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.[6]

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.[9]

[10]

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.