### Saturation (color theory)

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In colorimetry and color theory, **colorfulness**, **chroma**, and **saturation** are related but distinct concepts referring to the perceived intensity of a specific color. *Colorfulness* is the degree of difference between a color and gray. *Chroma* is the colorfulness relative to the brightness of another color that appears white under similar viewing conditions. Saturation is the colorfulness of a color relative to its own brightness.^{[1]} Though this general concept is intuitive, terms such as *chroma*, *saturation*, *purity*, and *intensity* are often used without great precision, and even when well-defined depend greatly on the specific color model in use.

A highly colorful stimulus is vivid and intense, while a less colorful stimulus appears more muted, closer to gray. With no colorfulness at all, a color is a “neutral” gray (an image with no colorfulness in any of its colors is called *grayscale*). With three attributes—colorfulness (or chroma or saturation), lightness (or brightness), and hue—any color can be described. **
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## Contents

## Saturation

Saturation is one of three coordinates in the HSL and HSV color spaces. Note that virtually all computer software implementing these spaces use a very rough approximation to calculate the value they call "saturation", such as the formula described for HSV and this value has little, if anything, to do with the description shown here.

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 (most saturated) color is achieved by using just one wavelength at a high intensity, such as in laser light. If the intensity drops, then as a result the saturation drops. To desaturate a color of given intensity in a subtractive system (such as watercolor), one can add white, black, gray, or the hue's complement.

Various correlates of saturation follow.

- CIELUV
- The
*chroma*normalized by the lightness:

- $s\_\{uv\}=\backslash frac\{C^*\_\{uv\}\}\{L^*\}=13\; \backslash sqrt\{(u\text{'}-u\text{'}\_n)^2+(v\text{'}-v\text{'}\_n)^2\}$

where (*u*′_{n}, *v*′_{n}) is the chromaticity of the white point, and chroma is defined below.^{[2]}

By analogy, in CIELAB this would yield:

- $s\_\{ab\}=\backslash frac\{C^*\_\{ab\}\}\{L^*\}=\backslash frac\{\backslash sqrt\{L^*\}$

The CIE has not formally recommended this equation since CIELAB has no chromaticity diagram, and this definition therefore lacks direct correlation with older concepts of saturation.^{[3]} Nevertheless, this equation provides a reasonable predictor of saturation, and demonstrates that adjusting the lightness in CIELAB while holding (*a**, *b**) fixed does affect the saturation.

But the following formula is in agreement with the human perception of saturation:
The formula proposed by Eva Lübbe is in agreement with the verbal definition of Manfred Richter: Saturation is the proportion of pure chromatic color in the total color sensation.^{[4]}

- $S\_\{ab\}=\backslash frac\{C^*\_\{ab\}\}\{\backslash sqrt\; 100\%$

where *S*_{ab} is the saturation, *L** the lightness and *C**_{ab} is the chroma of the color.

- CIECAM02
- The square root of the
*colorfulness*divided by the*brightness*:

- $s=\backslash sqrt\{M/Q\}$

This definition is inspired by experimental work done with the intention of remedying CIECAM97s's poor performance.^{[5]}^{[6]} *M* is proportional to the chroma *C* (*M* = *CF*_{L}^{0.25}), thus the CIECAM02 definition bears some similarity to the CIELUV definition. An important difference is that the CIECAM02 model accounts for the viewing conditions through the parameter *F*_{L}.^{[5]}

## Excitation purity

The **excitation purity** (purity for short) of a stimulus is the difference from the illuminant's white point to the furthest point on the chromaticity diagram with the same hue (dominant wavelength for monochromatic sources); using the CIE 1931 color space:^{[7]}

- $p\_e\; =\; \backslash sqrt\{\backslash frac\{(x\; -\; x\_n)^2\; +\; (y\; -\; y\_n)^2\}\{(x\_I\; -\; x\_n)^2\; +\; (y\_I\; -\; y\_n)^2\}\}$

where (*x*_{n}, *y*_{n}) is the chromaticity of the white point and (*x*_{I}, *y*_{I}) is the point on the perimeter whose line segment to the white point contains the chromaticity of the stimulus. Different color spaces, such as CIELAB or CIELUV may be used, and will yield different results.

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

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The naïve definition of saturation does not specify its response function. In the CIE XYZ and RGB color spaces, the saturation is defined in terms 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 nonlinear in terms 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 (lightness, 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**_{ab}, *h*_{ab}) is given by:

- $C\_\{ab\}^*\; =\; \backslash sqrt\{a^\{*2\}\; +\; b^\{*2\}\}$
- $h\_\{ab\}\; =\; \backslash arctan\; \backslash frac\{b^\{*\}\}\{a^\{*\}\}$

and analogously 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 terms 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 models

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 LCh (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.