An Advanced Mathematical Model Fitted

An advanced model with no attention to the duration factor is constructed here. Both the 2°

and the 8° dataset was pooled to fit the mathematical model to be introduced. We will see later that separating the two conditions (2 and 8 seconds exposure) does not end up with a significantly better model performance.

Lightness perception shift is modelled first. As noted in the statistical evaluation of Section 3, the perceived lightness of the near-immersive colour was only affected by the original lightness of the test colours and a mixed effect of this, with the duration of observation. In Figure I/14, the mean matching lightness values of the 21 large test colours are plotted against their measured (colorimetric) lightness values. The line of the identical function shows those positions where the mean values should lie if the lightness perception of the large near-immersive colour and the small (2°) matching colour coincided. To approximate the trend, a linear function was fitted, similar to Eq. (I/4) in the first series.

Forcing the trend curve to cross the (100, 100) point is mathematically equivalent to calculating a linear regression curve passing through the origin of the co-ordinate system for the data points transformed to the 3^rd quadrate by shifting (100 units both vertically and horizontally). In this case no constant is added only the slope of the line is calculated in the model. After the transformation and the application of the linear regression in SPSS for factor mL the value of 0.631986 arose and the determination coefficient equalled r²=0.949.

The magnitude of the mL factor agreed well with the previous findings from the first series of experiments (see Table I/5). On the confidence of the regression model and the calculated coefficient see Table I/8. The accepted rule dictates that the number of the digits of the coefficient should be provided so that the half of the value of the smallest digit should be approximately the same as the width of the confidence interval. In this case this width is 0.014 (the upper bound minus the lower bound in Table I/8), therefore the coefficient is rounded to the second digit. The proposed formula for lightness correction concerning the uncertainty of the visual data similarly to Eq. (I/4) looks as follows:

* *

imm = 100 + 0.63( st-100)

L L , (I/11)

where L*imm stands for the equivalent lightness of the colour stimulus viewed immersed, L*st is the measurable lightness of the immersive sample stimulus (‘st’ subscript is for standard, as in Eq. (I/4)).

Table I/8. Detailed statistics on the calculated coefficients of the lightness perception observations vs. the original corresponding values of the 21 test colours. Separate markers mean different lightness levels (square: L*=30, diamond: L*=50, triangle: L*=70). For the equations of the best fitting lines see the text. Confidence interval of the measured points can be traced in Figure I/13.

Modelling the equivalent chromatic appearance is the second step. The statistical evaluation suggested an interactive effect of lightness and hue on the colour perception of the large colour and therefore lightness information was included in the chroma correction.

As the duration (2 s or 8 s) was not significant, this effect was discarded. It is hard to detect systematic changes either chroma or hue in Figure I/13, if any. There is, however, a well recognizable trend of how the data points are spread much noticeable if the lightness of the test colours is also considered in a plot depicting the equivalent a* or b* values of the 21 immersive test samples against their measured a* or b* values. In Figure I/15 this composition is visualized with the average a* (left) and b* (right) coordinates of the

matching small colours plotted versus the measured a* and b* value of the 21 test colours grouped by the original lightness level of the large stimuli. Though, due to the definition of the attributes at fixed lightness in CIELAB space (a*, b*) coordinates are fully equivalent mathematically with the (C*ab, hab) cylindrical representation, this approach seems to yield less sophisticated formulae. Therefore, instead of making the size effect correction dependent directly on hue angle (hab), chroma correction is given in terms of the CIELAB a* and b* coordinates, as it occurred for Experiment I.

Equations of the best fitting linear curves plotted in Figure I/15 have the following

where a*imm (b*imm) means the “equivalent” CIELAB a* (b*) value of the perceived colour of the large test stimuli and a*st (b*st) means the colorimetric (measured) a* (b*) coordinate of the test stimulus. The value of the mx,L and cx,L coefficients (x=a or b, L=30, 50, or 70) can be further approached by the linear function of the measured lightness level:

(

) (

)

Checking the correlation of the model with the experimental data and the significance of the c_xn (x=a, b; n=1, …, 4) coefficients linear regression analysis was performed with the following predictors: a*st, L*st, and a*st× L*st (the multiplication of them), and for the b*

values respectively. The result of the analysis is listed in Table I/9.

As it is observable from the significance values of the selected variables, they significantly contribute to predict the chromatic changes of the immersive field. Applying again the accepted principle on the sufficient number of the provided digits due to the uncertainty of the experimental data set, the following model is derived:

( )

( )

To test the effectiveness of the above mathematical model, the overall mean values of the observers’ perceptions for the 21 test colours were compared with the model predictions for these 21 colours in terms of the ∆E*ab colour difference formula. The original average of ∆E*ab=17.9 (STD=5.6) was reduced to an average of ∆E*ab=2.9 (STD=1.45). A paired-sample t-test between the original 21 colour differences (colour difference of the measured data and the visual observations) and the 21 colour differences after the model predictions (colour difference of the model output and the visual observations) showed a significant improvement (t=10.68, p<0.001). For details see the Appendix.

Table I/9. Detailed statistics on the calculated coefficients of the model of the chromatic changes. The low values in the ‘Sig.’ column mean that the selected variables highly background result of Experiment I was selected, as the conditions of this series coincided with that of the experiment discussed in the present section. The average CIELAB colour difference between the equivalent colours of the 16 immersive test stimuli in Experiment I and the predictions for those by applying the model based on Experiment II was ∆E*ab=7.1 (STD=3.61), which is a reduction in accuracy compared to the performance of the model based on the first dataset predicting the colour appearance of the immersive field (for the

model derived from the 2° changing background data of Experiment I average ∆E*ab was 5.33 – see Section 2.2.1). Though, a paired sample t-test comparison showed that the two model performances are significantly different (t=2.211, p=0.043) the original uncorrected

∆E*ab=13 difference between the equivalent and the measured colour of the immersive field in Experiment I was still almost halved using the model of Experiment II. (Regarding data are listed in the Appendices.)