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Chapter D: Colour Appearance, Elementary Colours, and Metrics, Main part DEAI

1. Introduction and Goals

The coordinates of colour spaces and of the colour differences are connected by line elements. For example Schroedinger (1925) and Stiles (1946) have tried to develop line elements for this connection.

Schroedinger, E. (1925), The relationship of Four-Colour Theory and Three-Colour Theory, Sitzungsberichte Kaiserl. Akad. Wiss., Wien, [IIa] 134, 471-490.
Stiles, W. S (1946), The line element in colour theory: a historical review, p. 1-25, in Color metrics, AIC/Holland, TNO Soesterberg.

The basis are special mathematical relations of integrals and derivattions.

Figure 1: Line elements of Stiles (1946) for three receptors L, M, and S
(or P=Protanop, D=Deuteranop, T=Tritanop).
For the download of this figure in the VG-PDF format, see CEA51-1N.PDF.

The derivation of the logarithmic F functions give the difference sensitivity delta_F which includes the ratio delta_L/L. For the value delta_F=1 at the sesitivity threshold a function between delta_L and L is determined. The letter L is used for example for the luminance L, the tristimulus value Y, the reflecion R, or a rezeptor-colour value L, M, or S.

In the following usually the relative tristimulus value x=Y/Yu or the relative luminance x=L/Lu is used. In this case Yu is the tristimulus value of the grey surround (often Yu=18), and Lu the luminance (often Lu=0,2*142cd/m^2=28cd/m^2 in the office with the illuminance 500 lux).


Figure 2: Example of a line enelement with a logarithmic function F(x).
For the download of this figure in the VG-PDF format, see CEA00-1N.PDF.

By derivation of the logarithmic function F(x)=ln(1+bx) a linear rationale function f(x)=1+bx with f'(x)=b is determined. For all functions of the form f'(x)/f(x) one can determine the logarithmic scaling function F(x). In applications the scaling factor a is chosen very different. A normalization on the surround u is used here. Then the factor a is omitted.

Figure 3: Example of a line elements with a logarithmic function Fu(x).
For the download of this figure in the VG-PDF format, see CEA00-2N.PDF.

In figure 3 the functions Fu(x) and fu(x), and their derivations are used for the comparison of different experimental results. These functions are independent of the scaling factor a.

Figure 4: Line-element equations for the tristimulus-value threshold delta_Y as function of Y.
For the download of this figure in the VG-PDF format, see CEA00-4N.PDF.

Figure 4 show the line-element equations for a 50% recognition of the grey samples. The equations use the constants according to CIE 230:2019. A relative lightness L*u(x) is calculated with the experimental data at the threshold t.
The equation [4] in the figure produces for all normalizations of Y the equation:

delta_Y = 1/(1+b)+[b/(1+b)]Y = 1/7,14+[6,14/7,14]Y

This equation includes the black threshold
t=0,14=1/7,14, and the slope
m=0,86=6,14/7,14 at the grey surround with the tristimulus value Yu=18.

Figure 5: Line-element equations for the tristimulus-value threshold delta_Y as function of Y.
For the download of this figure in the VG-PDF format, see CEA30-3N.PDF.

Figure 5 shows schematically the relative threshold luminance delta_Lr as function of Lr in blue.
A paper of Richter (2006) includes a model with a an explanation, see A/BAMAT.PDF. According to this paper the local effective luminance Leff of two adjacent greys is calculated by the equations:

log(Leff) = 0,5 [log(L1)+log(L2)]

Figure 5 shows the model result in yellow. The perceived lightness L*r of the two grey samples which are hard to distiguish is not by the factor 100, but only by the factor 10 larger.
The model result for the threhold results according to CIE 230:2019, and the CIELAB-lightness derivations according to ISO/CIE 11664-4:2019 will be discussed after figure 7.


Figure 6: Line-element equations for the LABJND-tristimulus value threshold delta_Y as function of Y.
For the download of this figure in the VG-PDF format, see CEA40-4A.PDF.

Figure 6 shows the LABJND-tristimulus value threshold according to CIE 230:219. The functions and constants are given in the figure, compare also figure 4.


Figure 7: Line-element equations for the CIELAB-"tristimulus value threshold" delta_Y as function of Y.
For the download of this figure in the VG-PDF format, see CEA70-4A.PDF.

Figure 7 shows the CIELAB-"tristimulus value threshold" according to ISO/CIE 11644-4:2019, which is calculated from the ligtness L*CIELAB. The functions and constants are given in the figure.

CIELAB is based on the Munsell colour system. The separated samples were viewed on a grey surround. The CIELAB colour difference of two neighbor samples is approximately delta_E*CIELAB=10. This correspond to about 30 thresholds in black-white direction.

According to the model in figure 5 the following slopes delta_Y as function of Y are expected:
mLABJND=0,86 (LABJND according to CIE 230:2019) mCIELAB=0,43 (model expectation for CIELAB according to ISO/CIE 11664-4:2019 However, figure 7 shows the slope mCIELAB=0,66 (CIELAB slope according to ISO/CIE 11664-4:2019).

Result und interpretation:
The slope mu=0,66 is for Yu in figure 7 the mean of the slope mu=0,86 in figure 6, and the expected model value mu=0,43 in figure 5.

The visual system forms many mean values of the sample and surround luminances. These mean values depend on
1. the sample distance d (adjacent or separated),
2. the presentation time t (0,1s until >20s),
3. the luminance of the grey surround.Lu.

Zum Beispiel ist die Steigung:
(m >1) für kurze Darbietungszeiten t<1s,
(m <1) für lange Darbietungszeiten t>10s.
For example the slope is for the contrast C = white : black:
approximately linear (m=0,86) for C=2:1 (aplication case daylight projector),
nonlinear (m=0,66) for C=25:1 (application case colour in the office),
more nonlinear (m=0,50) for C>288:1, (application case high dynamic range).

For example for the display-output test, the ISO-test chart AE49 with 1080 colours has been developed for different contrasts. The output questions for the 15 contrast steps are given in english (E), german (G), and french (F).
The ISO-test charts are on the ISO Standards Maintenance Portal in the file formats PDF, and PostScript (PS, TXT), see
http://standards.iso.org/iso/9241/306/ed-2/index.html.

For further information about visual threshold data according to CIE 230:219, see
CEAI.HTM.

The following part is under development.
It will include more text and images on the above topics.

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