Colors and Colorimetry

David Madore
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Table of contents

• Table of contents
• Physical and biological color
  • What is light?
  • What is color?
  • The three kinds of cones
  • Color blindness
  • What is white?
  • What is hue?
• Interlude: the CIE X,Y,Z color space
• Monitor gamut
• TODO

Physical and biological color

What is light?

Light is the region of the electromagnetic spectrum which the unaided human eye can perceive. On the blue (short wavelength) end, the limit is rather sharp and we can put it at a wavelength of 390nm (a frequency of 770THz) for extreme violet. On the red (long wavelength) end, the limit is not as clear-cut, but it can be set around 700nm (430THz) for extreme red. Waves with higher frequency than extreme violet are called ultraviolet, and those with lower frequency than extreme red are called infrared. The picture of the spectrum shown above represents the hues of the monochromatic lights (terms to be explained below) ranging from 400nm at the far right up to 800nm at the far left, on a uniform logarithmic scale (notice in particular how the spectrum does not at all appear “uniform”, with a very sharp yellow band compared to a rather spread out turquoise).

An elementary light stimulus is an arbitrary mixture of the pure lights of the spectrum. It can be composed of a finite number of sharp rays (“discrete spectrum”) or of a continuous mixture of wavelengths. Mathematically, the space of possible spectra is infinite-dimensional. Physically, the light can be decomposed in its elementary components, whose intensity can be measured by a spectrometer. For example, the incoming light from the Sun under the Earth's atmosphere, under precise conditions, can be divided in its constituent wavelengths, and the relative intensities can be measured and form a well-studied spectrum.

A light (a spectrum) is said to be monochromatic when it consists of radiation of a single wavelength: thus, it is the most discrete spectrum possible.

It is essential to note that no mixture of, say, red and green monochromatic light, will ever produce a monochromatic yellow: by definition, a monochromatic light is unmixed, so it cannot be achieved as a mixture. The best we can try to do is something which looks identical to the eye, and that is where color comes in.

What is color?

The eye perceives luminous sensations by means of four kinds of cells on the retina: one kind is called rods, while the other is called cones, which themselves are of three kinds—long, medium and short cones (color-blind people might not have all three). Rods are sensitive to much lower intensities than cones, so they are essentially used in night vision: in day vision, their input is not used at all since it saturates, so the cones take over. Cones are capable of distinguishing color (by the very definition of “color”) as well as intensity of light; this is why we are unable to discern color in very dim settings.

The color of a light is the cone cells' response to that stimulus. Since there are three kinds of cones, the space of colors is three-dimensional, contrary to the space of spectra, which is infinite-dimensional; so there exist many different spectra producing the same color, that is, many different ways of mixing monochromatic lights to produce the same effect on the cones.

The three kinds of cones

Long, medium and short cones are so called because their sensitivity peak is situated in long, medium and short wavelengths (respectively around 570nm, 543nm and 442nm). They do not play a symmetric role: long and medium cones participate in the sensation of brightness or luminosity, whilst short cones contribute much less, if at all, to this sensation. Brightness, a combination of the long and medium cones' response to a light, is what causes the pupil to close, for example; it can be measured fairly objectively because the rapid alternation of two lights of different brightness causes the impression of flickering which is not caused by the mere alternation of color (two lights having the same brightness). Thus we can define and measure a brightness response of the human eye. In mathematical terms, the three cone response functions are positive linear forms (defined up to multiplication by a positive constant) on the infinite-dimensional cone (pun unintended) of specra, and the brightness function is a linear combination of the long and medium cone response functions. For a given wavelength, the physiological measure of luminosity is proportional to the physical magnitude of intensity (power per unit solid angle, say); but the eye's response to different wavelengths, even for the same intensity, is, of course, different.

The picture shown left represents a gray background (we will explain later what “gray” means) with three circles, each corresponding to a 20% increase in response of one of the cones (and where the circles overlap, several cones show the same increase). The circle on the left corresponds to long cones, that on the right to medium cones, and the circle on the top corresponds to short cones; notice how the latter is barely discernible—it does not give any luminosity contrast, and hardly any change of color although the increase in stimulation is the same (20%, in other words, the same stimulation for that cone as would be obtained by a light having 20% greater intensity). Now stare at the center of the top circle and watch as it seems to disappear completely in the gray background: this experiment makes it obvious that, contrary to long and medium cones which are fairly equally distributed on the retina, short cones are not: they are absent from the centermost part of the retina (the fovea). (But because the pupil's vergence depends on wavelength and we focus for, say, yellow light, blue light is always blurred on the retina, so even if it there are no short cones on the fovea, blue light aimed there will always find some cones to stimulate.) Incidentally, short cones are much fewer in number than long and medium cones, the latter being approximately in ratio 2:1.

The following pictures show two sample images and the response of the long, medium and short cones when viewing these images:

Sample image
Long cones' response
Medium cones' response
Short cones' response
Brightness (luminosity)

It is tempting, and common, but misleading, to call the three cone kinds red, green and blue instead of long, medium and short. It is misleading because the long and medium cones actually have very similar responses; if we interpret their sensitivity curves as transmission spectra of filters (over a white light), then the long curves are more like yellow than red, with their peek at 570nm. Now it is not possible to stimulate a single cone kind alone, because every wavelength of visible light, even when produced monochromatically, stimulates to some extent at least two of the three cones. But we can stretch our imagination and “desaturate” the three primary colors to some extent, to obtain meaningful hues (what that means will be explained later): then the three hues corresponding to the three cone types are the hues of the three circles of the previous figure: some kind of sanguine red, aquamarine, and deep purple.

Color blindness

People can be color blind when they lack one of the three kinds of cones in the retina. When the long cones are missing, the individual is called protanope, then the medium cones are missing, deuteranope, and the the considerably rarer case when the short cones are missing is called tritanope. When the cones are not completely absent, but merely fewer in number, one is said to be respectively protanomalous, deuteranomalous or tritanomalous.

The following images attempt to capture (insofar as it can be done) what it feels for various kinds of color-blind people to view two sample images. This isn't really meaningful, because it isn't possible not to stimulate one or another kind of cones an a non-color-blind person's retina. However, it is possible to stimulate it proportionally to a white light of the same luminosity (as perceived by the color-blind person), so that white stays white: this is how we have computed the protanope, deuteranope and tritanope images, while the *anomalous images are simply a mix of those with the original images. The scientific value of these images is perhaps questionable, but their effect is certainly striking. At any rate, a protanope person should see the protanope image as identical to the original, and similarly for other kinds of complete color-blindness.

Original image
As seen by a protanope
As seen by a deuteranope
As seen by a tritanope
As seen by a protanomalous
As seen by a deuteranomalous
As seen by a tritanomalous

Certainly protanopes, deuteranopes and tritanopes can only see two (complementary) hues (a term that we will explain later). The alert reader can try to explain why the protanope and deuteranope images, above, use the two same hue pair to represent the blindness, while the tritanope image uses another pair. (Hint: this is because luminosity depends only on the long and medium cones' response.)

What is white?

So far we have not defined white, and have only passingly alluded to it. Interestingly enough, there is no scientific definition of white, it is only a matter of convention. Certainly a filter can be white, when it acts equally on all wavelengths. But what about white light? It is not a mixture of all wavelengths with equal intensity (per unit wavelength, or per unit frequency): such a spectrum would be grossly artificial. Well, actually, what our eyes perceive as white is essentially the light environment to which we are accustomed; and in fact we accustom ourselves very rapidly to a given white point (this is why our incandescent lights do not appear very red, whereas “actually” they are, in comparison with the light of the Sun). The light of the Sun (perhaps not the direct sunlight, but some average over many days and various atmospheric conditions) defines a usable white point; there are various standardized white points bearing such names as (illuminant) D65 (the standard for sRGB, and therefore in the computer industry) or (illuminant) D50 (a much yellower white, the standard in the print industry).

There is one possible definition of white that relies on a scientific phenomenon, and while it is typically not used directly, it serves as an important basis for more ad hoc definitions. A blackbody spectrum at a given temperature T is the spectrum emitted by a perfect Planckian radiator at temperature T (or, to a good approximation, any object heated at that temperature). The light coming to us from the sun follows approximately a blackbody law of 6000K; actually, 6500K, for some reason, appears “whiter”. The white point defined by a blackbody at a given temperature is often used as a comparison point for other white points. Illuminant D65 is very close to a blackbody at 6500K, and illuminant D50 to a blackbody at 5000K, for example. Many computer monitors will offer the choice between white points having temperatures of 5000K, 6500K or 9300K. The picture of the spectrum shown above represents the colors of the blackbody spectra ranging from 500K at the far left up to 64000K at the far right, on a uniform logarithmic scale.

We assume in all further (and past) discussion that a fixed white point has been chosen. All images shown on this page adhere to the sRGB standard and use D65 as white point.

What is hue?

The word color is ambiguous: we have defined it as the response of the three kinds of cone cells to an elementary stimulus (of light): this is a three dimensional space (because there are three kinds of cones). But we can also choose to ignore differences in intensity: consider two colors to be identical when they are produced by lights which have the same spectrum up to a constant factor in intensity. Leaving the color black aside (which is just an absence of light), we are left with a two-dimensional color space, the third dimension being just brightness. To emphasize this we also speak of the chromaticity for the point in the two-dimensional space. It is not entirely clear that this operation is justified: a color need not appear identical as another one which differs only in intensity (for example, “gray” is not considered the same as “white” by most people, although it is just a less intense white), but we have to order things somehow and this is the most reasonable beginning. (For example, the blackbody spectrum presented earlier in this page is only a representation of blackbody colors in the sense that luminosity is not preserved: indeed, for a given blackbody surface, a 64000K blackbody is so much more luminous than a 500K blackbody—one quarter a billion times more luminous, to be precise—that if brightness had been preserved in the spectrum, the left end would have had to be entirely black for the right end to be representable on a monitor. Instead, we have made every color as bright as it possibly could, and left out any further luminosity consideration.)

The two-dimensional color (chromaticity) space that we are now left with has some remarkable points on it: one is the white point (white was defined only up to intensity, but once we quotient out by intensity, white is a single point, hence the name white-point). Imagine this point as being somehow in the center. Then we have an arc of points which correspond to monochromatic radiations. Imagine those as vaguely forming a kind of horseshoe around the white point. Any color which makes physical sense is inside this arc (mathematically speaking, is in the convex hull of this arc). Indeed, a color makes physical sense when it can be realized by mixing monochromatic radiations, and on our two-dimensional diagram colors are mixed by drawing the segment between two points (do not think, however, that mixing colors in “equal proportions” will give you the segment's midpoint; people familiar with projective geometry will immediately understand the problem: basically “equal proportions” just doesn't mean anything). Somewhere out of the physical region, imagine three points corresponding to the imaginary colors of a pure cone stimulation for each of the three cones; the triangle they form corresponds to colors which could be conceived if we were to somehow stimulate the cones directly on the retina, but do not necessarily make physical sense (unless they are within the arc of monochromatic radiation). Now we have a good picture of this two-dimensional color space. Let us proceed to remove one further dimension.

Take a color other than white and draw the half-line starting from the white point that goes through this color. This line represents a succession of colors, one end of which is white and the others are getting progressively more “colorful” as they move away from the white point. We say that the light is a hue, and the property of being away from white (which we cannot measure directly, so far) is the saturation of a color. A hue, therefore, is what stays constant in a color when you add or remove white (and when you change the luminosity: that was the first dimension we removed). What is the most saturated (physically meaningful) color in a hue? In general, it is a color on the arc of monochromatics: then the hue is said to correspond to this or that wavelength, that of the monochromatic light. Sometimes, however, the half-line of the hue does not cross the arc of monochromatics but rather escapes between the two ends of the horseshoe (extreme red and extreme violet; their choice is more or less conventional, but the differences are small and unimportant): this means that no color of that hue is a monochromatic, the most saturated color in it is a mixture of extreme red and extreme violet. Then we say that the hue, or any color in that hue, is a purple. (The word “purple”, here, is a technical term: it designates not just one color or hue, but a whole class of them, namely those that correspond to no monochromatic hue.) On our two-dimensional diagram, purples form a triangle, one of whose vertices is the white point, and the other two are the extreme red and extreme violet points: all points within this triangle, which “closes” the horseshoe, are purples, and all other points within the horseshoe are physically realizable non-purples.

Each hue has a well-defined complementary hue: it is the hue of colors that is on the other side of the white point, symmetric with respect to it. Mixing a color and a color of the complementary hue can give white; and only when the hues are complementary can the mixture of two colors give white. Note that of the monochromatic hues, some have complementary hues which are purples, and some have complementary hues which are other monochromatics: essentially, hues of monochromatics between 494nm and 566nm have purple complementary hues, while hues corresponding to wavelengths less than 494nm (the complementary hue of extreme red) or greater than 566nm (the complementary hue of extreme purple) have a complementary hue which also has a wavelength. For example, the complementary hue to that of the monochromatic with wavelength 570nm (in the narrow yellow region of the spectrum) os wavelength 463nm (royal blue).

The story so far: Every (physical…) color is of two types: either it is not a purple, or it is a purple. If it is not a purple, then it can be achieved in exactly one way by mixing a certain intensity of a pure monochromatic radiation, and a certain intensity of pure white light. The wavelength of the pure monochromatic radiation determines the (non-purple) hue of the color; the total luminosity is the luminosity of the light under consideration; and saturation is something like the ratio of the monochromatic's luminosity to the total luminosity (although that is perhaps not the best possible definition). If the color is a purple, it cannot be achieved by adding a monochromatic to white, but it can be achieved by substracting a monochromatic (viz. its complementary hue) from white.

Here is a visual example. From our usual two images, we have shown the following: first, the color (the chromaticity, that is), represented as the most luminous color available (in the sRGB device's gamut) having that given color; second, the most saturated color available having the original luminosity and the original hue; third, the hue, represented as the most luminous most saturated color available of that hue. (It is clear that the pictures show many artefacts in these respects, revealing for example some classical JPEG damages. We are not concerned with this.) It is in the same way, with maximal possible saturation and luminosity, that we have represented the spectrum at the beginning of this page: the monochromatic colors are not available, but some desaturated version of the same is available in the device's gamut, and we choose the most saturated color possible. Every hue that was not represented in this spectrum is, of course, a purple (but they can be seen to be rare in real images, or certainly in our two examples).

Base image
Color (luminosity max-ed)
Saturated image (luminosity preserved)
Hue (saturated, luminosity max-ed)
Luminosity

Interlude: the CIE X,Y,Z color space

It would be possible to parametrize a color by the value of the three cone cells' responses (normalized with respect to some reference white, say). Such is not, however, the usual path: the precise cone response functions to stimuli have been known only quite recently. Instead, some older, conventional functions, are of use: the CIE (Commission Internationale de l'Éclairage)'s color matching functions, boringly called X, Y and Z. Function Y is supposed to be precisely the luminosity function. Function Z is (proportional to) the response of the short cones (this one is easy to determine, because it vanishes for sufficiently large wavelengths, certainly from 620nm on). And function X is a more or less arbitrary convention.

Typically one gets rid of the annoying dependence on luminosity, and moves to the two-dimensional color space, as follows: let x=X/(X+Y+Z), y=Y/(X+Y+Z) and z=Z/(X+Y+Z) (so that x+y+z=1 and z is redundant). Then a color is specified by its chromaticity value, (x,y), and its luminosity Y (if necessary). Often luminosity is unspecified or otherwise irrelevant, and we live fully in the two-dimensional (x,y) diagram. This is the very coordinate system we have used earlier to represent the two-dimensional chromaticity diagram (with x ranging from 0 to 1 on abscissa and y ranging from 0 to 1 on ordinate; of course, everything “lives” in the lower-left triangle, because z=1-x-y has to be nonnegative also).

Here are a few sample data points:

It needs to be added that the CIE X,Y,Z system, although it is universally used as the fundamental basis of all modern colorimetry, is actually very old (having been adopted by CIE in 1931), and stands on dubious scientific ground: the experimental protocol to establish the color matching functions was questionable because of the insufficient number of test candidates. In certain regions, the functions are known to be severely wrong (most importantly, the Y luminosity function is very wrong in the short wavelength region). Better data are now available thanks to the Color & Vision Research Labs, but because of the amount of work that is based on the 1931 standard CIE color matching function, it is not certain whether these newer data will actually be used to the full extent of their usefulness.

Monitor gamut

A monitor emits light by combining that emitted by three phosphors, each of which can be stimulated to various intensities, between zero (not at all) and some maximal value. So the monitor's gamut of displayable colors forms a parallelepiped in some linear space (say, CIE X,Y,Z space) where black (the origin) is one vertex, the three phosphors its adjacent vertices, and the opposite vertex of black is white, which is hopefully calibrated to some standard and useful value: in other words, the phosphors' maximal intensities are chosen so that adding them up gives the desired white. A little linear algebra, or projective geometry, will suffice to convince that it is sufficient, in order to determine the entire gamut up to homothecy, to give the (x,y) chromaticity coordinates of the three phosphors and of the white point (thus, eight real numbers, which determine the nine real numbers of the X,Y,Z coordinates of all three phosphors, except for one global multiplication factor).

Certainly a monitor cannot display a color outside of the triangle in the chromaticity plane formed by its three phosphors. This is the first bad news: no monitor is capable of displaying every color of the chromaticity diagram. The image here shows the gamut triangle of the sRGB standard color space within the CIE (x,y) chromaticity plane, with the curve corresponding to monochromatics. Obviously there seems to be a large area missing; that's not very meaningful, in fact, in projective geometry, but it is still a fact that a color monitor, or certainly an sRGB monitor, cannot accurately render the color of certain tropical sea greens. Certainly the monitor can represent every hue: but, within a given hue, there is a limit in saturation that can be reached, and, for each given saturation, a limit in luminosity; or, if we prefer to fix luminosity first, then for each luminosity there is a chromaticity gamut, which is the full chromaticity gamut only for luminosities less than that of any of the phosphors; and when the required luminosity is made to increase, the gamut shrinks, until it dissapears into a point, the white point, which is the only achievable color of maximal brightness. Compare the diagram above, which is the sRGB gamut for a brightness equal to 10% of the maximal brightness (only the blue tip of the triangle is ever so slightly clipped) with the following one that shows the considerably more restricted gamut for a luminosity of 80% of the maximal:

We can also fix a hue and draw a gamut diagram in the saturation and luminosity space for that hue. Compare the following two diagrams, for example:

These represent the mixing process of pure white light and pure monochromatic light of a given wavelength: 520nm on the left and 550nm on the right. The left edge of each diagram corresponds to pure white (saturation zero) and the right edge corresponds to pure monochromatic (saturation one) of the same intensity. So saturation is in abscissa, and luminosity is in ordinate (being zero at the bottom edge, and the maximal luminosity at the top). The diagrams are comparable: black regions are colors that are out of the sRGB gamut and cannot be represented. This shows that while the sRGB space presents an acceptable gamut for a hue of 550nm, it is much more severely clipped for 520nm. If you use the inspiration provided by the diagram on the right to mentally stretch the one on the left as far as its companion, you get a clear mental image of those colors which cannot be represented by the sRGB color space.

TODO

Say something about gamma correction. But in the mean time, here is a nice picture to let you test whether your system has the right value:

Let your computer display this image, and stand at about ten meters' distance from the screen: you should see the image as made up of eight vertical bars, not more (not vertically broken in their middle). Adjust your color temperature to 6500K and twiddle with the brightness knob until you pass this test, and then you'll have a decent approximation of sRGB color space.

David Madore (davidmadoreensfr)

Last modified: $Id: color.daml,v 1.8 2005/08/01 04:29:48 david Exp $