Color Naming. The earliest color labels are embedded in common speech. Every language has words to name colors, though in many primitive cultures only a handful of words are available, and the words make very gross distinctions between light and dark, or warm and cool colors. Thus the Black Sea is not black, but dark blue; its color name has been handed down from a time when the color ideas green, blue, dark and black were denoted by the same word. This doesn't mean that primitive speakers could not see the differences between those colors, just that the differences were not conceptualized as abstract color categories.

However, early color enthusiasts had no assurance that they had sampled the full range of possible colors — they did not understand the physiological limits of color vision, and therefore could not be sure that a new insect, coral, flower or gem might not reveal a hue that had never been seen before.

This uncertainty was swept away in 1704 when Isaac Newton published his Opticks, the first scientific analysis of color. Among the many stunning breakthroughs in this book was Newton's claim to have identified all fundamental colors — and by extension all possible mixed colors. Any color could be reduced either to a pure spectral hue or to a geometrically defined mixture of spectral hues. Newton symbolized this intellectual closure as a closed circle: his hue circle. So ended doubt about the range of possible colors.

However, the practical problem remained that precise spectral mixtures were difficult to achieve until the mid 19th century, object colors were typically duller and more complex than the colors of mixed spectral lights, and all those complex colors still had to be identified with a name and captured as a physical exemplar. In fact, the last two requirements were the most important and the easiest to fulfill. So naturalists had to derive their color identifications and color labels from a color atlas, a tradition solution that extends from A.G. Werner's Nomenclature of Colors (1774) up to Robert Ridgeway's Color standards and color nomenclature (1912). These provided a series of hand painted colored patches as a visual standard for each color, with the color's name and, in earlier versions, a list of flowers, insects or minerals that exemplified the color in nature. Similar tools are still used today in horticulture, agriculture and food processing.

Color Mixture. If naming the color was incidental to using it, then colors could be reduced to a recipe or proportional mixture of a few standard colorants or "primary" colors. The entrepreneurial printer Jakob Christoffel Le Blon first applied this approach to the industrial task of color printing, by showing that color images could be created from three primary or "primitive" colored inks, usually supplemented by a fourth black ink, one ink printed over another with separate, carefully aligned mezzotint plates (a technique developed earlier in woodblock printing of book illustrations). Multicolor printing systems similar to LeBlon's, though not commercially successful, continued up to the end of the 18th century; the multicolor printing method was revived when chromolithography was invented in the 1860's, which did not become widely used until the 1890's. But from that time until today color printing systems based on standardized ink colors and mixture recipes have been universally used by commercial printers for mass reproduction of color images. Modern color mixing systems include the four ink process (CMYK) color system (developed in 1934), the 14 ink Pantone™ color system, and the six ink Hexachrome™ color system, to name a few.

The Pantone and other printing systems are based on standardized printed color samples, each color reproducible as a carefully proportioned mixture of a limited number of "primary" inks. These are really "color cook books" tied to specific technologies of color reproduction, and they require the graphic artist to visually select from a limited (though sometimes very large) index of printed color samples. These books provide no way to visualize how all the colors fit together, in the same way that a city phone directory gives you no idea of how the city looks on a map.

It will be impossible to think of any nuance produced by a mixture of the five elements (blue, yellow, red, white and black) and not contained in this framework; nor can the whole system be represented by any other correct and complete figure. And since each nuance is given its correct relation to all the pure elements as well as to all mixtures, this sphere must be considered a universal chart, enabling anybody to orient himself as to the overall context of all colors.

Michel-Eugène Chevreul followed with his own "color hemisphere" in 1839, and abstract color systems became one of the minor achievements of 19th century European scientists and the artists who studied their works.  

Color Model Assumptions. The geometrical framework used in modern color models consists of the three colormaking attributes arranged to define a three dimensional space. The figure below shows how this is done. The vertical black to white dimension is the lightness or value of a color; the circumference of the horizontal disk perpendicular to the lightness dimension is the hue of a color; and the lateral distance or radius on this disk, from the center outward, is the chroma (approximately the saturation) of a color.
 

the geometry of colormaking attributes
in modern color models

 
Notice that this arrangement allows hues to change gradually from one to the next, but defines chroma and lightness as rulers or ratio scales starting from a definite zero point — black for lightness, and gray for saturation.

Color models differ in the specific assumptions that are used to define the color geometry and color labels and the specific measurement methods used to link the geometry to color stimuli and color exemplars. My selection of color models is intended to highlight these differences:

• Geometrical Uniformity - The perceptual facts of color are tremendously complex, and many early models emphasized a geometrical or logical framework that summarized color in a simplified way. These systems use visual color matching to measure colors, appeal to the phenomenology or subjective relationships among colors, and emphasize a framework for verbal color communication. The best modern example is the Swedish NCS color model.

• Perceptual Uniformity - Here the goal is to create a color geometry where the units of measurement stand for equal apparent color differences on all the colormaking dimensions. Perceived color differences determine the color geometry and exemplification as physical color samples. The classic modern example is the Munsell Color Order System; very different recent examples include the OSA uniform color scales and the CIE 1976 UCS (uniform chromaticity scales) diagram.

• Trichromatic Colorimetry - Increasingly, modern color models are linked to the electronic measurement of spectral emission or reflectance profiles, expressed as a mixture of three imaginary "primary" lights called the XYZ tristimulus values. This standardizes the "viewing eye" used to compare colors and allows automated color matching. The modern examples of this approach are CIELUV and CIELAB.

The alternative approach, which became dominant in European philosophy and visual arts, emphasized the logical description of color experience. Color was idealized as "pure" sensations, organized as verbal or geometrical models of color relationships. The Swedish Natural Color System (NCS), proposed by Tryggve Johansson in the 1930's and eventually realized by Sven Hesselgren and Anders Hård at the Scandinavian Colour Institute in the 1960's, is a modern color model that exemplifies this approach. The NCS is currently a color standard in several Scandinavian and EU countries.

The experimental pedigree of the NCS goes back to Ewald Hering's Das natürliche Farbsystem (1905), which describes colors as the result of opponent color processing and the mixture of four unique hues plus black and white. Hering's was a phenomenological or subjective approach to color: he tried to describe the universal patterns or laws of our color experience without specifying what kind of external stimulus or internal color receptors might produce it. This phenomenological approach can be traced from Hering back to the Romantic German philosophers Arthur Schopenhauer, Georg W.F. Hegel and ultimately to J.W. von Goethe — although those authors and their philosophical concerns were of little practical use to the artists of their time.  

Four Components of NCS. This phenomenological or "in the head" approach to color modeling is based on four components.

The first is a single, universal color specification formula:

color = C(u₁+u₂) + S + W

where C represents the apparent proportion of pure hue (chromaticness) in the color, S (German Schwartz) is the apparent proportion of black in the color, and W (German Weiss) is the apparent proportion of white. The chromatic ingredient consists either of a single unique hue, or more commonly of the mixture of two adjacent unique hues, shown as (u₁+u₂). Hering called any color mixed with either white or black (or both) a veiled color, and by these mixtures accounted for all dull, diluted and near neutral colors.

The basic formula has a long pedigree. The Western painting tradition used the technique of broken color to adjust pure pigments (high chroma) colors by mixture with white and/or black paints, a method described by Cennini, Alberti and Leonardo. The same mixture geometry is used in Michel-Eugène Chevreul's color hemisphere. Hering's color specification was also used by the German chemist and Nobel laureate Wilhelm Ostwald (1853-1932) in his Mathetische Farbenlehre (Quantitative color theory) published in 1917 and exemplified by the Farbkörper (Color Atlas) published in 1919. Ostwald's system was very influential among the artists and designers of De Stijl and the Bauhaus.

The second component of the phenomenological approach to color is a restricted version of the universal color geometry. The four Hering unique hues are arranged to create four cardinal points around the hue circle, as shown in the figure. Following Hering's original concept, the y/b and r/g opponent functions define two dimensions at right angles within the hue circle. Antagonistic unique hues, which are not allowed to mix in Hering's theory, are on opposite ends of the two dimensions. The third opponent dimension defined by white and black is placed perpendicular to the first two.
 

three color dimensions defined by the hering colors

 
The third component consists of two color mixing rules:

(1) Only unique hues on different opponent dimensions can be mixed; unique hues at opposite ends of the same dimension cannot be mixed. (That is, we are not allowed to make "yellowish blue" or "reddish green" mixtures.)

(2) White and black can mix with each other, and with any hue mixture, in any proportions; this provides the full range of shades, tones and tints for any hue.

The fourth and last component is a fixed sum for the color specification:

C(u₁+u₂) + S + W = 100

in the same way that mixtures in Tobias Mayer's color hexahedron must sum to 1. This forces the color geometry into a triangular (conical, hexahedral) form.

NCS Color Model. These four design components interconnect the six Hering colors as 13 elementary scales — a single gray scale (white to black), and three chromatic scales for each hue (e.g., red to white, red to black, and red to gray) — that measure off mixtures among the six cardinal colors.

The basic mixing formula color = C + S + W = 100 dictates that the model is most conveniently represented as vertical sections, called hue triangles, that include one pure hue or hue mixture, pure white and pure black as the three corners of an equilateral triangle. This is equivalent the geometry of two cones joined at their base that was used in Ostwald's system, as shown below for the 13 elementary scales and a purple (mixed red and blue) hue section.
 

geometrical framework of the swedish NCS color model

 
The NCS is exemplified as the NCS color atlas, a large book of standard color swatches. The atlas contains 40 of these hue triangles at equal intervals around the hue circle — nine steps (increments of 10% mixture) between each pair of adjacent unique hues. The atlas also includes a representation of the complete hue circle at maximum chroma, a lightness scale from white to black, and several "off white" color samples as a help to specifying pastel colors.  

Each hue triangle displays a wide selection color samples made from mixtures of the pure hue with black (S) and/or white (W). Lines of equal black mixture are shown as diagonal rows parallel to the top edge of the triangle; blackness increases from the top edge to the bottom corner. Lines of equal whiteness are shown as diagonal rows parallel to the bottom edge; whiteness increases from the bottom edge to the top corner. As a result, gradations in chroma (C) are represented by vertical columns through the triangle, from the left corner (the maximum chroma), which contains no white or black, to the gray scale along the vertical edge. Color samples for the pure hues, white and black are omitted due to the limitations of available pigments, and very dark or near neutral samples are usually omitted to reduce production costs and visual clutter.
 

a page from the NCS color atlas 96 (standard version)

 
The NCS notation or color labeling defines the visual appearance of the color and shows how to find it in the NCS color atlas 96 (1996), represented by the sample page above. The tonal mixture is stated before the hue. The first two digits give the percentage of black in the color, followed by the two digit percentage of pure color. (The percentage of white is omitted, as it is equal to the other two percentages subtracted from 100.) Next, hue mixtures are notated with letters for the two unique hues contributing to the mixture, bracketing the percentage of the second hue in the mixture: Y60R represents a red orange mixture of 60% red and (by subtraction) 40% yellow. Pure hues are notated simply R, Y, G or B. Thus, the full notation S 1060-Y60R represents (1) a pure hue that is mixed from 60% red and 40% yellow, and (2) a color that is mixed from 10% black and 60% pure hue (and 30% white). Achromatic (gray) tones are indicated simply with the proportion of black, followed by N for neutral: S 1500-N represents a light gray containing 15% black (and 85% white).

A significant feature of the NCS is that all the pure color standards are imaginary: the point of comparison for a pure white or pure yellow is visualized in the mind, and color mixtures are judged against these ideal standards. (This is why there are no color swatches at the points C, S or W in the sample hue triangle above: no pigment can match these pure, ideal colors.) This was consistent with Hering's original phenomenological color definitions, but it was also validated by experiments that showed people were just as accurate judging color mixtures from ideal (imaginary) color standards as from physical color samples.

Using and Interpreting the NCS. The NCS was created primarily for communicating about color in everyday and design applications, because its simple color geometry and direct relationship to visual color judgments makes color easy to describe. The only "measurement tool" necessary is a person with normal color vision, training in the NCS notation conventions and access to an NCS color atlas to use as a reference. (As with Munsell, viewing conditions must be carefully standardized if color judgments are to be comparable.)

In addition, for most parts of the color circle, opposite colors are not "mix to gray" visual complements, because the unique hues themselves are not visual complements. The visual complement of unique red is not unique green, but a blue green located at roughly G40B; the visual complement of unique blue is not unique yellow, but a deep yellow at roughly Y30R. Finally, the artist's mixing complements are also different from the NCS complements. And while the NCS establishes new complementary relationships, there is no evidence that these are more (or less) pleasing or effective than the complementaries defined in other ways.  

Most perplexing, the "pure" hues at maximum chroma vary significantly in lightness around the color circle: an intense yellow is much lighter than an intense blue, but in NCS they are all notated as containing no black and no white, and are placed at the same halfway point on the vertical white/black dimension. This means that the white/black mixture does not consistently describe the actual lightness of colors, and all the mixture intervals (columns and diagonal rows in the hue triangle) do not define comparable steps across different hues. For a light valued yellow hue, the 10 gradations in W (between yellow and white) are perceptually quite small, while the 10 gradations in S (between yellow and black) are quite large. For a dark valued blue violet, the reverse is true. This is entirely consistent with the Hering definition of black as a color rather than as the absence of light, but it makes the NCS noncomparable with other color systems based on lightness.

The benefit of this apparently arbitrary relationship is that different hues at the same location within their respective hue triangles — that is, hues identified with the same whiteness and blackness numbers (S1060 for the orange in the illustration above) — have the same nuance, and nuance is an effective way to equate colors, as described in the page on color harmonies.

Most color order systems that build on a simple subjective or geometrical framework have similar drawbacks. These are not serious as long as the models are used only as a way to label and communicate about individual color samples, but the NCS must not be used to model perceived color differences. And these drawbacks may even be an advantage in color design, for example when equating the apparent whiteness of various hues in pastel color schemes.

Munsell originally conceived of his system as a method to describe color for artists and teach color to children. He was aware of the European, geometrically regular color models of Chevreul, Wundt and others, and at first conceived (and patented) his system as a color sphere. But with a typically American empiricism, he convinced himself through analysis of paint samples that color space was not geometrically regular. Instead, Munsell sought to build his model on equal perceived color differences on each colormaking attribute, linked in some cases to measurable changes in color stimulus composition, in a freely branching geometry called a color tree.

Munsell was inspired by and heavily relied on the expertise of the American physicist and color researcher Ogden Rood, who showed Munsell how to use a color top to analyze complex colors into additive color mixtures of "primary" colors, and to develop gray scales based on proportional mixtures of black and white. With this method Munsell was able to simulate incremental steps in chroma (proportional mixtures with gray) or hue (by proportional mixtures of "primary" colors, or analogous colors such as red and yellow), and to identify visual complementary colors (saturated hues that additively mix to a pure gray). Over a decade of patient, incremental work, Munsell analyzed reflective or surface colors into their component lightness, hue and chroma, published in his atlas as equal apparent differences on each of these colormaking attributes.  

Finally, each hue was calibrated in equal chroma steps from zero (pure gray) to the maximum color intensity in the paints or inks used for each hue; in the idealized (aim) colors, chroma intervals are calculated out to the optimal color limits.

The approximately cylindrical framework of the Munsell system (below), which includes a value scale limited by the luminance factor of surfaces and chroma steps bounded by the optimal color limits for each hue. The dimensions are not otherwise connected, which corresponds to the separate perceptual measurement procedures used to define them.
 

conceptual framework of the munsell color order system (1905)

 
Colors are "named" through the standard notation sequence hue value/chroma. Thus, the pigment vermilion would be notated as 8.5R 5.5/12 — a hue of 8.5 in the R segment, at value 5.5 and chroma 12. Any physically realizable surface color (paint, ink or dye) can be located within the Munsell framework by using decimal increments on the hue, value and chroma dimensions. (Munsell apparently adapted his color terms from the French valeur and chromatisme, which he picked up while studying art in Paris in the 1880's.)

So how does one measure color with the Munsell system? At first (and commonly today) through human color judgments. The Munsell colors are standardized as carefully prepared color samples or paint chips, presented on separate pages of a reference catalog, the Munsell Book of Color. Each page contains color samples of a single hue, arranged in a two dimensional grid defined by value and chroma. Colors are identified by placing them next to the atlas samples until the nearest color match is found. Decimal color values are inferred by locating a sample color between two existing Munsell chips. These visual judgments are accurate only if the comparison color sample is as large as the atlas samples, and all colors are viewed on a gray background under the same daylight or incandescent illumination (usually, illuminant C).  

The Munsell Renotations. In the 1920's some of the 1915 Munsell samples were first measured with a spectrophotometer, and these were mapped into the CIE 2° Yxy color space shortly after it was developed in 1931.

The early measurements were made by the USA National Bureau of Standards, which took an interest in developing the Munsell system as a color standard. In the 1930's the Optical Society of America also began a technical study of the system. The entire 1929 Book of Color was measured spectrophotometrically in 1935 and published as Yxy values in 1940. These revealed major discrepancies between the 1915 and 1929 color samples, and major irregularities in the spacing of color samples on the Yxy chromaticity diagram, particularly in the chroma intervals but also in the lines of equal hue.

The OSA committee primarily focused on defining the Munsell value scale as a mathematical formula, based on visual judgments of gray samples by a large sample of viewers, and on judgmentally adjusting or "smoothing" the contours of Munsell hue and chroma on 1931 xy chromaticity diagrams. These calculations were used to revise the value scale, to equalize the hue and chroma intervals, to extend chroma intervals out to the optimal color limits, and to redefine material color specifications. This renotated system was published in 1943 as specific Yxy values for 2700 unique color locations, as a mathematical formula to calculate Munsell value from XYZ tristimulus values, and as several 1931 xy chromaticity charts used for the visual estimation of Munsell hue and chroma values from xy values. Subsequent editions of the Book of Color are blended to match these 1943 targets. The renotated system is the standard specification of Munsell colors used today.

Subsequently, the OSA committee working on uniform color scales (discussed below) revisited the renotation data, generally to examine the effects of imposing a genuinely three dimensional definition of color differences, but particularly to examine the effects of response compression on the chroma of different hues. The data for this re-renotation system were published in 1967 for more than 2900 color samples, and represent a major revision to the Munsell standards. It is not in general use.

Today all the Munsell colors are available as idealized tristimulus values or aim colors, and Yxy color locations or color spectrophotometric data can be translated into the Munsell notation entirely by computer software. Thus the color system designed for teaching color to children has ascended to the realm of digital exemplification and industrial color standards.  

The Asymmetry of Color Space. Munsell and his successors worked hard to produce color samples that were perceptually equidistant or equally different from neighboring colors on the individual dimensions of value, chroma and hue. Thus, the difference between a chroma of 2 and a chroma of 4 is intended to appear to the viewer as large as the chroma difference between 12 and 14. However, these measurement units are not comparable across dimensions; the principle of equal perceived distance applies within each dimension of the model separately, not between any combination of color attributes. This is obvious from the fact that 10 units encompass the entire range of values (from black to white), but only 10% of the total range of hues (for example, from yellow to orange) and, for most hues, only half the total range of chroma.

However, the empirical measurements that have been at the heart of the Munsell system from the beginning compelled Munsell to an important realization: the natural color space is highly irregular when it is represented without geometrical preconceptions. Thus, Munsell always conceived of his color model as a sphere, but allowed for unequal dimensions of chroma at different levels of lightness and across different hues. Two "pages" from the Munsell system at the complementary hues 5Y and 5PB show this clearly.
 

two pages from the munsell book of color (1929)

 
These hues are directly opposite each other on the Munsell hue circle, so they show a vertical "slice" through the Munsell color solid (sometimes called a color tree). This shows what is obvious when you think about it: yellow at its highest chroma has a much lighter value than does blue at its highest chroma. These lopsided relationships between value and chroma affect nearly all hues around the color circle. As a result, no simple geometrical form accurately represents perceptual color space. All color models based on triangles, circles, squares, pyramids, cones, spheres, cubes or cylinders must (and do) grossly distort perceived color relationships. Munsell was the first researcher to document this basic color fact as a color model.

In addition, Munsell's color studies and the subsequent renotation research revealed that the background or surround color significantly affects the response compression (contrast) in lightness or chroma; in effect, lightness and chroma are contrast perceptions of brightness and colorfulness. In addition, the Munsell workers documented poor chroma or lightness discrimination in very intense or very light valued colors, and uneven perception of hue differences, especially as hue differences become very large. By its focus on perceptual uniformity in the spacing of material color samples, the Munsell research uncovered the many geometrical complexities in color space.  

But even the glossy colors are incomplete. The figure at right shows the range of surface colors represented by the Munsell color solid as irregular colored shapes, enclosed within a white envelope that represents the optimal color stimuli or maximum chroma for surface colors of each hue. (Colors with a higher color intensity than the white boundary will appear fluorescent or self luminous.) The Munsell system represents the range of colors that can be achieved by high quality acrylic paints, or inks printed on highly reflective white paper; the range of colors possible with watercolor pigments is significantly smaller.

Comparison of the Munsell color range with the optimal color envelope shows that the yellow and red pigments do a good job of filling out the potential color space, while modern lightfast magenta (RP), purple (P), blue (B), blue green (BG) and green (G) pigments reach less than half the chroma potential for their hues. Thus, the Munsell color atlas, and in fact any color atlas in any color system, exemplifies an incomplete and skewed subset of the total asymmetrical color space. Note, for example, that the G dimension of the Munsell atlas is smaller than the RP dimension, although in the perceptual surface color space just the reverse is true.

The subsequent development of the colorimetric system did not resolve matters, as the standard chromaticity diagram also badly distorted perceived color differences — the chromaticity differences between greens are much too large, and differences between blues are much too small.

Methods to quantify perceived color differences were very important in the implementation of color measurement, for purposes as diverse as photographic dye patents and governmental standards, and for manufacturing color control in industries as dissimilar as textiles and automotives. These requirements stimulated an effort to develop a truly uniform color space, from the ground up, as a set of uniform color scales across the central area of moderately saturated colors. This task was taken up by a committee under the auspices of the Optical Society of America (OSA), in collaboration with the USA National Bureau of Standards, and led by Deane Judd and David MacAdam. The final UCS aim color tristimulus values, color scales, recommendations for their use and conclusions regarding the human color space, were published in the 1970's as the Optical Society of America's Uniform Color Scales (UCS).  

Rhombohedral Lattice. The first issue was the specification of the color geometry. The decision was made to shift from the "wheel and spoke" arrangement of Munsell to a cartesian or three dimensional set of dimensions, as in the CIE Yxy color space.

The color geometry should also support measurement of perceived color differences in all directions, unlike Munsell where intervals of chroma are not perceptually equal to intervals of value or hue. To assess perceived color differences in all directions of the color space using identical psychometric tasks, it is desirable to have diagonal dimensions added to the cartesian space.

The OSA committee found the solution to both problems in a rhombohedral lattice as the basic color geometry — the same geometry found in a carbon crystal (diamond or graphite). It is difficult to represent as a single image, but the logic behind it is straightforward.
 

The three perpendicular dimensions of the uniform color space are the lightness dimension L and two chromatic dimensions, j (for jaune, approximately a y/b opponent dimension) and g (for green, approximately a g/r opponent dimension). The mid valued gray that is the anchor of the system has a value of 0 on all dimensions; colors with negative values are darker, bluer or redder.
 

The various diagonal scales or cleavage planes through the color space were intended to identify palettes of complex color change that might be useful to graphic artists and color designers. They turned out to be wildly unpopular for that purpose. The legacy of the OSA UCS is as a research landmark and a uniform tiling of the color space that is useful to assess color models or imaging systems.  

Color Space Is Noneuclidean. Despite the extensive and patient research program and the elegant color model it produced, the OSA committee had already in 1967 come to a dispiriting conclusion:

We will affirm to the world that no regular rhombohedral lattice sampling of color space, with a fixed background, can exist; we will produce the best approximation to such a lattice for a neutral value 6 background that we can design.

In plain language: it is impossible to represent uniform color differences as a three dimensional color model. The human color space is noneuclidean, which means a three dimensional framework cannot describe it. Any color difference formula or color model that represents color in only three dimensions (such as lightness, hue and hue purity) gives a distorted picture of color differences and color relationships.  

The second problem involves chromatic contrast, in particular crispening, which makes small color differences, or differences judged against a similar background, appear relatively larger. Most of us are familiar with the "reading magnifiers" that enlarge words when rested on a printed page. Crispening acts like a magnifying bubble inside the color space, enlarging the apparent contrast between similar colors on lightness, chroma and hue. As a result, all small color differences are magnified in comparison to large color differences, and "small distance" color difference formulas tend to overestimate the perceived color difference between colors that are very different.

Just as a reading magnifier can be shifted around a page, the crispening locus can migrate around the color space, depending on the color and lightness of the background (surround) used to judge colors, the chromaticity of the illuminant. To control this, color vision experiments and color atlas viewing standards use a mid valued gray surround in color presentations, so that crispening occurs as an increased chroma sensitivity around near neutral colors.

The third problem is that hue superimportance is not equal in all directions: chroma compression is strongest in yellow and green hues and weakest in red and blue hues. It also changes across different values, becoming more extreme in lighter valued (more luminous) colors.

The final problem, related to the previous one, is the unequal spacing of hues, which is explored on a previous page. Some hues produce a greater number of perceptible color gradations than others; in particular, color discriminations are much more acute for warm hues than for green hues.
 

• The first is that all color models we can see cannot represent colors relationships accurately, and color models that represent color relationships accurately cannot be seen (physically constructed or represented as an image).

• The second is that all color difference formulas are arbitrary, in the sense that they must include many judgmental corrections for specific discrepancies, can only provide difference estimates that are approximately correct for small (or large) color differences, and provide difference estimates whose accuracy depends on where in the color space the color difference is calculated.

The primary color paradoxes can be explained as resulting from the overlap in three cone sensitivity curves. The color model paradoxes can be generally explained by the fact that the perceptual color space changes continually under the influence of many contextual factors. It's not that change cannot be expressed as a static euclidean model. It's that the perceptual space adapts to different color stimuli, color differences, contrast backgrounds and light sources in different ways, depending on which specific combination of color stimuli is being used to measure it.

Color Matching and Color Differences. Since the early 1920's, a major goal of colorimetry was the prediction of an unrelated color response through the photometric measurement of a physical color stimulus. The approach that seemed most efficient was to measure the spectral power distribution of the light emitted or reflected by the color stimulus, translate this into photometric units by means of a luminous efficiency function or a standard set of color matching functions, then use these quantities to estimate the subjective brightness or chromaticity of the stimulus when it is viewed as an isolated patch of color.

However, the XYZ tristimulus values that came out of color matching tasks literally only predict color identity, that is, whether two colors will appear the same. The whole problem of how much two colors will appear different was left undetermined. The CIE xy chromaticity diagram does represent the relative difference between two colors — that one color sensation is redder, or brighter, or duller than another, if both colors are measured in the same way. But the desired next step was to subtract one set of XYZ tristimulus values from another, and use the numerical difference to predict the perceived color difference between them.

Projective transformations were proposed independently by Deane Judd and David MacAdam in the 1930's; Judd's version is shown below. It radically compresses the "green" part of the chromaticity space while expanding the "blue" and "red". The white point is located close to the R–G mixing line, because the L and M cones dominate the perception of whiteness (compare with my cone chromaticity diagrams).
 

CIELUV does even worse at predicting corresponding colors, or colors that match under different colors of illumination (illuminants). The problem is that CIELUV represents relative color differences in relation to the white point as a difference or simple subtraction (e.g., u'_c – u'_w), rather than as a ratio or proportion. This means that if the white point (the illuminant) is changed, or is strongly chromatic, all colors are shifted in a parallel direction by a constant amount (as if printed on a moveable transparency), regardless of where they are located in the chromaticity space.

If colors are shifted from yellow toward blue, the compressed chroma intervals in the yellow green to orange part of the u*v* chromaticity diagram do not change, so CIELUV grossly overestimates chroma differences wherever these boundaries are relocated. If colors are shifted from blue to yellow (for example, the color of a gem viewed under tungsten rather than halogen light), CIELUV thrusts the predicted color locations completely outside the spectrum locus, which is perceptually impossible.

As corresponding color predictions are a minimal requirement for modern color appearance models, CIELUV has been discarded as the basis for further work.

The CIELAB Dimensions. The CIELAB formula comprises four transformation steps, which are intuitive when examined separately.

Recall that the color geometry common to all modern color models consists of a brightness/lightness dimension that is perpendicular to (separate from) the chromaticity plane or hue circle. This geometry is implicit in the XYZ tristimulus values: the Y value represents the photopic luminosity function of the specified color. We start with the color's Y tristimulus value (Y_c) as the measure of its luminance or luminance factor (labeled L, as in CIELUV), and use the X_c and Z_c values as measures of chromaticity (hue and hue purity).

The X value increases primarily as the L cone response to a color becomes relatively large, so it is the "red" anchor for an r/g dimension (labeled a); the Z value increases with S cone response, so it is given a negative sign to represent the complementary "yellow" of a y/b (b) dimension:

L = Y_c
a = X_c
b = –Z_c.

The first transformation step is an adjustment of the tristimulus values to represent a chromatic adaptation to the illuminant (color of illumination). The correct method (a von Kries transform) divides the cone excitations produced by the color area by the cone excitations produced by the illuminant. CIELAB instead uses the "wrong" ratio between the tristimulus values X_wY_wZ_w of the illumination. These are a standard both for the luminance upper limit represented by a perfectly reflecting "white" surface (Y_w is always equal to 100) and for the chromaticity inherent in the light itself (in the values of X_w and Z_w relative to 100).

L = Y_c/Y_w
a = X_c/X_w
b = –Z_c/Z_w.

The tristimulus values of a perfectly white or equal energy illuminant are defined as X_w = Y_w = Z_w = 100. An illuminant with a perceptible "warm" (yellow or red) chromaticity has values of X_w greater than 100 and values of Z_w less than 100; "cool" or blue illuminants have low X_w and high Z_w. The adaptation ratios reduce or increase the values of X_c and Z_c in proportion.

In the second step, CIELAB applies an exponential compression to all three adaptation ratios. This inflates the perceived chromaticity and lightness differences among dark colors, and inflates the chromaticity of yellows relative to blues. At the time CIELAB was published, different compression fractions had been tried in color difference formulas, and the cube root seemed to perform about as well as any:

L = (Y_c/Y_w)^1/3
a = (X_c/X_w)^1/3
b = (Y_c/Y_w)^1/3.

L = (Y_c/Y_w)^1/3
a = (X_c/X_w)^1/3 – (Y_c/Y_w)^1/3
b = (Y_c/Y_w)^1/3 – (Z_c/Z_w)^1/3.

This creates two undulating curves for the difference values of spectral hues (diagram, right).

An explanation: when applied to quantities of the same thing, such as dollars or gallons, subtraction just identifies the quantity of the difference. When applied to quantities of dissimilar things measured within a similar range of values, subtraction purges any information the two measures have in common (in this case, information about the luminance) and leaves as a single number all the unique information in both (the combined information about hue and hue purity). As the X_cY_cZ_c values are already standardized on the white point (by the adaptation transform), the subtraction also centers the values of a and b at zero for a surface color that exactly matches the chromaticity of the illuminant: a pale yellow surface under a pale yellow light will appear to be a "pure" white.

Finally, the L, a and b dimensions are expanded by different factors so that a unit scale value on any combination of CIELAB dimensions represents an approximately equal and just noticeable perceptual difference. As the Y–Z curve (b dimension) has a larger amplitude than the X–Y curve (a dimension, diagram above right), it is multiplied by a smaller amount. The slope and intercept of the L function is also adjusted to match the units of Munsell value:

L* = 116*(Y_c / Y_w)^1/3 – 16
a* = 500*[(X_c / X_w)^1/3 – (Y_c / Y_w)^1/3]
b* = 200*[(Y_c / Y_w)^1/3 – (Z_c / Z_w)^1/3].

These formulas take an alternate form, not shown here, for extremely dark colors (where any adaptation ratio is less than 0.009). For details see the references at the end of this section.

CIELAB yields a reliable correlate (relative measurement) of chroma as the euclidean distance between the chromaticity of the color and the achromatic point:

C_ab = [a*² + b*²]^1/2

... and a numerical definition of hue as the hue angle:

h_ab = arctan[b*/a*]

... and its own color difference score ΔE_ab, obtained as the euclidean distance between the L*, a* and b* values of two similar colors:

ΔE_ab = [(L*₁–L*₂)²+(a*₁–a*₂)²+(b*₁–b*₂)²]^1/2

where "similar" means colors that are separated by a ΔE_ab value of around 10 or less. As a rule of thumb, 10 units of CIELAB lightness exactly matches 1 unit of Munsell value, and 10 units of CIELAB chroma approximately matches 2 units of Munsell chroma. The average ΔE_ab for the first chroma step in Munsell aim colors across all hues and values is about 5.8. Note that ΔE_ab is approximately in units of just noticeable difference, so color differences at or below 1 are generally not visible.  

The Uniform Color Space. The CIELAB transformations yield approximately perpendicular dimensions that are often represented as a cylindrical geometry. That is, the L* dimension is the axis of the cylinder, through the achromatic point; h_ab represents the hue angle or circumferential direction of the color, and C_ab the distance of the color from the cylinder axis.
 

geometry of the CIELAB color model

 
The diagram is somewhat misleading. In practice, the color difference (euclidean distance) score is based on the cartesian (a*b*) coordinates and not cylindrical radians or degrees; and the distribution of surface colors in CIELAB tapers toward the achromatic center at very high or low lightness.

The point is that CIELAB color relationships are not distorted by forcing them into a conceptual geometry, such as the ipsative hue triangles of NCS. Color measurements are limited vertically by the luminance factor of surfaces, and the chroma extent is limited by the domain of real surface colors or lights, but these are boundaries imposed by color perception, not arbitrary geometry or logic.

The illustration below shows the location of 700 commercial watercolor paints in the CIELAB space, with color icons grouped onto separate lightness planes for clarity. These define a roughly ellipsoidal or "football shaped" color domain inside the CIELAB color space. 
 

Because they are based on difference scores, CIELAB chromaticity dimensions do not redistribute or normalize the luminance information across the chromaticity coordinates: they expunge it. As a result, the CIELAB color metrics cannot be expressed directly as the additive mixture of three "primary" colors of real or imaginary lights. When narrowband optimal colors are represented in this cone opponent space, they produce an erratic, double lobed circuit, with the spectrum ends diving toward the achromatic origin (right). This is recognizably an upside down image of the cone excitation space, pinched and skewed by the cube root transformation. Note also the exaggerated spacing of the blue hues.

CIELAB measures chroma, the chromatic content of a color independent of its luminance; the a*b* plane does not represent saturation. Unlike the straight line, additive mixture between any two colors of light in the CIE UCS (or any chromaticity diagram), color mixtures can take curving or unpredictable paths on the CIELAB a*b* plane. This is not a result of the subtractive mixture of the two physical colorants (paints or dyes) but is basic to the CIELAB space.  

Current Use of CIELAB. Like CIELUV, color specification in CIELAB is tied entirely to spectrophotometric measurements of color; it can't be used routinely without a color measurement device, and there is no standard CIELAB color atlas available. However, the wide use and practicality of the CIELAB system have spawned a number of proprietary color order systems or color atlases based in whole or part on the CIELAB geometry, including the American Colorcurve System, the German RAL Design System and the English Eurocolor Atlas.

Fairchild (2005) reports validation studies that pit several color appearance models against each other, using datasets that measured color matching under illuminant adaptation (corresponding colors) or preferred contrast and chroma in color image reproductions viewed at different luminance levels. Overall, these studies showed that different models performed well in predicting some types of color judgments but not others. In every case, however, CIELAB typically performed well against more complex models and occasionally was as good as the best; CIELUV was often the worst.

A major problem is that CIELAB uses what is called a wrong von Kries transform to model chromatic adaptation. This computes the contrast ratio between a color and the white standard or illuminant directly on the XYZ tristimulus values, rather than on cone fundamentals. In addition, the cube root compression is not optimal when applied to the tristimulus values, and results in chroma intervals that can be misplaced in some parts of the hue plane: in particular, yellow chroma is not compressed enough, and blue chroma is compressed too much.

These failings aside, from the same standing start as CIELUV, CIELAB has become the standard colorimetric space worldwide. CIELAB remains the most practical and widely implemented color model available. Indeed, several extensions of or revisions to the basic CIELAB framework, including S-CIELAB and RLAB, have been published to improve its performance and suitability for a wider range of color modeling problems. Colors are not forced into an arbitrary geometric shape that distorts perceived color relationships. These extensions are possible because the CIELAB geometry does not impose an arbitrary symmetry on chroma, hue or lightness. The calculation from XYZ values to Lab values is simple and easily inverted (from Lab values back into XYZ), and can be replaced by a variety of chromatic adaptation transforms.

The original or revised CIELAB color difference formulas are widely used for automated color quality control in areas such as textiles, plastics, architectural and automotive coatings, printing, imaging and art materials, and in ASTM documents. Color samples from any other color model, including Munsell and NCS, can be conveniently located and compared in the CIELAB space, so it has become a convenient representation space for the evaluation of color models in general.

CIELAB is a useful color framework for painters who want to understand color perception and color mixing problems in paints, not least because its lightness and chroma metrics are, on average, ten times the same attributes in the Munsell color space. In 1996 I used CIELAB pigment locations as the foundation of my original artist's color wheel. However some of the "bad" hue and chroma spacing problems, which I had corrected by hand using the Munsell renotation data, later decided me in favor of the CIECAM02 color relationships for the current (2006) version of the color wheel.

My principal source for information about CIELAB and CIECAM is Mark Fairchild's Color Appearance Models (2nd ed.) (Addison Wesley, 2005); see also the updates and information at Fairchild's web site.

Artists may be interested in the average location of watercolor pigments on the CIELAB a*b* plane, also available as a PDF file.

The Adobe Technical Guides have a good overview of color models in general, the CIE series, and the groundbreaking Munsell color system.

Daniel Smith has republished their Watercolor Paint Guide, which includes a map of their watercolor paint colors on the CIELAB a*b* plane, as "The Study of Color", available online at their InkSpot archive and reprinted occasionally in their catalog.

CIECAM color appearance model

The CIECAM02 Dimensions. The CIECAM calculations are quite complex when compared to CIELAB. CIECAM also requires more starting information (refer to graphic at right):
 

• Photometric information: the average luminance of the visual environment or surround L_a, in nits (cd/m²).

• Context parameters: (1) contrast factors F and N_c, which both have values of 1.0 when the luminance (reflected illuminance) in the surround matches the illuminance on the color area (documents or prints under workplace lighting), 0.9 when the surround illuminance is dimmed compared to the image (computer displays or overhead transparencies), and 0.8 when the surround illuminance is dark compared to the luminance of the image (cinema, slide or video images); and (2) an exponent c that modulates the response compression in image lightness, brightness and chroma caused by background lightness (the Bartleson Breneman effect), empirically set at values of 0.69, 0.59 or 0.525 for the same viewing contexts. These parameters are not measured in the scene, but are determined by visual judgment and then read from a graph.

• Response Compression Factors: (1) a luminance adaptation factor D, which increases toward a limit value of 1 as the color area appears reflective rather than emitting (that is, the surround luminance L_a increases, and/or F increases — the luminance of the surround matches the luminance of the image); (2) a power function F_L that models the increase in brightness and colorfulness caused by brighter viewing environments (higher levels of surround luminance L_a); (3) chromatic contrast factors N_bb and N_cb, derived from the ratio Y_b/Y_w, that are used to calculate the increase in chroma, lightness and brightness caused by darker backgrounds; and (4) exponents z and n, also derived from Y_b/Y_w, that define the response compression on lightness and chroma caused by background lightness.

Note that the color area may be part of a larger image, but all adaptation or contrast factors operate on the color area in isolation. CIECAM only represents shifts due to lightness contrast with the background or the effects of illuminance levels or luminance adaptation. It does not model chromatic contrast, chromatic assimilation or effects of spatial frequency between components of an image.  

Chromatic Adaptation. Much of the development work leading up to CIECAM involved the choice of chromatic adaptation calculations that gave accurate and invertible results.

• Adjustments for luminance adaptation (D) are made on the color and illuminant RGB values, resulting in post adaptation values for the color area (R_cG_cB_c) and the illuminant. This is the point where the illuminant tristimulus values are used to perform a von Kries transform for chromatic adaptation, shown below for the R value:

R_c = [(100D/R_w)+1–D]R

The same adjustment is performed on the G_c and B_c values. This step corresponds to the adaptation ratios in CIELAB, which are done on the tristimulus values directly and are therefore wrong von Kries transforms. The CIECAM formula has zero effect (R_c = R) in cases where (1) colors are viewed under an equal energy illuminant (the R_wG_wB_w values are all 100), and (2) there is complete adaptation to the illuminant (D = 1). Otherwise this transform decreases or increases the cone output to match the chromaticity of the illuminant, in proportion to the degree that adaptation only partly occurs — that is, D < 1 when the surround is darker than the image, and the difference between image and surround luminance prevents complete adaptation. This becomes especially significant when the surround luminance is below 300 cd/m².

• The post adaptation R_cG_cB_c values are converted back into XYZ tristimulus values. Parallel transformations are done to retrieve the illuminant XYZ values.

The same adjustment is made to the illuminant R'_wG'_wB'_w values. This transformation has the double effect of amplifying cone response differences (contrast) at very low tristimulus values (the curve has a much steeper slope at low R' values), and increasing the overall response contrast in brightness and chroma as luminance increases (the maximum value of R'_a and the overall slope increases). Because wavelengths at the tails of a cone fundamental produce a much lower response (tristimulus value) than wavelengths near the peak, and the transformation increases relative response at low tristimulus values, the transformed cone fundamentals have substantially raised tails and broad, rounded shoulders around the peaks (see diagram below). These calculations yield post adaptation, response compressed cone fundamentals denoted R'_aG'_aB'_a and R'_awG'_awB'_aw.

These circuitous calculations result in a "correct" von Kries transform for chromatic adaptation, adjusted in relation to the degree of chromatic adaptation, and a response compression scaled to surround luminance (luminance adaptation).

Lightness/Brightness. Correlates of lightness (J) and brightness (Q) are computed from the post adaptation, response compressed cone fundamentals R'_aG'_aB'_a, the background contrast factor N_bb, and the surround luminance compression factor F_L, as follows:

A_w is computed from the illuminant R'_awG'_awB'_aw values. A is essentially a luminosity function with rounded peak and raised tails that is adjusted by the contrast factors c, z and F_L to represent compression effects caused by background contrast and surround luminance.  

Opponent Dimensions. A set of preliminary opponent dimensions are calculated as:

a = [R'_a+(B'_a/11)]–(12G'_a/11)
b = (R'_a+G'_a–2B'_a)/9

Note the similarity between these opponent dimensions and the fundamental dimensions of a cone excitation diagram, for example the dimensions L–M and L+M–S of a trilinear mixing triangle. The main differences are in the small amount of S (B'_a) output added to the L (R'_a) cone output in the a contrast, and in the better relative weighting of the cone outputs.

These preliminary ab dimensions are used to derive the color's hue angle (h), which in turn is used to derive an eccentricity factor e that adjusts the scaling of the a and b dimensions to represent the differences in chroma compression that occur around the hue circle.

Hue Purity Correlates. Next, chroma (C) is computed as the euclidean distance from the origin on the response compressed ab dimensions. I present the formula to show how intricate it is:

The formula can be parsed as follows: (1) the basic chromaticity distance is defined as the euclidean distance on a and b between the color and the achromatic point; (2) the chromaticity is multiplied by the eccentricity factor e, which produces a different chroma scaling along each hue angle of the ab chromaticity plane; (3) the rescaled chroma is multiplied by the chromatic induction factors N_bb and N_cb, which increase chroma as the background becomes darker; (4) the contrast scaled chroma values are normalized on the sum of response compressed cone outputs (R'_aG'_aB'_a); and (5) the normalized chroma values are adjusted to compensate for the color's lightness (J) and the background contrast exponent (n).

Modern chroma metrics in CIECAM and other color models are highly complex, but this is necessary to achieve equal chroma spacing around the hue circle, appropriately normalize the luminance information across an equal chromaticity plane, and fit the correct response compression in relation to background contrast, scene luminance and color lightness. The cylindrical coordinates of lightness/brightness, hue angle and chroma are the final form of the CIECAM system.

The cylindrical coordinates are used to compute the remaining appearance attributes saturation (s) and colorfulness (M). Trigonometric functions are then used to transform the color space into cartesian coordinates Jab, whose chromaticity units are in terms of chroma, saturation or colorfulness.

Finally, there is no color difference formula optimized for the CIECAM model: the sequence of color models that began in color difference formulas ended in a model without one. However, very good results have been obtained by using the standard euclidean distance between similar colors:

ΔE₀₂ = [(L₁–L₂)²+(a_C1–a_C2)²+(b_C1–b_C2)²]^1/2

where "similar" means colors that are separated by a ΔE₀₂ value of around 10 or less. The average CIECAM ΔE₀₂ for the first chroma step in Munsell aim colors across all hues and values is about 7.7, slightly larger than in CIELAB. Note that the ΔE₀₂ is approximately in units of just noticeable difference, so color differences at or below 1 are generally not visible.

Graphical Analysis of CIECAM. It is hardly possible to understand a color model as complex as CIECAM from the calculation stream alone. The effect of the calculations (and parameter values) on color specifications must always be examined graphically.  

Response Compression. Probably the single most significant feature of CIECAM in comparison to CIELAB and all previous color models is the pervasive use of