Editing Primary color (section)

=== Imaginary primaries ===
Color space primaries are derived from canonical colorimetric experiments that represent a standardized model of an observer (i.e., a set of ''color matching functions'') adopted by [[International Commission on Illumination|Commission Internationale de l'Eclairage]] (CIE) standards. The abbreviated account of color space primaries in this section is based on descriptions in ''Colorimetry - Understanding The CIE System''.<ref name="Schanda2007">{{cite book |editor1-last=Schanda |editor1-first=János |title=Colorimetry : understanding the CIE system |date=2007 |publisher=CIE/Commission internationale de l'eclairage |location=[Vienna, Austria] |isbn=978-0-470-04904-4}}</ref>

The [[CIE 1931]] standard observer is derived from experiments in which participants observe a foveal secondary bipartite [[field of view|field]] with a dark surround. Half of the field is illuminated with a monochromatic ''test stimulus'' (ranging from 380&nbsp;nm to 780&nbsp;nm) and the other half is the ''matching stimulus'' illuminated with three coincident monochromatic primary lights: 700&nbsp;nm for red (R), 546.1&nbsp;nm for green (G), and 435.8&nbsp;nm for blue (B).<ref name="Schanda2007"/>{{rp|29}} These primaries correspond to [[CIE RGB|CIE RGB color space]]. The intensities of the primary lights could be adjusted by the participant observer until the matching stimulus matched the test stimulus, as predicted by Grassman's laws of additive mixing. Different standard observers from other color matching experiments have been derived since 1931. The variations in experiments include choices of primary lights, field of view, number of participants etc.<ref>{{cite journal |last1=Li |first1=Jiaye |last2=Hanselaer |first2=Peter |last3=Smet |first3=Kevin A. G. |title=Impact of Color-Matching Primaries on Observer Matching: Part I – Accuracy |journal=LEUKOS |date=17 February 2021 |volume=18 |issue=2 |pages=104–126 |doi=10.1080/15502724.2020.1864395|doi-access=free }}</ref> but the presentation below is representative of those results.

Matching was performed across many participants in incremental steps along the range of test stimulus wavelengths (380&nbsp;nm to 780&nbsp;nm) to ultimately yield the color matching functions: <math>\overline{r}(\lambda)</math>, <math>\overline{g}(\lambda)</math> and <math>\overline{b}(\lambda)</math> that represent the relative intensities of red, green, and blue light to match each wavelength (<math>\lambda</math>). These functions imply that <math>[C]</math> units of the test stimulus with ''any'' spectral power distribution, <math>P(\lambda)</math>, can be matched by {{math|[R]}}, {{math|[G]}}, and {{math|[B]}} units of each primary where:<ref name="Schanda2007"/>{{rp|28}}

{{NumBlk|:|<math>
[C] =
\int_{380\text{ nm}}^{780\text{ nm}} \overline{r}(\lambda) P(\lambda)\,d\lambda \cdot [R] +
\int_{380\text{ nm}}^{780\text{ nm}} \overline{g}(\lambda) P(\lambda)\,d\lambda \cdot [G] +
\int_{380\text{ nm}}^{780\text{ nm}} \overline{b}(\lambda) P(\lambda)\,d\lambda \cdot [B].
</math>|{{EquationRef|Eq. 1}}}}

Each integral term in the above equation is known as a ''tristimulus value'' and measures amounts in the adopted units. No set of real primary lights can match another monochromatic light under additive mixing so at least one of the color matching functions is negative for each wavelength. A negative tristimulus value corresponds to that primary being added to the test stimulus instead of the matching stimulus to achieve a match.

The negative tristimulus values made certain types of calculations difficult, so the CIE put forth new color matching functions <math>\overline{x}(\lambda)</math>, <math>\overline{y}(\lambda)</math>, and <math>\overline{z}(\lambda)</math> defined by the following [[linear map|linear transformation]]:<ref name="Schanda2007"/>{{rp|30}}

{{NumBlk|:|<math>
\begin{bmatrix}
 \overline{x}(\lambda) \\
 \overline{y}(\lambda) \\
 \overline{z}(\lambda)
\end{bmatrix}
=
\begin{bmatrix}
 2.768892 & 1.751748 & 1.130160 \\
 1.000000 & 4.590700 & 0.060100 \\
 0 & 0.056508 & 5.594292 \\
\end{bmatrix}
\begin{bmatrix}
 \overline{r}(\lambda) \\
 \overline{g}(\lambda) \\
 \overline{b}(\lambda)
\end{bmatrix}.
</math>|{{EquationRef|Eq. 2}}}}

These new color matching functions correspond to ''imaginary'' primary lights X, Y, and Z ([[CIE XYZ|CIE XYZ color space]]). All colors can be matched by finding the amounts {{math|[X]}}, {{math|[Y]}}, and {{math|[Z]}} analogously to {{math|[R]}}, {{math|[G]}}, and {{math|[B]}} as defined in {{EquationNote|Eq. 1}}. The functions <math>\overline{x}(\lambda)</math>, <math>\overline{y}(\lambda)</math>, and <math>\overline{z}(\lambda)</math> based on the specifications that they should be nonnegative for all wavelengths, <math>\overline{y}(\lambda)</math> be equal to [[luminous efficiency function|photometric luminance]], and that <math>[X]=[Y]=[Z]</math> for an equienergy (i.e., a uniform spectral power distribution) test stimulus.<ref name="Schanda2007"/>{{rp|30}}

Derivations use the color matching functions, along with data from other experiments, to ultimately yield the ''cone fundamentals'': <math>\overline{l}(\lambda)</math>, <math>\overline{m}(\lambda)</math> and <math>\overline{s}(\lambda)</math>. These functions correspond to the response curves for the three types of color [[photoreceptor cell|photoreceptors]] found in the human retina: long-wavelength (L), medium-wavelength (M), and short-wavelength (S) [[cone cell|cones]]. The three cone fundamentals are related to the original color matching functions by the following linear transformation (specific to a 10° field):<ref name="Schanda2007"/>{{rp|227}}
 
{{NumBlk|:|<math>
\begin{bmatrix}
 \overline{l}(\lambda) \\
 \overline{m}(\lambda) \\
 \overline{s}(\lambda)
\end{bmatrix}
=
\begin{bmatrix}
 0.192325269 & 0.749548882 & 0.0675726702 \\
 0.0192290085 & 0.949098496 & 0.113830196 \\
 0 & 0.0105107859 & 0.991427669 \\
\end{bmatrix}
\begin{bmatrix}
 \overline{r}(\lambda) \\
 \overline{g}(\lambda) \\
 \overline{b}(\lambda)
\end{bmatrix}.
</math>|{{EquationRef|Eq. 3}}}}

[[LMS color space]] comprises three primary lights (L, M, and S) that stimulate only the L-, M-, and S-cones respectively. A real primary that stimulates only the M-cone is impossible, and therefore these primaries are imaginary. The [[LMS color space]] has significant physiological relevance as these three photoreceptors mediate trichromatic color vision in humans.

Both XYZ and LMS color spaces are ''complete'' since all colors in the gamut of the standard observer are contained within their color spaces. Complete color spaces must have imaginary primaries, but color spaces with imaginary primaries are not necessarily complete (e.g. [[ProPhoto RGB color space]]).