Lab color space

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The CIE 1976 (L*, a*, b*) color space (CIELAB), showing only colors that fit within the sRGB gamut (and can therefore be displayed on a typical computer display). Each axis of each square ranges from -128 to 128.

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 (e.g. CIE XYZ color space) coordinates. The terminology originates from the three dimensions of the Hunter 1948 color space, which are L, a, and b.[1][2] However, Lab is now more often used as an informal abbreviation for the L-a-b representation of the CIE 1976 color space (or CIELAB, described below). The difference between the original Hunter and CIE color coordinates is that the CIE coordinates are based on a cube root transformation of the color data, while the Hunter coordinates are based on a square root transformation. Other examples of color spaces with Lab representations include the CIE 1994 color space and the CIE 2000 color space.

The L*a*b* color space includes all perceivable colors, which means that its gamut exceeds those of the RGB and CMYK color models (for example, ProPhoto RGB includes about 90% all perceivable colors). One of the most important attributes of the L*a*b*-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 L*a*b* color space is used 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. The space itself is a three-dimensional Real number space, that contains an infinite 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.

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.[3] 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.[4] 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, 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).[5]

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

An example of color enhancement using LAB colorspace in Photoshop. The left side of the photo is enhanced, while the right side is normal.

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 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 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.

In addition, 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 into image editing applications, will pick the closest in-gamut approximation, changing lightness, chroma, 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.[6]

Differentiation

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

CIELAB

CIE L*a*b* (CIELAB) is the most complete[citation needed] 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.[11] 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[12] 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,[13] culminated with CIECAM02.

Perceptual 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.[14]

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{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"). Under Illuminant D65, the values are

$X_n=0.95047, Y_n=1.00000, Z_n=1.08883$

The division of the domain of the f(t) function into two parts 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 intercept f(0) = b was chosen so that L* would be 0 for Y = 0: 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.[15]

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

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

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 that 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 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.[16]

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[citation needed] 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:

$K_a\approx\frac{175}{198.04}(X_n+Y_n)$
$K_b\approx\frac{70}{218.11}(Y_n+Z_n)$

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

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.[17] 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 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.

References

1. ^ Hunter, Richard Sewall (July 1948). "Photoelectric Color-Difference Meter". JOSA 38 (7): 661. (Proceedings of the Winter Meeting of the Optical Society of America)
2. ^ Hunter, Richard Sewall (December 1948). "Accuracy, Precision, and Stability of New Photo-electric Color-Difference Meter". JOSA 38 (12): 1094. (Proceedings of the Thirty-Third Annual Meeting of the Optical Society of America)
3. ^ A discussion and proposed improvement, Bruce Lindbloom
4. ^ Explanation of this history, Bruce MacEvoy
5. ^ a b International Color Consortium, Specification ICC.1:2004-10 (Profile version 4.2.0.0) Image technology colour management — Architecture, profile format, and data structure, (2006).
6. ^ a b Margulis, Dan (2006). Photoshop Lab Color: The Canyon Conundrum and Other Adventures in the Most Powerful Colorspace. Berkeley, Calif. : London: Peachpit ; Pearson Education. ISBN 0-321-35678-0.
7. ^ The Lab Color Mode in Photoshop, Adobe TechNote 310838
8. ^ TIFF: Revision 6.0 Adobe Developers Association, 1992
9. ^ Color Consistency and Adobe Creative Suite
10. ^ Adobe Acrobat Reader 4.0 User Guide "The color model Acrobat Reader uses is called CIELAB…"
11. ^ 3D representations of the L*a*b* gamut, Bruce Lindbloom.
12. ^ CIE-L*C*h Color Scale
13. ^ Fairchild, Mark D. (2005). "Color and Image Appearance Models". Color Appearance Models. John Wiley and Sons. p. 340. ISBN 0-470-01216-1.
14. ^ Jain, Anil K. (1989). Fundamentals of Digital Image Processing. New Jersey, United States of America: Prentice Hall. pp. 68, 71, 73. ISBN 0-13-336165-9.
15. ^ János Schanda (2007). Colorimetry. Wiley-Interscience. p. 61. ISBN 978-0-470-04904-4.
16. ^ Hunter Labs (1996). "Hunter Lab Color Scale". Insight on Color 8 9 (August 1–15, 1996). Reston, VA, USA: Hunter Associates Laboratories.
17. ^ Adams, E.Q. (1942). "X-Z planes in the 1931 I.C.I. system of colorimetry". JOSA 32 (3): 168–173. doi:10.1364/JOSA.32.000168.