Color difference explained

In color science, color difference or color distance is the separation between two colors. This metric allows quantified examination of a notion that formerly could only be described with adjectives. Quantification of these properties is of great importance to those whose work is color-critical. Common definitions make use of the Euclidean distance in a device-independent color space.

Euclidean

sRGB

As most definitions of color difference are distances within a color space, the standard means of determining distances is the Euclidean distance. If one presently has an RGB (red, green, blue) tuple and wishes to find the color difference, computationally one of the easiest is to consider R, G, B linear dimensions defining the color space.

A very simple example can be given between the two colors with RGB values (0, 64, 0) and (255, 64, 0) : their distance is 255. Going from there to (255, 64, 128) is a distance of 128.

When we wish to calculate distance from the first point to the third point (i.e. changing more than one of the color values), we can do this:

$$\backslash text\; =\; \backslash sqrt.$$

When the result should be computationally simple as well, it is often acceptable to remove the square root and simply use

$$\backslash text^2\; =\; (R\_2\; -\; R\_1)^2\; +\; (G\_2\; -\; G\_1)^2\; +\; (B\_2\; -\; B\_1)^2.$$

This will work in cases when a single color WAS to be compared to a single color and the need is to simply know whether a distance is greater. If these squared color distances are summed, such a metric effectively becomes the variance of the color distances.

There have been many attempts to weigh RGB values to better fit human perception, where the components are commonly weighted (red 30%, green 59%, and blue 11%), however, these are demonstrably worse at color determinations and are properly the contributions to the brightness of these colors, rather than to the degree to which human vision has less tolerance for these colors. The closer approximations would be more properly (for non-linear sRGB, using a color range of 0 - 255):

where:

One of the better low-cost approximations, sometimes called "redmean", combines the two cases smoothly:^[1]

There are a number of color distance formulae that attempt to use color spaces like HSV or HSL with the hue represented as a circle, placing the various colors within a three-dimensional space of either a cylinder or cone, but most of these are just modifications of RGB; without accounting for differences in human color perception, they will tend to be on par with a simple Euclidean metric.

Uniform color spaces

CIELAB and CIELUV are relatively perceptually-uniform color spaces and they have been used as spaces for Euclidean measures of color difference. The CIELAB version is known as CIE76. However, the non-uniformity of these spaces were later discovered, leading to the creation of more complex formulae.

A uniform color space is supposed to make a simple measure of color difference, usually Euclidean, "just work". Color spaces that improve on this issue include CAM02-UCS, CAM16-UCS, and J_za_zb_z.^[2]

Rec. ITU-R BT.2124 or ΔE_ITP

In 2019 a new standard for WCG and HDR was introduced, since CIEDE2000 was not adequate for it: CIEDE2000 is not reliable below 1 cd/m² and has not been verified above 100 cd/m²; in addition, even in BT.709 blue primary CIEDE2000 is underpredicting the error.^[3] ΔE_ITP is scaled so that a value of 1 indicates the potential of a just noticeable color difference. The ΔE_ITP color difference metric is derived from display referenced IC_TC_P, but XYZ is also available in the standard. The formula is a simply scaled Euclidean distance:^[4]

$$\backslash Delta\; E\_\backslash text\; =\; 720\; \backslash sqrt,$$

where the components of this "ITP" is given by

I = I,

T = 0.5 C_T,

P = C_P.

Other geometric constructions

The Euclidean measure is known to work poorly on large color distances (i.e. more than 10 units in most systems). A hybrid approach where a taxicab distance is used between the lightness and the chroma plane, $\backslash Delta\; E\_\; =\; \backslash sqrt\; +\; \backslash left|\; L\_2-L\_1\; \backslash right|$, is shown to work better on CIELAB.^[5]

CIELAB ΔE*

The International Commission on Illumination (CIE) calls their distance metric (also inaccurately called,, or "Delta E") where delta is a Greek letter often used to denote difference, and E stands for Empfindung; German for "sensation". Use of this term can be traced back to Hermann von Helmholtz and Ewald Hering.^{[6] [7]}

Perceptual non-uniformities in the underlying CIELAB color space have led to the CIE refining their definition over the years, leading to the superior (as recommended by the CIE) 1994 and 2000 formulas.^[8] These non-uniformities are important because the human eye is more sensitive to certain colors than others. CIELAB metric is used to define color tolerance of CMYK solids. A good metric should take this into account in order for the notion of a "just noticeable difference" (JND) to have meaning. Otherwise, a certain may be insignificant between two colors in one part of the color space while being significant in some other part.^[9]

All formulae are originally designed to have the difference of 1.0 stand for a JND. This convention is generally followed by other perceptual distance functions such as the aforementioned . However, further experimentation may invalidate this design assumption, the revision of CIE76 JND to 2.3 being an example.

CIE76

The CIE 1976 color difference formula is the first formula that related a measured color difference to a known set of CIELAB coordinates. This formula has been succeeded by the 1994 and 2000 formulas because the CIELAB space turned out to be not as perceptually uniform as intended, especially in the saturated regions. This means that this formula rates these colors too highly as opposed to other colors.

Given two colors in CIELAB color space, $$ and $$, the CIE76 color difference formula is defined as:

$$\backslash Delta\; E\_^*\; =\; \backslash sqrt.$$

$\backslash Delta\; E\_^*\; \backslash approx\; 2.3$ corresponds to a JND (just noticeable difference).^[10]

CMC l:c (1984)

In 1984, the Colour Measurement Committee of the Society of Dyers and Colourists defined a difference measure based on the CIE L*C*h color model, an alternative representation of L*a*b* coordinates. Named after the developing committee, their metric is called CMC l:c. The quasimetric (i.e. it violates symmetry: parameter T is based on the hue of the reference

h _{1} alone) has two parameters: lightness (l) and chroma (c), allowing the users to weight the difference based on the ratio of l:c that is deemed appropriate for the application. Commonly used values are 2:1^[11] for acceptability and 1:1 for the threshold of imperceptibility.

The distance of a color $(L^*\_2,C^*\_2,h\_2)$ to a reference $(L^*\_1,C^*\_1,h\_1)$ is:^[12]

$$\backslash Delta\; E^*\_\; =\; \backslash sqrt$$

CMC l:c is designed to be used with D65 and the CIE Supplementary Observer.^[13]

CIE94

The CIE 1976 color difference definition was extended to address perceptual non-uniformities, while retaining the CIELAB color space, by the introduction of application-specific parametric weighting factors k_L, k_C and k_H, and functions S_L, S_C, and S_H derived from an automotive paint test's tolerance data.^[14]

As with the CMC I:c, ΔE (1994) is defined in the L*C*h* color space and likewise violates symmetry, therefore defining a quasimetric. Given a reference color

(L ^{*} _{1 ,} a ^{*} _{1 ,} b ^{*} _{1 )} and another color (L ^{*} _{2 ,} a ^{*} _{2 ,} b ^{*} _{2 )} , the difference is^{[15] [16] [17]}

$$\backslash Delta\; E\_^*\; =\; \backslash sqrt,$$

where

and where k_C and k_H are usually both set to unity, and the parametric weighting factors k_L, K₁ and K₂ depend on the application:

	graphic arts	textiles
k _{L}	1	2
K _{1}	0.045	0.048
K _{2}	0.015	0.014

Geometrically, the quantity

\Delta H ^{*} _{ab} corresponds to the arithmetic mean of the chord lengths of the equal chroma circles of the two colors.^[18]

CIEDE2000

Since the 1994 definition did not adequately resolve the perceptual uniformity issue, the CIE refined their definition with the CIEDE2000 formula published in 2001, adding five corrections:^{[19] [20]}

• A hue rotation term (R_T), to deal with the problematic blue region (hue angles in the neighborhood of 275°):^[21]
• Compensation for neutral colors (the primed values in the L*C*h differences)
• Compensation for lightness (S_L)
• Compensation for chroma (S_C)
• Compensation for hue (S_H)

$$\backslash Delta\; E\_^*\; =\; \backslash sqrt$$

Note: The formulae below should use degrees rather than radians; the issue is significant for R_T.

The parametric weighting factors k_L, k_C, and k_H are usually set to unity.

$$\backslash Delta\; L^\backslash prime\; =\; L^*\_2\; -\; L^*\_1$$

$$\backslash bar\; =\; \backslash frac\; \backslash quad\; \backslash bar\; =\; \backslash frac\; \backslash quad\; \backslash mbox\; C^*\_1\; =\; \backslash sqrt,\; \backslash quad\; C^*\_2\; =\; \backslash sqrt,\; \backslash quad$$

$$a\_1^\backslash prime\; =\; a\_1^*\; +\; \backslash frac\; \backslash left(1\; -\; \backslash sqrt\; \backslash right)\; \backslash quad\; a\_2^\backslash prime\; =\; a\_2^*\; +\; \backslash frac\; \backslash left(1\; -\; \backslash sqrt\; \backslash right)$$

$$\backslash bar^\backslash prime\; =\; \backslash frac\; \backslash mbox\; \backslash Delta=C\text{'}\_2-C\text{'}\_1\; \backslash quad\; \backslash mbox\; C\_1^\backslash prime\; =\; \backslash sqrt\; \backslash quad\; C\_2^\backslash prime\; =\; \backslash sqrt\; \backslash quad$$

$$h\_1^\backslash prime=\backslash text\; (b\_1^*,\; a\_1^\backslash prime)\; \backslash mod\; 360^\backslash circ,\; \backslash quad\; h\_2^\backslash prime=\backslash text\; (b\_2^*,\; a\_2^\backslash prime)\; \backslash mod\; 360^\backslash circ$$

Note: The inverse tangent (tan^-1) can be computed using a common library routine atan2(b, a′) which usually has a range from -π to π radians; color specifications are given in 0 to 360 degrees, so some adjustment is needed. The inverse tangent is indeterminate if both a′ and b are zero (which also means that the corresponding C′ is zero); in that case, set the hue angle to zero. See .

Note: The example above expects the parameter order of atan2 to be atan2(y, x). See implementation in ^[22]

Note: When either C′₁ or C′₂ is zero, then Δh′ is irrelevant and may be set to zero. See .

Note: When either C′₁ or C′₂ is zero, then ′ is h′₁+h′₂ (no divide by 2; essentially, if one angle is indeterminate, then use the other angle as the average; relies on indeterminate angle being set to zero). See stating most implementations on the internet at the time had "an error in the computation of average hue".

$$T\; =\; 1\; -\; 0.17\; \backslash cos\; (\backslash bar^\backslash prime\; -\; 30^\backslash circ)\; +\; 0.24\; \backslash cos\; (2\backslash bar^\backslash prime)\; +\; 0.32\; \backslash cos\; (3\backslash bar^\backslash prime\; +\; 6^\backslash circ)\; -\; 0.20\; \backslash cos\; (4\backslash bar^\backslash prime\; -\; 63^\backslash circ)$$

$$S\_L\; =\; 1\; +\; \backslash frac\; \backslash quad\; S\_C\; =\; 1+0.045\; \backslash bar^\backslash prime\; \backslash quad\; S\_H\; =\; 1+0.015\; \backslash bar^\backslash prime\; T$$

$$R\_T\; =\; -2\; \backslash sqrt\; \backslash sin\; \backslash left[60^\backslash circ\; \backslash cdot\; \backslash exp\; \backslash left(-\backslash left[\; \backslash frac\{\backslash bar\{H\}\text{'}-275^\backslash circ\}\{25^\backslash circ\}\; \backslash right]^2\; \backslash right)\; \backslash right]$$

CIEDE 2000 is not mathematically continuous. The discontinuity stems from calculating the mean hue $\backslash Delta\; H^\backslash prime$ and the hue difference $\backslash Delta\; h\text{'}$. The maximum discontinuity happens when the hues of two sample colors are about 180° apart, and is usually small relative to ΔE (less than 4%).^[23] There is also a negligible amount of discontinuity from hue rollover.

Sharma, Wu, and Dalal has provided some additional notes on the mathematics and implementation of the formula.^[24]

Tolerance

See also: Uniform color space.

Tolerancing concerns the question "What is a set of colors that are imperceptibly/acceptably close to a given reference?" If the distance measure is perceptually uniform, then the answer is simply "the set of points whose distance to the reference is less than the just-noticeable-difference (JND) threshold". This requires a perceptually uniform metric in order for the threshold to be constant throughout the gamut (range of colors). Otherwise, the threshold will be a function of the reference color—cumbersome as a practical guide.

In the CIE 1931 color space, for example, the tolerance contours are defined by the MacAdam ellipse, which holds L* (lightness) fixed. As can be observed on the adjacent diagram, the ellipses denoting the tolerance contours vary in size. It is partly this non-uniformity that led to the creation of CIELUV and CIELAB.

More generally, if the lightness is allowed to vary, then we find the tolerance set to be ellipsoidal. Increasing the weighting factor in the aforementioned distance expressions has the effect of increasing the size of the ellipsoid along the respective axis.^[25]

See also

• CIELAB
• Color coding in data visualization

Footnotes

References

Further reading

• Historical development of CIE recommended color difference equations. Alan R.. Robertson. Color Research & Application. 1990. 15. 3. 167–170. 10.1002/col.5080150308.
• Uniformity of some recent color metrics tested with an accurate color-difference tolerance dataset . M. . Melgosa . Quesada . J. J. . Hita . E.. December 1994. Applied Optics. 33. 34. 8069–77. 10.1364/AO.33.008069. 20963027. 1994ApOpt..33.8069M .
• Book: Colour Physics for Industry . Roderick . McDonald . 0-901956-70-8. . 1997 . 2nd.

External links

• Bruce Lindbloom's color difference calculator. Uses all CIELAB metrics defined herein.
• The CIEDE2000 Color-Difference Formula, by Gaurav Sharma. Implementations in MATLAB and Excel.
• Explore the Spectrum with Colors in Between, by Bettie M. Cobb.

Notes and References

1. Web site: Colour metric . Compu Phase.
2. Li . Changjun . Li . Zhiqiang . Wang . Zhifeng . Xu . Yang . Luo . Ming Ronnier . Cui . Guihua . Melgosa . Manuel . Brill . Michael H. . Pointer . Michael . 3. Comprehensive color solutions: CAM16, CAT16, and CAM16-UCS . Color Research & Application . December 2017 . 42 . 6 . 703–718 . 10.1002/col.22131.
3. Web site: What Is ICtCp – Introduction? . live . Dolby . Version 7.1 . https://web.archive.org/web/20160508234602/http://www.dolby.com/us/en/technologies/dolby-vision/ICtCp-white-paper.pdf . 2016-05-08 .
4. Web site: Recommendation ITU-R BT.2124-0 . Objective metric for the assessment of the potential visibility of colour differences in television . International Telecommunication Union . January 2019 . BT Series: Broadcasting service (television).
5. Abasi . Saeedeh . Amani Tehran . Mohammad . Fairchild . Mark D. . Distance metrics for very large color differences . Color Research & Application . April 2020 . 45 . 2 . 208–223 . 10.1002/col.22451. 209914019 .
6. Book: Color Vision: Perspectives from Different Disciplines. Backhaus . W. . Kliegl . R. . Werner . J. S. . 1998 . Walter de Gruyter . 9783110154313 . 188 . 2014-12-02.
7. Book: Valberg, A. . Light Vision Color . 2005. Wiley . 9780470849026 . 278 . 2014-12-02.
8. Book: Fraser . Bruce . Bunting . Fred . Murphy . Chris . Real World Color Management . Pearson Education . 2004 . 9780132777957 . 2nd .
9. https://web.archive.org/web/20080705054344/http://www.aim-dtp.net/aim/evaluation/cie_de/index.htm Evaluation of the CIE Color Difference Formulas
10. Book: Sharma, Gaurav . . Digital Color Imaging Handbook . 2003 . 0-8493-0900-X. 1.7.2.
11. Meaning that the lightness contributes half as much to the difference (or, identically, is allowed twice the tolerance) as the chroma
12. Web site: Lindbloom . Bruce Justin . Delta E (CMC) . 2009-04-16 . Brucelindbloom.com.
13. 1–15 October 1996 . CMC . Insight on Color . 8 . 13 . https://web.archive.org/web/20060312161957/http://www.hunterlab.com/appnotes/an10_96ar.pdf . 2006-03-12.
14. Web site: Delta E: The Color Difference . Colorwiki.com . 2009-04-16.
15. Web site: Bruce Justin . Lindbloom . Delta E (CIE 1994) . Brucelindbloom.com . 2011-03-23.
16. Web site: Colour Difference Software by David Heggie . Colorpro.com . 1995-12-19 . 2009-04-16.
17. Colorimetry - Part 4: CIE 1976 L*a*b* Colour Space . Draft Standard . CIE . CIE DS 014-4.3/E:2007 . 2007.
18. Book: Klein, Georg A. . Industrial Color Physics . limited . 147 . 978-1-4419-1196-4 . 2010-05-18 . Springer .
19. The CIEDE2000 color-difference formula: Implementation notes, supplementary test data, and mathematical observations . Color Research & Application . . Gaurav . Sharma . Wencheng . Wu . Edul N. . Dalal . 30 . 1 . 21–30 . 10.1002/col.20070 . 2005 . .
20. Web site: Bruce Justin . Lindbloom . Delta E (CIE 2000) . Brucelindbloom.com . 2009-04-16.
21. http://www.brucelindbloom.com/MunsellCalcHelp.html#BluePurple The "Blue Turns Purple" Problem
22. Web site: Sharma . Gaurav . The CIEDE2000 Color-Difference Formula . 2023-10-24 . "Excel spreadsheet" hyperlink.
23. Sharma . Gaurav . Wu . Wencheng . Dalal . Edul N. . Celik . Mehmet U. . Mathematical Discontinuities in CIEDE2000 Color Difference Computations . Color and Imaging Conference . 1 January 2004 . 12 . 1 . 334–339 . 10.2352/CIC.2004.12.1.art00058 . free.
24. Sharma . Gaurav . Wu . Wencheng . Dalal . Edul N. . The CIEDE2000 color-difference formula: Implementation notes, supplementary test data, and mathematical observations . Color Research & Application . February 2005 . 30 . 1 . 21–30 . 10.1002/col.20070.
25. Web site: A guide to Understanding Color Tolerancing . 14 January 1998 . Susan Hughes . 2014-12-02 . dead . https://web.archive.org/web/20151010064956/http://www.xrite.com/documents/literature/en/L10-024_Color_Tolerance_en.pdf . 10 October 2015.