Color Similarity. If we think in terms of the visible spectrum, it seems obvious that yellow and green are more similar than red and green: "yellow" light is closer to "green" light in the spectrum band. Spectral closeness seems a reasonable way to judge the similarity among colors.

But ask yourself, which hue is more similar to red: green, or purple? Although purple is at the opposite end of the spectrum from red, most people would answer that purple and red are more similar.
 

which hue is more similar to red: green, or purple?

 
Why? Because of the way color information is coded by the eye. The extreme "blue violet" end of the spectrum seems also to contain some "red" light. The reflectance profiles of magenta or purple paints show that these colors stimulate both the L and S cones in higher proportions than the M cones.
 

Newton's Opticks. This light mixing discovery was made by the English physicist and mathematician Isaac Newton (1642-1726), who first described the many optical and color experiments he performed in the late 1660's to the English Royal Society in 1672, and after an interval of more study and contentious public discussion published them in his Opticks of 1704. (That's the English edition; a scholarly Latin translation appeared in 1706.) Newton had experience grinding his own lenses and was attempting to understand the problem of chromatic aberration that introduces fringe colors into optical images. He knew of the "celebrated Phenomena of Colours" described in experiments with pentahedral (triangular in cross section) prisms performed decades earlier by Guido Scarmiglioni, Thomas Harriot and Marcus Marci. But he pursued these studies much further in experimental thoroughness, observational precision and logical clarity.  

The revolutionary aspect of Newton's work was his emphasis on keeping separate the physical and sensory (mathematical and psychological) aspects of color — a scientific strategy advanced in the treatises of Galileo Galilei. Newton demonstrated that each spectral hue has a unique and measurable refrangibility or angle of refraction when passed through a lens or prism. In his Experimentum crucis ("decisive experiment") and variations of it, he showed that this characteristic of light remains constant even if the light is blended by a lens, sent through multiple prisms, or passed through a colored filter.

He could not explain this refrangibility, which arises from the wave properties of light, but he concluded that the sensation of color was both separate from it yet intimately related to it. He first defined refrangibility and color in parallel terms:

The Light whose Rays are all alike Refrangible, I call Simple, Homogeneal, and Similar; and ... The Colours of Homogeneal Lights, I call Primary, Homogeneal and Simple.

and then, in a famous passage, he declared that color is a sensation in the viewer's mind, not a property of the light itself or of the materials illuminated by the light:

If at any time I speak of Light and Rays as coloured or endued with Colours, I would be understood to speak not philosophically [scientifically] and properly, but grossly, and according to such Conceptions as vulgar [uneducated] People in seeing all these Experiments would be apt to frame. For the Rays to speak properly are not coloured. In them there is nothing else than a certain Power and Disposition to stir up a Sensation of this or that Colour. ... So Colours in the Object are nothing but a Disposition to reflect this or that sort of Rays more copiously than the rest; in the Rays they are nothing but their Dispositions to propagate this or that motion into the Sensorium; and in the Sensorium they are Sensations of those Motions under the Forms of Colours. [Book One, Part II]

But he found they could be mixed. In particular, he discovered a region of new colors that were known in isolated natural exemplars such as gems and flowers but that did not appear in a prismatic spectrum. Instead, these extraspectral hues — "purple" and "red violet" (magenta) — appeared by overlapping the "red" and "violet" ends of two separate spectrums (right). And he found that recombining three or four colors of the spectrum through a lens reconstituted the original "white" color of sunlight.  

The Hue Circle. Newton summarized these striking observations as an ingenious new color model: he joined the "red" and "violet" ends of the spectrum to create a hue circle. This circle still shows the spectrum as a continuous gradation of color from red to violet, but now red is joined to violet, via the "artificial" or mixed colors carmine, magenta and purple.
 

In Newton's diagram (above), the small circles underneath each color name indicate the varying quantities or "weights" of each spectral color that might contribute to a color mixture: large amounts of red, orange and yellow, small amounts of "blew", indigo and violet. The color located at "Z", the average of them all, is the unsaturated red orange color that results. And, to close the circle, Newton observed that:

If the point Z fall in or near the line OD, the main ingredients being red and violet, the Colour compounded shall not be any of the prismatick Colours, but a purple inclining to red or violet, accordingly as the point Z lieth on the side of the line DO towards E or towards C, and in general the compounded violet is more bright and more fiery than the uncompounded.

Newton emphasized that the hue circle only applies to light mixtures. Pigment mixtures would not depend on the proportional weights or quantities of the pigments in a mixture, but on "the quantities of the Lights reflected from them." He verified that colors arise in pigments and in natural surfaces because they reflect some spectral colors and absorb others.  

The Analysis of White. Newton explored pigment mixtures at length, using the artists' primaries red, yellow, green and blue (as the pigments carmine lake, orpiment, verdigris and bremen blue) to explore the origin of white or gray mixtures:

All colour'd Powders do suppress and stop in them a very considerable Part of the Light by which they are illuminated. For they become colour'd by reflecting the Light of their own Colours more copiously, and that of all other colours more sparingly, and yet they do not reflect the Light of their own Colours so copiously as white Bodies do. ... And therefore by mixing such Powders, we are not to expect a strong and full White, such as is that of Paper, but some dusky obscure one, such as might arise from a Mixture of Light and Darkness, or from white and black, that is, a grey, or dun, or russet brown, such as are the Colours of a Man's Nail, or of a Mouse, or Ashes, of ordinary Stones, of Mortar, of Dust and Dirt in High-ways, and the like. And such a dark white I have often produced by mixing colour'd Powders. ... Now, considering that these gray and dun Colours may also be produced by mixing Whites and Blacks, and by consequence differ from perfect Whites, not in Species of Colours, but only in degree of Luminousness, it is manifest that there is nothing more requisite to make them perfectly white than to increase their Light sufficiently.

... and this Newton accomplished by illuminating the gray mixtures with a single beam of sunlight in a darkened room, or by comparing the sunlit mixtures to a shaded piece of white paper.

These mixing experiments with pigments and lights reveal Newton's fundamental preoccupation with the analysis of "white" into the compounding of different primary colors. Newton's hue circle implied that different combinations and proportions of primary colors would equivalently mix to white; white (or gray, in pigments) could be created by many different combinations of spectral light.  

Complementary Mixtures. Newton's play with "white" mixtures led to one of his most intriguing insights: two hues on opposite sides of the hue circle could create a near neutral color if mixed in the right proportions:

If only two of the primary Colours which in the circle are opposite to one another be mixed in an equal proportion, the point Z shall fall upon the center O, and yet the color compounded of those two shall not be perfectly white, but some faint anonymous [diluted and unnameable] Colour.  

This is the origin of the idea of complementary colors and the "mix to white" criterion for identifying them. But the passage reveals a few points of confusion. For starters, Newton admitted he could not create a pure "white" by mixing two or three "primary" colors, but he apparently did not grasp that this was because he was mixing broad sections of the spectrum as a single color, rather than mixing single wavelengths. (That is, his hue circle works only if the calculations are made on the physical attribute of "refrangibility" and not on the psychological categories of "homogeneal color", those wide bands of seven spectral hues.)

Newton also did not explain that some spectral hues are more luminous or have a higher tinting strength than others, which means they have inherently more weight in color mixtures and should be either located farther from the "white" center of the hue circle (as they are in some chromaticity diagrams) or overweighted in the "center of gravity" calculations. Finally, Newton's parallel experiments with pigments and lights misled many readers into thinking that pigments and lights mix in the same way, which only confused 18th century "color theorists".  

Newton's Legacy. It is difficult from a modern perspective to grasp the extraordinary originality of Newton's color theory. Perhaps the most important breakthrough in the Opticks is the use of geometry — the circle — to explain color mixtures, not as a medieval symbol of completeness or unity, but as the framework for a mathematical analysis of hues in a color mixture. This linkage between physical quantities and sensory qualities is the essential principle of psychophysics.

According to Newton, at least seven spectral hues or "simple" colors of light were necessary to describe all color mixtures. The arbitrary distinction between "real" and "illusory" color is swept away: all color arises in mixtures of light, no matter whether the light comes from a prism or the reflectance qualities of colored powders. These ideas went far beyond the 17th century dyer's lore that three primary colored paints or dyes defined color mixing.

Newton's book was intended for an audience of Enlightenment naturalists, and it was widely read. The English mathematician and perspective theorist Brook Taylor (1685-1731), who was also a talented amateur painter, pointed out in his New Principles of Linear Perspective (1719) that "the knowledge of this theory may be of great use in painting", and then explained in general terms how to apply it to mixing paints — though Newton had cautioned that his circle only applied to mixtures of light. From this misconception all artists' color wheels have branched and bloomed.

1. Every color sensation may be analyzed into three mathematically determinable elements: the hue, the brightness, and the brightness of the intermixed white.

2. If one of two mixed lights is continuously altered while the other light remains constant, the appearance of the mixed light is also continuously changed.

3. Two colors, both of which have the same hue and the same proportion of intermixed white, also give identical mixed colors, no matter of what monochromatic lights they may be composed.

4. The total luminance of any light mixture is the sum of the luminances of the lights mixed.

From these principles, Grassmann was able to prove an important corollary:

For every hue of monochromatic light, there is another monochromatic color which, when mixed with it, gives colorless light.

Grassmann clarified the principle that mixtures of light are additive in hue, brightness and chroma (mixture with "white" light), and therefore that color sensations are related to light mixtures according to quantitative principles. He also showed that every monochromatic [homogeneous] hue has a monochromatic complementary color — or (a point Grassmann missed) a mixture of extraspectral "red" and "violet" hues — which produce "white" when mixed together (diagram, right).

Today these principles, in a more algebraic version, are known as Grassmann's Laws. They formed the theoretical framework for color experiments by Helmholtz and James Clerk Maxwell in the 1850's. These established the trichromatic model of color mixture, and the modern study of color perception.

By the 1850's this anecdotal and speculative approach to color was largely displaced by psychophysics or the quantitative study of stimulus and sensation. This discipline emerged in Germany, through the work of Ernst Weber (1795-1878), Gustav Fechner (1801-1887), Wilhelm Wundt (1832-1920), Hermann von Helmholtz and others. Psychophysicists developed the experimental methods and mathematical equations necessary to link the intensity of a basic sensation to quantitative units of the physical stimulus — weight in relation to mass, brightness in relation to light intensity, pitch in relation to frequency of vibration, and so on. At the same time, biologists developed the laboratory and dissection methods necessary to understand the physiology of sense organs and the nervous system. As a result the sensory and biological structure of perception began to be viewed as an integrated mechanism obeying basic laws that quantify the connection between mind and matter.

This is the historical moment in which Bohemian physiologist Ewald Hering (1834-1918) launched a new defense of subjective color experience and color antagonism, published as the monograph Zur Lehre vom Lichtsinne (On a Theory of the Light Sense) in 1874 and as a book in 1878.  

Hering's Urfarben. Hering's strategy was to start with color experience and from that attempt to deduce color physiology. He pointed out that the trichromatic theory advocated by Helmholtz, Arthur König and others, which postulated three types of nerve excitations produced by three kinds of receptor cells, could not explain several well established observations in color experience. Hering noted that:

•  yellow seems to be psychologically just as basic as the trichromatic red, green or blue, yet does not seem to contain any of those colors

• dichromats, who lack a "red" or "green" cone, cannot see either a red or green color yet still can see yellow

• deuteranopia (green colorblindness, caused by a lack of the M cones) does not cause the point of maximum spectral luminance to shift from "green" wavelengths toward "red"

• most colors of the spectrum seem to shift in hue as they brighten or darken (the Bezold-Brücke effect), but these shifts do not appear in a pure blue, green or yellow.  

Based on clues of this kind, Hering asserted the perceptual primacy of four Urfarben or "primordial colors", known in the USA as the four unique hues: red, yellow, green and blue. He conjectured that they were produced by visual substances or processes located somewhere in the visual system outside the retina. He was vague about the physiology but precise about the "pure" form of the unique hues, which he equated with the color of monochromatic lights at 470, 500, 570 and 700 nm.  

Hering's Opponent Processes. Hering then turned to color mixtures. He observed that light or surface colors can produce a sensation of red mixed with yellow (orange) or red mixed with blue (purple), but never create the sensation of red mixed with green ("reddish green" or "greenish red") or yellow mixed with blue ("yellowish blue" or "bluish yellow"). This proved to Hering that the visual substances were organized as antagonistic or opponent processes.

In one process assimilation of visual substance produced the red sensation and dissimilation produced the green sensation; the other process produced the opponent sensations of yellow and blue. When the visual substances were neutralized or in balance, both hues associated with that opponent process disappeared.

Hering also observed that we commonly lose color vision at night, yet still see light and dark, which convinced him that luminosity is separate from hue as a color vision process. So he proposed two additional "colors", white and black, that create the perception of brightness or lightness and also affect color chromatic purity by mixing with any of the unique hues to create less intense (desaturated) colors across the complete range of shades, tones and tints — which Hering called veiled colors. However, he conceded that yellow and red had an "inherent brightness", and green and blue an "inherent darkness", which mingled perceptions of hue and luminosity, and that white and black did not disappear at balance but produced the positive sensation of gray, which meant that black and white are not linked as opponent processes.

In summary, Hering proposed there are six fundamental color processes that are arranged as three visual contrasts, including two opponent processes. They are:

The achromatic sensation of middle gray results when all the substances and opponent processes are in balance.

Hering took pains to fit his theory to Newton's hue circle. Roughly half the hues on the circle can be described as containing some yellow, and the other half some blue; these semicircles overlap with two opposing spans of red and green, oriented perpendicular to the yellow and blue like four hue compass points. Binary mixtures of any two neighbor unique hues (one hue each from the two opponent processes) would explain all spectral hues between them, including the extraspectral purples between red and blue.
 

hue circle explained by two opponent processes
combining two illustrations from Ewald Hering (1920), reversed to match modern color wheel orientation

 
The outer ring shows overlapping quantities of neighbor opponent processes. These tapering color bands show that neighboring unique hues can be blended in any proportions. Opposing hues — red vs. green, and yellow vs. blue — do not overlap in the outer circle: they cannot mix. A line through the outer ring indicates the relative proportions of two unique colors necessary to create the inner hue circle of color swatches. Hering's examples (in the lower left corner) show that a mixture of 25% unique blue and 75% unique red produces a crimson red, while a mixture of 75% blue and 25% red produces a blue violet, and a 50%/50% mixture produces purple. Other mixtures produce scarlet, orange, yellow green and blue green to complete the circle.

Although Hering did not describe a complete color system, his color writings suggested the basic geometry for the Swedish NCS color model.

As explained in the next section, opponent contrasts have proven to be fundamental to color vision. So it is ironic that most of Hering's specific explanations of colorblindness and color perception have turned out to be incorrect, or the data he relied on flawed or incomplete. His concept of "visual substances" directly causing conscious sensations and working in chemical opponency, like acid and base, is wrong in several respects. The opponent contrasts that do underlie color vision are not anchored on the unique hues. Opponent contrasts can be identified in neural pathways only between the retina and visual cortex, not in brain areas associated with color recognition or color naming. Hering was an innovator who uncovered important and valid general principles despite a flawed interpretation of the facts.  

The Opponent Functions. For several decades after the publication of Hering's ideas, Hering and Helmholtz, and then their descendant partisans, disputed the theoretical mechanisms and research evidence for the trichromatic and opponent theories. The trichromatic theory and the opponent process theory were usually seen as incompatible explanations of color vision.

By the middle of the 20th century, researchers had concluded that both theories were necessary to explain the physiological processes of color perception. These hybrid models were first suggested by Johannes von Kries and G.E. Müller in the 1890's and were definitively reformulated by Müller in 1930 as a zone theory of color vision. Zone theories derive the opponent dimensions from the trichromatic retinal responses by a step by step transformation or recoding across three or more stages or zones. In 1949 D.B. Judd specified Müller's physiological theory as testable equations based on cone fundamentals of the CIE standard observer. Today these opponent dimensions are the foundation for most modern color appearance models.

recoding opponent dimensions from trichromatic outputs

 
In the 1950's by Leo Hurvich and Dorothea Jameson developed a quantitative hue cancellation method to measure the opponent functions by decomposing monochromatic hues into a mixture of two of the four unique hues. Their method uses the same color matching techniques developed to measure the cone fundamentals championed by Helmholz.

In this procedure, either the viewers or the experimenters chose four wavelengths of light to match Hering's unique hues. (A mixture of "red" and "violet" wavelengths must be used to produce a unique red.) The intensity of each light was then adjusted so that equal parts of red and green matched the brightness of unique yellow, and equal parts of yellow and blue matched the brightness of unique green.

Next, a fifth wavelength was chosen as the test color. Viewers used a light mixing apparatus to blend the test light with the two unique hues that were nearest complementary colors (opposite on the hue circle) to the test light, until the mixture produced a pure "white" light. This was repeated for test lights at regular intervals across the spectrum.
 

opponent functions in spectral hues
chromatic response functions for the CIE 1964 standard observer; from Hurvich & Jameson (1955); the unique hues are located at B, G and Y (unique red is composed of a mixture of "red" and "violet" light)

 
As shown here, the horizontal zero line represents the balance between the two unique hues in a single opponent function. The distance of a curve above or below the line represents the relative quantity of each unique hue required to match (rather than cancel) the spectral hue at each wavelength (i.e., they are tetrachromatic color matching functions).

The y/b opponent function is represented by the yellow/blue curve. The area where the yellow curve is above the horizontal line, with a maximum in "yellow green" wavelengths, indicates colors that appear to contain some yellow. The area where the blue curve is below the line, with a maximum in "blue violet", indicates hues that contain some blue. This y/b function contrasts the long and short wavelength ends of the spectrum.

The r/g opponent function (red/green curve) codes for hues that appear to contain some red (the areas where the red curve is above the horizontal line, with peaks in "blue violet" and "scarlet" wavelengths) as opposed to hues that contain some green (the area where the green curve is below the line, with a maximum in "middle green"). This r/g function contrasts the middle wavelengths from both ends of the spectrum.  

The unique hues appear at any point where one opponent function is at the zero line where it cannot bias or alter the hue created by the other opponent function. Thus, unique yellow appears at the long wavelength balance point in the r/g opponent function, at about 573 nm, and unique blue at the short wavelength balance point around 472 nm. Similarly, unique green appears at the single balance point in the y/b function, at about 492 nm. (The exact location of these unique hues varies across individuals.)

What about unique red? There is no point where the y/b contrast crosses the neutral line a second time in the visible spectrum — both contrasts taper toward neutral as the spectrum fades to invisibility, and near infrared "red" light (beyond 700 nm) actually appears more yellow (scarlet) as the wavelength increases to 900 nm. To produce unique red from monochromatic lights, a small amount of "violet" must be added to spectral "red" light, to neutralize the yellow hue. (This is one reason for the S cone input to the r/g opponent coding.)

Finally, Hurvich and Jameson adopted the photopic sensitivity function (gray curve) to represent the whiteness/blackness or w/k opponent function. Procedurally, they equated the w/k curve with the quantity of "white" light produced by the neutralizing mixture of spectral and complementary opponent hues. But this curve glosses over several problems. It does not account for the additivity failures in spectral mixtures, and blackness is not a color sensation created by isolated color mixtures — it arises from the luminance contrast between a color stimulus and the color area around it. Luminance perception is actually structured as a brightness/blackness (b/k) opponent dimension: white is its neutral value.
 

These curves recognizably correspond to the general form of the three chromatic response functions: curve 1 is the w/k function, curve 2 is the y/b function, and curve 3 is the r/g function. There are discrepancies in the exact shape and crossover points of the curves (especially in function 1), but these discrepancies can be attributed to four factors: (1) reflectance curves are a physical specification of the color stimulus, not a perceptual specification via the cone fundamentals; (2) reflectance curves define the color independent of any light source, whereas the human opponent functions are optimized for chromatic adaptation along the dimensions of chromatic variation in the daylight phases; (3) the reflectance variations in the Munsell color samples are constrained to some degree by the limited number of pigments used to manufacture them; and (4) by far the largest proportion of natural surface colors are weakly chromatic and rather dark, which means the Munsell system contains a disproportionately large number of saturated and light valued colors.

Note the heavy weighting of the S cone outputs necessary to match the y/b opponent function estimated from hue cancellation experiments. This demonstrates the perceptual overweighting of S cones in comparison both to their impact on the opponent dimensions and to their sparse numbers among the much more numerous rod, L and M photoreceptor cells.  

All this implies that the wavelength locations of the unique hues are a byproduct of the opponent dimensions and not the other way around. Thus, the fact that very different sets of opponent functions, derived from statistical reflectance analyses or hue cancellation experiments, can produce the same hue circle, implies that the orientation of the oppponent dimensions 3 the location of the unique hues — is of secondary importance. And in fact there are large individual differences in the choice of unique hues, which strongly suggests the hues are not structurally basic to color vision in the way the cone fundamentals are.

This problem can be explored with graphical data analysis, which has been used by multivariate statisticians for over a century. It lets us examine complex mathematics by means of pictures rather than numbers, and allows us to show complex transformations in simple geometrical terms.

By making graphical adjustments to the color space — changing the length of color dimensions, rotating our view of the dimensions, adjusting the angle between dimensions — we can see the transformations from cones to colors as explicit sculptural steps.  

A Geometrical Color Standard. To start we require a graphical standard for the geometrical pattern we expect to find in the color appearance of color samples that have been correctly transformed from the light excitation produced in the cones.

Since the 1920's, the most commonly used standard in color research has been the Munsell Color System. The Munsell system can be used to define colors that are perceived to be evenly spaced in hue and chroma across equal value (lightness or gray scale) differences. In addition, the Munsell hues 5R, 5Y, 5G and 5B approximately locate the Hering unique hues red, yellow, green and blue.

The Munsell system arranges colors along equally spaced, radial hue angles and circular chroma levels from the achromatic point at each lightness level. This ideal geometry is most recognizable when viewed in the orientation shown in the diagram below, which displays all hues within the chroma range typical of artist's paints and at the two Munsell values 4 and 8.
 

Each Munsell value plane is perpendicular to the L,M plane, so that the projections of all colors onto the plane form a series of lines across the luminosity function (gray lines, above), but the hue planes cross both the luminosity function and the achromatic axis at oblique angles.

Next we orient the colors at values 4 and 8 to match the Munsell ideal standard (above). To do this, the space must be turned or rotated so that the achromatic axis is aligned with the direction of view, then turned on its head to place yellow at the top, as shown below.
 

This chromaticity space standardizes all values on the luminance Lum (L+M) response and gives all three cone responses equal weight. Some color spaces double the M response to the red/green contrast (L–2M).

The Lum and L–M dimensions are created by a single orthogonal rotation of about 43° around the S axis of the cone excitation space, which aligns the luminosity axis with the direction of view toward the color samples. These new dimensions are simply weighted sums of the original cone excitation dimensions, as follows:

W =	0.665*LΔ + 0.610*MΔ + 0.431*SΔ
L–M =	0.676*LΔ – 0.737*MΔ
Lum–S =	0.317*LΔ + 0.291*MΔ – 0.903*SΔ

This procedure also forces a new distinction between the luminosity and achromatic axes. If the Lum (L+M) dimension includes any S output, then a second rotation of about 26° around the L–M dimension has been applied to align the achromatic points with the direction of view. This defines a whiteness/blackness or lightness induction dimension (W).

The innovations here are that the cone excitations have been combined according to specific proportions that establish contrast, priority and independence among the cone signals; and that four distinct quantities — Lum, W, L–M and Lum–S are in play.

Separating Chromaticity from Lightness. The final adjustment addresses the oblique angles between each hue/chroma plane and the achromatic and luminosity axes (see diagram above, right). These angles remain even after the previous transformations have been applied, as shown below in the opponent dimensions of response compressed, rotated cone excitation space. (In the original XYZ tristimulus values, the XZ hue/chroma planes are already perpendicular to the luminance Y and achromatic axes, so adjustment of L+M+S is not required.)
 

Wp =	1.031*LΔ + 0.511*MΔ + 0.006*SΔ
L–M =	5.946*LΔ – 6.482*MΔ
Lum–S =	1.215*LΔ + 1.113*MΔ – 3.454*SΔ

The large weights given to the L–M contrast are necessary to overcome the narrow elliptical shape of the color space. The diagram (right) shows the spectral profile of the chromatic postreceptor opponent functions. The recognition features are the absence of positive "violet" hue coefficients on the L–M curve, and the very large negative "violet" hue coefficients on the L+M–S curve.

The diagram below shows the hue/chroma distribution of surface colors. The L+M–S null line is placed at a distinctive yellow green (leaf green) and a blue violet (approximately Munsell 5GY and 7.5P). The L–M null line is placed at blue green and magenta, at Munsell 7.5BG and 10R. These "balance hues" do not correspond to those at the spectrum locus, especially in the placement of unique yellow (Munsell 5Y): we expect this because human vision is adapted to perceive surfaces, not spectral lights.
 

The distribution of surface colors (below) shows a much more circular outline in the chroma rings, and better placement of the yellow and blue violet colors in relation to the vertical null line. However, there is still significant irregularity in the chroma contours, visible as the "wedge" exaggeration of chroma distances in the yellow green and red directions, and in the crowded hue spacing in yellow greens, reds and blue greens and the exaggerated hue spacing in greens.
 

The correction for these deviations is probably a more nuanced specification of the compression exponent. There are at least three different problems to solve. The first is the increased, eventually linear compression of chroma spacing along the zero S (=Z) line for hues between 7.5G to 7.5YR at chroma above 8. This is clearly visible as the flat chroma boundary across the yellow part of the color distribution in the x,z and opponent spaces (diagrams above and right).

The second problem is the spike of inflated b+ chroma values, which is due to the peak values of the Lum (L+M) sum where they cannot be moderated by subtracting S values near zero. The effect disappears across the yellow orange and orange hues because these colors are substantially less luminous. This implies a smaller exponent (a larger chroma deduction) across the yellow green hues.

The third variation, the g–m+ and g+m– deviations, suggests a third exponent operating in relation to the sign of both opponent dimensions: the exponent produces a relatively smaller compression for hues where the sign product of the ab dimensions is negative (greens or purples), and a higher compression when the sign product is positive (oranges and blues).

Perhaps the most significant themes to emerge in this discussion of undefined additional corrections is the regulatory importance of the S cones in the specification of an opponent color space, and the complex interplay of different compression, rotation and normalizing steps necessary to reach an approximately circular scaling of perceived chroma.

As mentioned earlier, many of the problems described above are minimized by direct use of XYZ tristimulus values, because these already redistribute the S ("violet") content to the other dimensions — so that the "red" (X) dimension is actually a magenta, the hue/chroma planes are already perpendicular to the luminosity dimension, and the exponent compressed dimensions form nearly circular chroma rings. But this means that many of the necessary transformation steps are "precooked" into the XYZ values; alternately, that the XYZ values must be "uncooked" (transformed or rotated) to get back to the (theoretically) primitive LMS values.

Starting from the LMS values displays the essential information processing steps more clearly, and shows that color vision is a tightly integrated, precisely balanced process. It also confirms that current color models do not yet describe it accurately.

Measuring Perceptual Discrimination. The perceptual geometries of vision are identified through judgments of color difference — by comparing two colors that are different along one or more of the colormaking attributes, while holding the other attributes constant. The kinds of questions addressed in this way are:

• What is the smallest stimulus difference we can perceive on each of the three colormaking attributes?

• Does discrimination ability change as the physical properties or intensity of the stimulus changes?

• How does change on one colormaking attribute affect color appearance on another colormaking attribute?

All these questions concern related colors — one color seen in contrast to beside another. As a bridge between the realm of color discrimination and the colorimetric definitions of color matching, I open each discrimination section with the color geometry predicted by standard chromaticity diagrams, which describe unrelated colors or colors perceived in isolation.  

This defines a quantity of stimulus change T, known as the difference threshold or 1 jnd. When the starting quantity is the stimulus zero value (complete absence of the stimulus), then the first jnd value T is called the absolute threshold Q₀. This is the minimum stimulus quantity necessary to produce any sensation.

The process can be repeated by using the previous value Q as the new starting value, increasing the stimulus intensity again until a new difference appears, and taking this new value of T as a difference threshold at that stimlus intensity. By repeating this procedure across incremental increases in the stimulus quantity, the subjective sensation produced by any physical attribute can be measured off using the "yardstick" of the just noticeable difference.  

The Weber Fraction (ΔQ/Q). At each step the jnd is the smallest detectable increase T in the stimulus, given the starting value of the stimulus. It turns out that this increase is often a constant proportional increase (k) in the stimulus intensity, which is defined by the Weber fraction (pronounced Veybur):

k = T/(Q + Q₀) or equivalently ΔQ/(Q + Q₀)

Weber's law asserts that the ratio k is constant across a wide range of stimulus quantities, although the value of k is different for different sensory domains or types of stimulus. If this is assumed to be true, then Fechner's Law allows the estimation of the sensory intensity (S) from the stimulus quantity, as:

S = a + b*log[Q + Q₀]

where a and b are applied to remove negative log values and match the stimulus units to the perceptual quantities. It is now, especially in color vision research, typical to express this relationship as a power function, known as Steven's Law, where the exponent 1/e is typically less than 1 (e=2 or 1/2 is the square root)

S = b*[Q + Q₀]^1/e

where b is used to fit the stimulus units to a practical perceptual scale. Similar power functions occur in all sensory domains — vision, taste, hearing, touch, smell, pain and muscular contraction (sensations of weight or effort) — although the exponent in each domain is different. In fact, log or power functions similar to Fechner's Law or Steven's law are used in many scales related to energetic quantities — stellar luminances in magnitude, sound in decibels, earthquakes on the Richter scale, ion ratios on the pH scale, and so on.

I should also mention a generic formula that is used to model the depletion of photopigment by light (derived from an enzyme kinetics curve):

where R is the response as a proportion of the maximum possible response R_max, L is the stimulus luminance (or retinal illuminance), σ₅₀ is the luminance value that produces a 50% response (the half saturation value), and n is the response compression exponent (usually around 0.70). (Zero is the assumed minimum response.) The implications of this function are discussed in the section on luminance adaptation mechanisms.  

The Power Function in Lightness. What is the effect of the Weber fraction on perception? The relationship between the perceived lightness of a surface color and its reflectance or luminance factor (luminance as a proportion of the luminance of an ideal "white" surface) shows the perceptual geometry very clearly.
 

lightness as a function of reflectance
relationship between surface reflectance (luminance relative to the luminance of a white standard) and perceived lightness (CIE L* scale; divide L* by 10 to get the Munsell value V)

 
This power function or nonlinear psychophysical relationship between lightness and relative luminance is approximately a square root function for reflectances above ~7% (that is, 1/e is about 0.43). The curve for brightness discrimination is approximately the cube root (1/e = 0.33) of absolute luminance.

Underlying these sensory power functions is the common sensory property of response compression. Fundamentally, response compression represents a compromise between the physical limitations of an organism and the enormous range of a real world stimulus. In effect, the organism experiences an increase in the stimulus as a sensory change proportional to the remaining sensory response — which becomes smaller as the sensation approaches its physiological upper limit.
 

"Less" and "more" luminance is always relative to our light adaptation around an average luminance (which is usually a 19% luminance factor for the "middle gray" used in graphic arts but is 10% to 13% for the "gray" luminance used to meter photographic exposures). For illumination levels above a few lux the subjective effects of response compression are remarkably consistent.

The sensory response compression displayed in psychophysical power functions is the single most important asymmetry in color vision. Its effect throughout the visual system is to create curvatures or nonlinearities in stimulus intensities across steps of perceptual difference that appear to us as evenly spaced "just noticeable differences".  

If we consider difference thresholds in the brightness of two lights viewed side by side, in the Weber fraction (ΔL/L) similarly changes across luminance levels from .0001 to 10,000 cd/m² (equivalent to light intensity or illuminance on a white surface of about .0003 to 30,000 lux), as shown below.
 

There are a few details where the opponent dimensions diverge from the experimentally observed hue discrimination curves. But the overall fit is close enough to suggest that the opponent dimensions, not the cone fundamentals, define the geometry of hue discrimination.  

The unequal spacing occurs because the results are shown along a nanometer scale, which corresponds to the visual spacing of hues in a diffraction grid spectrum. This is remedied by respacing the spectrum to correspond to equal changes on the hue coefficients (the proportional mixtures of two unique hues). As a result, some parts of the spectrum are perceptually expanded while others are greatly compressed (diagram below). This new spacing is hue discrimination acuity predicted from the unique hues.
 

The cone responses also show a very large initial increment in saturation in relation to the chromatic signal, a jump that is too large to be traced in detail by the Munsell chroma intervals. This indicates that the visual system is especially sensitive to small chroma differences from the