# CIE Physiologically Based Color Matching Functions and Chromaticity Diagrams

**DOI:**https://doi.org/10.1007/978-1-4419-8071-7_326

## Synonyms

## Definition

Because each of the long-, middle-, and short-wavelength-sensitive (L, M, and S) cone types responds univariantly to light, human color vision and human color matches are trichromatic. Trichromatic color matches depend on the spectral sensitivities of the three cones, which are also known as the fundamental color matching functions (or CMFs): \( \overline{l}\left(\lambda \right) \), \( \overline{m}\left(\lambda \right) \), and \( \overline{s}\left(\lambda \right) \). The spectral sensitivity of each cone reflects how its sensitivity changes with wavelength. Measured at the cornea, the L-, M-, and S-cone quantal spectral sensitivities peak at approximately 566, 541, and 441 nm, respectively. These fundamental CMFs are the physiological bases of other measured CMFs, all of which should be linear transformations of the fundamental CMFs.

The CIE [1] has now explicitly defined a standard set of physiologically based fundamental CMFs (or cone fundamentals) by adopting the estimates of Stockman and Sharpe [2] for 2- and 10-deg vision. These estimates were based on psychophysical measurements made in normal trichromats, red-green dichromats, blue-cone monochromats, and tritanopes all of known genotype; and from a direct analysis of the color matching data of Stiles and Burch [3].

The 10-deg cone fundamentals are defined as linear combinations of the 10-deg CMFs of Stiles and Burch [3] with some adjustments to \( \overline{s}\left(\lambda \right) \) at longer wavelengths. The 2-deg cone fundamentals are similarly defined, but have also been adjusted to be appropriate for 2-deg vision.

The CIE cone fundamentals are physiologically based in the sense that they reflect the spectral sensitivities of the cone photoreceptors, the initial physiological transducers of light. In principle, any set of CMFs can be linearly transformed back to the fundamental CMFs. The popular CIE 1931 CMFs, however, are substantially flawed especially at shorter wavelengths, so they cannot be used to accurately model the cone photoreceptors or indeed human color vision. One of the many advantages of using physiologically relevant functions is that they can be easily extended to represent the postreceptoral transformation of the cone signals to chromatic (L-M and S-[L + M]) and achromatic (L + M) signals.

In addition to \( \overline{l}\left(\lambda \right) \), \( \overline{m}\left(\lambda \right) \), and \( \overline{s}\left(\lambda \right) \), the CIE standard also defines the photopic luminous efficiency function [*V*(*λ*) or \( \overline{y}\left(\lambda \right) \)] for 2-deg and 10-deg vision as linear combinations of \( \overline{l}\left(\lambda \right) \) and \( \overline{m}\left(\lambda \right) \). This facilitates the further transformation of \( \overline{l}\left(\lambda \right) \), \( \overline{m}\left(\lambda \right) \), and \( \overline{s}\left(\lambda \right) \) to physiologically relevant versions of the more familiar CMFs: \( \overline{x}\left(\lambda \right) \), \( \overline{y}\left(\lambda \right) \), and \( \overline{z}\left(\lambda \right) \), for which \( \overline{y}\left(\lambda \right) \) is the luminous efficiency function and \( \overline{z}\left(\lambda \right) \) is a scaled version of \( \overline{s}\left(\lambda \right) \).

## Overview

**RGB**(red-green-blue) primaries of 645, 526, and 444 nm. Each CMF defines the amount of that primary required to match monochromatic test lights of equal energy. CMFs, such as these, can be determined directly. CMFs can be linearly transformed to any other set of real primary lights and to imaginary primary lights, such as the

**LMS**cone fundamental primaries (or “Grundempfindungen” – fundamental sensations) shown in the bottom right-hand panel of Fig. 1, which are the physiologically relevant cone spectral sensitivities, or to the still popular

**XYZ**CMFs shown in the top panel. The three fundamental primaries correspond to the three imaginary primary lights that would uniquely stimulate each of the three cones and yield the \( \overline{l}\left(\lambda \right) \), \( \overline{m}\left(\lambda \right) \), and \( \overline{s}\left(\lambda \right) \) fundamental CMFs. All other CMF sets depend on the fundamental CMFs and should be a linear transformation of them.

A definition of the fundamental CMFs requires two things: first, an accurate set of representative \( \overline{r}\left(\lambda \right) \), \( \overline{g}\left(\lambda \right) \), and \( \overline{b}\left(\lambda \right) \) CMFs that can be linearly transformed to give the \( \overline{l}\left(\lambda \right) \), \( \overline{m}\left(\lambda \right) \), and \( \overline{s}\left(\lambda \right) \) CMFs and, second, a knowledge of the coefficients of the transformation from one to the other. Stockman and Sharpe [2] obtained the coefficients of the transformation primarily by fitting linear combinations of \( \overline{r}\left(\lambda \right) \), \( \overline{g}\left(\lambda \right) \), and \( \overline{b}\left(\lambda \right) \) to spectral sensitivity measurements made in red-green dichromats, blue-cone monochromats, and normals and in the case of the S-cones also by analyzing the CMFs themselves (see below).

## Choice of “Physiologically Relevant” RGB CMFs

Of critical importance in the definition of the cone fundamentals is the choice of CMFs from which they are transformed. The ones that are available vary considerably in quality. The most widely used, the CIE 1931 2-deg CMFs [4], are the least secure. Based only on the *relative* color matching data of Wright [5] and Guild [6], these CMFs were reconstructed by assuming that their linear combination must equal the 1924 CIE V(λ) function [4, 7]. Not only is this assumption unnecessary, since CMFs can be measured directly, but the CIE V(λ) curve used in the reconstruction is far too insensitive at short wavelengths. Thus, the CIE 1931 CMFs are a poor choice for defining the cone fundamentals.

By contrast, the Stiles and Burch 2-deg [8] and 10-deg [3] CMFs are directly measured functions. Although referred to by Stiles as “pilot” data, the 2-deg CMFs are the most extensive set of directly measured data for 2-deg vision available, being averaged from matches made by ten observers. They are used as an intermediate step in the derivation of the cone fundamentals (see below).

The most secure and comprehensive set of directly measured color matching data are the large-field, centrally viewed 10-deg CMFs of Stiles and Burch [3]. They were measured in 49 subjects from approximately 390–730 nm (and in nine subjects from 730 to 830 nm). Consequently, the 10-deg CMFs of Stiles and Burch have been chosen as the basis for defining the “physiologically relevant” cone fundamentals. The downside of using 10-deg CMFs to model 2-deg spectral sensitivity data is that the spectral sensitivities must be corrected for the differences in preretinal filtering and in photopigment optical density between a 2-deg and 10-deg viewing field. However, such adjustments are straightforward once the spectral sensitivities are known (for details and formulae, see [9]).

Note that the Stiles and Burch [3] 10-deg CMFs are preferable to the large-field 10-deg CIE 1964 CMFs, which, although based mainly on the 10-deg CMFs of Stiles and Burch [3], were compromised by the inclusion of the Speranskaya [10] 10-deg data and by several adjustments carried out by the CIE (see [2]).

## Spectral Sensitivity Measurements

The upper panel of Fig. 3, below, shows the mean S-cone spectral sensitivity measurements (blue circles) made in three blue-cone monochromats, who lack L- and M-cones, and at wavelengths shorter than 540 nm in five normal subjects by Stockman, Sharpe, and Fach [12]. In normals, an intense yellow background field selectively adapted the M- and L-cones, so revealing the S-cone response at wavelengths up to 540 nm.

These spectral sensitivity measurements were then used to find the linear combinations of \( \overline{r}\left(\lambda \right) \), \( \overline{g}\left(\lambda \right) \), and \( \overline{b}\left(\lambda \right) \) that best fit each of the three cone spectral sensitivities, \( \overline{l}\left(\lambda \right) \), \( \overline{m}\left(\lambda \right) \), and \( \overline{s}\left(\lambda \right) \), allowing adjustments in the densities of pre-receptoral filtering and photopigment optical density in order to account for differences in the mean densities between different populations and to account for differences in the retinal area (see [9]).

*k*

_{ l }(or \( 1/{\overline{l}}_B \)),

*k*

_{ m }(or \( 1/{\overline{m}}_B \)), and

*k*

_{ s }(or \( 1/{\overline{s}}_B \)) remain unknown, but are typically chosen to scale three functions in some way, for example, so that \( {k}_l\overline{l}\left(\lambda \right) \), \( {k}_m\overline{m}\left(\lambda \right) \), and \( {k}_s\overline{s}\left(\lambda \right) \) peak at unity.

## L- and M-cone Fundamentals

The four M- and L-cone unknowns in Eq. 3, \( {\overline{l}}_R/{\overline{l}}_B \), \( {\overline{l}}_G/{\overline{l}}_B \), \( {\overline{m}}_R/{\overline{m}}_B \), and \( {\overline{m}}_G/{\overline{m}}_B \), can be estimated by fitting CMFs to the cone spectral sensitivity data shown in Fig. 2. However, since the cone spectral sensitivity data are defined for 2-deg viewing conditions and the CMFs for 10-deg, we employed an intermediate step of fitting the 2-deg data to the Stiles and Burch [8] 2-deg CMFs. Figure 2 shows the linear combinations of the Stiles and Burch 2-deg CMFs that best fit the mean L(ser^{180}) deuteranope data (red circles), L(ala^{180}) deuteranope data (yellow and orange squares), and L1M2/L2M3 protanope data (green diamonds) of Sharpe et al. [11]. An overall population mean for the L-cone spectral sensitivity function was derived by averaging the L(ser^{180}) and L(ala^{180}) fits after weighting them in ratio of 62 L(ser^{180}) to 38 L(ala^{180}), which is the ratio believed to correspond to normal population incidences (see Table 1 of Reference 2).

Having defined the mean L- and M-cone fundamentals in terms of the 2-deg Stiles and Burch CMFs, they were next defined in terms of linear combinations of the Stiles and Burch [3] 10-deg CMFs corrected to 2-deg. These were derived by a curve-fitting procedure in which the linear combinations of the Stiles and Burch 10-deg CMFs found that, after adjustment to 2-deg macular, lens and photopigment densities best fit the Stiles and Burch-based 2-deg L- and M-cone fundamentals. The coefficients are given in Eq. 4.

In one final refinement, the relative weights of the blue CMF were fine-tuned for consistency with tritanopic color matching data [13], from which the S-cones are excluded (for further details, see [2]). This final adjustment is important because of the inevitable uncertainties that arise at short wavelengths owing to individual differences in preretinal filtering.

## S-Cone Fundamental

The coefficients for the transformation to the S-cone fundamental require knowledge of just one unknown, \( {\overline{s}}_G/{\overline{s}}_B \), which can similarly be estimated by fitting CMFs to the cone spectral sensitivity data. The upper panel of Fig. 3 shows the mean central S-cone spectral sensitivities (blue circles) measured by Stockman, Sharpe, and Fach [12] averaged from normal and blue-cone monochromat data below 540 nm and from blue-cone monochromat data alone from 540 to 615 nm. Superimposed on the threshold data is the linear combination of the Stiles and Burch 2-deg \( \overline{b}\left(\lambda \right) \) and \( \overline{g}\left(\lambda \right) \) CMFs that best fits the data below 565 nm with best-fitting adjustments to the lens and macular pigment densities.

The unknown value, \( {\overline{s}}_G/{\overline{s}}_B \), can also be derived directly from the color matching data [14]. This derivation depends on the longer-wavelength part of the visible spectrum being tritanopic for lights of the radiances typically used in color matching experiments. Thus, target wavelengths longer than about 560 nm, as well as the red primary, are invisible to the S-cones. In contrast, the green and blue primaries are both visible to the S-cones. Targets longer than 560 nm can be matched for the L- and M-cones by a mixture of the red and green primaries, but a small color difference typically remains, because the S-cones detect the field containing the green primary. To complete the match for the S-cones, a small amount of blue primary must be added to the field opposite the green primary. The sole purpose of the blue primary is to balance the effect of the green primary on the S-cones. Thus, the ratio of green to blue primary should be negative and fixed at \( {\overline{s}}_G/{\overline{s}}_B \), the ratio of the S-cone spectral sensitivity to the two primaries.

The lower left panel of Fig. 3 shows the Stiles and Burch [8] green, *g*(*λ*), and blue, *b*(*λ*), 2-deg chromaticity coordinates (blue squares). As expected, the function above ~555 nm is a straight line. It has a slope of −0.01625, which implies \( {\overline{s}}_G/{\overline{s}}_B \) = 0.01625, and is the same as the value obtained from the direct spectral sensitivity measurements, 0.0163 (upper panel). The lower right panel of Fig. 3 shows the Stiles and Burch [3] green, *g*(*λ*), and blue, *b*(*λ*), 10-deg chromaticity coordinates and the line that best fits the data above 555 nm, which has a slope of −0.0106 (light blue diamonds). Thus, the color matching data suggest that \( \overline{b}\left(\lambda \right) \) +0.0106 \( \overline{g}\left(\lambda \right) \) is the S-cone fundamental in the Stiles and Burch [3] 10-deg space. The differences between the 2-deg (left panel) and 10-deg (right panel) coefficients are consistent with changes in preretinal filtering and in photopigment optical density with eccentricity.

## LMS Transformation Matrix

The 2-deg cone fundamentals are the 10-deg functions adjusted in photopigment optical density and macular pigment density according to the expected differences in those densities for 2-deg and 10-deg viewing conditions. Because the CMFs are conventionally given in energy units, this transformation yields cone fundamentals in energy units. To convert cone fundamentals in energy units to quantal units, which are more practical for vision science, multiply by \( {\lambda}^{-1} \). The values of *k*_{ l }, *k*_{ m }, and *k*_{ s } in Eq. 3 depend on the desired normalization and on the units (energy or quanta). For further details, see [2]. Tabulated functions can be downloaded from http://www.cvrl.org.

## XYZ Transformation Matrix

By making a few simple assumptions, the cone fundamental CMFs, \( \overline{l}\left(\lambda \right) \), \( \overline{m}\left(\lambda \right) \), and \( \overline{s}\left(\lambda \right) \), can be linearly transformed to the more familiar colorimetric variants: \( \overline{x}\left(\lambda \right) \), \( \overline{y}\left(\lambda \right) \), and \( \overline{z}\left(\lambda \right) \), a form still in common use.

First, the \( \overline{y}\left(\lambda \right) \) CMF is assumed to be the 2-deg or 10-deg version of the luminous efficiency functions proposed by Sharpe et al. [15], which are linear combinations of \( \overline{l}\left(\lambda \right) \) and \( \overline{m}\left(\lambda \right) \) (defined in Eqs. 5 and 6, below). Second, the \( \overline{z}\left(\lambda \right) \) CMF is assumed to be the \( \overline{s}\left(\lambda \right) \) cone fundamental scaled to have an equal integral to the \( \overline{y}\left(\lambda \right) \) CMF for an equal energy white. Lastly, the definition of the \( \overline{x}\left(\lambda \right) \) CMF, which owes much to the efforts of Jan Henrik Wold for the TC 1–36 committee, is based on a series of requirements: (i) like the other CMFs, the values of \( \overline{x}\left(\lambda \right) \) are all positive; (ii) the integral of \( \overline{x}\left(\lambda \right) \) for an equal energy white is identical to the integrals for \( \overline{y}\left(\lambda \right) \) and \( \overline{z}\left(\lambda \right) \); and (iii) the coefficients of the transformation that yields \( \overline{x}\left(\lambda \right) \) are optimized to minimize the Euclidian differences between the resulting \( \overline{x}\left(\lambda \right) \), \( \overline{y}\left(\lambda \right) \), and \( \overline{z}\left(\lambda \right) \) chromaticity coordinates and the CIE 1931 \( \overline{x}\left(\lambda \right) \), \( \overline{y}\left(\lambda \right) \), and *z*(*λ*) chromaticity coordinates.

*x*(

*λ*) and

*y*(

*λ*), where:

*x*,

*y*chromaticity diagram.

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