Editing Color balance (section)

===Scaling X, Y, Z===
If the image may be transformed into [[CIE 1931 color space|CIE XYZ tristimulus values]], the color balancing may be performed there.  This has been termed a "wrong von Kries" transformation.<ref name=Terstiege1972>{{Cite journal
 | author = Heinz Terstiege
 | title = Chromatic adaptation: a state-of-the-art report
 | year = 1972
 | journal = Journal of Color Appearance
 | volume = 1
 | issue = 4
 | pages = 19–23 (cont. 40)
}}</ref><ref name="Fairchild1998">Mark D Fairchild, ''Color Appearance Models.'' Reading, MA: Addison-Wesley, 1998.</ref> Although it has been demonstrated to offer usually poorer results than balancing in monitor RGB, it is mentioned here as a bridge to other things. Mathematically, one computes:

:<math>\left[\begin{array}{c} X \\ Y \\ Z \end{array}\right]=\left[\begin{array}{ccc}X_w/X'_w & 0 & 0 \\ 0 & Y_w/Y'_w & 0 \\ 0 & 0 & Z_w/Z'_w\end{array}\right]\left[\begin{array}{c}X' \\ Y' \\ Z' \end{array}\right]</math>

where <math>X</math>, <math>Y</math>, and <math>Z</math> are the color-balanced tristimulus values; <math>X_w</math>, <math>Y_w</math>, and <math>Z_w</math> are the tristimulus values of the viewing illuminant (the white point to which the image is being transformed to conform to); <math>X'_w</math>, <math>Y'_w</math>, and <math>Z'_w</math> are the tristimulus values of an object believed to be white in the un-color-balanced image, and <math>X'</math>, <math>Y'</math>, and <math>Z'</math> are the tristimulus values of a pixel in the un-color-balanced image. If the tristimulus values of the monitor primaries are in a matrix <math>\mathbf{P}</math> so that:

:<math>\left[\begin{array}{c} X \\ Y \\ Z \end{array}\right]=\mathbf{P}\left[\begin{array}{c}L_R \\ L_G \\ L_B \end{array}\right]</math>

where <math>L_R</math>, <math>L_G</math>, and <math>L_B</math> are the un-[[gamma correction|gamma corrected]] monitor RGB, one may use:

:<math>\left[\begin{array}{c} L_R \\ L_G \\ L_B \end{array}\right]=\mathbf{P^{-1}}\left[\begin{array}{ccc}X_w/X'_w & 0 & 0 \\ 0 & Y_w/Y'_w & 0 \\ 0 & 0 & Z_w/Z'_w\end{array}\right]\mathbf{P}\left[\begin{array}{c}L_{R'} \\ L_{G'} \\ L_{B'} \end{array}\right]</math>