Colorimetry/Chapter 1

by I. Nimeroff

1. General Considerations

1.1. Introduction

It is common practice to regard color as a property of objects, and in a limited sense this view is justified. We have color comparators for solutions in which the color is taken as an index of the composition of the solution; and in applying a suitably prepared set of color standards in a color comparator, the color of the unknown behaves as if it were a property of the solution itself, just as the concentrations of the constituents which it indicates. However, this document deals with color for its own sake; and, for this purpose, a broader view is useful. Lights have colors as well as objects. The flame of a Bunsen burner can be changed from bluish purple to orange by the introduction of sodium. And since even objects lose their colors and become invisible unless they reflect, scatter, or transmit radiant energy, or form a part of an illuminated scene, this broader view is that color is a property of light, and of light alone.

As it is possible to measure with a spectrophotometer the spectral energy distribution of any light beam, and as the color oi a light correlates closely with its spectral composition, some of the more physically minded people have contended that color is a physical property of radiant energy; but this is not the most useful view. The color change of the Bunsen flame from bluish purple to orange can be shown by a purely physical measurement to be caused by a change in the spectral composition of the emitted energy, but it takes more than physics to decide whether this flame has the same color as the light reflected from the peel of a given citrus fruit. Application of the spectrophotometer to the orange peel will show that the spectral composition of the light reflected from it under daylight illumination is radically different from that emitted by the sodium flame. It has a continuous spectrum relatively strong in the long-wave (550 to 770 nm)^[1] portion of the visible spectrum (380 to 770 nm) . The visible energy of the sodium flame is nearly all confined to two narrow bands (589.0 and 589.6 nm) . Physically, therefore, the two lights are different, but they have closely the same color. The two lights must therefore be identical in some other respect. This identity consists in some aspect of the response made by a normal observer to the sodium flame being the same as the corresponding aspect of the response to the peel of the citrus fruit. The broader view of color must, therefore, include not only the spectral composition of the radiant energy reaching the eye of the observer, but the properties of the observer as well. These properties have been evaluated by finding equivalent stimuli, like the energy of the sodium flame and that reflected from an orange peel, which have different spectral compositions but still manage to stimulate the same color response to the normal observer. Such equivalent stimuli are called metamers. In this chapter there will be presented the standard method for finding by computation whether or not any two lights form equivalent stimuli. The fundamental method of color specification based upon equivalent stimuli plus spectrophotometry will be described in detail together with other methods of obtaining the same numbers. And finally some discussion will be given on the use of the Munsell color system and color dictionaries such as the ISCC-NBS (Inter-Society Color Council-National Bureau of Standards) Method of Designating Color and a Dictionary of Color Names, National Bureau of Standards Circular 553, 1956 [79].^[2]

1.2. Definition of Color

The most widely accepted technical definition of color is that given by the Committee on Colorimetry of the Optical Society of America [15]: "Color consists of the characteristics of light other than spatial and temporal inhomogeneities; light being that aspect of radiant energy of which a human observer is aware through the visual sensations which arise from the stimulation of the retina of the eye." It will be noted that this definition relates color and light to radiant energy only in so far as the energy produces a visual effect within an observer. On this account color and light are said to be psychological entities that can be evaluated by means of psychophysical quantities, and in their evaluation it is ordinarily not necessary to pay attention to energy of wavelength shorter than 380 nm, nor longer than 770 nm because the eye is relatievly insensitive to such energy.

1.3. Tridimensionality of Color

If an observer with normal color vision attempts to adjust one element of his visual field whose color is under his control so that it matches a neighboring element, he will ultimately discover that three independent adjustments have to be made. If he is using the red, yellow, and blue paints frequently found in primary grade schools, only by chance will he obtain a match from a mixture of two of them. Even a brown color requires blue in addition to red and yellow. Within the color gamut of the three paints, an exact match for a given paint color is easily possible, but three is the irreducible minimum. Similarly, if he is trying to color-match one spot of light by shining several spotlights of different colors onto the same neighboring spot of a screen, he finds, in general, that either three lights of fixed spectral composition are required, or, if only two lights be added together, not only the amounts of both but also the spectral composition of at least one has to be adjustable. The same rule applies to rotary mixture on a sector disk; four sectors, giving three independent adjustments, are necessary and sufficient.

As the color vision of a normal observer is at least tridimensional, it follows that there must be at least three independent excitations in the optic nerve fibers corresponding to each patch of the visual field. Theories of color vision have been, derived mostly from speculation as to the character of these excitations. It also follows that a color specification is expressible by three numbers. For normal observers three numbers are necessary; for partially color-blind observers only two numbers are necessary; and for totally color-blind observers only one is necessary.

In the examples given (paints, spotlights, sector disks), the observer by adjustment of three variables obtains a color match, that is, he has to set up a second stimulus equivalent to the first. Except by accident, however, the ternary or binary mixture does not match the unknown in spectral composition. In the usual case the mixture is equivalent to the unkonwn in color but not in spectral composition, and the unknown and the mixture are said to form a metameric pair. There are, however, degrees of difference in spectral composition. If one painted panel be matched by a mixture of red, yellow, and blue paints, the degree of metamerism is likely to be only moderate; but if the paint panel illuminated by daylight be matched by shining on a white card three spotlights each of which contains energy restricted to a narrow wavelength band (such as spectrum red, green, and blue), the degree of metamerism will ordinarily be large.

1.4. Metamerism and the Macular Pigment

Studies of extremely metameric pairs in which mixtures of two parts of the spectrum are set up to color-match other two-part spectrum mixtures have yielded our most valuable knowledge regarding the properties of the average normal eye [1, 44, 84, 99, 158]. An outstanding fact derived from observation of such metamers is that the center of the retina (fovea centralis) has somewhat different properties from that part of the retina immediately surrounding it; that is, a color match set up for the central 2 or 3 deg of the retina becomes an easily detectable mismatch if the eye be turned so as to allow the stimuli forming the metameric pair to affect a portion of the retina, say, 6 deg from the fovea. Furthermore, if the metamers are compared in large patches so as to subtend 6 deg or more at the eye of the observer, this mismatch causes a central spot, known as the Maxwell spot [98], to appear temporarily on either field even though the field is physically uniform. After the spot has faded away, change of fixation to the other field will renew the spot. This dependence of metamerism on the portion of the retina used arises chiefly from the existence of a spot of brownish or yellowish pigment irregularly covering and interpenetrating the central 3 or 4 deg of the normal, retina; it is called the macula lutea or sometimes the yellow spot (see fig. 1). Figure 1 shows a horizontal cross section of the eye. Light enters the tear-film ff, passes through the cornea aa, the aqueous humor B, the pupil bb, the crystalline lens A, the vitreous humor C, and the macula p before reaching the retina i. The macular pigment acts as Fig. 1.Horizontal cross-section of the normal human eye. a selective filter interposed between the vitreous humor C and the retina i. Metamers set up for one normal observer usually fail to hold strictly for anyone else. This failure is ascribable to variations in amount of pigmentation of the eye media (cornea, lens, humors, macula), the macular pigment being one of the chief variables. The properties of the normal eye derived from a small-field study of these extreme metamers therefore refer only to the central 2 deg of the retina, and they refer to an hypothetical average eye. Nobody has been found whose eye differs so little from this average eye that the differences could not be detected. Practically speaking, therefore, nobody has an eye that is colorimetrically normal.

1.5. Grassman's Law

From a knowledge of spectral metamers, it has been possible to summarize concisely the properties of the average normal eye. This summary is made in accord with the principle known as Grassman's law [42] foreshadowed by Newton's laws of color mixture. If a light composed of known amounts of three components (called primaries) is equivalent in color to an unknown light, the three known amounts may be used as a color specification for this light. These amounts are called the tristimulus values of the color. Grassman's law states that, when equivalent lights are added to equivalent lights, the sums are equivalent. Thus, if an unknown spot of color were matched by shining on the same spot of a white screen two component spotlights of tristimulus values, $X_{1}$, $Y_{1}$, $Z_{1}$, and $X_{2}$, $Y_{2}$, $Z_{2}$, respectively, by Grassman's law, the tristimulus values, $X$, $Y$, $Z$, of the unknown spot of color would be simply:

${\begin{alignedat}{5}X&=&X_{1}&+&X_{2}\\Y&=&Y_{1}&+&Y_{2}\\Z&=&Z_{1}&+&Z_{2}\\\end{alignedat}}$

(1)

Any beam of light, whether it originates from a self-luminous body or comes, by transmission, scattering, or reflection, from a nonself-luminous object, may be considered as made up of a large number of portions of the spectrum. The amounts of these various portions may be determined by spectrophotometry. The spectral values, ${\overline {x}}(\lambda )$, ${\overline {y}}(\lambda )$,${\overline {z}}(\lambda )$, of each of these portions have been determined for a number of normal observers, and average values are given in table 1 in arbitrary units for a spectrum of unit spectral irradiance.

The principle expressed in Grassman's law has been established by repeated experiment over a wide middle range of retinal illuminances. It breaks down for very high retinal illuminances [159] that begin to approach those sufficient to do the eye permanent harm, and it breaks down if the illumination of the whole retina continues for several minutes to be so slight that vision by the retinal rods (twilight vision) intrudes significantly [84]. Between these two extremes, however, Grassman's law holds independently of the adaptive state of the eye. Thus, if two stimuli of different wavelength distributions of energy be found that are once responded to alike by the eye, they will be seen alike even after exposure of the eye to another stimulus sufficient to change considerably the appearance of the two equivalent stimuli. For example, if a portion of the spectrum near 640 nm (red) be superposed on a portion near 550 nm (yellowish green), it will be found possible to obtain the color of this combination from an intermediate portion of the spectrum, say, 590 nm (orange). If the retina of the eye be highly illuminated by light of wavelength near 640 nm, and its sensitivity to radiant flux of this wavelength region considerably reduced in this way, it is found that, although neither of the equivalent stimuli any longer appears orange, they still give identical colors; for example, they may yield identical yellows or identical greenish yellows. The eye thus cannot be trusted to yield the same color perception from a given stimulus; simultaneous and successive contrast affect it profoundly. But the eye is still a satisfactory null instrument and obeys Grassman's law.

By Grassman's law it is possible to test whether any two beams of light of differing spectral composition form a metameric pair. The condition for metamerism of two beams of light of spectral irradiance. $E_{1}$ and $E_{2}$, is that simultaneously:

${\begin{aligned}\sum _{0}^{\infty }(E_{1})_{\lambda }{\overline {x}}(\lambda )\Delta \lambda &=\sum _{0}^{\infty }(E_{2})_{\lambda }{\overline {x}}(\lambda )\Delta \lambda \\\sum _{0}^{\infty }(E_{1})_{\lambda }{\overline {y}}(\lambda )\Delta \lambda &=\sum _{0}^{\infty }(E_{2})_{\lambda }{\overline {y}}(\lambda )\Delta \lambda \\\sum _{0}^{\infty }(E_{1})_{\lambda }{\overline {z}}(\lambda )\Delta \lambda &=\sum _{0}^{\infty }(E_{2})_{\lambda }{\overline {z}}(\lambda )\Delta \lambda \\\end{aligned}}$

(2)

where ${\overline {x}}(\lambda )$, ${\overline {y}}(\lambda )$ and ${\overline {z}}(\lambda )$ characterize the observers' spectral responses. (These equations are written in accord with the CIE notation adopted in 1963. Symbols with subscripted $\lambda )$, as $E_{(}\lambda )$, indicate spectral concentration, while symbols with parenthetical $\lambda$, as ${\overline {x}}(\lambda )$, indicate other spectral relationships not critically dependent on choice of wavelength interval.) The wavelength interval, $\Delta \lambda$, to be used in these summations depends upon the irregularity of the curve of spectral irradiance with wavelength; intervals of 20 nm are sometimes sufficient to yield a significant result; intervals of 10 nm often do; and intervals of 5 nm are usually sufficient except for discontinuous spectral distributions such as those characterising gaseous discharge lamps. De Kerf [27] has reported instances where 1-nm intervals have been required.

The summations of eq (2) form the tristimulus values of the color and are customarily given the symbols $X$, $Z$, $Z$, respectively, so that the condition for a color match would ordinarily be written:

${\begin{aligned}X_{1}&=X_{2}\\Y_{1}&=Y_{2}\\Z_{1}&=Z_{2}\end{aligned}}$

(2a)

and would mean that the two colors are identical since their tristimulus values are identical. Thus, to match color one requires $X_{1}$ parts of the $X$ primary, $Y_{1}$ parts of the $Y$ primary, and $Z_{1}$ parts of the $Z$ primary; and because to match color two requires the same amounts of the same primaries as does color one, the two colors are the same.

Any three lights may be used as primaries in a system of tristimulus color specifications, provided only that no one of them is equivalent to a combination of the other two. Tristimulus specifications $X,Y,Z,$ expressed relative to one set of primaries, may be transformed into specifications $R,G,B,$ relative to any other set, by transformation equations of the form:

${\begin{aligned}R&=K_{1}X+K_{2}Y+K_{3}Z\\G&=K_{4}X+K_{5}Y+K_{6}Z\\R&=K_{7}X+K_{8}Y+K_{9}Z\\\end{aligned}}$

(3)

The constants $K_{1}$ to $K_{3}$ may take on any arbitrary values, positive, negative, or zero, provided they are not such as to make one of the new primaries identical to a combination of the other two; that is, provided that:

${\begin{matrix}K_{1}&K_{2}&K_{3}\\K_{4}&K_{5}&K_{6}\\K_{7}&K_{8}&K_{9}\\\end{matrix}}\neq 0$

(3a)

As the exceptions that cause the determinant of the system to vanish are trivial, the choice of coordinate system is very wide. The primaries do not even have to correspond to physically realizable lights. Imaginary lights defined by spectral compositions having negative values for some parts of the spectrum are admissible, and indeed are preferred for routine colorimetry because by their use the computation of tristimulus values from spectrophotometric data is somewhat simplified.

1. The unit nanometers (nm), 10⁻⁹ meters, was formerly called millimicrons (mμ).
2. Figures in brackets indicate the literature references at the end of this paper.