Lab color space



A Lab color space is a color-opponent space with dimension L for lightness and a and b for the color-opponent dimensions, based on nonlinearly compressed CIE XYZ color space coordinates.

The coordinates of the Hunter 1948 L, a, b color space are L, a, and b. However, Lab is now more often used as an informal abbreviation for the CIE 1976 (L*, a*, b*) color space (also called CIELAB, whose coordinates are actually L*, a*, and b*). Thus the initials Lab by themselves are somewhat ambiguous. The color spaces are related in purpose, but differ in implementation.

Both spaces are derived from the "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 which 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 visual importance. 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 ICC 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.

The L*a*b* color space includes all perceivable colors which means that its gamut exceeds those of the RGB and CMYK color models. One of the most important attributes of the L*a*b*-model is the device independency. This means that the colors are defined independent of their nature of creation or the device they are displayed on. The L*a*b* color space is used e.g. in Adobe Photoshop when graphics for print have to be converted from RGB to CMYK, as the L*a*b* gamut includes both the RGB and CMYK gamut. Also it is used as an interchange format between different devices as for its device independency.

Advantages of Lab
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. It can thus 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 only be done with the help of appropriate blend modes in the editing application.

Because Lab space is much larger than the gamut of computer displays, printers, or even human vision, 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 was mostly limited to storing and manipulating 8 bit/channel bitmaps, converting an RGB image to Lab and back was a lossy operation. With 16 bit/channel support now common, this is no longer such a problem.

Additionally, many of the "colors" within Lab space fall outside the gamut of human vision, and are therefore purely imaginary; these "colors" cannot be reproduced in the physical world. Though color management software, such as that built in to image editing applications, will pick the closest in-gamut approximation, changing lightness, colorfulness, and sometimes hue in the process, author Dan Margulis claims that this access to imaginary colors is useful, going between several steps in the manipulation of a picture.

Which "Lab"?
Some specific uses of the abbreviation in software, literature etc.
 * In Adobe Photoshop, image editing using "Lab mode" is CIELAB D50.
 * 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.

CIE 1976 (L*, a*, b*) color space (CIELAB)
CIE L*a*b* (CIELAB) is the most complete 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 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 only be represented properly 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. CIELUV can also be expressed in cylindrical form (CIELCH), 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, although not designed as can be seen as a simple example, culminated with CIECAM02.

Measuring differences
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 need to 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 L*a*b* 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.

The forward transformation

 * $$\begin{align}

L^\star &= 116 f(Y/Y_n) - 16\\ a^\star &= 500 \left[f(X/X_n) - f(Y/Y_n)\right]\\ b^\star &= 200 \left[f(Y/Y_n) - f(Z/Z_n)\right] \end{align}$$

where
 * $$f(t) = \begin{cases}

t^{1/3} & \text{if } t > (\frac{6}{29})^3 \\ \frac13 \left( \frac{29}{6} \right)^2 t + \frac{4}{29} & \text{otherwise} \end{cases}$$

Here Xn, Yn and Zn are the CIE XYZ tristimulus values of the reference white point (the subscript n suggests "normalized").

The division of the f(t) function into two domains was done to prevent an infinite slope at t = 0. f(t) 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{align}

t_0^{1/3} &= a t_0 + b & \text{ (match in value)}\\ \tfrac13 t_0^{-2/3} &= a & \text{ (match in slope)} \end{align}$$

The slope was chosen to be b = 16/116 = 4/29. The above two equations can be solved for a and t0:
 * $$\begin{align}

a &= \tfrac13\delta^{-2} &= 7.787037\ldots\\ t_0 &= \delta^3 &= 0.008856\ldots \end{align}$$

where δ = 6/29. Note that the slope at the join is b = 4/29 = 2δ/3.

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


 * $$\begin{align}

Y &= Y_n f^{-1}\left(\tfrac{1}{116}\left(L^*+16\right)\right)\\ X &= X_n f^{-1}\left(\tfrac{1}{116}\left(L^*+16\right) + \tfrac{1}{500}a^*\right)\\ Z &= Z_n f^{-1}\left(\tfrac{1}{116}\left(L^*+16\right) - \tfrac{1}{200}b^*\right)\\ \end{align}$$

where


 * $$f^{-1}(t) = \begin{cases}

t^3 & \text{if } t > \tfrac{6}{29} \\ 3\left(\tfrac{6}{29}\right)^2\left(t - \tfrac{4}{29}\right) & \text{otherwise} \end{cases}$$

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


 * $$L=100\sqrt{Y / Y_n}$$

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 and which 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 $$100 \sqrt{25/100} = 100 \cdot 1/2$$

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


 * $$a=K_a\left(\frac{X/X_n-Y/Y_n}{\sqrt{Y/Y_n}}\right)$$

where $$K_a$$ is a coefficient which depends upon the illuminant (for D65, Ka 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:


 * $$b=K_b\left(\frac{Y/Y_n-Z/Z_n}{\sqrt{Y/Y_n}}\right)$$

where Kb is a coefficient which 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 which have the same chromaticity coordinates as the specified white objects (i.e., achromatic, grey, objects).

Approximate formulas for Ka and Kb
In the previous version of the Hunter Lab color space, Ka was 175 and Kb was 70. Apparently, 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 formulæ are:


 * $$K_a\approx\frac{175}{198.04}(X_n+Y_n)$$


 * $$K_b\approx\frac{70}{218.11}(Y_n+Z_n)$$

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

The Hunter Lab Color Space 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:


 * $$L=100\sqrt{Y / Y_n}$$

and, as the uniform chromaticity coordinates:


 * $$c_a=\frac{X/X_n}{Y/Y_n}-1=\frac{X/X_n-Y/Y_n}{Y/Y_n}$$


 * $$c_b=k_e\left(1-\frac{Z/Z_n}{Y/Y_n}\right)=k_e\frac{Y/Y_n-Z/Z_n}{Y/Y_n}$$

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


 * $$a=K\cdot L\cdot c_a=K\cdot 100\frac{X/X_n-Y/Y_n}{\sqrt{Y/Y_n}}$$

and


 * $$b=K\cdot L\cdot c_b=K\cdot 100 k_e \frac{Y/Y_n-Z/Z_n}{\sqrt{Y/Y_n}}$$

which is identical to the Hunter Lab formulae 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.