Editing CIELAB color space (section)

===From CIE XYZ to CIELAB===

:<math>\begin{align}
  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{align}</math>

where ''t'' is <math>X/X_\mathrm{n},</math> <math>Y/Y_\mathrm{n},</math> or <math>Z/Z_\mathrm{n}</math>: 
:<math>\begin{align}
  f(t) &= \begin{cases}
    \sqrt[3]{t} & \text{if } t > \delta^3 \\[4mu]
    \tfrac13 t \delta^{-2} + \tfrac{4}{29} & \text{otherwise}
  \end{cases} \\
  \delta &= \tfrac{6}{29}
\end{align}</math>

{{mvar|X}}, {{mvar|Y}}, and {{mvar|Z}} describe the color stimulus considered and {{math|''X''<sub>n</sub>}}, {{math|''Y''<sub>n</sub>}}, {{math|''Z''<sub>n</sub>}} describe a specified white achromatic reference illuminant. for the CIE 1931 (2°) standard colorimetric observer and assuming normalization where the reference white has {{math|1=''Y'' = 100}}, the values are:

For [[Illuminant D65|Standard Illuminant D65]]:
:<math>\begin{align}
X_{\mathrm{n}}&=95.0489,\\
Y_{\mathrm{n}}&=100,\\
Z_{\mathrm{n}}&=108.8840
\end{align}</math>

For [[Standard illuminant#Illuminant series D|illuminant D50]], which is used in the printing industry: 
:<math>\begin{align}
X_{\mathrm{n}}&=96.4212,\\
Y_{\mathrm{n}}&=100,\\
Z_{\mathrm{n}}&=82.5188
\end{align}</math>

The division of the domain of the {{mvar|f}} function into two parts was done to prevent an infinite slope at {{math|1=''t'' = 0}}. The function {{mvar|f}} was assumed to be linear below some {{math|1=''t'' = ''t''<sub>0</sub>}} and was assumed to match the <math>\sqrt[3]t</math> part of the function at {{math|''t''<sub>0</sub>}} in both value and slope. In other words:
:<math>\begin{align}
                \sqrt[3]{t_0} &= m t_0 + c & \text{ (match in value)}\\[3mu]
  \tfrac13 \left(t_0\right)^{-2/3} &= m & \text{ (match in slope)}
\end{align}</math>

The intercept {{math|1=''f''(0) = ''c''}} was chosen so that {{math|''L''*}} would be 0 for {{math|1=''Y'' = 0}}: {{math|1=''c'' = {{sfrac|16|116}} = {{sfrac|4|29}}}}. The above two equations can be solved for {{math|''m''}} and {{math|''t''<sub>0</sub>}}:
:<math>\begin{align}
    m &= \tfrac13\delta^{-2} &= 7.787037\ldots\\
  t_0 &= \delta^3 &= 0.008856\ldots
\end{align}</math>

where {{math|1=''δ'' = {{sfrac|6|29}}}}.<ref>
{{cite book
 | title = Colorimetry
 | author = János Schanda
 | publisher = Wiley-Interscience
 | year = 2007
 | isbn = 978-0-470-04904-4
 | page = 61
 | url = https://books.google.com/books?id=uZadszSGe9MC&q=lab+color+6-29+16-116&pg=PA61
}}</ref>

<ref>{{Cite web|title=CIE 1976 L*a*b* colour space {{!}} eilv|url=http://eilv.cie.co.at/term/157|archive-url=https://web.archive.org/web/20191228145700/http://eilv.cie.co.at/term/157|url-status=dead|archive-date=2019-12-28|access-date=2020-12-12|website=eilv.cie.co.at}}</ref>