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Chapter C: Colour Spaces, Colour Differences, and Line Elements, Main part CEAI

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.

A special case of a colour difference is the threshold colour difference. At threshold the colour difference of two adjacent colours are recognized with a probability of 50%. Usually after adaptaion and fixation of the two adjacent colours the observer detects a difference for 50% of the viewing time and detects NO difference for 50% of the viewing time.


Figure 1: Visual threhold delta_Y of achromatic and chromatic samples in a grey surround with a white frame.
For the download of this figure in the VG-PDF format, see CEA01-3N.PDF.

The figure shows that the threshold delta_Y increases approximately linear with Y with the slope 0,86. This slope is near the value 1. In this case the so called Weber-Fechner ratio Y/delta_Y is approximately constant within the range 10 <= Y <= 100. For the very smal tristimulus values Y < 0,4 it is valid delta_Y=0,012.In applications for all values Y < 0,4 the two central field colours appear as a uniform deep black.

The results in Fig. 1 are the basis of the colour-difference formula LABJND for Just Noticeable Differences (JNDs) in CIE 230:2019.

The performance of LABJND_PF is calculated in Table 9 and 11 of CIE 230 for the scope range 0 <= delta_E*ab <=2 of the CIELAB colour-difference formula, and for the 8 available CIE datasets, see
http://files.cie.co.at/TC181_Datasets.zip.

The extension "_PF" describes a modification of the LABJND-colour difference which produces the value (delta E*)LABJND = 1 at threshold. A "power function correction PF", for example with an exponent n=0,5, does not change the colour difference. However, for 100 threshold differences for example an exponent n=0,5 reduces the colour difference from the value 100 to the value 10:

(delta E*)LABJND_PF = 100^0,5 = 10

A paper of Richter (2006) includes a model with an explanation, see A/BAMAT.PDF. According to this paper the local effective relative luminance (Leff) of two adjacent greys is given by the equation:

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

If for example the relative luminance L1 is equal to the relative surround luminance Lu=1 and L2=100Lu, then it is valid:

log(Leff) = 0,5 [log(Lu)+log(100Lu)] = 0,5 [0 + 2] = 1 = log(10)

The visual effective relative luminance at the border of the two samples is Leff=10 and NOT L2=100. This produces half of the slope with a value n=0,5 instead of 1. For the threshold the calculated colour difference is equal for the values n=1 and 0,5.

In CIE 230:2019 the PF-value n was optimized for the following five colour difference formulae:
CIELAB_PF (n=0,55),
CMC_PF (n=0,66),
CIE94 (n=0,70),
CIEDE2000_PF (n=0,70), and
LABJND_PF (n=0,35).

The PF-value n (in brackets) is the smallest for the LABJND-threshold formula with the smallest colour-difference unit. Eight datasets for small colour differences were studied and used in CIE TC1-81. TC1-81 has produced the publication CIE 230.
The LABJND_PF colour difference formula shows the best performance for 5 out of 8 CIE datasets. CIELAB_PF, CMC_PF, and CIEDE2000_PF show the best performance for one out of 8 CIE datasets.

In 2016 CIE TC1-63 has produced in WD11 results for large colour differences. The comparison of the five formulae for Large (LCD) and Extra Large (ELCD) colour differences shows about the same performance for all formulae, see
YE370-7N.PDF.
However, for Extra Large Colour Differences (ELCD) with ratio and interval scaling the performance is different, see
YE330-7N.PDF.
The performance of LABJND_PF is better than for all other formulae.

The STRESS value according to CIE 217 is reduced from about 35 to 40 of the four above formulae to about 25 for LABJND_PF. Smaller values describe a better performance.
According to the terms of reference for CIE TC1-63 the results shall be compared for the scope range 5 < delta_E*ab < 199. In 2016 the CIE TC1-63 members could not agree to proceed with the draft WD11 for a CIE Technical Report on Large (LCD) and Extra Large (ELCD) colour differences.

The unexpected and conflicting result compared to the above recommended CIE colour difference formulae may be a reason.

The results of
1. in WD11 of CIE TC1-63 and
2. on this web site
seem to be a promising basis for improved results in the area of:
Colour Spaces, Colour Differences, and Line Elements.

For more information about "line elements" and "metrics", see
DEAI.HTM.

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

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