Encyclopedia of Color Science and Technology

2016 Edition
| Editors: Ming Ronnier Luo

CIE Special Metamerism Index: Change in Observer

Reference work entry
DOI: https://doi.org/10.1007/978-1-4419-8071-7_322



Special Metamerism Index: Change in Observer refers to a method for evaluating the average of and the range of color mismatches for metameric color pairs when test observers with normal color vision are substituted for a reference observer (i.e., a CIE standard colorimetric observer). This method was proposed in 1989 by the CIE Technical Committee 1-07 [1]. The method, principally based on the works of Nayatani et al. [2] and Takahama et al. [3], was intended to evaluate both the average values and the range of color mismatches.


Two color stimuli with different spectral characteristics within the visible spectral range can have identical tristimulus values for a given illuminant and the reference observer (standard colorimetric observer). These stimuli are said to be metameric. Such metameric color match breaks down on changing the illuminant (illuminant metamerism) or the observer (observer metamerism). This is depicted in Fig. 1. While for observer A (reference observer), spectra 1 and spectra 2 (shaded curves) result in identical tristimulus values, they are no longer identical in the case of observer B (test observer), since this observer’s color-matching functions differ from those of observer A. If the spectral characteristics of the primary colorants of two color reproduction devices are not the same, any color match made on these devices is metameric in nature and thus may not hold when one observer is replaced by another.
CIE Special Metamerism Index: Change in Observer, Fig. 1

A graphical depiction of the phenomenon of observer metamerism. For observer A (the reference observer), the tristimulus values resulting from spectrally integrating the color-matching functions of observer A and the two spectra are identical. However, that is no longer the case for observer B, whose color-matching functions are different from those of observer A. Thus, spectra 1 and 2 are metameric (Image courtesy: Dr. Laurent Blondé)

Given the fact that the concept of metamerism is fundamentally important in the science of colorimetry, it was recognized long ago that a suitable index to quantify this effect would be of high relevance. It is relatively straightforward to formulate specific indices for illuminant metamerism, based on the color differences of a metameric pair under various sources of illumination. However, it is quite challenging to define a general index for observer metamerism. This is due to the immense variability that exists in the spectral sensitivities of the cone photoreceptors of individual observers having normal color vision. This variability is manifested in the measured color-matching functions and thus in color mismatches in a metameric pair when evaluated by individual observers.

Observer metamerism can be evaluated using experimental procedure involving actual observers with normal color vision or through mathematical modeling, using a set of color-matching functions that is representative of individual variations among color-normal observers. In one of the first attempts to model the uncertainties involved in the color-matching data, Nimeroff et al. [4] proposed a statistical model they termed as complete standard colorimetric observer system. The model included the mean of the color-matching functions of various observers as well as variance and covariance of these functions derived from the intra- and interobserver variability. On the other hand, Wyszecki and Stiles [5] attempted to define an index for observer metamerism by using the color-matching functions of 20 individual observers from the large-field color-matching experiment of Stiles and Burch [6]. They computed color differences for a given metameric pair as perceived by each of the 20 observers and used the mean color difference as the degree of observer metamerism. However, this method was not proposed with industrial applications in mind.

In 1989, a method was formulated by the CIE for evaluating observer metamerism. The method was detailed in a technical report titled “CIE Special Metamerism Index: Change in Observer,” prepared by the CIE Technical Committee (TC) 1-07. The committee came under CIE Division 1 (Vision and Colour) and was chaired by Prof. N. Ohta. This method was analogous to the one for evaluating illuminant metamerism, proposed by the CIE in 1986: “Special Metamerism Index: Change in Illuminant” [7]. This method, described hereafter as CIE observer metamerism index, assessed the degree of color mismatch for metameric color pairs (object colors or illuminant colors), resulting from substitution of the reference observer by a test observer. The reference observer is either the CIE 1931 standard colorimetric observer or the CIE 1964 supplementary standard colorimetric observer. The test observers are assumed to be a number of actual observers with normal color vision and are represented by four deviation functions characterizing the variations of color-matching functions of color-normal observers.

Standard Deviate Observer

Computation of the CIE observer metamerism index requires the use of color-matching functions for a standard deviate observer. Thus, the definition of the standard deviate observer and an overview of various propositions for its formulation would be relevant here.

The concept of standard deviate observer is a mathematical construct that was first proposed by Allen in 1970 [8]. This observer has color-matching functions differing from the reference observer by amounts equal to standard deviations among a defined set of color-normal observers. Allen and subsequently other researchers used the 20 individual Stiles and Burch observers [6] for deriving the standard deviate observer.

In the method proposed by Allen [8], starting from the standard deviations \( \Delta \overline{x}\left(\lambda \right) \), \( \Delta \overline{y}\left(\lambda \right) \) and \( \Delta \overline{z}\left(\lambda \right) \) of color-matching functions of the 20 Stiles and Burch observers, differential tristimulus values ΔX, ΔY, and ΔZ were computed for any metameric pair. These values represented the root mean square differences in tristimulus values for a metameric pair as perceived by the observers. The tristimulus and delta tristimulus values could then be used in a color-difference formula to calculate ΔE, which was the desired observer metamerism index. Allen’s method allowed for negative differences in tristimulus values.

In a different statistical approach, Nayatani et al. [9] performed a singular value decomposition analysis on the 20 observer data and derived three deviation functions whose linear combinations were used to reconstitute the color-matching functions of the 20 Stiles and Burch observers. Further, the authors showed that the first deviation function was similar to Allen’s standard deviate observer. Adding the first deviation function to the reference observer (e.g., CIE 1964 supplementary standard colorimetric observer) yielded the test observer. Observer metamerism indices derived by using the first deviation function showed high correlation to the average metamerism index of the 20 Stiles and Burch observers. These results were obtained by using two sets of metameric spectral reflectance values of 12 and 68 metamers.

A subsequent study by Takahama et al. [3] expanded the method by using the first deviation to evaluate the index of observer metamerism. All four deviation functions were used to construct the confidence ellipsoids of tristimulus values defining the range of mismatches expected for a given pair of metamers, viewed by actual observers with normal color vision but different from the reference.

In an independent study, Ohta [10] performed a nonlinear optimization of the 20-observer data to formulate a standard deviate observer model. The model was close to the one obtained by Nayatani [9] and was assessed to well represent the original 20 observers.

Procedure for Computing CIE Observer Metamerism Index

Tristimulus values for a pair of object colors metameric for a reference observer with color-matching functions \( \overline{x}\left(\lambda \right) \), \( \overline{y}\left(\lambda \right) \), and \( \overline{z}\left(\lambda \right) \) and an illuminant with spectral power distribution S(λ) are given by Eq. 1.
$$ \left[\begin{array}{l}{X}_{ref,i}\hfill \\ {}{Y}_{ref,i}\hfill \\ {}{Z}_{ref,i}\hfill \end{array}\right]={\displaystyle \sum_{\lambda }{\rho}_i\left(\lambda \right)\;S\left(\lambda \right)}\;\left[\begin{array}{l}\overline{x}\left(\lambda \right)\hfill \\ {}\overline{y}\left(\lambda \right)\hfill \\ {}\overline{z}\left(\lambda \right)\hfill \end{array}\right] \varDelta \lambda $$
where ρ i (λ) is the spectral reflectance of the i-th object color (i = 1, 2) of the metameric pair.
Since the object colors are metameric, this can be written as:
$$ \begin{array}{l}{X}_{ref,1}={X}_{ref,2}\cong {X}_{ref}\hfill \\ {}{Y}_{ref,1}={Y}_{ref,2}\cong {Y}_{ref}\hfill \\ {}{Z}_{ref,1}={Z}_{ref,2}\cong {Z}_{ref}\hfill \end{array} $$
Further, S(λ) is normalized so that it has a luminance of 100, as shown in Eq. 3.
$$ {\displaystyle \sum_{\lambda }S\left(\lambda \right) \overline{y}\left(\lambda \right)\Delta \lambda =100} $$
Four sets of deviation functions \( {\overline{x}}_j\left(\lambda \right) \), \( {\overline{y}}_j\left(\lambda \right) \), and \( {\overline{z}}_j\left(\lambda \right) \), where j denotes the set number, are used in this method and have been tabulated in the report of CIE TC 1-07.
The first set of deviation functions denotes the differences in color-matching functions of CIE standard colorimetric observer and the standard deviate observer [denoted by \( {\overline{x}}_{dev}\left(\lambda \right) \), \( {\overline{y}}_{dev}\left(\lambda \right) \), and \( {\overline{z}}_{dev}\left(\lambda \right) \)]. Thus, the color-matching functions of the standard deviate observer are obtained by using Eq. 4.
$$ \begin{array}{l}{\overline{x}}_{dev}\left(\lambda \right)=\overline{x}\left(\lambda \right)+\Delta {\overline{x}}_1\left(\lambda \right)\hfill \\ {}{\overline{y}}_{dev}\left(\lambda \right)=\overline{y}\left(\lambda \right)+\Delta {\overline{y}}_1\left(\lambda \right)\hfill \\ {}{\overline{z}}_{dev}\left(\lambda \right)=\overline{z}\left(\lambda \right)+\Delta {\overline{z}}_1\left(\lambda \right)\hfill \end{array} $$
From these color-matching functions, the tristimulus values Xdev,i, Ydev,i, and Zdev,i of metameric object colors (i = 1 or 2) corresponding to the standard deviate observer can be obtained as shown in Eq. 5.
$$ \left[\begin{array}{l}{X}_{dev,i}\hfill \\ {}{Y}_{dev,i}\hfill \\ {}{Z}_{dev,i}\hfill \end{array}\right]={\displaystyle \sum_{\lambda }{\rho}_i\left(\lambda \right)S\left(\lambda \right)}\left[\begin{array}{l}{\overline{x}}_{dev}\left(\lambda \right)\hfill \\ {}{\overline{y}}_{dev}\left(\lambda \right)\hfill \\ {}{\overline{z}}_{dev}\left(\lambda \right)\hfill \end{array}\right] \Delta \lambda $$
The CIE observer metamerism index (Mobs) for the pair of metameric object colors is expressed by Eq. 6.
$$ {M}_{obs}=\Delta {E}_{obs}^{*}\left[\left({X}_{dev, 1},{Y}_{dev, 1},{Z}_{dev, 1}\right), \left({X}_{dev,2},{Y}_{dev,2},{Z}_{dev,2}\right)\right] $$
where ΔE obs * is the color difference between the metameric object colors as evaluated by the standard deviate observer, calculated in a uniform color space. CIE TC 1-07 recommends CIE 1976 (L*, u*, v*) or (L*, a*, b*) as uniform color space [7].
When the tristimulus values of the samples do not exactly match, they are not strictly metameric (instead, they are parameric). In such a scenario, the tristimulus values of the first stimulus are defined as reference (X ref , Y ref , and Z ref ), as in Eq. 7.
$$ \begin{array}{l}{X}_{ref}={X}_{ref,1}\left(\ne {X}_{ref,2}\right)\hfill \\ {}{Y}_{ref}={Y}_{ref,1}\left(\ne {Y}_{ref,2}\right)\hfill \\ {}{Z}_{ref}={Z}_{ref,1}\left(\ne {Z}_{ref,2}\right)\hfill \end{array} $$
The tristimulus values of the second stimulus obtained by Eq. 5 are then corrected using Eq. 8.
$$ \begin{array}{l}{X}_{dev,2}^{\prime }={X}_{dev,2}\left(\frac{X_{ref,1}}{X_{ref,2}}\right)\hfill \\ {}{Y}_{dev,2}^{\prime }={Y}_{dev,2}\left(\frac{Y_{ref,1}}{Y_{ref,2}}\right)\hfill \\ {}{Z}_{dev,2}^{\prime }={Z}_{dev,2}\left(\frac{Z_{ref,1}}{Z_{ref,2}}\right)\hfill \end{array} $$

Deriving 95 % Confidence Ellipse

Resultant tristimulus values of metameric colors inevitably mismatch when evaluated by a test observer. These tristimulus values spread within a certain range in the three-dimensional color space. This range is characterized in a chromaticity diagram by a statistical confidence ellipse encompassing 95 % of the spread. In this way, evaluation of observer metamerism allows an estimation of tolerances in color-difference judgments for various metameric color pairs.

All four deviate observer functions are used to estimate the confidence ellipse containing 95 % of color mismatches. First, a set of tristimulus value deviations for a pair of metameric object colors is defined using Eq. 9.
$$ \left[\begin{array}{c}\hfill {\Delta}^2{X}_j\hfill \\ {}\hfill {\Delta}^2{Y}_j\hfill \\ {}\hfill {\Delta}^2{Z}_j\hfill \end{array}\right]={\displaystyle \sum_{\lambda}\left[{\rho}_2\left(\lambda \right)-{\rho}_1\left(\lambda \right)\right]S\left(\lambda \right)\left[\begin{array}{c}\hfill {\overline{x}}_j\left(\lambda \right)\hfill \\ {}\hfill {\overline{y}}_j\left(\lambda \right)\hfill \\ {}\hfill {\overline{z}}_j\left(\lambda \right)\hfill \end{array}\right]} \Delta \lambda $$
Here, the spectral reflectance factors of the object colors are represented by ρ1(λ) and ρ2(λ), and j refers to one of the four deviation functions. As before, the spectral power distribution of the illuminant is denoted by S(λ).
In case of metameric illuminant colors, the spectral power distributions S1(λ) and S2(λ) of the illuminant pair are normalized similar to Eq. 3. Instead of Eq. 9, Eq. 10 is used.
$$ \left[\begin{array}{c}\hfill {\Delta}^2{X}_j\hfill \\ {}\hfill {\Delta}^2{Y}_j\hfill \\ {}\hfill {\Delta}^2{Z}_j\hfill \end{array}\right]={\displaystyle \sum_{\lambda}\left[{S}_2\left(\lambda \right)-{S}_1\left(\lambda \right)\right]\left[\begin{array}{c}\hfill {\overline{x}}_j\left(\lambda \right)\hfill \\ {}\hfill {\overline{y}}_j\left(\lambda \right)\hfill \\ {}\hfill {\overline{z}}_j\left(\lambda \right)\hfill \end{array}\right] }\Delta \lambda $$
Next, two matrices D and V are defined as shown in Eqs. 11 and 12.
$$ D=\frac{1}{{\left({X}_{ref}+15{Y}_{ref}+3{Z}_{ref}\right)}^2}\left[\begin{array}{ccc}\hfill 60{Y}_{ref}+12{Z}_{ref}\hfill & \hfill -60{X}_{ref}\hfill & \hfill -12{X}_{ref}\hfill \\ {}\hfill -9{Y}_{ref}\hfill & \hfill 9{X}_{ref}+27{Z}_{ref}\hfill & \hfill -27{Y}_{ref}\hfill \end{array}\right] $$
$$ V={\displaystyle \sum_{i=1}^4\left[\begin{array}{ccc}\hfill {\left({\Delta}^2{X}_i\right)}^2\hfill & \hfill {\Delta}^2{X}_i\cdot {\Delta}^2{Y}_i\hfill & \hfill {\Delta}^2{Z}_i\cdot {\Delta}^2{X}_i\hfill \\ {}\hfill {\Delta}^2{X}_i\cdot {\Delta}^2{Y}_i\hfill & \hfill {\left({\Delta}^2{Y}_i\right)}^2\hfill & \hfill {\Delta}^2{Y}_i\cdot {\Delta}^2{Z}_i\hfill \\ {}\hfill {\Delta}^2{Z}_i\cdot {\Delta}^2{X}_i\hfill & \hfill {\Delta}^2{Y}_i\cdot {\Delta}^2{Z}_i\hfill & \hfill {\left({\Delta}^2{Z}_i\right)}^2\hfill \end{array}\right]} $$
The variance-covariance matrix ∑ of the (u′, v′) chromaticity coordinates of all the color matches evaluated by test observers is given by Eq. 13.
$$ \sum =DV{D}^t $$
where Dt is the transpose of matrix D. If the elements of inverse matrix ∑−1 are denoted as ∑ mn where m is the row and n is the column of a given element, the confidence ellipse containing 95 % of the color mismatches is given by Eq. 14.
$$ {\sum}^{11}{\left(\Delta {u}^{\prime}\right)}^2+2{\sum}^{12}\left(\Delta {u}^{\prime}\right)\;\left(\Delta {v}^{\prime}\right)+{\sum}^{22}{\left(\Delta {v}^{\prime}\right)}^2={\chi}^2\left(2,0.05\right)=5.991 $$
Here, χ2(2, 0.05) is the 5 % of the χ2 distribution for two degrees of freedom.

The center of the ellipse is given by (uref, vref), which corresponds to the reference observer.

It should be noted that the two-dimensional ellipse in the chromaticity diagram does not contain information about the psychometric lightness of the stimuli. Thus, the ellipse cannot be used for comparing the degree of observer metamerism between two different metameric pairs with varying lightness.

Effect of Age on Color Mismatches

The CIE TC 1-07 report [1] also proposes a method that accounts for the effect of age on the index of observer metamerism. Mainly, aging of the lens (pigmentation) is considered as a contributing factor. A pair of object colors which are metameric for an average observer with an age N1 and a reference illuminant will mismatch for an observer with an age N2. The deviation functions \( \Delta {\overline{x}}_1\left(\lambda, N\right) \), \( \Delta {\overline{y}}_1\left(\lambda, N\right) \), and \( \Delta {\overline{z}}_1\left(\lambda, N\right) \) for age N (20 ≤ N ≤ 60) are computed using Eqs. 15 and 16. These values are used instead of those obtained from Eq. 4.
$$ \left[\begin{array}{l}\Delta {\overline{x}}_1\left(\lambda, N\right)\hfill \\ {}\Delta {\overline{y}}_1\left(\lambda, N\right)\hfill \\ {}\Delta {\overline{z}}_1\left(\lambda, N\right)\hfill \end{array}\right]=L(N)\cdot \left[\begin{array}{l}\Delta {\overline{x}}_1\left(\lambda \right)\hfill \\ {}\Delta {\overline{y}}_1\left(\lambda \right)\hfill \\ {}\Delta {\overline{z}}_1\left(\lambda \right)\hfill \end{array}\right] $$
$$ L(N)=0.064\cdot N-2.31 $$

Evaluation of the CIE Observer Metamerism Index

Following the introduction of CIE observer metamerism index [1], several researchers evaluated the model with independent experimental data. North and Fairchild [11] conducted a Maxwell-type color-matching experiment using a cathode ray tube (CRT) display and a tungsten-halogen lamp on two halves of a 2° bipartite field. The authors estimated the color-matching functions of each observer through a mathematical model, starting from experimental data obtained at seven wavelengths. They concluded that the interobserver variability in the experimental data was much larger than what was predicted by the CIE model [1].

Alfvin and Fairchild [12] conducted another visual experiment on color matches between color prints or transparencies and a CRT display. They used an equilateral glass prism to allow the observers to view simultaneously both the soft and hard copy stimuli in a vertically symmetric bipartite field. In analyzing the data, they arrived at the same conclusion: the interobserver variability was significantly larger than the prediction of the CIE observer metamerism index [1]. As an example, Fig. 2 shows bivariate 95 % confidence ellipses containing the range of CIELAB (Δa*, Δb*) color mismatches for cyan-transparency sample. The plot includes bivariate ellipse calculated using the CIE method [1] as well as experimentally determined bivariate ellipses for intra- and interobserver color matches. According to these results, the interobserver variability is significantly underpredicted by the CIE observer metamerism index [1].
CIE Special Metamerism Index: Change in Observer, Fig. 2

Comparison of 95 % confidence regions for measured and predicted ranges of color mismatch for the cyan transparency. Measured data are shown for Alfvin and Fairchild [12] and Nimeroff et al. [4], along with confidence region predicted by the CIE observer metamerism index (CIE standard deviate observer). All data are shown in CIELAB Δa* − Δb* plane (Reproduced from Alfvin and Fairchild [12])

Oicherman et al. [13] conducted an asymmetric color-matching experiment where eleven observers were asked to match the colors displayed on a CRT and an LCD to the colors of two achromatic and eight chromatic paint samples placed inside a light booth. Even in this study, the authors reported a significant underprediction of the observer variations of color-matching data by the CIE observer metamerism index [1], accounting for only 15 % of interobserver variability.

Why did the CIE observer metamerism index not perform well? The suggested explanations include the exclusion of some of the Stiles and Burch observers from the analysis that led to the development of the CIE observer metamerism index [11] and improper mathematical treatment of the original color-matching data [12]. Looking from the point of view of practical industrial applications, in particular hard copy vs. soft copy color matching, some researchers [13] have questioned the purpose and usefulness of an index of observer metamerism and a standard deviate observer. They suggested that individual variability in these conditions is governed by mechanisms of chromatic discrimination and could be modeled by advanced color-difference formulas with suitably adjusted parametric coefficients.

Future Directions

In 2006, CIE’s Technical Committee 1-36 published a report [14] on the choice of a set of color-matching functions and estimates of cone fundamentals for the color-normal observer. Starting from the 10° color-matching functions of 47 observers from Stiles and Burch’s large-field color-matching experiment [6], the model defines 2° and 10° fundamental observers and provides a convenient framework for calculating average cone fundamentals for any field size between 1° and 10° and for an age between 20 and 80. This model provides a theoretical framework for quantifying observer metamerism resulting from age variations.

Adopting a different approach to address the issue of observer metamerism in applied colorimetry, Sarkar et al. [15] introduced the concept of colorimetric observer categories. In this work, eight of such categories were derived through statistical analyses of a combined dataset of experimental and physiological color-matching functions. Physiological data were obtained from the mathematical model proposed by CIE TC 1-36 [14], while experimental data consisted of color-matching functions of 47 Stiles and Burch observers [6]. The authors developed and implemented an experimental method to classify a color-normal human observer as belonging to one of these categories, based on his or her color vision. Subsequently, a compact proof-of-concept prototype for conducting such experiments was developed.

At the time of writing this essay, more studies are being undertaken to further investigate the possibility of establishing such observer categories, but the results are yet to be published. If these studies validate the concept of observer categories and the method of observer classification as proposed by Sarkar et al. [15], this can eventually lead to an alternative method for quantifying observer metamerism in applied colorimetry as well as for providing practical solution to the issue of observer metamerism in various color-critical industrial applications.



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Authors and Affiliations

  1. 1.SurfaceMicrosoft CorporationRedmondUSA