User:Lovibond/CIELAB

CIELAB, the CIE 1976 L*, a*, b* Uniform Color Space, is an example of an opponent color space, in which the axes have at either end a pair of colors which cannot simultaneously appear in a color stimulus. It is also an Adams chromatic value color space, in that cone fundamentals, or approximations thereof, are non-linearly transformed so that gray stimuli approximately track the Munsell value (lightness) scale. One of the axes, L*, pronounced as "L-star," has white at one end (L*=100) and black at the other (L*=0). Another axis, a*, has a magenta-red at one end (positive a*) and a green at the other (negative a*). The third axis, b*, has a slightly orangish yellow at one end (positive b*) and a blue at the other (negative b*). The axes are orthogonal, or perpendicular.

History
In a JOSA paper, Elliot Adams proposed a color space which he termed "chromatic value" in which cone fundamental signals (which he assumed, for expedience, were the XYZ tristimulus values) were non-linearly transformed so neutrals were mapped to their Munsell value in all three channels. The transformed luminance signal was taken as the lightness signal. The difference between the transformed X and Y signals was taken as the red-green opponent signal, and the difference between the transformed Z and Y signals was taken as the yellow-blue opponent signal.

After some optimization and adjustment, including the reversal of the sign of the yellow-blue opponent signal, the color space became known as "ANLAB-40," where the inital "A" stood for "Adams," the "N" stood for Dorothy Nickerson, and "LAB" stood for the three axes.

L* and Munsell Value computation
The Munsell Value function, which was applied to all three tristimulus values, was defined in closed form only in the opposite direction needed to compute ANLAB 40 coordinates. In 1955, Ladd and Pinney of the Eastman Kodak company explored a number of possibile alternatives, two of which were especially faithful to the existing relationship. The first used an exponent selected as a fitting parameter, the relationship was:


 * $$V \approx 2.217 Y^{0.352}-1.324$$

where $$V$$ was on a scale of 0 (black) to 10 (white) and Y was on a scale of 0 to 100. Recognizing the optimized exponent was close to one-third, they computed:


 * $$V \approx 2.468 Y^{1/3}-1.636$$

Because the latter formula had a deviation of no greater than one-tenth Munsell Value unity (for Munsell Values between 0.5 and 10), Reilly and his associates adopted a similar formula in the construction of their "cube root" color space:


 * $$L^* = 25.29 \cdot Y^{1/3} - 18.38$$

where $$Y$$ was normalized so that Magnesium oxide had a Y tristimulus value of 100. NB: L* has a range of 0 to 100, so it is essentially ten times Munsell Value.

That this has a real root at approximately $$Y = 0.384$$ and negative values are assigned to luminances lower than this was not seen at the time as a practical problem, as this occurs only for very dark stimuli.

In 1964, as part of the first uniform color space it adopted, U*V*W*, CIE used a slightly simplified version, following a suggestion by Wyszecki:


 * $$W^* = 25 Y^{1/3} - 17$$

where the range of $$W^*$$ is 0 to 100. This was specifically restricted to the range $$1 \le Y \le 100$$.

In 1976, when CIELAB was adopted, the following formula was adopted:


 * $$L^* = 116 \left(\frac{Y}{Y_n}\right)^{1/3} - 16$$

where $$Y_n$$ is the Y tristimulus value of a "specified white object" and was subject to the restriction $$Y/Y_n > 0.01$$.

Pauli of Ciba-Geigy, who wished to remove this restriction, computed a linear extension which mapped $$Y/Y_n=0$$ to $$L^*=0$$ and was tangent to the formula above at the point at which the linear extension took effect. First, the transition point was determined to be $$Y/Yn=(6/29)^3 \approx 0.008856$$, then the slope of $$(29/3)^3 \approx 903.3$$ was computed.

Today, Pauli's suggestion is used not only for L*, but for the other "cube roots" as well. The function:



f(u)= \begin{cases} \frac{841}{108}u + \frac{4}{29}, & u \le (6/29)^3 \\ \\ u^{1/3}, & u > (6/29)^3 \end{cases} $$

is applied to all three tristimulus ratios $$X/X_n$$, $$Y/Y_n$$, and $$Z/Z_n$$; the CIELAB coordinates L*, a*, and b* are computed as:


 * $$\begin{array}{rcl}

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