CIELAB ΔE* Color Difference

CIELAB ΔE*

The International Commission on Illumination (CIE) calls their distance metric ΔE^*_ab (also called ΔE*, or, inaccurately, dE*, dE, or "Delta E") where delta is a Greek letteroften 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.^[2][3]

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.^[4] These non-uniformities are important because the human eye is more sensitive to certain colors than others. A good metric should take this into account in order for the notion of a "just noticeable difference" to have meaning. Otherwise, a certain ΔE that may be insignificant between two colors in one part of the color space while being significant in some other part.^[5]

CIE76

The 1976 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, ${\displaystyle ({L_{1}^{*}},{a_{1}^{*}},{b_{1}^{*}})}$ and ${\displaystyle ({L_{2}^{*}},{a_{2}^{*}},{b_{2}^{*}})}$, the CIE76 color difference formula is defined as:

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

${\displaystyle \Delta E_{ab}^{*}\approx 2.3}$ corresponds to a JND (just noticeable difference).^[6]

CIE94

The 1976 definition was extended to address perceptual non-uniformities, while retaining the CIELAB color space, by the introduction of application-specific weights derived from an automotive paint test's tolerance data.^[7]

ΔE (1994) is defined in the L*C*h* color space with differences in lightness, chroma and hue calculated from L*a*b* coordinates. Given a reference color^[8] ${\displaystyle (L_{1}^{*},a_{1}^{*},b_{1}^{*})}$and another color ${\displaystyle (L_{2}^{*},a_{2}^{*},b_{2}^{*})}$, the difference is:^[9][10][11]

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

where:

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

${\displaystyle C_{1}^{*}={\sqrt {{a_{1}^{*}}^{2}+{b_{1}^{*}}^{2}}}}$

${\displaystyle C_{2}^{*}={\sqrt {{a_{2}^{*}}^{2}+{b_{2}^{*}}^{2}}}}$

${\displaystyle \Delta C_{ab}^{*}=C_{1}^{*}-C_{2}^{*}}$

${\displaystyle \Delta H_{ab}^{*}={\sqrt {{\Delta E_{ab}^{*}}^{2}-{\Delta L^{*}}^{2}-{\Delta C_{ab}^{*}}^{2}}}={\sqrt {{\Delta a^{*}}^{2}+{\Delta b^{*}}^{2}-{\Delta C_{ab}^{*}}^{2}}}}$

${\displaystyle \Delta a^{*}=a_{1}^{*}-a_{2}^{*}}$

${\displaystyle \Delta b^{*}=b_{1}^{*}-b_{2}^{*}}$

${\displaystyle S_{L}=1}$

${\displaystyle S_{C}=1+K_{1}C_{1}^{*}}$

${\displaystyle S_{H}=1+K_{2}C_{1}^{*}}$

and where k_C and k_H are usually both unity and the 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. ^[12]

CIEDE2000

Since the 1994 definition did not adequately resolve the perceptual uniformity issue, the CIE refined their definition, adding five corrections:^[13][14]

• A hue rotation term (R_T), to deal with the problematic blue region (hue angles in the neighborhood of 275°):^[15]
• 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)

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

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

The k_L, k_C, and k_H are usually unity.

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

${\displaystyle {\bar {L}}={\frac {L_{1}^{*}+L_{2}^{*}}{2}}\quad {\bar {C}}={\frac {C_{1}^{*}+C_{2}^{*}}{2}}}$

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

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

${\displaystyle h_{1}^{\prime }={\text{atan2}}(b_{1}^{*},a_{1}^{\prime })\mod 360^{\circ },\quad h_{2}^{\prime }={\text{atan2}}(b_{2}^{*},a_{2}^{\prime })\mod 360^{\circ }}$

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 Sharma 2005, eqn. 7.

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

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

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

Note: 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".

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

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

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

CMC l:c (1984)

In 1984, the Colour Measurement Committee of the Society of Dyers and Colourists defined a difference measure, also based on the L*C*h color model. Named after the developing committee, their metric is called CMC l:c. The quasimetric 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^[16] for acceptability and 1:1 for the threshold of imperceptibility.

The distance of a color ${\displaystyle (L_{2}^{*},C_{2}^{*},h_{2})}$ to a reference ${\displaystyle (L_{1}^{*},C_{1}^{*},h_{1})}$ is:^[17]

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

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

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

CMC l:c is designed to be used with D65 and the CIE Supplementary Observer.^[18] The formula is not a metric but rather a quasimetric because it violates symmetry, parameter T is based on the hue of the ${\displaystyle h_{1}}$ alone. This reference color means that the color difference from the first color to the second color is different than the color distance from the second color is to the first.

Tolerance

A MacAdam diagram in the CIE 1931 color space. The ellipses are shown ten times their actual size.

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.^[19]