A Mathematical Model Fitted To Present Dataset

The aim of modelling the colour perception changes of different visual fields is to predict the way the same colour stimulus will look if presented subtending different angles of view compared to a standard sample size (e.g. 2°). Since the size effect appears to cause changes directly in terms of the perceptual attributes of colours, the correction seems reasonable to be carried out on this level primarily. Concerning the previous work found in literature on mathematical modelling, in order to predict the colour appearance in a real room, a

correction based upon CIELAB lightness and chroma proved to be a good approximation of the situation, although, the overall performance of the model was a mean of ∆Eab*=8.7 for the experimental data collected including mixed illumination in the calculations.⁴² (In that work, performance of the model was defined as the average difference between the model-predicted colorimetric data and the result of the visual experiment, i.e. how observers perceived the wall-sized colours.) Since in those experiments mainly lightness and chroma shifts were detected the model uses a quadratic polynomial distortion for the CIELAB L* and Cab* values of the room size colours (as the function of the originally measured L* and Cab*) to approximate the observers’ perception and leaves hue (CIELAB hab) unchanged, as follows:

2

S= L R L R 100(1 100 L L)

L α L +β L + − α −β , (I/1)

2

S = C R C R

C α C +β C , (I/2)

S= R

h h , (I/3)

where subscript S stands for the size effect corrected colour attributes, subscript R is for measured values of the room size colours and αX and βX parameters (X=L or C, where L refers to the lightness attribute and C to chroma attribute) are to be optimised minimising the quadratic error between the predictions and the visual data. For the current dataset, due to the similarity of the two scenes of the fully painted rooms and the immersion achieved by the mirror-tiled viewing booth the previously mentioned model⁴² (Eqs. (I/1)-(I/3)) seemed the most appropriate to check whether it is a good descriptor for the present near-immersive self-luminous experimental conditions and situation, too. Optimising that model to the present dataset, however, resulted in greater overall mean colour difference between the model-manipulated colours and the visual observations (∆E*ab =28.9) than the original mean colour difference of the size phenomenon (between the large field colours and the observers’ settings for the 2° fixed background matching – ∆E*ab =17.5) showing that the model is not suitable for the present data.

The size dependence of the perceived attributes of colour stimuli was also modelled with more or less success from 2° up to 50° in Ref. No. 43. Similar to the room appearance correction, here, also the lightness and chroma value of stimuli subtending different visual

fields were corrected that were found to be affected by size. Lightness perception was found to increase linearly with size, while for chroma, a second order polynomial relationship was found between measured and visual data. Converting the CIE XYZ values of visual data of the same experiment into LMS cone signals, another mathematical model was derived suggesting that it is at the level of cone responses where the phenomenon colours in case of the 2° fixed background matching plotted against the measured values of the large stimuli. Trend lines and equations are described in the text. Error bars stand for 95% confidence intervals.

Applying the compensation technique in LMS space with the suggested parameters described in Ref. 44 to the 2° fixed background results of the present dataset and choosing the visual angle of 85° [this was the visual angle under which the plasma display appeared without the booth] again the overall mean colour difference between the compensated colours and the observations (∆E*ab =20.9) is larger than the effect originally (∆E*ab

=17.5). The more the visual angle as input parameter is increased the more the mean colour difference will be. Using the same equations but parameters optimized directly to the obtained near-immersive self-luminous dataset the overall mean difference will be

∆E*ab=12.8 which is an advance compared to the original difference (again ∆E*ab =17.5) but it will be shown later that further improvements are reachable.

As can be seen from the above, though the visual results of the experiment discussed in this section show systematic output (see Figure I/6), none of the previously published models affords a solution to predict the perceptual changes of display colours at an acceptable level if stimuli are presented fully immersed to the observers. In Figure I/6, the mean CIELAB L*, a*, and b* values of the visual result of the observations are plotted for the 16 observed colours in case of the 2° fixed background condition. In each plot, data points are represented along the axes of the measured value of the observers’ mean adjustments against the measured original value of the large stimulus. The 45° dotted lines (identical function) on all three graphs represent the locations where the observers’

adjustments should lie if the perception of the large scene entirely matched that of the standard size patches. As can be seen, the deviation of the mean points from the identical functions follows certain trends in each graph. Fitting an appropriate function on the data points (thick lines) can give a mathematical description of the phenomenon and serve as a base for the prediction of the appearance of colours compared to standard situations.

Following formulae were used for the trend curves in Figure I/6. For the lightness,

* *

imm = 100 + L( st-100)

L m L , (I/4)

where L^*imm stands for the lightness of a visually matching small stimulus if the colour stimulus is viewed immersed (actually, it is an equivalent lightness for a measurable situation), L^*st is the measurable lightness of the immersive colour stimulus (the same lightness as if the same stimulus was viewed and measured under standard conditions, e.g.

2° here for Figure I/6) and mL is the modifying constant to be optimized. The benefit of Eq.

(I/4) instead of using a simple linear equation is to force the curve to cross the (100,100) point preventing the model from predicting an L* value larger than 100.⁴³ In literature, sometimes observations report L*>100 values⁵⁶ (e.g. metallic colours, fluorescent or glossy materials) but this was not considered applicable for the present paradigm using self-luminous stimuli presented on computer-controlled displays. Metallic gloss and fluorescence are phenomena related to typically surface colours or reflective circumstances. (Using an unrestricted linear trend with two parameters did not end up significantly better overall performance of the model, indeed.) The trend for CIELAB a*

was approximated by using the formula

* *

imm = 1,a* st 2,a*

a c a +c , (I/5)

where, similar to Eq. (I/4), a^*imm is the “equivalent” a^* coordinate for the large stimulus, a^*st is the standard (measurable) a^* value on the plasma display and c1,a* and c2,a* are the constants that determine the deviation of the trend compared to the identical function. For b^*, also the same formula was used in the following form:

* *

imm = 1,b* st + 2,b

b c b c , (I/6)

where, again, b^*imm means the “equivalent” b^* value of the large colour, b^*st is the measurable value for the same immersive stimulus, and cb*,n (n=1, 2) are the parameters of the line. The optimal values of mL and cx,n (x= a^* or b^*, n=1, 2) constants were found by minimising a quadratic error between the observers’ mean results and the model predictions for the 16 studied colours. The mean error between the model predictions and the result of the visual observations for the experimental data decreased to ∆E*ab=5.3 CIELAB units, and this is a good improvement compared to the original ∆E*ab =17.5 difference and the ∆E*ab≤6.1 inter-observer variability. The 95% confidence intervals of the optimized parameters for the case of the less apparent a* and b* shifts are listed in Table I/4. In both cases the identical function (dotted line in Figure I/6) is not inside the calculated ranges, which means that the shift of the data point is significant on the provided level.

Table I/4. Detailed statistics on the optimized coefficients of the 2° fix background case.

The low values in the ‘Sig.’ column mean that the independent variables (ast and bst) highly contribute to the models.

The same optimization was carried out for the dataset of the other conditions (background and matching stimulus size), too. The estimated parameters and the mean error of the models for the different paradigms are listed in Table I/5. Originally, the experimental results of the 10° matching on changing background showed the least size effect (∆E*ab

=11.4) and, by using the correction of Eqs. (I/4)-(I/6) on the direct measurements on the large stimuli with the parameters described in Table I/5, this condition represented the appearance of the 16 colours most accurately (∆E*ab =4.2). As can be seen from the results for all conditions the mean error of the models deviate around 5 CIELAB units and they are almost all within the just noticeable difference in CIELAB (1 unit). If a so called

‘winner’ is required to be chosen from all, for the present dataset it might be the 10°

matching on changing background, but its performance is not significantly different from that of the others. It might also be affirmed that, according to the results, the displayed

colour appearance of this condition represents the best that of the immersive scene, though the ANOVA procedure at the beginning of Section 2.2 tells only that the conditions have effect on the average colour difference adjusted.

After this discussion, the size of the corresponding stimuli in the experimental set-up of the 2^nd series (see Section 3) seems a bit contradicting because 2° was used. This size was chosen for two practical reasons, namely, that (i) in real life situations generally a rather small sample is provided as the part of a swatch collection and (ii) as we saw previously this case also yielded a systematic and predictable dataset.