# Saturation

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In color theory, saturation or purity refers to the intensity of a specific hue. A highly saturated hue has a vivid, intense color, while a less saturated hue appears more muted and grey. With no saturation at all, the hue becomes a shade of grey. Saturation is one of three coordinates in the HSV color space.

The saturation of a color is determined by a combination of light intensity and how much it is distributed across the spectrum of different wavelengths. The purest colour is achieved by using just one wavelength at a high intensity such as in laser light. If the intensity drops the saturation also drops. To desaturate a color in a subtractive system (such as watercolor), you can add white, black, gray, or the hue's complement.

### Purity in CIE 1931 XYZ color spaceEdit

In the CIE XYZ color space, the purity or saturation is the Euclidean distance between the position of the color $(x, y)$ and the illuminant's white point $(x_{I}, y_{I})$ on the CIE xy projective plane, divided by the same distance for a pure (monochromatic, or dichromatic on the purple line) color with the same hue $(x_{P}, y_{P}) = \rho_\mathrm{max} (x - x_{I}, y - y_{I}) + (x_{I}, y_{I})$:

$p = \sqrt{\frac{(x - x_{I})^2 + (y - y_{I})^2}{(x - x_{P})^2 + (y - y_{P})^2}}$

and $\rho_\mathrm{max}$ maximal within the boundary of the chromaticity diagram.

### Saturation in RGB color spaceEdit

In an RGB color space, saturation can be thought of as the standard deviation σ of the color coordinates R(red), G(green), and B(blue). Letting μ represent the brightness defined as the mean of R, G, and B, then

$\sigma = \sqrt{ (R - \mu)^2 + (G - \mu)^2 + (B - \mu)^2 \over 3}$

An example of saturation in layman's terms in the RGB color model is that you will have maximum saturation if you have 100% brightness in (for instance) the red channel while having 0% brightness in the other channels. And you would have no saturation if all the color channels are equal. Thus, saturation is the difference between the values of the channels.

In term of absolute colorimetry, this simple definition in the RGB color space exhibits several problems. The RGB color space is not an absolute colorimetric space, and therefore the value of saturation is arbitrary, depending on the choice of the color primaries and the white point illuminant. For example, the RGB colorspace does not necessarily have a unitary Jacobian in term of absolute colorimetry.

Although the above math would make a lot more sense, most (all?) software that represents a saturation value to the user returns this value:

$\max{ r,g,b } - \min{ r,g,b } \over \mu$

### Chroma in CIE 1976 L*a*b* and L*u*v* color spacesEdit

The naïve definition of saturation does not specify its response function. In the CIE XYZ and RGB color spaces, the saturation is defined in term of additive color mixing, and has the property of being proportional to any scaling centered at white or the white point illuminant. However, both color spaces are not linear in term of psychovisually perceived color differences. It is also possible, and sometimes desirable to define a saturation-like quantity that is linearized in term of the psychovisual perception.

In the CIE 1976 L*a*b* and L*u*v* color spaces, the unnormalized chroma is the radial component of the cylindrical coordinate CIE L*C*h (luminance, chroma, hue) representation of the L*a*b* and L*u*v* color spaces, also denoted as CIE L*C*h(a*b*) or CIE L*C*h for short, and CIE L*C*h(u*v*). The transformation of $(a^{*}, b^{*})$ to $(C^{*}, h)$ is given by:

$C^{*} = \sqrt{a^{*2} + b^{*2}}$
$h = \arctan \frac{b^{*}}{a^{*}}$

and analogeously for CIE L*C*h(u*v*).

The chroma in the CIE L*C*h(a*b*) and CIE L*C*h(u*v*) coordinates has the advantage of being more psychovisually linear, yet they are non-linear in the in term of linear component color mixing. And therefore, chroma in CIE 1976 L*a*b* and L*u*v* color spaces is very much different from the traditional sense of "saturation".

### Chroma in color appearance modelsEdit

Another, psychovisually even more accurate, but also more complex method to obtain or specify the saturation is to use the color appearance model, like CIECAM. The chroma component of the JCh (lightness, chroma, hue) coordinate, and becomes a function of parameters like the chrominance and physical brightness of the illumination, or the characteristics of the emitting/reflecting surface, which is also psychovisually more sensible.