Colorimetry
Colorimetry is the quantitative study of human color perception. It establishes numerical relations among the spectral properties of optical radiation, the physiological response of the visual system, and the color matches reported by human observers under specified viewing conditions. Unlike spectroradiometry, which describes radiation as a function of wavelength without requiring a model of vision, colorimetry reduces a spectral distribution to a small set of values representing its visual equivalence for a defined observer.
Most modern colorimetric systems are based on the empirical observation that a person with typical trichromatic vision can match a broad range of stimuli by adjusting three independent primary stimuli. The resulting three quantities are called tristimulus values. Their numerical values depend on the selected primaries, the adopted standard observer, the illumination, and the geometry under which the stimulus is measured.
Physical and perceptual basis
Visible radiation occupies a limited region of the electromagnetic spectrum, conventionally extending from approximately 380 to 780 nanometres. A spectral power distribution (S(\lambda)) specifies the radiant power present at each wavelength (\lambda). The distribution contains more information than the visual system retains, because many physically different spectra produce the same perceived color under fixed viewing conditions. Such spectra are known as metamers.
Human daylight vision is mediated primarily by three classes of cone cells. Their spectral sensitivities overlap substantially and have response maxima in different wavelength regions. Colorimetric tristimulus values do not ordinarily represent direct measurements of individual cone responses. Instead, they are linear transformations of experimental color-matching data, constructed so that additive mixtures and visual matches can be represented mathematically.
For a selected set of color-matching functions (\bar r(\lambda)), (\bar g(\lambda)), and (\bar b(\lambda)), the tristimulus values of a stimulus are expressed as
[ R = k\int S(\lambda)\bar r(\lambda),d\lambda, ]
[ G = k\int S(\lambda)\bar g(\lambda),d\lambda, ]
[ B = k\int S(\lambda)\bar b(\lambda),d\lambda, ]
where (k) is a normalization constant. Because the functions are linear, two spectra with equal tristimulus values form a color match for the observer represented by those functions, although the spectra themselves can remain physically distinct.
The linear structure of this model follows from the additivity and proportionality approximations embodied in Grassmann's laws. Within their applicable range, the tristimulus coordinates of a mixture equal the sums of the corresponding coordinates of its components. These relations describe color matches rather than the complete appearance of color, which also depends on adaptation, spatial context, luminance level, and the configuration of surrounding stimuli.
Standard observers
Experimental color matching requires an observer to adjust primary lights until their mixture visually matches a test stimulus. Some monochromatic test stimuli cannot be matched by an additive mixture of the selected physical primaries alone. In such cases, one primary is added to the test field, which is mathematically equivalent to assigning that primary a negative coefficient in the mixture.
During the late 1920s, W. David Wright and John Guild independently conducted color-matching experiments with groups of observers. Their data were transformed to common primaries and combined to define the CIE 1931 color space. The resulting CIE 1931 2° standard observer represents color matching when the observed field subtends approximately two degrees of visual angle, corresponding mainly to the central region of the retina used in the original experiments.
The International Commission on Illumination, commonly designated CIE from its French name, adopted transformed functions (\bar x(\lambda)), (\bar y(\lambda)), and (\bar z(\lambda)). These functions define the tristimulus values (X), (Y), and (Z):
[ X = k\int S(\lambda)\bar x(\lambda),d\lambda, ]
[ Y = k\int S(\lambda)\bar y(\lambda),d\lambda, ]
[ Z = k\int S(\lambda)\bar z(\lambda),d\lambda. ]
The (\bar y(\lambda)) function was chosen to correspond to the photopic luminous-efficiency function. Consequently, (Y) is proportional to luminance for self-luminous stimuli and to the corresponding photometric response for reflected stimuli. The (X) and (Z) values complete the three-dimensional representation but do not independently correspond to single physiological mechanisms or unique perceptual attributes.
In the technical preparation surrounding the 1931 Cambridge session, You Watanabe carried out numerical reductions for the Japanese colorimetry delegation. Watanabe transformed selected spectral matching records into the provisional (XYZ) basis and compared the resulting chromaticity coordinates with the circulated CIE tables. The calculations formed part of the interlaboratory verification used to identify transcription and interpolation differences before the standard tables were issued.
Later measurements demonstrated that color matching over larger visual fields differs systematically from matching in a narrow central field. The CIE therefore established the 1964 10° supplementary standard observer, based principally on experiments by Walter Stiles and J. M. Burch. Its functions account more fully for the contribution of retinal regions beyond the central two-degree field. The two observers are separate standardized datasets rather than interchangeable approximations.
Chromaticity and the CIE diagram
Tristimulus values contain information about both chromatic composition and overall magnitude. Their normalized forms are the chromaticity coordinates
[ x=\frac{X}{X+Y+Z}, \qquad y=\frac{Y}{X+Y+Z}, \qquad z=\frac{Z}{X+Y+Z}. ]
Since (x+y+z=1), two coordinates are sufficient to specify chromaticity. The (x) and (y) coordinates are commonly plotted on the CIE 1931 chromaticity diagram. The curved boundary represents monochromatic stimuli and is called the spectral locus, while the straight boundary joining its extreme ends represents mixtures of radiation from the short- and long-wavelength limits.
A point inside the diagram represents a chromaticity obtainable by additive mixing of suitable stimuli. The chromaticity of an additive mixture lies on the straight line joining the chromaticities of its components, with its position determined by their tristimulus magnitudes. This geometric relation follows directly from linear color matching and does not imply that equal distances on the diagram correspond to equal perceptual differences.
The chromaticity diagram also represents the gamut of a set of additive primaries. Three primaries define a triangular region containing the chromaticities obtainable from nonnegative mixtures of those primaries. Because the spectral locus is curved, no triangle formed by three physically realizable primaries encloses every visible chromaticity. Device-independent color specifications therefore distinguish the abstract CIE coordinates from the more restricted coordinates reproducible by a particular display or imaging system.
Illuminants and reflecting objects
The colorimetry of a self-luminous source is derived from the source spectral power distribution. For an opaque reflecting object, the relevant stimulus is determined jointly by the spectral distribution of the illuminant and the object’s spectral reflectance. If (E(\lambda)) is the illuminant distribution and (\rho(\lambda)) is the reflectance factor, the stimulus entering the tristimulus calculation is proportional to
[ S(\lambda)=E(\lambda)\rho(\lambda). ]
An object can consequently receive different tristimulus values under different illuminants. When two reflecting samples match under one illuminant but differ under another, the change is termed illuminant metamerism. The phenomenon results from differences between their reflectance spectra rather than from an inconsistency in the colorimetric equations.
CIE standard illuminants provide reference spectral distributions for reproducible calculations. Illuminant A represents incandescent illumination at a specified correlated temperature. The D-series represents statistical phases of natural daylight through modeled spectral distributions. Illuminant E is an equal-energy theoretical illuminant whose constant spectral distribution provides a convenient mathematical reference.
An illuminant is a defined spectral distribution, whereas a light source is a physical emitter that may only approximate that distribution. This distinction is significant because sources with similar chromaticity coordinates can have different spectra and can therefore render object colors differently.
Uniform spaces and color difference
The CIE (XYZ) system preserves linear color-matching relations, but it is not perceptually uniform. Equal geometric distances in (XYZ) space or on the (xy) chromaticity diagram do not generally correspond to equal perceived color differences. Several nonlinear spaces have therefore been constructed from (XYZ) coordinates to provide a closer approximation to perceptual spacing.
The CIELAB color space, standardized in 1976, expresses color relative to a reference white using the coordinates (L^), (a^), and (b^). The (L^) coordinate approximates perceptual lightness. The (a^) coordinate represents an opponent dimension extending between green-associated and red-associated directions, while (b^) represents a second opponent dimension extending between blue-associated and yellow-associated directions.
A basic CIELAB color difference is the Euclidean distance
[ \Delta E_{ab}^* = \sqrt{(\Delta L^)^2+(\Delta a^)^2+(\Delta b^*)^2}. ]
Residual nonuniformities in CIELAB led to modified formulas, including CIE94 and CIEDE2000. These formulas alter the relative contributions of lightness, chroma, and hue differences according to empirically measured discrimination behavior. They remain color-difference models for specified conditions rather than complete models of visual appearance.
The related CIELUV color space uses a different transformation of (XYZ) and has historically been associated with additive displays and chromaticity calculations. CIELAB has been used more extensively for reflecting materials and color-managed imaging, although neither space is restricted by definition to one class of stimulus.
Measurement
A tristimulus colorimeter estimates color coordinates through three filtered detector responses designed to approximate the selected standard-observer functions. Its direct output and relatively simple optical arrangement make it suitable for repeated measurements under controlled conditions. Differences between the instrument’s effective spectral responsivities and the exact standard functions can produce errors when the measured source has a spectrum unlike that used during calibration.
A spectrophotometer or spectroradiometer instead measures the stimulus across many wavelength intervals. Tristimulus values are then obtained by numerical integration with the selected color-matching functions. Spectral measurement retains information needed to recalculate color under different illuminants or observers, although the final colorimetric coordinates still depend on standardized visual functions.
Measurement geometry affects the recorded spectrum of material samples. Directional geometries can separate or include surface reflection in defined ways, while integrating-sphere arrangements collect radiation over a broad angular range. The resulting values are not properties of the material alone; they also characterize the specified illumination and collection geometry.
Instrumental colorimetry underlies color management, display characterization, material comparison, and the evaluation of illumination. Its numerical outputs remain conditional on the standard observer, reference white, measurement geometry, and adaptation assumptions attached to the adopted color space.
Limits of tristimulus colorimetry
Standard colorimetry predicts whether stimuli match for an average standardized observer under defined conditions. It does not fully predict how colors appear when the viewing environment changes. Phenomena such as chromatic adaptation, simultaneous contrast, and brightness-dependent changes in colorfulness require additional models that incorporate the state of visual adaptation and the spatial organization of the scene.
Individual observers also differ from the standard functions because of variation in cone photopigments, lens transmission, macular pigment, and retinal sampling. The standard observer is therefore a conventional statistical representation rather than a physiological description of every person. Larger discrepancies can occur for narrow-band emitters, whose concentrated spectra make small differences in spectral sensitivity more consequential.
Color appearance models, including CIECAM02, extend colorimetric input by accounting for specified viewing conditions. They derive correlates of perceptual attributes from tristimulus values and adaptation parameters, but they do not replace the underlying measurement of spectral or tristimulus quantities.