Color models

color circle of hue and saturationA color model is an abstract mathematical model describing the way colors can be represented as tuples of numbers, typically as three or four values or color components. The resulting set of colors is called color space. This section describes ways in which human color vision can be modeled.

Tristimulus color space

The human tristimulus color space.
One can picture this space as a region in three-dimensional Euclidean space if one identifies the x, y, and z axes with the stimuli for the long-wavelength (L), medium-wavelength (M), and short-wavelength (S) receptors. The origin, (S,M,L) = (0,0,0), corresponds to black. White has no definite position in this diagram; rather it is defined according to the color temperature or white balance as desired or as available from ambient lighting. The human color space is a horse-shoe-shaped cone such as shown here (see also CIE chromaticity diagram below), extending from the origin to, in principle, infinity. In practice, the human color receptors will be saturated or even be damaged at extremely-high light intensities, but such behavior is not part of the CIE color space and neither is the changing color perception at low light levels

The most saturated colors are located at the outer rim of the region, with brighter colors farther removed from the origin. As far as the responses of the receptors in the eye are concerned, there is no such thing as "brown" or "gray" light. The latter color names refer to orange and white light respectively, with an intensity that is lower than the light from surrounding areas. One can observe this by watching the screen of an overhead projector during a meeting: one sees black lettering on a white background, even though the "black" has in fact not gotten darker than the white screen on which it is projected before the projector was turned on. The "black" areas have not actually become darker but appear "black" relative to the higher intensity "white" projected onto the screen around it. See also color constancy above.

The human tristimulus space has the property that additive mixing of colors corresponds to the adding of vectors in this space. This makes it easy to, for example, describe the possible colors (gamut) that can be constructed from the red, green, and blue primaries in a computer display.

Continuing with our mathematical description of light using the wave equation, a good model for the way our receptors work can be explained in terms of Hilbert spaces and orthogonal projections. Indeed, as mentioned previously, the light going through point (x,y,z) in space is a signal. It is useful to think of this signal as some function in L2, the space of square-integrable functions. This space is a (infinite dimensional!) Hilbert space, which means that it has a useful notion of orthogonal projection.

Each receptor can be thought of as a unit vector. For instance, the red receptor would be some vector r of light, whose Fourier transform would be large in the 405 to 480 THz interval, and smaller elsewhere. If we take the Fourier transform of v and plot its absolute value, we obtain what is called the frequency response curve of the human red receptor.

Then, the amount of "red" present in any color will be the orthogonal projection onto the axis generated by the vector r. In fact, only the magnitude of the orthogonal projection onto r is measured by our receptors. There are two more vectors, one for blue and one for green. Therefore, our color perception is in fact limited to a three-dimensional subspace of the infinite dimensional space of all possible colors.

CIE XYZ color space

CIE (http://www.cie.co.at/framepublications.html)1964 Supplementary Standard Colorimetric Observer functions X, Y, and Z functions between 380 nm and 780 nm (at 5 nm intervals).One of the first mathematically defined color spaces is the CIE XYZ color space (also known as CIE 1931 color space), created by the International Commission on Illumination in 1931. These data were measured for human observers and a 2-degree field of view. In 1964, supplemental data for a 10-degree field of view were published.

It must be noted that the tabulated sensitivity curves have a certain amount of arbitrariness in them. The shapes of the individual X, Y and Z sensitivity curves can be measured with a reasonable accuracy. However, the overall luminosity curve (which in fact is a weighted sum of these three curves) is subjective, since it involves asking a test person whether two light sources have the same brightness, even if they are in completely different colors. Along the same lines, the relative magnitudes of the X, Y, and Z curves are arbitrary. One could as well define a valid color space with an X sensitivity curve that has twice the amplitude. This new color space would have a different shape. The sensitivity curves in the CIE 1931 and 1964 xyz color space are scaled to have equal areas under the curves.

Here,x and y are related to the X, Y, and Z tristimulus values under Human tristimulus color space above according to:

x = X/(X + Y + Z),
y = Y/(X + Y + Z).
Mathematically, x and y are projective coordinates and the colors of the chromaticity diagram occupy a region of the real projective plane. Because the CIE sensitivity curves have equal areas under the curves, light with a flat energy spectrum corresponds to the point (x,y) = (0.333,0.333).

The values for X, Y, and Z are obtained by integrating the product of the spectrum of a light beam and the published color-matching functions. Blue and red wavelengths do not contribute strongly to the luminosity, which is illustrated by the following example:

red green blue red+green green+blue red+blue red+green+blue zero light

For someone with normal color vision, green is brighter than red, which is brighter than blue. Even though the pure blue appears to be very dark and hardly discernible from black when observed from a distance, blue has a strong coloring power when mixed with green or red.

With some forms of "red-green color blindness" the green is very slightly brighter than the blue, and the red is so dark it can barely be made out. Red traffic lights in bright daylight appear broken (no light). The green traffic light appears dirty white and hard to distinduish from night street lights.

The CIE-xyz color space is a prism, as opposed to the cone-shaped tristimulus space above. In the two-dimensional xy representation, all possible additive mixtures of two colors A and B form a straight line. However, the additive mixture of two colors does generally not lie on the mid-point of this line.

RGB color space
Media that transmit light (such as television) use additive color mixing with primary colors of red, green, and blue, each of which stimulates one of the three types of the eye's color receptors with as little stimulation as possible of the other two. This is called "RGB" color space—see also RGB color model. Mixtures of light of these primary colors cover a large part of the human color space and thus produce a large part of human color experiences. This is why color television sets or color computer monitors need only produce mixtures of red, green and blue light. See Additive color.

Other primary colors could in principle be used, but with red, green and blue the largest portion of the human color space can be captured. Unfortunately there is no exact consensus as to what loci in the chromaticity diagram the red, green, and blue colors should have, so the same RGB values can give rise to slightly different colors on different screens.

CMYK color model

It is possible to achieve a large range of colors seen by humans by combining cyan, magenta, and yellow transparent dyes/inks on a white substrate. These are the subtractive primary colors. Often a fourth black is added to improve reproduction of some dark colors. This is called "CMY" or "CMYK" color space.

The cyan ink will reflect all but the red light, the yellow ink will reflect all but the blue light and the magenta ink will reflect all but the green light. This is because cyan light is an equal mixture of green and blue, yellow is an equal mixture of red and green, and magenta light is an equal mixture of red and blue.

HLS color space

HLS Color SpaceThe HLS color space, also called HSL, stands for "Hue, Saturation, Lightness". While HSV (Hue, Saturation, Value) can be viewed graphically as a color cone or hexcone, HSL is drawn as a double cone or double hexcone. Both systems are non-linear deformations of the RGB color cube. The two apexes of the HLS double hexcone correspond to black and white. The angular parameter corresponds to hue, distance from the axis corresponds to saturation, and distance along the black-white axis corresponds to lightness.

HSV color space
The RGB and CMYK color spaces are most useful for technical reproduction of color scenes. A color space used in computer graphics that more closely models the human experience is the HSV color space which arranges colors in a cylinder, somewhat similar to the CIE-xyz space discussed above. The cross-section of the cylinder is a color wheel, but instead of pure spectral colors, the edge consists of additive mixtures of red, green, and blue. In the HSV color space, every color is specified by its hue (position on the circle), saturation (distance from the circle's center) and value (luminosity). The basic idea of the HSV color space was already used by 19th century physiologist Ewald Hering, although the modern definition dates from the 1970s. The HSV color space is also sometimes referred to as the HSB (hue-saturation-brightness) color space.

Color systems
There are various types of color systems that classify color and analyse their effects. The Munsell color system is a famous classification that organises various colors into a color solid based on hue, saturation and value.


Reproduction of color

A colored photo of a sunsetTwo different light spectra which have the same effect on the three color receptors in the human eye will be perceived as the same color. This is exemplified by the white light that is emitted by fluorescent lamps, which typically has a spectrum consisting of a few narrow bands, while daylight has a continuous spectrum. The human eye cannot tell the difference between such light spectra just by looking into the light source, although reflected colors from objects can look different.

Similarly, most human color perceptions can be generated by a mixture of three colors called primaries. This is used to reproduce color scenes in photography, printing, television, and other media.

No mixture of colors, though, can produce a fully pure color perceived as completely identical to a spectral color, although one can get very close for the longer wavelengths, where the chromaticity diagram above has a nearly straight edge. For example, mixing green light (530 nm) and blue light (460 nm) produces cyan light that is slightly desaturated, because response of the red color receptor would be greater to the green and blue light in the mixture than it would be to a pure cyan light at 485 nm that has the same intensity as the mixture of blue and green.

Because of this, and because the primaries in color printing systems generally are not pure themselves, the colors reproduced are never perfectly saturated colors, and so spectral colors cannot be matched exactly. However, natural scenes rarely contain fully saturated colors, thus such scenes can usually be approximated well by these systems. The range of colors that can be reproduced with a given color reproduction system is called the gamut. The CIE chromaticity diagram can be used to describe the gamut.

Another problem with color reproduction systems is connected with the acquisition devices, like cameras or scanners. The characteristics of the color sensors in the devices are often very far from the characteristics of the receptors in the human eye. In effect, acquisition of colors that have some special, often very "jagged", spectra caused for example by unusual lighting of the photographed scene can be relatively poor.

Species that have color receptors different from humans, e. g. birds that may have four receptors, can differentiate some colors that look the same to a human. In such cases, a color reproduction system `tuned' to a human with normal color vision may give very inaccurate results for the other observers.

Pigments and reflective media
When producing a color print or painting a surface, the applied paint changes the surface; if the surface is then illuminated with white light (which consists of equal intensities of all visible wavelengths), the reflected light will have a spectrum corresponding to the desired color. If a dab of paint looks red in white light, that is because the reflection of all non-red wavelengths is interrupted by the pigment, such that only red light is reflected into one's eye.

Structural color is a property of some surfaces that are scored with fine parallel lines, formed of many thin parallel layers, or otherwise composed of periodic microstructures on the scale of the color's wavelength, to make a diffraction grating. The grating reflects some wavelengths more than others due to interference phenomena, causing white light to be reflected as colored light. Variations in the pattern's spacing often give rise to an iridescent effect, as seen in peacock feathers, films of oil, and mother of pearl, because the reflected color depends upon the viewing angle.

Structural color is studied in the field of thin-film optics. A layman's term that describes particularly the most ordered structural colors is iridescence.

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