Lab color space

The Lab color space describes mathematically all perceivable colors in the three dimensions L for lightness and a and b for the color components green–red and blue–yellow. The terminology “Lab” originates from the Hunter 1948 color space. Nowadays “Lab” is frequently mis-used as abbreviation for CIEL*a*b* 1976 color space (also CIELAB); the asterisks/stars distinguish the CIE version from Hunter’s original version. The difference from the Hunter Lab coordinates is that the CIELAB coordinates are created by a cube root transformation of the CIE XYZ color data, while the Hunter Lab coordinates are the result of a square root transformation. Other, less common examples of color spaces with Lab representations make use of the CIE 1994 color difference and the CIE 2000 color difference.

The Lab color space exceeds the gamuts of the RGB and CMYK color models (for example, ProPhoto RGB includes about 90% of all perceivable colors). One of the most important attributes of the Lab model is device independence. This means that the colors are defined independent of their nature of creation or the device they are displayed on. The Lab color space is used when graphics for print have to be converted from RGB to CMYK, as the Lab gamut includes both the RGB and CMYK gamut. Also it is used as an interchange format between different devices as for its device independency. The space itself is a three-dimensional real number space, that contains an infinite number of possible representations of colors. However, in practice, the space is usually mapped onto a three-dimensional integer space for device-independent digital representation, and for these reasons, the L*, a*, and b* values are usually absolute, with a pre-defined range. The lightness, L*, represents the darkest black at L* = 0, and the brightest white at L* = 100. The color channels, a* and b*, will represent true neutral gray values at a* = 0 and b* = 0. The red/green opponent colors are represented along the a* axis, with green at negative a* values and red at positive a* values. The yellow/blue opponent colors are represented along the b* axis, with blue at negative b* values and yellow at positive b* values. The scaling and limits of the a* and b* axes will depend on the specific implementation of Lab color, as described below, but they often run in the range of ±100 or −128 to +127 (signed 8-bit integer).

Both the Hunter and the 1976 CIELAB color spaces were derived from the prior “master” space CIE 1931 XYZ color space, which can predict which spectral power distributions will be perceived as the same color (see metamerism), but which is not particularly perceptually uniform. Strongly influenced by the Munsell color system, the intention of both “Lab” color spaces is to create a space that can be computed via simple formulas from the XYZ space but is more perceptually uniform than XYZ. Perceptually uniform means that a change of the same amount in a color value should produce a change of about the same perceptual distance. When storing colors in limited precision values, this can improve the reproduction of tones. Both Lab spaces are relative to the white point of the XYZ data they were converted from. Lab values do not define absolute colors unless the white point is also specified. Often, in practice, the white point is assumed to follow a standard and is not explicitly stated (e.g., for “absolute colorimetric” rendering intent, the International Color Consortium L*a*b* values are relative to CIE standard illuminant D50, while they are relative to the unprinted substrate for other rendering intents).

The lightness correlate in CIELAB is calculated using the cube root of the relative luminance.

Advantages
Unlike the RGB and CMYK color models, Lab color is designed to approximate human vision. It aspires to perceptual uniformity, and its L component closely matches human perception of lightness, although it does not take the Helmholtz–Kohlrausch effect into account. Thus, it can be used to make accurate color balance corrections by modifying output curves in the a and b components, or to adjust the lightness contrast using the L component. In RGB or CMYK spaces, which model the output of physical devices rather than human visual perception, these transformations can be done only with the help of appropriate blend modes in the editing application.

Because the Lab space is larger than the gamut of computer displays and printers and because the visual stepwidths are relatively different to the color area, a bitmap image represented as Lab requires more data per pixel to obtain the same precision as an RGB or CMYK bitmap. In the 1990s, when computer hardware and software were limited to storing and manipulating mostly 8-bit/channel bitmaps, converting an RGB image to Lab and back was a very lossy operation. With 16-bit/channel and floating-point support now common, the loss due to quantization is negligible.

CIELAB is copyright and license-free: as it is fully mathematically defined, the CIELAB model is public domain, it is in all respects freely usable and integrable (also systematic Lab / HLC color value tables).

A big portion of the Lab coordinate space cannot be generated by spectral distributions, it therefore falls outside the human vision and such Lab values are not “colors”.

Differentiation
Some specific uses of the abbreviation in software, literature etc.

In Adobe Photoshop, image editing using “Lab mode” is CIELAB D50.
In Affinity Photo, Lab editing is achieved by changing the document’s Colour Format to “Lab (16 bit)”
In ICC profiles, the “Lab color space” used as a profile connection space is CIELAB D50.
In TIFF files, the CIELAB color space may be used.
In PDF documents, the “Lab color space” is CIELAB.
In Digital Color Meter on OS X, it is described as “L*a*b*”
In the open source non-destructive-editing software RawTherapee, an entire tab with many controls is dedicated to the CIE Color Appearance Model

CIELAB
CIE L*a*b* (CIELAB) is a color space specified by the International Commission on Illumination (French Commission internationale de l’éclairage, hence its CIE initialism). It describes all the colors visible to the human eye and was created to serve as a device-independent model to be used as a reference.

The three coordinates of CIELAB represent the lightness of the color (L* = 0 yields black and L* = 100 indicates diffuse white; specular white may be higher), its position between red/magenta and green (a*, negative values indicate green while positive values indicate magenta) and its position between yellow and blue (b*, negative values indicate blue and positive values indicate yellow). The asterisk (*) after L, a and b are pronounced star and are part of the full name, since they represent L*, a* and b*, to distinguish them from Hunter’s L, a, and b, described below.

Since the L*a*b* model is a three-dimensional model, it can be represented properly only in a three-dimensional space. Two-dimensional depictions include chromaticity diagrams: sections of the color solid with a fixed lightness. It is crucial to realize that the visual representations of the full gamut of colors in this model are never accurate; they are there just to help in understanding the concept.

Because the red-green and yellow-blue opponent channels are computed as differences of lightness transformations of (putative) cone responses, CIELAB is a chromatic value color space.

A related color space, the CIE 1976 (L*, u*, v*) color space (a.k.a. CIELUV), preserves the same L* as L*a*b* but has a different representation of the chromaticity components. CIELAB and CIELUV can also be expressed in cylindrical form (CIELCH and CIELCHuv, respectively), with the chromaticity components replaced by correlates of chroma and hue.

Since CIELAB and CIELUV, the CIE has been incorporating an increasing number of color appearance phenomena into their models, to better model color vision. These color appearance models, of which CIELAB is a simple example, culminated with CIECAM02.

Perceptual differences
This topic is covered in more detail at Color difference.

The nonlinear relations for L*, a*, and b* are intended to mimic the nonlinear response of the eye. 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
There are no simple formulas for conversion between RGB or CMYK values and L*a*b*, because the RGB and CMYK color models are device-dependent. The RGB or CMYK values first must be transformed to a specific absolute color space, such as sRGB or Adobe RGB. This adjustment will be device-dependent, but the resulting data from the transform will be device-independent, allowing data to be transformed to the CIE 1931 color space and then transformed into L*a*b*.

Range of coordinates
As mentioned previously, the L* coordinate ranges from 0 to 100. The possible range of a* and b* coordinates is independent of the color space that one is converting from, since the conversion below uses X and Y, which come from RGB.

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

${\begin{aligned}f(t)&={\begin{cases}{\sqrt[{3}]{t}}&{\text{if }}t>\delta ^{3}\\{\frac {t}{3\delta ^{2}}}+{\frac {4}{29}}&{\text{otherwise}}\end{cases}}\\\delta &={\frac {6}{29}}\end{aligned}}$
Here, $X n$ , $Y n$ and $Z n$ are the CIE XYZ tristimulus values of the reference white point (the subscript n suggests “normalized”).

Under Illuminant D65 with normalization Y = 100, the values are

${\begin{aligned}X_{\mathrm {n} }&=95.047,\\Y_{\mathrm {n} }&=100.000,\\Z_{\mathrm {n} }&=108.883\end{aligned}}$
Values for illuminant D50 are

${\begin{aligned}X_{\mathrm {n} }&=96.6797,\\Y_{\mathrm {n} }&=100.000,\\Z_{\mathrm {n} }&=82.5188\end{aligned}}$
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 t1/3 part of the function at t0 in both value and slope. In other words:

${\begin{aligned}t_{0}^{1/3}&=mt_{0}+c&{\text{ (match in value)}}\\{\frac {1}{3}}t_{0}^{-2/3}&=m&{\text{ (match in slope)}}\end{aligned}}$
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:

${\begin{aligned}m&={\frac {1}{3}}\delta ^{-2}&=7.787037\ldots \\t_{0}&=\delta ^{3}&=0.008856\ldots \end{aligned}}$
where δ = 6/29.

Reverse transformation
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

$f^{-1}(t)={\begin{cases}t^{3}&{\text{if }}t>\delta \\3\delta ^{2}\left(t-{\frac {4}{29}}\right)&{\text{otherwise}}\end{cases}}$
and where δ = 6/29.

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

$L=100{\sqrt {Y/Y_{\mathrm {n} }}}$
where Yn 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, since {\displaystyle 100{\sqrt {25/100}}=100\cdot 1/2} $100{\sqrt {25/100}}=100\cdot 1/2$

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

$a=K_{\mathrm {a} }\left({\frac {X/X_{\mathrm {n} }-Y/Y_{\mathrm {n} }}{\sqrt {Y/Y_{\mathrm {n} }}}}\right)$
where Ka is a coefficient that depends upon the illuminant (for D65, Ka is 172.30; see approximate formula below) and Xn 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:

$b=K_{\mathrm {b} }\left({\frac {Y/Y_{\mathrm {n} }-Z/Z_{\mathrm {n} }}{\sqrt {Y/Y_{\mathrm {n} }}}}\right)$
where Kb is a coefficient that depends upon the illuminant (for D65, Kb is 67.20; see approximate formula below) and Zn is the Z tristimulus value of the specified white object.

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).

The name for the system is an attribution to Richard S. Hunter.

Approximate formulas for Ka and Kb
In the previous version of the Hunter Lab color space, Ka was 175 and Kb 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:

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

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.

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

$L=100{\sqrt {Y/Y_{\mathrm {n} }}}$
and, as the uniform chromaticity coordinates:

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

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

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

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

$b=K\cdot L\cdot c_{\mathrm {b} }=K\cdot 100k_{\mathrm {e} }{\frac {Y/Y_{\mathrm {n} }-Z/Z_{\mathrm {n} }}{\sqrt {Y/Y_{\mathrm {n} }}}}$
which is identical to the Hunter Lab formulas given above if we select K = Ka/100 and ke = Kb/Ka. Therefore, the Hunter Lab color space is an Adams chromatic valence color space.

Cylindrical representation: CIELCh or CIEHLC
The CIELCh color space is a CIELab cube color space, where instead of Cartesian coordinates a*, b*, the cylindrical coordinates C* (chroma, relative saturation) and h° (hue angle, angle of the hue in the CIELab color wheel) are specified. The CIELab lightness L* remains unchanged.

The conversion of a* and b* to C* and h° is done using the following formulas:

$C^{\star }={\sqrt {a^{\star \,2}+b^{\star \,2}}},\qquad h^{\circ }=\arctan \left({\frac {b^{\star }}{a^{\star }}}\right)$
Conversely, given the polar coordinates, conversion to Cartesian coordinates is achieved with:

$a^{\star }=C^{\star }\cos(h^{\circ }),\qquad b^{\star }=C^{\star }\sin(h^{\circ })$
The LCh color space is not the same as the HSV, HSL or HSB color models, although their values can also be interpreted as a base color, saturation and lightness of a color. The HSL values are a polar coordinate transformation of what is technically defined RGB cube color space. LCh is still perceptually uniform.

Further, H and h are not identical, because HSL space uses as primary colors the three additive primary colors red, green, blue (H = 0, 120, 240°). Instead, the LCh system uses the four colors yellow, green, blue and red (h = 90, 180, 270, 360°). Regardless the angle h, C = 0 means the achromatic colors, that is, the gray axis.

The simplified spellings LCh, LCH and HLC are common, but the latter presents a different order. HCL color space (Hue-Chroma-Luminance) on the other hand is a commonly used alternative name for the L*C*h(uv) color space, also known as the cylindrical representation or polar CIELUV.

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