CIELAB color space

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The CIELAB color space, also referred to as L*a*b*, is a color space defined by the International Commission on Illumination (abbreviated CIE) in 1976. It expresses color as three values: L* for perceptual lightness and a* and b* for the four unique colors of human vision: red, green, blue and yellow. CIELAB was intended as a perceptually uniform space, where a given numerical change corresponds to a similar perceived change in color. While the LAB space is not truly perceptually uniform, it nevertheless is useful in industry for detecting small differences in color.

Like the CIEXYZ space it derives from, CIELAB color space is a device-independent, "standard observer" model. The colors it defines are not relative to any particular device such as a computer monitor or a printer, but instead relate to the CIE standard observer which is an averaging of the results of color matching experiments under laboratory conditions.

Coordinates

The CIELAB space is three-dimensional and covers the entire gamut (range) of human color perception. It is based on the opponent model of human vision, where red and green form an opponent pair and blue and yellow form an opponent pair. This makes CIELAB a Hering opponent color space. The nature of the transformations also characterizes it as a chromatic value color space. The lightness value, L* (pronounced "L star"), defines black at 0 and white at 100. The a* axis is relative to the green–red opponent colors, with negative values toward green and positive values toward red. The b* axis represents the blue–yellow opponents, with negative numbers toward blue and positive toward yellow.

The lightness value, L* in CIELAB is calculated using the cube root of the relative luminance with an offset near black. This results in an effective power curve with an exponent of approximately 0.43 which represents the human eye's response to light under daylight (photopic) conditions.

The a* and b* axes are unbounded and depending on the reference white they can easily exceed ±150 to cover the human gamut. Nevertheless, software implementations often clamp these values for practical reasons. For instance, if integer math is being used it is common to clamp a* and b* in the range of −128 to 127.

CIELAB is calculated relative to a reference white, for which the CIE recommends the use of CIE Standard illuminant D65. D65 is used in the vast majority of industries and applications, with the notable exception being the printing industry which uses D50. The International Color Consortium largely supports the printing industry and uses D50 with either CIEXYZ or CIELAB in the Profile Connection Space, for v2 and v4 ICC profiles.

While the intention behind CIELAB was to create a space that was more perceptually uniform than CIEXYZ using only a simple formula, CIELAB is known to lack perceptual uniformity, particularly in the area of blue hues.

The sRGB gamut (left) and optimal color solid under D65 illumination (right) plotted within the CIELAB color space. a and b are the horizontal axes; L is the vertical axis.

The asterisks (*) after L*, a*, and b* are pronounced star and are part of the full name to distinguish L*a*b* from Hunter's Lab, described below.

Since the L*a*b* model has three axes, it requires a three-dimensional space to be represented completely. Also, because each axis is non-linear, it is not possible to create a two-dimensional chromaticity diagram. Additionally, the visual representations shown in the plots of the full CIELAB gamut on this page are an approximation, as it is impossible for a monitor to display the full gamut of LAB colors.

Perceptual differences

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.

RGB and CMYK conversions

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 contains these needed data, which are then used to perform the conversions.

Range of coordinates

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 color space. 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.

Advantages

Unlike the RGB and CMYK color models, CIELAB is designed to approximate human vision. The L* component closely matches human perception of lightness, though it does not take the Helmholtz–Kohlrausch effect into account. CIELAB is less uniform in the color axes, but is useful for predicting small differences in color.

The CIELAB coordinate space represents the entire gamut of human photopic (daylight) vision and far exceeds the gamut for sRGB or CMYK. In an integer implementation such as TIFF, ICC or Photoshop, the large coordinate space results in substantial data inefficiency due to unused code values. Only about 35% of the available coordinate code values are inside the CIELAB gamut with an integer format.

Using CIELAB in an 8-bit per channel integer format typically results in significant quantization errors. Even 16-bit per channel can result in clipping, as the full gamut extends past the bounding coordinate space. Ideally, CIELAB should be used with floating-point data to minimize obvious quantization errors.

CIE standards and documents are copyrighted by the CIE and must be purchased; however, the formulas for CIELAB are available on the CIE website.

Converting between CIELAB and CIE XYZ coordinates

From CIE XYZ to CIELAB

{\begin{aligned}L^{\star }&=116\,f{{\bigl (}Y/Y_{\mathrm {n} }{\bigr )}}-16,\\[5mu]a^{\star }&=500{\bigl (}f(X/X_{\mathrm {n} })-f(Y/Y_{\mathrm {n} }){\bigr )}\\[5mu]b^{\star }&=200{\bigl (}f(Y/Y_{\mathrm {n} })-f(Z/Z_{\mathrm {n} }){\bigr )}\end{aligned}}

where t is $X/X_{\mathrm {n} },$ $Y/Y_{\mathrm {n} },$ or $Z/Z_{\mathrm {n} }$ :

{\begin{aligned}f(t)&={\begin{cases}{\sqrt[{3}]{t}}&{\text{if }}t>\delta ^{3}\\[4mu]{\tfrac {1}{3}}t\delta ^{-2}+{\tfrac {4}{29}}&{\text{otherwise}}\end{cases}}\\\delta &={\tfrac {6}{29}}\end{aligned}}

$X$ , $Y$ , and $Z$ 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 the reference white has $Y = 100$ , the values are:

For Standard Illuminant D65:

{\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:

{\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 $t = t 0$ and was assumed to match the ${\sqrt[{3}]{t}}$ part of the function at $t 0$ in both value and slope. In other words:

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

The intercept $f (0) = c$ was chosen so that $L *$ would be 0 for $Y = 0$ : $c = .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠16/116⁠ = ⁠4/29⁠$ . The above two equations can be solved for $m$ and $t 0$ :

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

where $δ = ⁠ 6 / 29 ⁠$ .

From CIELAB to CIEXYZ

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

{\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

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 $δ = ⁠ 6 / 29 ⁠$ .

Cylindrical model

The sRGB gamut (left) and optimal color solid under D65 illumination (right) plotted within the CIELCHab color space. L is the vertical axis; C is the cylinder radius; h is the angle around the circumference.

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