The colours of objects need to be described and quantified for the purposes of describing the difference in colour between them. Using a colour space such as CIELAB allows this difference to be expressed in the form of a numerical index. This index should be able to predict, and represent, the visually perceived difference between colours. In a visually uniform colour space the difference between two colours (1 and 2) should, in theory, equate to the distance between their positions within the space, which is referred to as the Euclidean distance. For CIELAB the formula for this difference, denoted ΔE*ab - with E standing for Empfindung,
the German for ‘sensation’ (Berns, 2000) - , is:
߂ܧכ
ൌ ඥ߂ܮכଶ ߂ܽכଶ ߂ܾכଶ
Equation 2.7 As well as the total colour differences indicated by ΔE*ab, the colour differences can be
30 ߂ܮכൌ ܮכ ଵെܮכଶ Equation 2.8 ߂ܥכ ൌ ቆටܽכଵଶ ܾכଵଶቇ-ቆටܽכଶଶ ܾכଶଶቇ Equation 2.9 ߂ܪכ ൌ ඥሺ߂ܧכሻଶെ ሺ߂ܮכሻଶെ ሺ߂ܥכሻଶ Equation 2.10 ΔH*abis the Euclidean rather than the angular difference in hue. Therefore ΔH*abwill increase with increasing ΔC*abwhile hab remains the same (Berns, 2000).
Alternative formulae, for calculating ΔH*ab directly, are given in Berns (2000), and MacDougall (2002a).
In practice, the correlation between visually perceived and Euclidean colour differences is poor. In the assessment of visual differences a standard colour is paired with a selection of ‘test’ samples of near colours, and the degree of the visual difference between the standard and sample in each pair is made relative to the difference between the samples in an ‘anchor pair’, representing a defined instrumental colour difference. A test sample is ‘passed’ or ‘accepted’ if the difference between it and the standard is the same, or smaller, than the difference displayed by the anchor pair. This process can be repeated for other (standard) colours. When the measured colours of the accepted, test samples are plotted in colour space about each relevant standard, not only does the size of the distributions differ between the standard colour ‘centres’, but the distributions are ellipsoidal rather than spherical. Referring to the comparisons of the anchor pair with the standard/sample pair, this indicates that pairs with the same visual difference do not necessarily have the same measured difference. Further, the differences between colour centres indicate that the disparity between visual and measured colours is a function of the colour centre, while the ellipsoids point to lightness, chroma and hue differences being affected in different ways. These effects are the result of colour spaces not being truly
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visually uniform; in reality, this is very difficult to achieve given wide observer variability, factors affecting perception, and that colour spaces very likely to need be described by more than three dimensions (Berns, 2000).
Several weighted colour difference equations derived from CIELAB have since been developed taking into account the above effects, and have been used in a variety of applications, with some preferred by specific industries. Weighted equations for total colour difference include:
The CMC(l:c) Colour-Difference Equation (Colour Measurement Committee, Society of Dyers and Colourists) for difference designated ΔECMC(l:c), standard for textiles:
߂ܧெሺǣሻൌ ඨ൬ ߂ܮכ ݈ܵ൰ ଶ ൬߂ܥכ ܿܵ ൰ ଶ ൬߂ܪכ ܵு ൰ ଶ Equation 2.11 The CIE94 equation, for difference designated ΔE*94, used preferentially by industries
such as the textile industry for accurate colour difference measurements related to perception and acceptability (MacDougall, 2002a):
߂ܧכ ଽସൌ ඨ൬ ߂ܮכ ݇ܵ൰ ଶ ൬߂ܥכ ݇ܵ ൰ ଶ ൬߂ܪכ ݇ுܵு൰ ଶ Equation 2.12 The CIEDE2000 Colour-Difference Formula, difference ΔE00, a further improvement
on earlier formulae (Luo et al., 2001a), now a new CIE/ISO standard (Melgosa, 2013); CIEDE2000 applies to colour-uniform samples with colour differences below five CIELAB units ([CIE] International Commission on Illumination, 2001):
߂ܧൌ ඨ൬ ߂ܮᇱ ݇ܵ൰ ଶ ൬߂ܥᇱ ݇ܵ൰ ଶ ൬߂ܪᇱ ݇ுܵு൰ ଶ ்ܴ൬ ߂ܥᇱ ݇ܵ൰ ൬ ߂ܪᇱ ݇ுܵு൰ Equation 2.13
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Corresponding formulae for lightness, hue and chroma differences are also available for
ΔE00.
2.3.4.1.Experimental conditions and the effects of texture
In the weighted colour-difference equations, the l, c, kL, kC and kH coefficients are adjusting constants which represent the effects of experimental conditions (parametric effects), and the resulting relative influence of lightness, chroma and hue, on perceived total colour difference. The S coefficients are positional functions which account for the lack of visual uniformity in CIELAB. The convention is to align the parametric constants for lightness and chroma (l, c, kL and kC) relative to that of hue (denoted kH in CIE94 and CIEDE2000 but undesignated in CMC(l:c)) which is set to unity. An l or a kL of 2 for example would indicate that experimental conditions were such that hue had twice the influence of lightness on the perceived difference (Berns, 2000). The values of kL and kC that are used depend on how much experimental conditions deviate from a set of reference conditions, in which a pair of colour-uniform samples are placed in edge contact against a background of L*=50, and viewed by observers with normal colour vision under illuminant D65 at a viewing angle of at least four degrees; under these conditions kL=kC=kH=1. Details of the individual formulae for SL, SC and SH will not be given here, but can be found in Berns (2000) and in Luo et al. (2001a).
In the textile industry it is common practice to use kL=2 for CIEDE2000 colour differences. Despite this, the correlation between objectively-measured CIEDE2000 (2:1:1) differences displayed by knitted polyester yarn samples of different textures (coarseness) and the corresponding visual differences was found by Gorji Kandi et al. (2008) to be poor, with an r
value of 0.56 across all the textures. Visual differences were much better correlated (r=0.94) with a measure of texture structure known as the Gabor texture difference (GTD), said to be more closely related to the processing by the primary visual cortex of the brain, though r values did vary among the individual colours, being as low as -0.10 for Yellow and 0.15 for Blue (Gorji Kandi et al., 2008). By controlling viewing geometry and illumination conditions to maximise the perceptions of differences in colour, and in the visual texture attributes coarseness
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and glint, and together with an instrument capable of coarseness, glint, as well as reflectance measurements, Huang et al. (2010) developed two formulae that could predict the total visual difference, ΔT, of pairs of metallic coating samples from their measured colour and texture. Two formulae were proposed: one for diffuse illumination conditions, which includes terms for differences in measured colour (CIEDE2000) and coarseness, and the other for directional illumination, based on total colour and glint differences.
2.3.4.2.Tolerance levels
Limits have been set on the sizes of calculated (total) colour differences beyond which a colour match between samples is deemed unsatisfactory. These limits are referred to as instrumental colour tolerances and in various industry applications usually indicate when the colour difference is predicted to become a perceptible difference. Perceptible differences include those which are just-perceptible or just-noticeable (‘threshold’) and those just above perceptible (‘supra-threshold’); for surface colours, perceptibility judgements are made relative to the difference between the samples in a standard pair. Examples of colour tolerances used in various industries, or reported in different studies, based on either CIELAB or HunterLab, are given in Table 2.4. Tolerance levels are seen to vary according to the application and some are above threshold. Perceptibility tolerances can be increased by a commercial factor if colour differences encountered exceed those that are just perceptible (Berns, 2000; MacDougall, 2002a). Tolerances of 2.4 and 0.7 CMC2:1 units were used by two different textiles companies
producing exactly the same product in similar colours, and both companies were meeting their customer needs, indicating that in these cases tolerances were related largely to commercial requirements (Gay and Hirschler, 2002).
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Table 2.4 A selection of colour tolerance limits applied in different industries, or that have been used in various studies.
Industry/application Difference formula
Tolerance level, and comments
Threshold CMC(1:1)
CIELAB
0.3 – 1.4 units
0.4 – 1.8 units (Melgosa et al., 1992)1 Automotive CIELAB < 1 unit
Paint, plastics, textiles Unspecified Lab space
= 1 unit for approximate commercial match
(noticeable difference at ΔE=2 units, unacceptable at
ΔE=3 units) (Francis and Clydesdale, 1975)
Textiles CMC(1:1)
CIELAB
0.7 – 2.3 units
1.0 – 2.8 units (Gay and Hirschler, 2002)2 Food and beverage
Red wine CIELAB = 3 units (Martínez et al., 2001)
Muffins CIELAB = 3 units (Baixauli et al. (2008), referencing Francis and Clydesdale (1975))
Carrot puree with added green food colouring
ΔE, HunterLab 0.1 to 0.15 units (allowed ranking of samples) (Huang et al., 1970b)
Squash puree ΔE, HunterLab ~ 0.2 units (allowed ranking of samples) (Huang et al., 1970a)
Pale beers CIELAB 1.5 units (Hutchings, 1999)
1Across surface colours, and experimental threshold results using visual colorimeters (coloured lights)
2
Across seven different textiles companies; individual values dependent on product range and marketing situation of companies
Tolerance levels, and the difference formulae used in their calculation, vary also according to how well they correlate with visual assessments. Of several formulae tested (including CIELAB and CIE94) tolerance levels calculated using CMC (2:1) were in best agreement with visual evaluations of textiles (Gay and Hirschler, 2002), while CMC(1:1) was the best performing formula with respect to visual threshold differences (Melgosa et al., 1992). For the purpose of comparing tolerances within and across studies only CMC(1:1) and CIELAB results are shown in Table 2.4; CMC(2:1) was not used in the study by Melgosa et al. (1992) while in the study by Gay and Hirschler (2002) the range of tolerance levels was the same for both CMC(2:1) and CMC(1:1), though the individual levels within each range were different. For colour differences larger than threshold, such as those displayed between Munsell colours, CIELAB gave the better results (Melgosa et al., 1992). Tolerances can also change with different surface textures and colours (Gay and Hirschler, 2002), consistent with the variable effects of surface texture, colour and their interaction on measured and perceived colours, described earlier. Colour tolerances (for example, lightness tolerances in the textile industry), increase with sample surface texture.
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