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this difference in the R /G ratio between the average male observer and the average female observer is statistically significant.
The inter-group and intra-group variances were studied using a parametric test (ANOVA) and a non-parametric test (Kruskal-Wallis). The null hypothesis was the absence of differences between groups split by sex, and therefore the recognition that the potential 50% of the female population whose retinas have both types of L-cone photopigments present behaviour that is equiparable to the average of the two populations whose retinas have potentially only one or the other type of photopigment. The ANOVA test needs to check three prerequisites – randomness, normality, and homoskedasticity of the sample – before being applied, but it is robust enough for use with samples like these that are theoretically bimodal and trimodal. We also performed the Kruskal-Wallis non-parametric test. The result of the two tests was the same (Table 1): the differences between men and women recorded in the ratios of the red/green mixture were indeed statistically significant (P<0.05), the null hypothesis could be rejected, and the alternative hypothesis that the two groups could be treated as distinct populations could be accepted.
4. DIFFERENCES IN THE MODEL OF COLOUR VISION
In the previous section, we saw that there exist significant differences in colour perception between different groups of observers in their Rayleigh metameric matches. It remains to be seen whether these differences are general, or just apparent in matching tests. To test this idea, a psychophysical experiment was devised based on obtaining each observer's isobrightness curve.
The CIE in 1978 in its publication No. 41 [56] called attention to the difference between the luminous efficiency function Vλ and the data obtained from direct heterochromatic luminosity matches, with the efficiency of the latter being greater than that of the former at both ends of the visible spectrum. This highlighted the need to obtain a luminous efficiency curve that would provide the luminosity perceived by an observer in response to a monochromatic stimulus (isobrightness curve).
In 1981, a luminous efficiency curve of this type was obtained for a 2° field, based on the data of 19 observers. Finally, in 1985, more data were added for the function corresponding to a 2° field, corresponding to a population of 63 observers between 18 and 50 years in age, and a photopic light level of between 50 and 500 trolands. The result is shown in Fig. 11 together with the 1924 CIE luminous efficiency curved as amended in 1951 by Judd [18] and in 1978 by Vos [25].
To obtain this curve, monochromatic stimuli were used displayed on a bipartite field in which one of the hemifields acted as reference and the other was matched by the observer. The data from which this curve was constructed were obtained by different researchers whose methods may have differed. For instance, for the field of observation, some used a white reference hemifield while others used a spectral stimulus of a particular wavelength. The time of exposure to the stimuli also varied from worker to worker, from 0.004 seconds to an unlimited time.
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Figure 11. Isoluminance vs. Isobrightness curve.
The principal drawback of this brightness efficiency curve is that it should not be used with light sources of a compound spectrum since the additivity law is not satisfied. It can, however, provide us with information on each individual's underlying colour vision model, as we shall see below.
To perform the experiment, the aim was to attempt to simulate observing conditions similar to those of everyday life. To this end, we introduced a surrounding adaptive field of white with luminance 10 cd/m², which ensures the adaptation of the visual system to the photopic level and minimizes the effects of auto-adaptation to the test stimulus itself. The exposure time was unlimited. Luminosity matches were made after some seconds of exposure to ensure a stable level of photopigment bleaching.
Since this experiment had a twofold objective – obtaining a luminosity efficiency curve and using those data to study the colour vision models – the L, M, and S values were recorded together with the luminance of spectral stimuli over a large part of the visible spectrum (400– 700 nm) whose common characteristic is that the observer perceives them with the same luminosity as the surrounding field. In particular, the matching performed was heterochromatic by direct comparison. To this end, the observer was asked, starting from a spectral stimulus of 570 nm, to fix its luminance level so that it was perceived to have the same luminosity as the adaptive field. This was done for the two plates of the visual field. Then the wavelength of the spectral stimulus of each plate began to be varied alternately in steps of 5 nm.
To facilitate the observer's evaluation of the luminosity, he or she was told to use both referents – the luminosity of the achromatic stimulus of the surrounding field, and the luminosity of the adjacent plate – but priority was given to the surrounding field.
This test was performed with three observers in sessions of about an hour, until completing a total of five sessions per observer each performed on different days. Observer 1 was a 25 year old man without previous experience in the topic. Observer 2 was a 29 year old
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woman also without previous experience. Observer 3 was a 40 year old woman with experience in tasks of evaluating chromatic stimuli.
Below are the luminous efficiency curves together with the LMS tristimulus curves obtained from the experimental data for these three observers.
Figure 12. Luminance curve which gives the same luminosity as the white reference for the three observers.
For the three observers, the luminance curve which gives the same luminosity as the white reference appears in figure 12. In these three curves, one observes a very pronounced absolute maximum at around 580 nm, and a second lower local maximum at longer wavelengths.
To continue with the analysis of the results of this experiment, we will now consider the LMS tristimulus values obtained for each match of brightness. For the three observers, the L values are shown in fig. 13. The plot of the M value curves for the three observers appear in fig 14, and finally, we complete the data obtained from this experiment with the curves of the S values, which are shown in the figure 15 for the three observers.
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Figure 13. Normalized L-Fundamental values of stimuli which gives the same brightness as the white reference for the three observers.
In this graphical analysis of the results, there stand out the differences between the three observers. A priori, this would indicate major differences in their colour vision. The plots were displayed normalized so that each observer's absolute maximum was of value unity. In order to determine the magnitude of these differences in the intensity of the peaks, we must refer to the un-normalized data. One observes in this data that there is agreement between LMS values of Observers 1 and 2 over the entire range of values of the visible spectrum studied. This is not the case, however, for Observer 3, for whom the short wavelength LMS values are much higher than for Observers 1 and 2. This difference declines with increasing wavelength, but for the zone of the spectrum corresponding to wavelengths near 470 nm the differences reach about four times the L and M values obtained for Observers 1 and 2, and even a factor of six for the S values.
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Figure 14. Normalized M-Fundamental values of stimuli which gives the same brightness as the white reference for the three observers.
That Observer 3's LMS values were higher, especially for short wavelengths, indicates that this observer's visual system requires more light in that part of the spectrum than the other two observers (the LMS units are the number of photons effectively captured per second, per m², and per steradian in each type of cone). This type of behaviour could be the result of either a yellowing of the intraocular media which filter the radiation incident on the eye, especially in zone close to the ultraviolet, or a high optical density of the macular pigment (Fig. 16).
It is known that there is great variation in yellowing of the lens, expressed as the optical density of the intraocular lenses, together with the optical density of the macular pigment in the population [57] and in the same individual over the course of his or her life [58]. Figure 16 shows the spectral variation of the optical density due to the macular pigment and the intraocular lenses, and the sum of the two used by Stockman & Sharpe [26]:
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Figure 15. Normalized S-Fundamental values of stimuli which gives the same brightness as the white reference for the three observers.
On this last point, since the fundamental response curves reflect the different densities of the intraocular media (lenses and macular pigment), and this optical density is known in the case of Stockman & Sharpe's fundamentals, we can try to disentangle this information, and obtain the optical density of the photopigments of each type of cone. Based on the fundamentals of Stockman & Sharpe and the optical densities of the lenses and macular pigment that they also obtained (Fig. 16), one can extract the information corresponding to the photopigment response curves. Plotting them on a logarithmic scale of wavelengths, and shifting the curves so that they all have their maxima at the same point, one obtains the curves shown in Fig. 17:
It is clear that, once the contribution of the intraocular media has been discounted, and the response of the three cones has been rescaled and shifted on the logarithmic scale of wavelength so that all their maxima coincide, the three types of photopigment have essentially the same behaviour in their response curves.
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Figure 16. Optical density of intraocular media.
Figure 17. Optical density of photopigments normalized and centred in a Log scale.
This figure shows that the spectral sensitivity curve of the photopigment found in each type of cone is the same except for the scaling conditions. With these facts as a basis, we can approach the task of generating fundamental response curves for each observer, using for this purpose the intraocular lens and macular pigment optical densities and the wavelengths of maximum sensitivity of each of the three types of cone.
In 1991, the CIE set up the technical committee TC 1-36 with the following Terms of Reference: "Establish a fundamental chromaticity diagram of which the coordinates
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correspond to physiologically significant axes". In 2006, this TC published a report [59] with estimates of cone fundamentals for the normal observer, ranging in viewing angle from 1° to 10°. The report starts with the choice of the 10° CMFs of Stiles & Burch (1959). Then, following the ideas put forward by Stockman & Sharpe [26], by application of König’s hypothesis, and using the most recent data on the spectral sensitivity functions of dichromats, this is followed by the derivation of the spectral sensitivity functions of the long-wave sensitive (L-), medium-wave sensitive (M-) and short-wave sensitive (S-) cones, measured in the corneal plane for a 10° viewing field – the so called "cone fundamentals".
Next, by correcting these functions for the absorption of the ocular media and the macular pigment, and taking into account the optical densities of the cone visual pigments, all for a 10° viewing field, the low density absorbance functions of these pigments were derived. Using these low density absorbance functions one can derive from the absorption of the ocular media and the macula, and taking into account the densities of the visual pigments for a 2° viewing field, the 2° cone fundamentals. Using the same procedure one can derive cone fundamentals for every viewing angle between 1° and 10°. Effects of age can also be incorporated by application of the relationship of the absorption of the lens as a function of age [60].
In our experiment, Observer 3 was significantly older than the other two observers, and the difference in the observed LMS values was particularly acute for those obtained from the S fundamental. Recall that the LMS values were obtained using the fundamentals of Stockman & Sharpe, and that these fundamentals were obtained using average values of the macular pigment and intraocular media optical densities. Assuming that the large difference in the LMS values of Observer 3 relative to the other two observers was due to a greater optical density of the intraocular lenses, and modifying the fundamentals of Stockman & Sharpe to take this into account, one obtains new LMS values for Observer 3, as shown in the following curves.
The normalized curve of L, M & S values obtained after correcting the fundamentals of Stockman & Sharpe for the optical density of the intraocular media is presented in the figure 18.
As can be seen in the above figures, applying this correction leads to the LMS curves of Observer 3 having many more similarities with the same curves of Observers 1 and 2 than before.
Before continuing with the data analysis, we must first reflect on the nature of those data. The psychophysical experiment gave some points of perception space of different hues and saturation, but the same luminosity as the surrounding achromatic referent. This makes the variation in the data as one moves along the spectrum to be mainly in hue, then in saturation, and finally in luminance, since the test was at equal luminosity.
Principal component analysis [61] seeks linear combinations of the variables comprising the data set that account for as much of the variability in the data as possible. It is an appropriate technique when the original variables are strongly correlated, as in the present case. When this is so, much of the information contained in the data set overlaps among several variables, thus hindering its identification. Principal component analysis constructs a new set of uncorrelated variables to avoid repetition of the information.
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Figure 18. Normalized L,M,S- Fundamental rebuilt values for observer 3
The first principal component is the linear combination which accounts for as much of the variability as possible. The second is the linear combination uncorrelated with the first principal component that accounts for as much of the remaining variability as possible. The third is the linear combination uncorrelated with the first two principal components that accounts for as much of the remaining variability as possible, and so on.
Another interesting point is the possibility of working with other new variables which are rotations of the original variables and better explain the variability of the data. It has to be borne in mind that the principal components depend on the scaling of the original variables so that, for the data resulting from the principal component analysis to be valid, one must maintain the scaling conditions. Although this statistical tool was designed to eliminate variables with little influence on the event under study, in the present case we shall apply it to the data obtained in our experiment for the three observers in order to obtain a linear combination of the LMS components that reveals the underlying colour vision model.
There was a question as to whether the data should be subjected to a principal component analysis or to a common (principal) factor analysis. This latter is applied assuming that not all the variance can be explained by the variables involved in the study, so that there is always a reduction in the number of dimensions of the resulting space. In the present case, there
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were no reasons that would allow us to make this assumption, so that we opted for principal component analysis.
Before beginning with the principal component analysis, it is advisable to standardize the data, especially if the starting variables represent values of a different nature. In the present case, although the nature of the data is the same since they indicate the effective capture of photons in the three types of cone, their behaviour is very different for each type of cone. We therefore proceeded to standardize the data making the mean of each variable 0 and the variance 1.
There are several ways of performing a principal component analysis according to whether the basis is the correlation matrix or the covariance matrix, and according to the normalization applied to the results. In our case, we used the correlation matrix and the Hotelling normalization [62]. This gives equal importance to all three initial LMS variables assuming that each has a variance of 1 and the total variance of the system is 3. Also, the resulting components are scaled such that each retains the proportion of the observed variance while maintaining the variance of 3 for the complete system.
Some workers, including Boynton et al. [63] and Hunt [64], consider that wavelengths shorter than 442 nm or longer than 613 nm do not provide optimal chromatic stimuli since in these regions of the visible spectrum the rate of change of hue with wavelength falls rapidly. This variation in the scaling of the hue versus wavelength may affect the method used in the previous analysis, since the experimental data on which it was based surpassed the upper limit of 613 nm, reaching 680 nm. This was not the case, however, for the other end of the spectrum, because the experimental data were obtained only down to 470 nm. Performing the principal component analysis on the experimental data only in the range 470–615 nm, we obtained the results described in tables 2-4.
Table 2. Total variance explained for Observer 1.
Component |
Initial eigenvalues |
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Sums of the squares of the saturations |
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Total |
% Variance |
Cumulative % |
Total |
% Variance |
Cumulative % |
1 |
2.280 |
75.992 |
75.992 |
2.280 |
75.992 |
75.992 |
2 |
0.548 |
18.283 |
94.275 |
0.548 |
18.283 |
94.275 |
3 |
0.172 |
5.725 |
100.000 |
0.172 |
5.725 |
100.000 |
Extraction method: Principal component analysis
Table 3. Total variance explained for Observer 2.
Component |
Initial eigenvalues |
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Sums of the squares of the saturations |
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Total |
% Variance |
Cumulative % |
Total |
% Variance |
Cumulative % |
1 |
2.183 |
72.751 |
72.751 |
2.183 |
72.751 |
72.751 |
2 |
0.548 |
18.253 |
91.003 |
0.548 |
18.253 |
91.003 |
3 |
0.270 |
8.997 |
100.000 |
0.270 |
8.997 |
100.000 |
Extraction method: Principal component analysis
In tables 2-4, Component 1 accounts for more thant 70% of the total variance. Component 2 accounts for 18% of the variance, and Component 3 accounts for the remaining 6-10% for the three observers.