CIELAB color space

The nonlinear relations for L*, a* and b* are intended to mimic the nonlinear response of the visual system. Furthermore, uniform changes of components in the L*a*b* color space aim to correspond to uniform changes in perceived color, so the relative perceptual differences between any two colors in L*a*b* can be approximated by treating each color as a point in a three-dimensional space (with three components: L*, a*, b*) and taking the Euclidean distance between them.[10]

In order to convert RGB or CMYK values to or from L*a*b*, the RGB or CMYK data must be linearized relative to light. The reference illuminant of the RGB or CMYK data must be known, as well as the RGB primary coordinates or the CMYK printer's reference data in the form of a color lookup table (CLUT).

In color managed systems, ICC profiles contain these needed data, which are then used to perform the conversions.

As mentioned previously, the L* coordinate nominally ranges from 0 to 100. The range of a* and b* coordinates is technically unbounded, though it is commonly clamped to the range of −128 to 127 for use with integer code values, though this results in potentially clipping some colors depending on the size of the source colorspace. The gamut's large size and inefficient utilization of the coordinate space means the best practice is to use floating-point values for all three coordinates.

${\displaystyle {\begin{aligned}L^{\star }&=116\ f\!\left({\frac {Y}{Y_{\mathrm {n} }}}\right)-16\\a^{\star }&=500\left(f\!\left({\frac {X}{X_{\mathrm {n} }}}\right)-f\!\left({\frac {Y}{Y_{\mathrm {n} }}}\right)\right)\\b^{\star }&=200\left(f\!\left({\frac {Y}{Y_{\mathrm {n} }}}\right)-f\!\left({\frac {Z}{Z_{\mathrm {n} }}}\right)\right)\end{aligned}}}$

where, being t = X/X_n, Y/Y_n, or Z/Z_n:

${\displaystyle {\begin{aligned}f(t)&={\begin{cases}{\sqrt[{3}]{t}}&{\text{if }}t>\delta ^{3}\\{\dfrac {t}{3\delta ^{2}}}+{\frac {4}{29}}&{\text{otherwise}}\end{cases}}\\\delta &={\tfrac {6}{29}}\end{aligned}}}$

X, Y, Z describe the color stimulus considered and X_n, Y_n, Z_n describe a specified white achromatic reference illuminant. for the CIE 1931 (2°) standard colorimetric observer and assuming normalization where reference white = Y = 100, the values are:

For Standard Illuminant D65:

${\displaystyle {\begin{aligned}X_{\mathrm {n} }&=95.0489,\\Y_{\mathrm {n} }&=100,\\Z_{\mathrm {n} }&=108.8840\end{aligned}}}$

For illuminant D50, which is used in the printing industry:

${\displaystyle {\begin{aligned}X_{\mathrm {n} }&=96.4212,\\Y_{\mathrm {n} }&=100,\\Z_{\mathrm {n} }&=82.5188\end{aligned}}}$

The division of the domain of the f function into two parts was done to prevent an infinite slope at t = 0. The function f was assumed to be linear below some t = t₀ and was assumed to match the ³√t part of the function at t₀ in both value and slope. In other words:

${\displaystyle {\begin{aligned}{\sqrt[{3}]{t_{0}}}&=mt_{0}+c&{\text{ (match in value)}}\\{\tfrac {1}{3}}\left(t_{0}\right)^{-{\frac {2}{3}}}&=m&{\text{ (match in slope)}}\end{aligned}}}$

The intercept f(0) = c was chosen so that L* would be 0 for Y = 0: c = 16/116 = 4/29. The above two equations can be solved for m and t₀:

${\displaystyle {\begin{aligned}m&={\tfrac {1}{3}}\delta ^{-2}&=7.787037\ldots \\t_{0}&=\delta ^{3}&=0.008856\ldots \end{aligned}}}$

where δ = 6/29.[11]

[12]

The reverse transformation is most easily expressed using the inverse of the function f above:

${\displaystyle {\begin{aligned}X&=X_{\mathrm {n} }f^{-1}\left({\frac {L^{\star }+16}{116}}+{\frac {a^{\star }}{500}}\right)\\Y&=Y_{\mathrm {n} }f^{-1}\left({\frac {L^{\star }+16}{116}}\right)\\Z&=Z_{\mathrm {n} }f^{-1}\left({\frac {L^{\star }+16}{116}}-{\frac {b^{\star }}{200}}\right)\\\end{aligned}}}$

where

${\displaystyle f^{-1}(t)={\begin{cases}t^{3}&{\text{if }}t>\delta \\3\delta ^{2}\left(t-{\tfrac {4}{29}}\right)&{\text{otherwise}}\end{cases}}}$

and where δ = 6/29.

L is a correlate of lightness and is computed from the Y tristimulus value using Priest's approximation to Munsell value:

${\displaystyle L=100{\sqrt {\frac {Y}{Y_{\mathrm {n} }}}}}$

where Y_n is the Y tristimulus value of a specified white object. For surface-color applications, the specified white object is usually (though not always) a hypothetical material with unit reflectance that follows Lambert's law. The resulting L will be scaled between 0 (black) and 100 (white); roughly ten times the Munsell value. Note that a medium lightness of 50 is produced by a luminance of 25, due to the square root proportionality.

a and b are termed opponent color axes. a represents, roughly, Redness (positive) versus Greenness (negative). It is computed as:

${\displaystyle a=K_{\mathrm {a} }\left({\frac {{\frac {X}{X_{\mathrm {n} }}}-{\frac {Y}{Y_{\mathrm {n} }}}}{\sqrt {\frac {Y}{Y_{\mathrm {n} }}}}}\right)}$

where K_a is a coefficient that depends upon the illuminant (for D65, K_a is 172.30; see approximate formula below) and X_n is the X tristimulus value of the specified white object.

The other opponent color axis, b, is positive for yellow colors and negative for blue colors. It is computed as:

${\displaystyle b=K_{\mathrm {b} }\left({\frac {{\frac {Y}{Y_{\mathrm {n} }}}-{\frac {Z}{Z_{\mathrm {n} }}}}{\sqrt {\frac {Y}{Y_{\mathrm {n} }}}}}\right)}$

where K_b is a coefficient that depends upon the illuminant (for D65, K_b is 67.20; see approximate formula below) and Z_n is the Z tristimulus value of the specified white object.[26]

Both a and b will be zero for objects that have the same chromaticity coordinates as the specified white objects (i.e., achromatic, grey, objects).

In the previous version of the Hunter Lab color space, K_a was 175 and K_b was 70. Hunter Associates Lab discovered that better agreement could be obtained with other color difference metrics, such as CIELAB (see above) by allowing these coefficients to depend upon the illuminants. Approximate formulae are:

${\displaystyle K_{\mathrm {a} }\approx {\frac {175}{198.04}}(X_{\mathrm {n} }+Y_{\mathrm {n} })}$

${\displaystyle K_{\mathrm {b} }\approx {\frac {70}{218.11}}(Y_{\mathrm {n} }+Z_{\mathrm {n} })}$

which result in the original values for Illuminant C, the original illuminant with which the Lab color space was used.

Adams chromatic valence color spaces are based on two elements: a (relatively) uniform lightness scale and a (relatively) uniform chromaticity scale.[27] If we take as the uniform lightness scale Priest's approximation to the Munsell Value scale, which would be written in modern notation as:

${\displaystyle L=100{\sqrt {\frac {Y}{Y_{\mathrm {n} }}}}}$

and, as the uniform chromaticity coordinates:

${\displaystyle c_{\mathrm {a} }={\frac {\frac {X}{X_{\mathrm {n} }}}{\frac {Y}{Y_{\mathrm {n} }}}}-1={\frac {{\frac {X}{X_{\mathrm {n} }}}-{\frac {Y}{Y_{\mathrm {n} }}}}{\frac {Y}{Y_{\mathrm {n} }}}}}$

${\displaystyle c_{\mathrm {b} }=k_{\mathrm {e} }\left(1-{\frac {\frac {Z}{Z_{\mathrm {n} }}}{\frac {Y}{Y_{\mathrm {n} }}}}\right)=k_{\mathrm {e} }{\frac {{\frac {Y}{Y_{\mathrm {n} }}}-{\frac {Z}{Z_{\mathrm {n} }}}}{\frac {Y}{Y_{\mathrm {n} }}}}}$

where k_e is a tuning coefficient, we obtain the two chromatic axes:

${\displaystyle a=K\cdot L\cdot c_{\mathrm {a} }=K\cdot 100{\frac {{\frac {X}{X_{\mathrm {n} }}}-{\frac {Y}{Y_{\mathrm {n} }}}}{\sqrt {\frac {Y}{Y_{\mathrm {n} }}}}}}$

and

${\displaystyle b=K\cdot L\cdot c_{\mathrm {b} }=K\cdot 100k_{\mathrm {e} }{\frac {{\frac {Y}{Y_{\mathrm {n} }}}-{\frac {Z}{Z_{\mathrm {n} }}}}{\sqrt {\frac {Y}{Y_{\mathrm {n} }}}}}}$

which is identical to the Hunter Lab formulas given above if we select K = K_a/100 and k_e = K_b/K_a. Therefore, the Hunter Lab color space is an Adams chromatic valence color space.