the relation between the brightness perception data and brightness Q of the CAM97u model. The horizontal axis of the graph represents the perceived magnitude of the brightness, while the vertical axis of the figure presents the Q values of CAM97u for the 50 stimuli.
The correlation coefficient between the magnitude of the perceptual brightness and the brightness Q estimated by CAM97u was 0.84, which implied a good correlation between these two scales. The coefficient of determination (r2) was 0.71 and the adjusted r2 was 0.70. F-test was performed at significance level 0.05, and p-value was 0.00 (F = 117.33). As the p-value is much less than 0.05, the brightness Q of CAM97u fit the perceptual data well. However, the performance of the model was worse than those of the above models derived by the logarithmic and power functions using the observer data.
Figure 22 Comparison between perceived brightness and estimated brightness derived by CAM97u
Performance of Revised CIECAM02 for Brightness
Figure 23 illustrates the relation between the perceived brightness and brightness Q of the revised CIECAM02 model. The horizontal axis of the graph represents the perceived brightness, while the vertical axis of the figure presents the Q values of revised CIECAM02 for the 50 stimuli. The correlation coefficient between the magnitude of the perceived brightness and the brightness Q estimated by CIECAM02 was 0.91, which implied a good correlation between these two scales. The coefficient of determination (r2) was 0.82 and the adjusted r2 was 0.82. F-test was performed at significance level 0.05, and p-value was 0.00 (F = 224.51). As the p-value is much less than 0.05, there is a significant relationship between the brightness Q of revised CIECAM02 and the perceptual data. The performance of the model was better than those of the above derived two models and CAM97u.
Figure 23 Comparison between perceived brightness and estimated brightness derived by revised CIECAM02 model