# Colorimetry

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In colorimetry and color theory, colorfulness, chroma, and saturation are related concepts referring to the intensity of a specific color. More technically, colorfulness is the perceived difference between the color of some stimulus and gray, chroma is the colorfulness of a stimulus relative to the brightness of a stimulus that appears white under similar viewing conditions, and saturation is the colorfulness of a stimulus 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. With three attributes—colorfulness (or chroma or saturation), lightness (or brightness), and hue—any color can be described.

## Saturation

Saturation is one of three coordinates in the HSL and HSV color spaces.

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

Various correlates for saturation follow.

CIELUV
The chroma normalized by the luminance:
$s_{uv}=13 \sqrt{(u'-u'_n)^2+(v'-v'_n)^2}=\frac{C^*_{uv}}{L^*}$

where $(u'_n,v'_n)$ is the chromaticity of the white point, and the L*C*h color space is defined below.[2]

CIECAM02
The square root of the colorfulness divided by the brightness:
$s=\sqrt{M/Q}$

This definition is inspired by experimental work done with the intention of remedying CIECAM97s's poor performance.[3][4] It should be noted that 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 $F_L$.[3]

## Excitation purity

The excitation purity (purity for short) of a stimulus is its difference from the illuminant's white point relative to the furthest point on the chromaticity diagram with the same hue (dominant wavelength for monochromatic sources); using the CIE 1931 color space:[5]

$p_e = \sqrt{\frac{(x - x_n)^2 + (y - y_n)^2}{(x_I - x_n)^2 + (y_I - y_n)^2}}$

where $(x_I,y_I)$ is the chromaticity of the white point and $(x_n,y_n)$ 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

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 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 (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 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 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 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 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.

## References

1. Mark D. Fairchild. “Color Appearance Models: CIECAM02 and Beyond”. Slides from a tutorial at the IS&T/SID 12th Color Imaging Conference. 9 November 2004. Retrieved 19 September 2007.
2. 3.0 3.1 Moroney, Nathan; Fairchild, Mark D.; Hunt, Robert W.G.; Li, Changjun; Luo, M. Ronnier; Newman, Todd (November 12 2002). "The CIECAM02 Color Appearance Model". IS&T/SID Tenth Color Imaging Conference, Scottsdale, Arizona: The Society for Imaging Science and Technology. ISBN 0-89208-241-0.
3. Juan, Lu-Yin G. (June 2002). "Magnitude estimation for scaling saturation" in 9th Congress of the International Colour Association. Proceedings of SPIE 4421: 575-578. DOI:10.1117/12.464511.
4. Stroebel, Leslie D. (1993). The Focal Encyclopedia of Photography, 3E, Focal Press.