Colorfulness

Original image, with relatively muted colors

L*C*h (CIELAB) chroma increased 50%

HSL saturation increased 50%; notice that changing HSL saturation also affects the perceived lightness of a color

CIELAB lightness preserved, with a* and b* stripped, to make a grayscale image

Colorfulness, chroma, pure color and saturation are related but distinct concepts in colorimetry and color theory, referring to the perceived intensity of a specific color. Colorfulness is the visual sensation according to which the perceived color of an area appears to be more or less chromatic.^[1] Chroma is the colorfulness relative to the brightness of a similarly illuminated area that appears to be white or highly transmitting. Therefore, chroma should not be confused with colorfulness.^[2] Saturation is the colorfulness of a color relative to its own brightness.^[3] Though the 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). Any color can be described using three color appearance parameters — colorfulness (or chroma or saturation), lightness (or brightness), and hue .

Saturation

Saturation scale (0% at bottom, corresponding to black and white).

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 (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:

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

where (u′_n, v′_n) is the chromaticity of the white point, and chroma is defined below.^[4]

By analogy, in CIELAB this would yield:

$s_{ab}=\frac{C^*_{ab}}{L^*}=\frac{\sqrt{{a^*}^2+{b^*}^2}}{L^*}$

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.^[5] 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.^[6]

$S_{ab}=\frac{C^*_{ab}}{\sqrt{{C^*_{ab} }^2+{L^*}^2}} 100\%$

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:

$s=\sqrt{M/Q}$

This definition is inspired by experimental work done with the intention of remedying CIECAM97s's poor performance.^[7][8] 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.^[7]

Excitation purity

Excitation purity is the relative distance from the white point. Contours of constant purity can be found by shrinking the spectral locus about the white point. The points along the line segment have the same hue, with p_e increasing from 0 to 1 between the white point and position on the spectral locus (position of the color on the horseshoe shape in the diagram) or (as at the saturated end of the line shown in the diagram) position on the line of purples.

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:^[9]

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

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

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:

$C_{ab}^* = \sqrt{a^{*2} + b^{*2}}$

$h_{ab} = \arctan \frac{b^{*}}{a^{*}}$

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 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 a color appearance model. Here, the chroma color appearance parameter might (depending on the color appearance model) be intertwined with e.g. the physical brightness of the illumination or the characteristics of the emitting/reflecting surface, which is more sensible psychovisually.

References

1. Fairchild, Mark (2013). Color Appearance Models. John Wiley & Sons., page 87.
2. Colorfulness Vs. Chroma
3. 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.
4. Schanda, János (2007). Colorimetry: Understanding the CIE System. Wiley Interscience. ISBN 978-0-470-04904-4., page 88.
5. Hunt, Robert William Gainer (1993). Leslie D. Stroebel, Richard D. Zakia, ed. The Focal Encyclopedia of Photography. Focal Press. p. 124. ISBN 0-240-51417-3.
6. Lübbe, Eva (2010). Colours in the Mind - Colour Systems in Reality- A formula for colour saturation. [Book on Demand]. ISBN 978-3-7881-4057-1.
7. Moroney, Nathan; Fairchild, Mark D.; Hunt, Robert W.G.; Li, Changjun; Luo, M. Ronnier; Newman, Todd (November 12, 2002). IS&T/SID Tenth Color Imaging Conference. The CIECAM02 Color Appearance Model (PDF). Scottsdale, Arizona: The Society for Imaging Science and Technology. ISBN 0-89208-241-0.
8. Juan, Lu-Yin G.; Luo, Ming R. (June 2002). Robert Chung, Allan Rodrigues, eds. Magnitude estimation for scaling saturation. 9th Congress of the International Colour Association. Proceedings of SPIE. 4421. pp. 575–578. doi:10.1117/12.464511.
9. Stroebel, Leslie D.; Zakia, Richard D. (1993). The Focal Encyclopedia of Photography (3E ed.). Focal Press. p. 121. ISBN 0-240-51417-3.