Color difference

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:

$${\displaystyle {\text{distance}}={\sqrt {(R_{2}-R_{1})^{2}+(G_{2}-G_{1})^{2}+(B_{2}-B_{1})^{2}}}.}$$

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

$${\displaystyle {\text{distance}}^{2}=(R_{2}-R_{1})^{2}+(G_{2}-G_{1})^{2}+(B_{2}-B_{1})^{2}.}$$

This will work in cases when a single color is 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, however these are demonstrably^{[citation needed]} 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):^[1] $${\displaystyle {\begin{cases}{\sqrt {2\Delta R^{2}+4\Delta G^{2}+3\Delta B^{2}}},&{\bar {R}}<128,\\{\sqrt {3\Delta R^{2}+4\Delta G^{2}+2\Delta B^{2}}}&{\text{otherwise}},\end{cases}}}$$

where: $${\displaystyle {\begin{aligned}\Delta R&=R_{1}-R_{2},\\\Delta G&=G_{1}-G_{2},\\\Delta B&=B_{1}-B_{2},\\{\bar {R}}&={\frac {1}{2}}(R_{1}+R_{2}).\end{aligned}}}$$

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

$${\displaystyle {\begin{aligned}{\bar {r}}&={\frac {1}{2}}(R_{1}+R_{2}),\\\Delta C&={\sqrt {\left(2+{\frac {\bar {r}}{256}}\right)\Delta R^{2}+4\Delta G^{2}+\left(2+{\frac {255-{\bar {r}}}{256}}\right)\Delta B^{2}}}.\end{aligned}}}$$(^[2])

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.^{[citation needed]}

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.

Uniform color space: a color space in which equivalent numerical differences represent equivalent visual differences, regardless of location within the color space. A truly uniform color space has been the goal of color scientists for many years. Most color spaces, though not perfectly uniform, are referred to as uniform color spaces, since they are more nearly uniform when compared to the chromaticity diagram.

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.^[4]

Other geometric constructions

The Euclidean measure is known to work poorly on large color distances (i.e. more than 10 units in most systems). On CIELAB, a hybrid approach where a taxicab distance is used between the lightness and the chroma plane, ${\textstyle \Delta E_{\text{HyAB}}={\sqrt {(a_{2}-a_{1})^{2}+(b_{2}-b_{1})^{2}}}+\left|L_{2}-L_{1}\right|}$, is shown to work better than Euclidean and CIEDE2000.^[7]

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, ${\textstyle ({L_{1}^{*}},{a_{1}^{*}},{b_{1}^{*}})}$ and ${\textstyle ({L_{2}^{*}},{a_{2}^{*}},{b_{2}^{*}})}$, the CIE76 color difference formula is defined as:

$${\displaystyle \Delta E_{ab}^{*}={\sqrt {(L_{2}^{*}-L_{1}^{*})^{2}+(a_{2}^{*}-a_{1}^{*})^{2}+(b_{2}^{*}-b_{1}^{*})^{2}}}.}$$

${\textstyle \Delta E_{ab}^{*}\approx 2.3}$ corresponds to a JND (just noticeable difference).^[13]

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 ${\displaystyle h_{1}}$ 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^[14] for acceptability and 1:1 for the threshold of imperceptibility.

The distance of a color ${\textstyle (L_{2}^{*},C_{2}^{*},h_{2})}$ to a reference ${\textstyle (L_{1}^{*},C_{1}^{*},h_{1})}$ is:^[15]

$${\displaystyle \Delta E_{CMC}^{*}={\sqrt {\left({\frac {L_{2}^{*}-L_{1}^{*}}{l\times S_{L}}}\right)^{2}+\left({\frac {C_{2}^{*}-C_{1}^{*}}{c\times S_{C}}}\right)^{2}+\left({\frac {\Delta H_{ab}^{*}}{S_{H}}}\right)^{2}}}}$$

$${\displaystyle S_{L}={\begin{cases}0.511&L_{1}^{*}<16\\{\frac {0.040975L_{1}^{*}}{1+0.01765L_{1}^{*}}}&L_{1}^{*}\geq 16\end{cases}}\quad S_{C}={\frac {0.0638C_{1}^{*}}{1+0.0131C_{1}^{*}}}+0.638\quad S_{H}=S_{C}(FT+1-F)}$$

$${\displaystyle F={\sqrt {\frac {C_{1}^{*^{4}}}{C_{1}^{*^{4}}+1900}}}\quad T={\begin{cases}0.56+|0.2\cos(h_{1}+168^{\circ })|&164^{\circ }\leq h_{1}\leq 345^{\circ }\\0.36+|0.4\cos(h_{1}+35^{\circ })|&{\mbox{otherwise}}\end{cases}}}$$

CMC l:c is designed to be used with D65 and the CIE Supplementary Observer.^[16]

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.^[12]

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^[a] ${\displaystyle (L_{1}^{*},a_{1}^{*},b_{1}^{*})}$ and another color ${\displaystyle (L_{2}^{*},a_{2}^{*},b_{2}^{*})}$, the difference is^[17][18][19]

$${\displaystyle \Delta E_{94}^{*}={\sqrt {\left({\frac {\Delta L^{*}}{k_{L}S_{L}}}\right)^{2}+\left({\frac {\Delta C^{*}}{k_{C}S_{C}}}\right)^{2}+\left({\frac {\Delta H^{*}}{k_{H}S_{H}}}\right)^{2}}},}$$

where

$${\displaystyle {\begin{aligned}\Delta L^{*}&=L_{1}^{*}-L_{2}^{*},\\C_{1}^{*}&={\sqrt {{a_{1}^{*}}^{2}+{b_{1}^{*}}^{2}}},\\C_{2}^{*}&={\sqrt {{a_{2}^{*}}^{2}+{b_{2}^{*}}^{2}}},\\\Delta C^{*}&=C_{1}^{*}-C_{2}^{*},\\\Delta H^{*}&={\sqrt {{\Delta E_{ab}^{*}}^{2}-{\Delta L^{*}}^{2}-{\Delta C^{*}}^{2}}}={\sqrt {{\Delta a^{*}}^{2}+{\Delta b^{*}}^{2}-{\Delta C^{*}}^{2}}},\\\Delta a^{*}&=a_{1}^{*}-a_{2}^{*},\\\Delta b^{*}&=b_{1}^{*}-b_{2}^{*},\\S_{L}&=1,\\S_{C}&=1+K_{1}C_{1}^{*},\\S_{H}&=1+K_{2}C_{1}^{*},\\\end{aligned}}}$$

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
${\displaystyle k_{L}}$	1	2
${\displaystyle K_{1}}$	0.045	0.048
${\displaystyle K_{2}}$	0.015	0.014

Geometrically, the quantity ${\displaystyle \Delta H_{ab}^{*}}$ corresponds to the arithmetic mean of the chord lengths of the equal chroma circles of the two colors.^[20]

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:^[21][22]

• A hue rotation term (R_T), to deal with the problematic blue region (hue angles in the neighborhood of 275°):^[23]
• 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)

$${\displaystyle \Delta E_{00}^{*}={\sqrt {\left({\frac {\Delta L'}{k_{L}S_{L}}}\right)^{2}+\left({\frac {\Delta C'}{k_{C}S_{C}}}\right)^{2}+\left({\frac {\Delta H'}{k_{H}S_{H}}}\right)^{2}+R_{T}{\frac {\Delta C'}{k_{C}S_{C}}}{\frac {\Delta H'}{k_{H}S_{H}}}}}}$$

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.

$${\displaystyle \Delta L^{\prime }=L_{2}^{*}-L_{1}^{*}}$$

$${\displaystyle {\bar {L}}={\frac {L_{1}^{*}+L_{2}^{*}}{2}}\quad {\bar {C}}={\frac {C_{1}^{*}+C_{2}^{*}}{2}}\quad {\mbox{where }}C_{1}^{*}={\sqrt {{a_{1}^{*}}^{2}+{b_{1}^{*}}^{2}}},\quad C_{2}^{*}={\sqrt {{a_{2}^{*}}^{2}+{b_{2}^{*}}^{2}}},\quad }$$

$${\displaystyle a_{1}^{\prime }=a_{1}^{*}+{\frac {a_{1}^{*}}{2}}\left(1-{\sqrt {\frac {{\bar {C}}^{7}}{{\bar {C}}^{7}+25^{7}}}}\right)\quad a_{2}^{\prime }=a_{2}^{*}+{\frac {a_{2}^{*}}{2}}\left(1-{\sqrt {\frac {{\bar {C}}^{7}}{{\bar {C}}^{7}+25^{7}}}}\right)}$$

$${\displaystyle {\bar {C}}^{\prime }={\frac {C_{1}^{\prime }+C_{2}^{\prime }}{2}}{\mbox{ and }}\Delta {C'}=C'_{2}-C'_{1}\quad {\mbox{where }}C_{1}^{\prime }={\sqrt {a_{1}^{'^{2}}+b_{1}^{*^{2}}}}\quad C_{2}^{\prime }={\sqrt {a_{2}^{'^{2}}+b_{2}^{*^{2}}}}\quad }$$

$${\displaystyle h_{1}^{\prime }={\begin{cases}0&b_{1}^{*}=0,a_{1}^{\prime }=0\\{\text{atan2}}(b_{1}^{*},a_{1}^{\prime })&{\text{atan2}}(b_{1}^{*},a_{1}^{\prime })\geq 0^{\circ }\\{\text{atan2}}(b_{1}^{*},a_{1}^{\prime })+360^{\circ }&{\text{otherwise}}\end{cases}}}$$

$${\displaystyle h_{2}^{\prime }={\begin{cases}0&b_{2}^{*}=0,a_{2}^{\prime }=0\\{\text{atan2}}(b_{2}^{*},a_{2}^{\prime })&{\text{atan2}}(b_{2}^{*},a_{2}^{\prime })\geq 0^{\circ }\\{\text{atan2}}(b_{2}^{*},a_{2}^{\prime })+360^{\circ }&{\text{otherwise}}\end{cases}}}$$

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 Sharma 2005, eqn. 7.

The example above expects the parameter order of atan2 to be atan2(y, x).^[24]

$${\displaystyle \Delta h'={\begin{cases}h_{2}^{\prime }-h_{1}^{\prime }&\left|h_{2}^{\prime }-h_{1}^{\prime }\right|\leq 180^{\circ }\\\left(h_{2}^{\prime }-h_{1}^{\prime }\right)-360^{\circ }&\left(h_{2}^{\prime }-h_{1}^{\prime }\right)>180^{\circ }\\\left(h_{2}^{\prime }-h_{1}^{\prime }\right)+360^{\circ }&\left(h_{2}^{\prime }-h_{1}^{\prime }\right)<-180^{\circ }\end{cases}}}$$

When either C′₁ or C′₂ is zero, then Δh′ is irrelevant and may be set to zero. See Sharma 2005, eqn. 10.

$${\displaystyle \Delta H^{\prime }=2{\sqrt {C_{1}^{\prime }C_{2}^{\prime }}}\sin \left({\frac {\Delta h^{\prime }}{2}}\right),\quad {\bar {h}}^{\prime }={\begin{cases}\left({\frac {h_{1}^{\prime }+h_{2}^{\prime }}{2}}\right)&\left|h_{1}^{\prime }-h_{2}^{\prime }\right|\leq 180^{\circ }\\\left({\frac {h_{1}^{\prime }+h_{2}^{\prime }+360^{\circ }}{2}}\right)&\left|h_{1}^{\prime }-h_{2}^{\prime }\right|>180^{\circ },h_{1}^{\prime }+h_{2}^{\prime }<360^{\circ }\\\left({\frac {h_{1}^{\prime }+h_{2}^{\prime }-360^{\circ }}{2}}\right)&\left|h_{1}^{\prime }-h_{2}^{\prime }\right|>180^{\circ },h_{1}^{\prime }+h_{2}^{\prime }\geq 360^{\circ }\\\end{cases}}}$$

When either C′₁ or C′₂ is zero, then h′ 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 Sharma 2005, eqn. 7 and p. 23 stating most implementations on the Internet at the time had "an error in the computation of average hue".

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

$${\displaystyle S_{L}=1+{\frac {0.015\left({\bar {L}}-50\right)^{2}}{\sqrt {20+{\left({\bar {L}}-50\right)}^{2}}}}\quad S_{C}=1+0.045{\bar {C}}^{\prime }\quad S_{H}=1+0.015{\bar {C}}^{\prime }T}$$

$${\displaystyle R_{T}=-2{\sqrt {\frac {{\bar {C}}'^{7}}{{\bar {C}}'^{7}+25^{7}}}}\sin \left[60^{\circ }\cdot \exp \left(-\left[{\frac {{\bar {h}}'-275^{\circ }}{25^{\circ }}}\right]^{2}\right)\right]}$$

CIEDE 2000 is not mathematically continuous. The discontinuity stems from the discontinuity of the mean hue ${\textstyle {\bar {h}}^{\prime }}$ and the hue difference ${\textstyle \Delta h'}$. 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%).^[25] There is also a negligible amount of discontinuity from hue rollover.^[26]

Sharma, Wu, and Dalal have provided additional notes on the mathematics and implementation of the formulae.^[26]

Notes

b. CIELAB is used to characterise the color of biological objects, and transformation of L*a*b* to Chroma and Hue Angle has been proposed as a better solution than just publishing raw L*a*b* values. A set of free tools, including calculation of DeltaE, is described at https://hal.science/hal-05479897

References