Hello,

Is it possible that you are referring to "Colour appearance model" or "colour perception model" rather than color model? 

RGB, HSV, HSL, HSI, CMYK do not attempt to model or describe colour perception or appearance, but describe the intensities of the colour channels of a device, e.g. the primaries of a CRT monitor or the sensors of a camera. This implies that they are device-dependent, i.e. the same values produce different colours depending on a device. In particular, CMYK refers to the four sources of pigments that are mixed to produce colours in printing.

Tristimulus Values (XYZ) and chromaticity coordinates (xyY) describe colour perception at the retinal level. The assume a standard observer (abstract from individual differences) to provide values that allow for colorimetry (measuring of color). The standard observers is inherent in the so-called "colour matching functions" that describe sensitivities of the channels (XYZ) across the spectrum. There are different kinds of colour matching functions: The one you mentioned (CIE 1931), Judd-, and Judd-Voss-corrected colour matching functions; the latter are more precise, but less common/less in use. Note that, Tristimulus values are - in principle - a linear transformation of the excitations of the photoreceptors (L-, M-, and S-cones), hence showing you the level of processes they represent. In principle means apart from measurement noise and other artefacts from differences in how colour matching functions and cone sensitivities are measured. For illustration, one implication of that (approximately) linear relationship between XYZ and LMS has been shown in the supplementary material of http://jov.arvojournals.org/Article.aspx?articleid=2346729. Chromaticity coordinates xy are just Tristimuls Values (XY) normalized to the overall intensity (X+Y+Z).

Note that device dependent colour spaces may be converted to the above perceptual/receptoral spaces by characterising the monitor, linearising the device space and then applying a linear transformation based on the chromaticities and luminance of the device (simple matrix inner product).

The so-called second-stages of colour perception, the cone-opponent channels, are also models of colour perception, but at a later stage than the above, namely the so called second-stage (e.g. Gegenfurtner & Kiper, 2003: http://invibe.net/biblio_database_dyva/woda/data/att/2190.file.pdf). Typical spaces that model this stage are Derrington-Krauskopf-Lennie (DKL) and MacLeod-Boynton space. 

Finally, colour appearance models aim at modelling how colours subjectively appear to the observer i.e. at the end of all colour processing. Early and widely used models are CIELAB and CIELUV, which can be simply calculated starting from Tristimulus (XYZ) or chromaticity coordinates and luminance (xyY). However, these spaces are only coarsely approximate (see e.g. Figure 15 in http://jov.arvojournals.org/article.aspx?articleid=2121380&resultClick=1 or Figure 4 in http://psycnet.apa.org/?&fa=main.doiLanding&doi=10.1037/xhp0000154). New models have been developed, such as CIECAM97 and CIECAM02.

For further reading on colour appearance models I suggest Mark Fairchild's book, the second edition of which is freely available (http://115.28.34.34/wp-content/uploads/2015/01/The-Wiley-IST-Series-in-Imaging-Science-and-Technology-Fa....pdf), or:

Hunt, R. W. G., & Pointer, M. R. (2011). Measuring Colour (4 ed.). Chichester, UK: John Wiley & Sons.

I hope that sheds some light on your question.

All the best,

Christoph

Thank you very much for your answer.

Hi Sibasish,

Although the reply from Christoph is technically correct, it does not really answer your initial question.

The reason for your question lies in the fact that most book and articles (even those that are not written for the layman, let alone all the internet scribbles by so-called experts) seldom stick to clear and standard terminology. Many people do indeed use the terms colour space when then mean colour model and vice versa.

1 - Colour models

As you might know, colour is not a primary physical property but a perceptual result that is said to have three perceptual dimensions: hue, chroma and lightness. In the digital world, however, colour is represented using global colour models and more specific colour spaces. Colour is often mathematically defined as a three-dimensional (3D) property.

However, a point in a 3D space can be defined using many different coordinate systems (CSs). In cartography, a CS defines a set of axes used (i.e. the dimensionality of the system) and the attributes of these axes (direction, name, abbreviation, units, order). As an example: a three-dimensional Cartesian XYZ system using meter. If any of these attributes changes, the coordinate system changes.

The same goes more or less for colour. As such, many CSs for colour exist, built upon three or four coordinate axes. A specific set of coordinate axes used to describe colour is called a colour model. Although a colour model is thus a particular CS to describe colour (e.g. RGB, CMYK), it only describes its dimensionality (three or four variables/axes) and the specific name, direction and order of its axes. Colour models can thus be thought of as systems that describe how colours can be mathematically represented. Most models use three different axes to represent one colour (like the RGB colour model which uses red, green and blue axes), while others such as CMYK use four components/axes.

However, a specific RGB triplet such as R:50 – G:25 – B:255 does not define a particular colour, but only indicates the ratio of the three components used. Although 255 is the maximum value for each R, G and B channel in an 8-bit image, a colour model does not specify how ‘vibrantly blue’ this maximum should be? In other words: a colour model is a mathematical system with no connection to any reference system of colour so that there is no description at all on how these values should be interpreted. Therefore, colour models such as RGB, CMYK (using the primaries cyan, magenta, yellow and blacK/Key), HSV (hue – saturation – value) and HSL (hue – saturation – lightness) are said to be relative.

To accomplish an unambiguous description of colours, the RGB values need to be defined with respect to a particular scale of reference, more specifically a well-defined colour standard. Generally, the CIE XYZ or CIE L*a*b* colour spaces are used as reference standard since they define all the colours an average human perceives. Thus, as soon as the colour model has an associated mapping function that maps its arbitrary values to absolute CIE XYZ values, a colour space is born. In practice, this means that the units of the axes are explicitly described. Since the exact description of a particular colour is accomplished by numbers that are given in relation to a specific colour space, colour spaces are absolute descriptions.

2 - Colour spaces

A colour space is thus much more specific than a colour model, as it is just one possible instance of the more general colour model. Using different mapping functions, various absolute colour spaces can be created. Without going into further detail, the following elements are essential for describing a colour space.

• well-defined primaries (e.g. specific RGB or CMYK primaries);
• a specific CIE illuminant (which represents the assumed illumination: CIE illuminants can be considered quantified and standardised illumination sources);
• a specific gamma-value.

The best-known colour spaces that use the RGB colour model are sRGB, Adobe RGB (1998) and ProPhoto RGB (also known as ROMM RGB). The attached figure compares all three in the CIE (x,y) chromaticity diagram. The chromaticity coordinates (x,y) are defined by

x = X / (X + Y + Z)

y = Y / (X + Y + Z)

so that the chromaticity of a colour can be described irrespective from its luminance. The three points that form one triangle correspond to the chromaticity coordinates of the three primaries of that particular colour space (these values can also be found in the attached figure). The area in the triangle is known as the colour gamut and encompasses all the colour values that can be created by mixing those three primaries. Values outside the triangle are said to be out-of-gamut for these primaries. Despite the criticism on such diagrams, they are still one of the most straightforward – although not very accurate – means to compare different gamuts on a sheet of paper.

From this figure, it can be inferred that a colour space such as sRGB has a rather limited gamut, while Adobe RGB [1998] and ProPhoto RGB have much wider gamuts. In other words: the ProPhoto RGB colour space can represent more colours than the Adobe RGB [1998] space which can in turn store a wider range of colours than the sRGB colour space. Despite these differences, all three are based on the RGB colour model to mathematically represent colours.

Dear Geert:

1. Every article and book I read would tell that you can create a color with know SPD by using CIE CMFs: (X, Y, Z) = sum(SPD x Illuminant x CMFs), then convert to RGB. Is this statement misleading, because this would mean you can use CIE CMFs to reproduce colors? In real situation, colors cannot be created by that way, right? In fact, I have come across an article which would prove it’s impossible to create or ‘reproduce’ colors using CMFs. In the following link, the Macbeth chart reproduced using CIE CMFs is completely mismatched with the original chart.

https://www.scratchapixel.com/lessons/digital-imaging/colors/color-space

2. It turns out that the CIE RGB color matching was just to bend the color spectrum into a curve so that all visible colors can be created by connecting and mixing the spectral colors on the curve. If that was the purpose or the finding, one can simply bend the color spectrum into a circle: (x-1)^2+(y-1)^2=1, instead of this odd horse shoe or tongue shape which has puzzled the world ever since. The representation would be even more efficient because x and y are dependent and only need one variable to specify a color. In fact, the CIE xy gamut is nothing but a hue and saturation plane, so CIE XYZ is just a HSV color model.

Thanks for your time to enlighten me through this.

Regards

Hi Dengsheng,

It is true that – given a certain input – the CIE XYZ Colour-Matching Functions (CMFs) can be used to convert any spectrum into a standard X, Y, Z tristimulus encoding. From these X, Y, Z tristimulus values, all other values such as CIE L*a*b* or sRGB can be computed. Notice that the CIE XYZ CMFs use imaginary primaries because no physical matching lights can be used to obtain these functions.

So, the CIE XYZ CMFs allow to calculate the CIE XYZ tristimulus values for any perceivable colour (and from the XYZ coordinates into any other set of colour coordinates). Since all possible viewable colours can only be achieved using three imaginary primitives, there is currently no physical way to create three fixed primaries so that humans see all possible colours they are capable of seeing. Until the emitting source is a tuneable laser, specific colours will always be out of reach of any three real primaries. In other words: a real display that directly responds to XYZ cannot be built (and thus reproducing colour using the CIE XYZ CMFs is indeed not possible).

Concerning the RGB CMFs. In 1931, the Commision Internationale De l’Éclairage (International Commission on Illumination, CIE) collected the results of the colour-matching experiments of Wright and Guild done in the 1920s and mathematically derived the spectral curves that represented the average human colour. This CIE 1931 2° Standard Observer is thus a statistical fiction and nothing more than a numerical representation of what an ‘average person’ sees. The ‘2°’ in the name comes from the fact that Wright and Guild had their participants look through a hole that allowed them to have a 2 degrees Field of View (FoV) during the colour-matching experiment. The spectral curves that define this CIE standard Observer are called the CIE 1931 2° RGB CMFs. Later, CIE also made these CMFs for a 10 degrees FoV and called them the CIE 1964 10° RGB CMFs.

Now, the horseshoe you refer to is the CIE (x,y) chromaticity diagram. Chromaticity is computed as follows: x = X / (X+Y+Z); y = Y / (X+Y+Z).

In this way, the chromaticity of a colour can be described irrespective of its luminance. So, the RGB CMFs were computed to understand human colour vision and enable accurate colour capture and production, not to bend colours inside a specific shape or to be used for colour mixing. This “bending into a diagram” is just a specific way to display all the colours a normal human can see, without taking luminance into account. The shape of the chromaticity diagram is just visualising the 1931 Standard Observer Data projected onto a plane (with luminance value equal to 1). When choosing two points inside the diagram and connecting them with a straight line, all the values that are found along the line can be created by additively mixing the colours represented by the two dots in different proportions.

Hope this helps.

Cheers, Geert