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. 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. 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. 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 .
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 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: \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 } S_L=\begin{cases} 0.511 & L^*_1 F = \sqrt{\frac{C^{*^4}_1}{C^{*^4}_1+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.
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
kL,
kC and
kH, and functions
SL,
SC, and
SH derived from an automotive paint test's tolerance data. As with the CMC I:c, Δ
E (1994) is defined in the L*C*h* color space and likewise violates symmetry, therefore defining a quasimetric. Given a reference color{{efn|1=Called such because the operator is not
commutative. This makes it a
quasimetric. Specifically, S_{\{C,H\}} both depend on C^*_1 only.}} (L^*_1, a^*_1, b^*_1) and another color (L^*_2, a^*_2, b^*_2), the difference is \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 \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
kC and
kH are usually both set to unity, and the parametric weighting factors
kL,
K1 and
K2 depend on the application: : Geometrically, the quantity \Delta H^*_{ab} corresponds to the arithmetic mean of the chord lengths of the equal chroma circles of the two colors.
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: • A hue rotation term (RT), to deal with the problematic blue region (hue angles in the neighborhood of 275°): • Compensation for neutral colors (the primed values in the L*C*h differences) • Compensation for lightness (SL) • Compensation for chroma (SC) • Compensation for hue (SH) \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
RT. The parametric weighting factors
kL,
kC, and
kH are usually set to unity. \Delta L^\prime = L^*_2 - L^*_1 \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 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) \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 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 360^\circ \\ \text{atan2} (b_1^*, a_1^\prime) + 360^\circ & \text{otherwise} \end{cases} 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 360^\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, ) 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 ''
and b
are zero (which also means that the corresponding '' is zero); in that case, set the hue angle to zero. See . The example above expects the parameter order of atan2 to be atan2(y, x). \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) When either '
1 or '2 is zero, then Δ is irrelevant and may be set to zero. See . \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 180^\circ, h_1^\prime + h_2^\prime \geq 360^\circ \\ \end{cases} When either '
1 or '2 is zero, then '
is '1+''''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 \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) 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 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 \bar{h}^\prime and the hue difference \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%). There is also a negligible amount of discontinuity from hue rollover. ==Tolerance==