The Use of Metamers to Compare
the Color Vision of Observers

J. M. Ezquerro,1* F. Carreño,1 J. M. Zoido,1
E. Bernabeu2

1 Escuela Universitaria de Optica, Universidad Complutense de Madrid, C/Arcos de Jalo´n s/n, Madrid 28037, Spain

2 Facultad de CC. Fı´sicas, Universidad Complutense de Madrid, Ciudad Unversitaria s/n, Madrid 28040, Spain

Received 18 September 1999; accepted 1 July 2000

Abstract: In this research we compare the colorimetric
behavior of several observers. For color centers recom-
mended by CIE we have produced large sets of spectral
distributions, which are metameric for the CIE 1931 stan-
dard observer. For each one of the color centers, we com-
pare the clouds of chromaticity coordinates with the chro-
maticity thresholds. We define a parameter that provides a
quantitative measure of the interobserver variability. This
parameter is used to arrange the observers by their degree
of likeness. A similar procedure has been used to compare
two real observers. It is shown how there is no reciprocity
between the colorimetric behavior of two real observers.
© 2001 John Wiley & Sons, Inc. Col Res Appl, 26, 262–269, 2001

Key words: color vision; colorimetry

INTRODUCTION

The colorimetric behavior of different observers is an im-
portant area of research in color science. Several studies
have been carried out in order to verify the suitability of CIE
1931 Standard Observer1–6 and CIE 1964 standard observ-
er.7–9 The design of adequate commercial devices requires
us to evaluate the differences among the behavior of differ-
ent color detection, or color reproduction, systems. Thus,
there is no doubt about the interest in obtaining a quantita-
tive measure of the differences in color performance be-
tween a given observer, or device, and a standard one.

A usual procedure when comparing two observers is to
analyze the similarity of their corresponding sets of color-
matching functions (cmf’s). This comparison can also be
carried out by studying the differences among the tristimu-

lus values associated with each observer when considering
a fixed set of color centers (see for instance Refs. 5, 10, 11).
It is well known that, due to interobserver variability, a
given physical stimulus evokes different color sensations in
each observer.

The conclusions derived from the study of two observers
depend on the particular method used when analyzing the
data. In this work, we propose a procedure to compare the
colorimetric behavior of two observers. The method pro-
posed by us can be applied in a systematic way when
different observers are considered.

We present the proposed method in Section II. In Section
III, we provide the results obtained for different sets of
cmf’s. We also provide criteria to establish an order of
likeliness among the different observers under consider-
ation. Section IV is devoted to analyzing a special case of
two observers for whom the cmf’s and the chromaticity
thresholds have been determined.10 Finally, the conclusions
are given in Section V.

OBSERVERS AND COLOR-MATCHING FUNCTIONS

Let { xi
k(l)} ( i 5 1, 2, 3) be the set of cmf’s that determine

the colorimetric behavior of a real observerk. The tristimu-
lus values associated with a spectral radiant fluxr(l) are
given by

Xi
k 5 K E

l1

l2

r~l! xi
k~l!dl, ~i 5 1, 2, 3!, (1)

with K being a constant that depends on the system of
primary stimuli used to specify the cmf’s, andI 5 [l1, l2]
is the spectral range considered. The spectral range is taken
asI 5 [360, 830] nm in thecase of the CIE 1931 standard
observer.13 In general, the intervalI depends on the spectral

* Correspondence to:Dr. J. M. Ezquerro, Escuela Universitaria de
Optica, Universidad Complutense de Madrid, C/Arcos de Jalo´n s/n,
E-28037 Madrid, Espan˜a (e-mail: ferpo@fis.ucm.es)
© 2001 John Wiley & Sons, Inc.

262 COLOR research and application



range considered when measuring the set {xi
k(l)}. Most

authors takel1 5 400 nm andl2 5 700 nm.
Any color stimulus perceived by a given observerk can

be specified in the corresponding color-representation sys-
tem as a point defined by the vectorXk 5 (X1

k, X2
k, X3

k).
Thus, the setRk 5 { Xk} defined by the tristimulus values
of all possible color stimuli provides the region that contains
all the possible color sensations perceived by the observerk.
We refer to the subsetRk as color-representation system
associated with the observerk.

In Eq. (1), cmf’s are specified with regard to a set of
primary stimuli {Ci}. If we consider another set of primary
stimuli {C*i} linearly related with the previous one, i.e.,
C*m 5 ¥n51

3 cnmCn, the relation between the corresponding
tristimulus values,X 5 (X1, X2, X3) and X* 5 (X91, X92,
X93), is given by

X 5 CX*, (2)

with C 5 { cmn}.
In general, the set of cmf’s of a real observer, {xi

k(l)},
differs from the set {xi

so(l)} associated with the standard
one. Thus, Eq. (1) provides different tristimulus values
when considering both cases. A real observer and a standard
one generate different color-representation systems.

Let us consider an ensemble ofQ observers, each one of
them being characterized by its corresponding set of cmf’s
{ xi

k(l)} ( k running from 1 toQ andi from 1 to 3). We take
one of them as the reference observer: CIE 1931 Standard
Observer in our case. The interobserver variability can be
estimated by analyzing the differences in the perception of
a given set ofR reference stimuli. For each one of these
color stimuli, we have produced a set ofS spectral radiant
power distributions, which are metameric for the reference
observer.

Let r j
l(l) be the j -th metameric spectral radiant power

distribution associated with thel -th reference stimulus,
wherel runs from 1 toR andj runs from 1 toS. For thek-th
observer, distributionr j

l(l) produces the tristimulus values

Xi, j
k,l 5 K E

l1

l2

xi
k~l!r j

l~l!dl,

i 5 1, 2, 3,

j 5 1, . . . , S,

k 5 1, . . . , Q,

l 5 1, . . . , R. (3)

For a distributionr j
l(l), it is expected that, if the set

{ xi
k(l)} does not differ strongly from the set {xi

so(l)}, Eq.
(3) should provide similar results for both sets. In this case,
the observerk has a colorimetric behavior similar to that of
the reference observer. In the following analysis, we try to
elucidate some conclusion about this subject.

By using the set of cmf’s {xi
k(l)}, we have computed the

chromaticity coordinates for theS metameric distributions
associated with color centerl . Due to interobserver vari-

ability, each spectral distributionr j
l(l) produces a different

point in the chromaticity diagram. In this way, we obtain a
cloud of points. If the behavior of both observers is similar,
this cloud should be close to the point defined by the
chromaticity coordinates perceived by the standard observer
for the color center under consideration. To check this
closeness, we compare the cloud of points with the chro-
maticity threshold experimentally determined for the ob-
serverk. If the distribution of points is inside the threshold,
we could conclude that there is no difference between the
colorimetric behavior of both observers for the considered
color center. If the cloud of points is outside the threshold,
the color sensation perceived by each observer differs. This
point was noted by Hitaet al.14 when considering what they
called “weak definition of metamerism.” To quantify this
difference, we compute the distanceda

k,l from xk,l to x#k,l, xk,l

being the center of the threshold andx#k,l being the point
defined as the average chromaticity coordinates of the cloud
of points. The straight line joining both points intersects the
ellipse representing the threshold at pointxth

k,l. The distance
from this point to the center of the threshold is labeled as
dth

k,l. In this case, quotient,

Vk,l 5
da

k,l

dth
k,l (4)

could provide a quantitative measure of interobserver vari-
ability when considering thel -th color center.

OBSERVERS METAMERISM AND
COLOR-DIFFERENCE THRESHOLDS

We have considered in this research the following sets of
cmf’s, all of them determined for a visual field of 2°:

● The mean observer obtained by averaging, SBM, the ten
sets of cmf’s provided by Stiles–Burch15 labeled asSBi
(wherei runs from 1 to 10).

● Standard observer CIE 1931 modified by VOS.16

● Observers MM, JAM and CF.10,11

Some of these sets of cmf’s take negative values when
they are specified in the CIE 1931 system of primary stim-
uli. To avoid the possible difficulties derived from this fact,
we have transformed them to a representation system pro-
posed in Refs. 10, 11. We refer to this system as G94. All
the above-mentioned sets of cmf’s take positive values in
G94. Tristimulus values,X, in this system are obtained from
those in the CIE 1931 system,X*, by introducing the matrix

CXYZ31
G94 5 S 0.9980 0.0020 0.0000

0.0000 1.0000 0.0000
0.2270 20.0428 0.8158

D, (5)

in expression (2).10,11 All the results obtained in this work
are specified in G94.

The reference color stimuli used in this study, labeled
A–E, are those suggested by the CIE.17 The tristimulus
values associated with these stimuli are given in Table I. For

Volume 26, Number 4, August 2001 263



the l -th color center we have produced a certain numbernl

of spectral distributions in the spectral rangeI 5 [400,
700] nmwith a sampling interval ofDl 5 10 nm. These
distributions are metameric for the CIE 1931 standard ob-
server. These procedure has been repeated for the different
reference stimuli.

Note that there are several methods to produce metamers.
In this study we have used a method developed by us.18,19In
brief, this method uses a linear-programming technique
based in the simplex algorithm.20 By taking into account
physical color properties, such as excitation purity and
dominant wavelength, an “objective function” is defined in
terms of cmf’s, and certain inequalities for the unknown
spectral distributionsrj

l(l) are proposed. The key of the
method is that these inequalities are given in terms of
certain bounds,Vj, for the values of the unknown spectral
distributions, i.e., 0, r j

l(lm) , Vj, wherer j
l(lm) is the

value of the spectral distribution sampling at pointlm. By
maximizing the objective function, we obtain the metamers.
The algorithm was complemented with criteria of softness19

to produce spectral distributions of any color stimulus. The
boundsVj are produced in a random way, thus this method
provides a large number of different metamers (see Refs.
18, 19 for more details).

In this case, integrals are replaced by sums in Eq. (3):

Xi, j
k,l 5 K O

m51

N

xi
k~lm!r j

l~lm!Dl, (6)

N is the number of wavelengths in which functionsr j
l(l)

have been sampled (see Refs. 18, 19).
In the following analysis we considerQ 5 5 observers,

R 5 5 reference color stimuli, andS 5 nl metameric
distributions for each reference stimulus. Note that the num-
bernl depends on the center considered:nA 5 1225,nB 5
1143, nC 5 875, nD 5 1224, andnE 5 700. These
numbers are different, because for some color centers the
algorithm used to produce the metamers requires a large
amount of computing time.

The color discrimination ellipses for the observers JAM
and CF have been measured by Martı´nez10 and Martı´nezet
al.11 around the five color centers suggested by the CIE.
Thus, we can compare the clouds of chromaticity coordi-
natesxi , j

k,l obtained for the above-mentioned sets of cmf’s
with the corresponding ellipses for JAM and CF. The results
obtained for the five color centers are shown in Fig. 1.

Note that the location of the clouds of points for observer

VOS is nearer to the ellipses than in the case of the other
observers. This effect is reproduced for all color centers.
The chromaticity coordinates computed for all the sets of
cmf’s are outside the corresponding ellipses. The clouds of
points associated with observers VOS and SBM are par-
tially inside the ellipses for color center E. The data for
observers JAM and MM are always far from the ellipses,
although they are close to each other.

In Tables II and III, quotients (4) are listed for all ob-
servers with regard to the ellipses measured for JAM and
CF. Note the low values obtained for observer VOS. In most
cases, quantityVk,l is greater than unity. This result seems
to indicate that the colorimetric behavior of the real observ-
ers clearly departs from that of the reference observer.

We can considerdth
k,l as the just noticeable difference in

the direction determined by the straight line joining points
xk,l andxth

k,l. In this way, high values ofVk,l indicate that the
color performances of thek-th observer and the standard
one are very different when considering thel -th color cen-
ter. Thus, from the previous analysis we can conclude that
the colorimetric behavior of real observers cannot be ap-
proximated by that of the reference observer. These results
are in good agreement with those provided in Ref. 5. Au-
thors of this work report significant displacements of the
center of the ellipses when several sets of cmf’s were
analyzed (see Table I in Ref. 5). Looking for a check of
these displacements, these authors calculated the tristimulus
values of a limited number of theoretical metameric spectral
reflectances, and they found a significant spreading of the
chromaticity coordinates for the observers under consider-
ation. Our results also agree with those reported in the
studies of interobserver variability by other methods.2,9,11

For a given observer,k, we compute the averageVk 5
¥ l51

5 Vk,l/5. This parameter can be used to obtain a quan-
titative measure of theglobal discrepancy between the
standard and thek-th observer. In this way, we can establish
an order of likeliness among the different observers with
regard to the standard one. The results are listed in Table IV
when considering the ellipse associated with observer CF.
The same order of likeness is obtained when using the
ellipse of JAM. The results obtained by using this method
are coincidental with those reported in Ref. 21.

INTEROBSERVER VARIABILITY

The results obtained in the previous section are a conse-
quence of the differences of cmf’s of a given observer with

TABLE I. Tristimulus values corresponding to the color centers recommended by the CIE.17 These data are
expressed in the CIE 1931 (labeled with primes) and in the G94 color-representation systems.

COLOR CENTER X91 X92 X93 X1 X2 X3

A (achromatic) 28.459 30.000 32.175 28.462 30.000 31.4246
B (red) 19.954 14.100 7.174 19.942 14.100 9.778
C (yellow) 62.823 69.300 29.793 62.836 69.300 35.600
D (green) 16.442 24.000 25.856 16.457 24.000 23.798
E (blue) 8.922 8.800 23.0185 8.922 8.800 20.427

264 COLOR research and application



regard to the cmf’s of the standard observer. In this
section, we intend to carry out a comparison restricted to
observers JAM and CF. Martı´nez10 and Martı´nez et
al.11,12have measured the cmf’s of the mentioned observ-
ers together with the thresholds corresponding to the five
CIE centers. We analyzed the similarities of both observ-

ers when perceiving these color centers. We used the
following procedure:

1. For a given observer, JAM for example, and each color
center (l ), we have produced a set of metameric spec-

FIG. 1. Chromaticity coordinates obtained for each observer around the color centers studied. The data are represented in
the chromaticity diagram associated to the CIE 1931 Standard Observer in the G94 system of primary stimuli.

Volume 26, Number 4, August 2001 265



tral distributions. The number of metamers is labeled as
nl

JAM (see column 1 in Table V).
2. We computed the chromaticity coordinates for these

distributions with cmf’s associated with the other ob-
server, CF in this case. The clouds of points together
with the threshold determined for CF are represented in
a common figure for each color center (see Fig. 2).

Steps 1 and 2 were repeated interchanging the role of the
observers. The data for observer CF are given in column 2
of Table V and in Fig. 3.

Note in Fig. 2 that, for color centers A and C, the points
are partially inside the ellipse. In Fig. 3, the clouds of points
are partially inside the ellipse when the color center A and
D are considered; however, in the case of color center B, all
the points are inside the ellipse. The case is different for the
rest of the color centers and each observer: the clouds of
points are outside the ellipses. To quantify the discrepancy
in evaluating the selected color centers, we have computed
quotientsVCF,l (taking into account the results in Fig. 2)
andVJAM,l (taking into account the results in Fig. 3). These
quotients are defined as in Eq. (4). The results are listed in
columns 2 and 3 of Table VI, respectively.

When comparing results shown in Figs. 2 and 3, and data
listed in Table VI, we deduce that color perception of observers
JAM and CF clearly differs for color centers B, C, D, and E. In
the case of the achromatic stimulus, A, the clouds of chroma-
ticity coordinates are inside the corresponding ellipses.

This analysis reveals an important discrepancy between
CF and JAM in evaluating color centers. We deduce that
cmf’s of the observers have a strong influence on the spec-
ification of individual colors. Note that according to Refs. 5,
12, the variability appearing in the threshold parameters is
not significant in most cases, although there was significant
displacement of the ellipses’ centers. In summary, the anal-
ysis reveals pronounced discrepancies between the two ob-

servers evaluating a particular color, but strong similarities
judging color differences.

Note that the results presented in Figs. 2 and 3, and Table VI
reveal a nonreciprocity behavior between CF and JAM, except
for color center A. This fact requires a brief description: the
metamers generated for one observer, JAM for example, pro-
duce the same color perception for this observer, whereas the
other observer, CF in this case, specifies each metamer as a
different color. The quotientsVCF,l are estimators for the
differences in the specification of the physical stimuli. This
behavior is reproduced when the roles of the observers are
interchanged. The surprising result is that quotientsVCF,l and
VJAM,l are clearly different between them for a given color
centerl, except for color centers A and C. This is because the
color representation systems spanned by two different sets of
cmf’s are not isomorphic (see Section II), i.e., the correspon-
dence betweenRCF andRJAM is not one-to-one: this point has
been treated extensively in Refs. 22, 23. The origin of this
behavior comes from the local differences in cmf’s of different
observers. In Ref. 6, a detailed statistical analysis of the cmf’s
of several observers was carried out. These authors introduced
a parameter labeled as VAF (see the section entitled “Overall
analysis of the cmf curve shape”), to test the similarity between
two sets of cmf’s. In Table IV of Ref. 6, it is shown how the
VAF parameter forx#l andz#l are lower than the value fory#l.
This fact confirms the nonisomorphic character between the
RCF andRJAM color-representation systems.

CONCLUSIONS

From the analysis of data shown in Fig. 1 and Tables II and
III, it can be concluded that the specification of a given
physical stimulus strongly depends on the set of color-
matching functions used. The clouds of points of chroma-

TABLE III. Quotients Vk,l computed for the specific
observers with regard to the ellipse measured for
JAM.

Color center MM CF JAM VOS SBM

A 3.6 1.1 4.4 0.7 1.0
B 7.0 7.5 8.0 2.2 5.9
C 2.4 4.3 2.9 2.3 3.8
D 13.0 2.3 16.1 5.9 6.4
E 10.7 2.1 12.7 0.3 0.5

TABLE IV. Quantity Vk for the different observers
when considering the threshold of CF. The pair of
observers are ordered by their degree of global like-
ness with regard to CIE 1931 Standard Observer.

Pair of
observers Vk Order

CIE31-VOS 2.24 1
CIE31-CF 3.04 2
CIE31-SBM 3.28 3
CIE31-MM 7.10 4
CIE31-JAM 8.56 5

TABLE V. The number of metamers produced for
each color center and observers JAM (column 2) and
CF (column 3).

Color center nl
JAM nl

CF

A 1225 1225
B 964 992
C 1225 1225
D 1134 1225
E 1225 1225

TABLE II. Quotients Vk,l computed for the specific
observers with regard to the ellipse measured for CF.

Color center MM CF JAM VOS SBM

A 7.7 1.3 9.5 1.5 1.6
B 6.4 6.8 7.3 2.0 5.4
C 2.1 3.7 2.7 2.2 3.4
D 10.2 2.1 12.6 4.8 5.1
E 9.1 1.3 10.7 0.7 0.8

266 COLOR research and application



ticity coordinates tell us that color perception of real ob-
servers is different from that of the standard observer when
specifying a certain stimulus. Thus, real observers are not
well represented by the set of cmf’s of the standard ob-
server. These results are in agreement with those reported in
Refs. 6, 11, 12. In Ref. 3, the authors reported similar results
concerning the displacement of the centers of the thresholds

when considering isomeric and metameric color-matchings.
They took this fact to be an indication of the failure in the
colorimetric additivity. In our opinion this “failure” is a
consequence of the nonisomorphic character between the
color-representation system associated with each observer
(see comments that follow).

QuantityVk provides a quantitative measure of the dif-

FIG. 2. Chromaticity coordinates obtained for observer CF for those distributions that are metamers for JAM. The
continuous line represents the ellipse of CF for each color center. The results are referred to the G94 system.

Volume 26, Number 4, August 2001 267



ference in the global colorimetric behavior of two observers.
When comparing a set of several observers, this magnitude
can be used to establish an order of likeness among them.
The results listed in Table III are in accordance with those
provided in Ref. 21. It has been shown in Section IV how
there is no reciprocity between the colorimetric behavior
predicted by the sets of cmf’s associated with two real
observers.

The problem underlying interobserver variability when
comparing different observers was discussed in an article by
Hita et al.14 In that work the causes of some anomalies
found when representing the chromaticity thresholds in CIE
1931 chromaticity diagram were analyzed. These anomalies
were considered to be a consequence of the distortion in-
troduced when one set of experimental data is transformed
from one color-representation system to another. To mini-

FIG. 3. Chromaticity coordinates obtained for observer JAM for those distributions that are metamers for CF. The
continuous line represents the ellipse of JAM for each color center. The results are referred to the G94 system.

268 COLOR research and application



mize the distortion, Hitaet al. proposed to work with a
unique system of primaries and with the same experimental
dispositive. This is a partial solution to the problem derived
from the nonisomorphic character between the color-repre-
sentation systems associated with a given pair of observ-
ers.22,23 This question together with the validity of estab-
lishing an adequate standard observer should be further
investigated.

ACKNOWLEDGMENTS

The authors are especially endebted to Dr. J. A. Martı´nez,
Dr. E. Hita, and Dr. M. Melgosa at Departamento de Optica
of Granada University for their continual advice and help.
They also want to express their gratitude to the anonymous
referees who comments helped us to improve the article.

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TABLE VI. Quantities VCF,l and VJAM,l. See text for
details.

Color center VCF,l VJAM,l

A 0.8 0.2
B 4.1 0.2
C 1.0 1.7
D 7.8 0.7
E 4.6 1.3

Volume 26, Number 4, August 2001 269