Theoretical Considerations on SmallColor Differences Ascribed to theStandard Observer Made on the Basisof Individual Color-Matching Functions

Fernando Carreno,* Jose Miguel Ezquerro, Jesus M. ZoidoEscuela Universitaria de Optica, Universidad Complutense de Madrid, C/Arcos de Jalon s/n, Madrid 28037, Spain

Received 8 February 2007; revised 19 February 2008; accepted 28 February 2008

Abstract: An analytical method to determine how color-matching functions influence the perception of chromatic-ity differences is proposed. We show that, as a conse-quence of the observer metamerism, a metameric color-match perceived by one observer may appear to be a sig-nificant mismatch to a different observer. It is also shownthat, on average, the differences between the color-match-ings made by two different observers can be estimated tobe in the order of 2 CIELAB units. � 2009 Wiley Periodicals,

Inc. Col Res Appl, 34, 194 – 200, 2009; Published online in Wiley Inter-

Science (www.interscience.wiley.com). DOI 10.1002/col.20493

Key words: color vision; variability of color-matchingfunctions; small color differences

INTRODUCTION

Given an observer a, the perception of color stimuli is

specified in terms of the corresponding color-matching

functions (cmf’s) and the spectral characteristics of the

spectral radiant flux q(k) which evokes the color sensa-

tion. It is well-known that, in color science, one usually

deals with tristimulus values.

Xai ¼ k

Z l2

l1

rðlÞxai ðlÞ dl; i ¼ 1; 2; 3; (1)

where, xai ðkÞ stands for the i-th color matching function

of observer a in the selected colorimetric system and k is

a normalization constant. In the CIE 1931 system, k ¼683 lm/W. The interval [k1, k2] is the visible spectral

range considered. Most authors take k1 ¼ 400 nm and k2

¼ 700 nm. Any color stimulus can be specified in terms

of a vector belonging to a three-dimensional space, in

such a way that

Ra ¼ Xa1 ;X

a2 ;X

a3

� �for any rðlÞ

� �: (2)

It is the subset of the space which contains the vectors

representing all the color stimuli perceived by the ob-

server a.

Let us consider two observers, labeled a and b. Usu-

ally, the sets of cmf’s associated with each observer will

be different between them. This fact is usually known as

variability in cmf’s. Taking into account this fact, a sim-

ple inspection of Eq. (1) reveals that a spectral distribu-

tion q(k) will produce different color sensations to both

observers. The variability in cmf’s is the origin of the dis-

crepancies in the absolute specification of color stimuli

when we are interested in the evaluation of color percep-

tion by a set of different observers. Interobserver variabil-

ity can be formally expressed by saying that the subset Ra

and Rb, associated with the observers a and b, respec-

tively, are nonisomorphic between them from a global

point of view. From earlier work, it was expected that the

comparison among the color stimuli as perceived by dif-

ferent observers will provide us valuable information

about the influence of the variability of cmf’s on the

absolute color perception. Several analyses have been car-

ried out to gain a further insight of this effect.1–4

On the other hand, the sensitivity of the visual system

to small color differences has been an area of intense ac-

tivity since the pioneering work of MacAdam.5 In that

work, the elliptical geometry was adopted for representing

the thresholds of chromatic discrimination. In subsequent

investigations, the thresholds were considered to be

ellipsoids in the experiments where luminance and

*Correspondence to: Fernando Carreno (e-mail: ferpo@fis.ucm.es).

VVC 2009 Wiley Periodicals, Inc.

194 COLOR research and application

chromaticity were both allowed to vary.6–8 See also Refs.

9–19 and references therein for a comprehensive overview

of the research activity in this field.

When a color-matching experiment is performed to

determinate the color discrimination threshold around a

given color center, it is expected to obtain different

results for different observers. From an experimental point

of view, the comparison of the differences among the

thresholds will allow us to analyze the interobserver vari-

ability in the perception of small color differences. In this

work, we propose a theoretical procedure which allows us

to obtain quantitative information about interobserver var-

iability in the perception of small differences in chroma-

ticity. This is an intriguing question, since although the

absolute specification of colors may differ among differ-

ent observers, there may be a certain degree of similarity

among them when judging chromaticity differences. We

partially addressed this problem earlier.20 In that work,

the interobserver variability when judging luminance

thresholds was analyzed from a theoretical point of view,

but it should be pointed out that the analysis was re-

stricted to the evaluation of difference among thresholds

associated to a single color attribute (luminance). In this

work, we carry out a complementary analysis to have

available a theoretical procedure that allows us to per-

form a complete study about the interobserver variability

in the perception of color thresholds. By assuming that

color stimuli that lie within the thresholds of color-dif-

ferences associated with a given observer are perceived

essentially as identical by the observer, we evaluate the

way in which these color-differences are equal or not for

a set of different observers. For these purposes, we will

use the set of data supplied in Ref. 19 since the authors

of that study have provided the complete set of data

required for our analysis. In the next section, we present a

description of the theoretical background. The numerical

results and the discussions are presented in the section enti-

tled ‘‘Inter-observer variability in the judgment of small

color-differences.’’ Finally, we discuss the major conclu-

sions.

THE EVALUATION OF COLOR-DIFFERENCES

Given a reference observer (a ¼ st), let us consider the

thresholds of color-differences around a set of S color

centers associated with this observer. In what follows,

each color center will be labeled with superindex m, run-

ning from 1 to S. The different thresholds are specified by

the tristimulus values of the considered color center

(center of the ellipsoid) and the matrix of covariances

derived from the analysis of the set of experimental

color-matching data. In this work, we will consider the

CIE 1931 standard observer as the reference observer.

For practical reasons, we use the main chromaticity

section of the ellipsoid. This section is obtained by pro-

jecting the ellipsoid onto the chromaticity diagram for a

selected value of the luminance. Usually, the chosen value

is the mean luminance Xst;m2;0 of the color center. Under

this assumption, the parameters needed to characterize the

m-th threshold of color-differences are the luminance

Xst;m2;0 , the chromaticity coordinates (xst;m1;0 , xst;m2;0 ) of the

center of the ellipse, the semiaxes am and bm of the

ellipse, and the angle ym which indicates the orientation

of the major axis with regard to the x1-axis of the chro-

maticity diagram. Note that, we use x1 ¼ x and x2 ¼ ywhen referring to the conventional chromaticity coordi-

nates.

In the following, we propose an analytical method to

evaluate how the thresholds of color-differences associ-

ated with the standard observer are perceived by a set of

different observers. These observers are not directly

involved in the color-matching experiments but we can

perform our analysis by using their corresponding set of

color-matching functions, which are known. The proposed

method proceeds as follows:

1. Let us consider the m with tristimulus values and the

corresponding threshold of color-differences, we con-

sider the elliptic section with mean luminance Xst;m2;0 .

This section is specified by the values am, bm, hm, and

by the chromaticity coordinates of the center (xst;m1;0 ,

xst;m2;0 ).

2. By using the software tool described in Refs. 21 and

22, a set consisting of five spectral power distributions

(SPD) qm;j0 (k) has been generated, where superindex jruns from 1 to 5. These SPD’s reproduce the tristimu-

lus values of the m color center for the reference ob-

server, i.e., condition

Xst;mi;0 ¼ k

Z l2

l1

rm;j0 ðlÞxsti ðlÞdl ði ¼ 1; 2; 3; j ¼ 1; � � � ; 5Þ

(3)

is satisfied.

3. We consider a set of Q straight lines which contain the

center of the ellipse. Each one of these lines has a dif-

ferent slope and is labeled with the subindex n (n ¼1 , � � � , Q). The intersections (xst;m1;n , xst;m2;n ) of n-th line

with the boundary of the ellipse are calculated. The

color-difference between each one of these points and

the color center (xst;m1;0 , xst;m2;0 ) should be nearly identical

for all of them. This fact follows from the definition of

color-difference threshold. The number Q of straight

lines can be selected by changing their slope chosen at

angles an with the major axis in constant steps. The

procedure is illustrated in Fig. 1. In our case, we have

changed the angle an in steps of 108 degrees in such a

way that the number Q of straight lines is 18 when the

whole ellipse is covered. We obtain the tristimulus val-

ues for each one of the points intersecting the bound-

ary of the ellipse by taking into account that we are

working in a plane of constant luminance Xst;m2;0 .

4. With the aid of the software tool previously men-

tioned, we have also generated for each point (xst;m1;n ,

xst;m2;n ) in the color center m five SPD qm;jn (k) which

Volume 34, Number 3, June 2009 195

reproduce the tristimulus values Xst;mi;n for the reference

observer as follows:

Xst;mi;n ¼ k

Z l2

l1

rm;jn ðlÞxsti ðlÞdl ði ¼ 1; 2; 3; j ¼ 1; � � � ; 5Þ:

(4)

5. We now consider a set of M observers whose sets of

color-matching functions xai ðkÞ are known (each ob-

server is labeled with superindex a which runs from 1

to M). For each color center m, we have computed the

tristimulus values

Xa;m;ji;0 ¼ k

XNs¼1

rm;j0 ðlsÞxai ðlsÞDl; (5)

obtained for the SPD rm;j0 (l) which reproduce the tristi-

mulus values of the color center for the reference ob-

server. We have also calculated the tristimulus values

Xa;m;ji;n ¼ k

XNs¼1

rm;jn ðlsÞxai ðlsÞDl; (6)

for each SPD rm;jn (l). In the two previous expressions,

we have approximated expression (1) by a sum, Dlbeing the spectral width of the sampling interval, s is

an index to specify the wavelengths at which the spec-

tral functions are sampled, and N is the number of

sampling points.

6. In the approximately visually uniform CIELAB color

representation space,23 for each one of the observers

a and for each color centers m, we have computed

the coordinates (La;m;j0 , aa;m;j0 , ba;m;j0 ) associated with

the tristimulus values of the color center given by (5)

and the coordinates (La;m;ln , aa;m;ln , ba;m;ln ) obtained from

the tristimulus values (6) (l,j ¼ 1 , � � � , 5). Since we

have generated five SPD for the center of the ellipse

and other set of five SPD for each one of the inter-

sections, we have a collection of 25 possible color

differences for each one of the straight lines depicted

in Fig. 1. From these data, we have evaluated the dif-

ferences:

DEa;m;j;l0;n ¼ ðLa;m;jn � La;m;l0 Þ2 þ ðaa;m;jn � aa;m;l0 Þ2

h

þðba;m;jn � ba;m;l0 Þ2i1=2

; j; l ¼ 1; . . . ; 5 ð7Þ

We perform the average over the 25 possible color

differences associated to each straight line and the

result is labeled as (shDiE)a;m0;n ). The standard devia-

tion for the color differences associated to each

straight line is also computed (s(DE)am0;n). In the par-

ticular case that a = st these differences should be

very small due to the fact that the SPD’s are meta-

meric for the reference observer. In addition, we will

get the following result:

sðDEÞst;m0;n ¼ 0:

7. We are interested in establishing the similarity between

the observer a and the reference observer st when eval-

uating color-differences at a given color center m, we

have carried out a statistical analysis of the sets of data

hDEist;m0;n and hDEia;m0;n . The mean value

DEa;m ¼ 1

Q

XQn¼1

hDEia;m0;n ; (8)

and the standard deviations sa,m of the differences (7)

obtained for each observer and each color center are

computed. Further statistical analysis is developed

to obtain descriptors which allow us to establish a global

comparison when evaluating color-differences between

the reference observer and each one of the a observers.

Note that the results stated in Eqs. (5) and (6) are de-

pendent upon the superindex j, whereas this is not the case

in Eqs. (3) and (4). This result arises from the fact that the

SPD’s are metameric for the reference observer, whereas

metamerism is not preserved for the rest of observers.

INTER-OBSERVER VARIABILITY IN THE

JUDGMENT OF SMALL COLOR-DIFFERENCES

The objective of this section is to analyze how different

observers evaluate the color-differences for a given set of

pairs of physical stimuli which are just noticeable for a

reference observer. To do it, we evaluate the differences

between each individual observer and a standard one

(CIE 1931 standard observer). The set of observers corre-

spond to the cmf’s of the 10 observers of Stiles-Burch.24

In this way, the number of observers is M ¼ 11 when the

reference observer (a ¼ st) is also considered. It should

be pointed out that these observers were not involved in

the experiments reported in Ref. 19. Note that all of these

sets of cmf’s were determined for visual fields of 28.The color centers used in our analysis are those

reported in Ref. 19. These centers are approximately the

same as those recommended by the CIE. Our analysis is

restricted to the luminance levels labeled as ‘‘level 5’’ in

FIG. 1. Color-difference ellipse. (x10st,m, x20

st,m) stands for thechromaticity coordinates of the center of the ellipsewhereas (x1n

st,m, x2nst,m) stands for the chromaticity coordi-

nates belonging to the boundary of the ellipse.

196 COLOR research and application

Ref. 19. With this choice, the number of color centers an-

alyzed is S ¼ 5 (m ¼ 1 , � � � , 5). The tristimulus (Xst;mi;0 )

values of each color center can be determined from the

data in columns 2, 3, and 8 of Table I in Ref. 19. By

using the data provided in Table II of Ref. 19 for ob-

server MP, we can determine the ellipse defining the

thresholds of differences in chromaticity and, from this,

the intersections (xst;m1;n , xst;m2;n ) of the straight lines contain-

ing the center with the boundary of the ellipse. We com-

pute the tristimulus values associated with the 2Q inter-

sections in each color center by considering the luminance

supplied in column 8 in Table I of Ref. 19. Note that the

difference DY (column 6 of Table II in Ref. 19) between

the luminance of the center and a point on the surface

of the ellipsoid with the same chromaticity coordinates

than the center is required for a proper determination of

TABLE I. Averaged color-differences hDEia,10,n, and standard deviations s(DE)a,m0,n for the achromatic center.

n SB1 SB2 SB3 SB4 SB5 SB6 SB7 SB8 SB9 SB10 CIE1931

1 1.97/0.83 1.37/0.47 1.91/0.78 1.76/0.73 2.10/0.76 1.78/0.87 1.92/0.81 2.14/0.96 1.79/0.69 2.33/1.06 1.03/0.002 1.99/0.82 1.37/0.47 1.94/0.78 1.80/0.74 2.15/0.78 1.80/0.93 1.95/0.83 2.17/0.97 1.81/0.69 2.32/1.04 1.03/0.003 2.02/0.83 1.39/0.46 1.99/0.79 1.86/0.74 2.21/0.82 1.85/0.98 2.00/0.85 2.21/1.00 1.85/0.70 2.33/1.03 1.06/0.004 2.06/0.85 1.44/0.45 2.05/0.81 1.94/0.74 2.29/0.86 1.92/1.03 2.08/0.87 2.26/1.02 1.91/0.71 2.35/1.03 1.14/0.005 2.11/0.88 1.50/0.43 2.12/0.83 2.04/0.73 2.36/0.90 2.00/1.06 2.16/0.88 2.33/1.05 1.98/0.73 2.39/1.03 1.24/0.006 2.11/0.97 1.54/0.44 2.08/0.88 2.01/0.71 2.27/0.94 2.02/1.04 2.11/0.89 2.29/1.10 1.91/0.76 2.44/1.07 1.35/0.007 2.11/0.99 1.61/0.44 2.11/0.89 2.10/0.72 2.32/0.98 2.09/1.06 2.17/0.90 2.32/1.11 1.95/0.77 2.43/1.06 1.45/0.008 2.13/1.09 1.65/0.41 2.12/0.97 2.13/0.71 2.40/1.05 1.95/0.86 2.17/0.84 2.32/1.16 1.96/0.80 2.36/1.10 1.53/0.009 2.15/1.12 1.72/0.43 2.16/0.98 2.19/0.72 2.45/1.07 2.01/0.84 2.24/0.84 2.36/1.17 2.02/0.82 2.39/1.13 1.59/0.0010 2.04/1.07 1.74/0.45 2.09/0.92 2.18/0.76 2.39/1.09 1.97/0.74 2.21/0.79 2.27/1.09 2.00/0.76 2.28/1.07 1.63/0.0011 2.05/1.07 1.78/0.47 2.12/0.91 2.20/0.78 2.41/1.10 2.01/0.71 2.25/0.79 2.31/1.08 2.04/0.76 2.31/1.08 1.63/0.0012 2.09/1.07 1.80/0.50 2.16/0.91 2.20/0.82 2.42/1.12 2.09/0.69 2.29/0.81 2.37/1.07 2.09/0.77 2.35/1.09 1.60/0.0013 2.10/1.05 1.79/0.52 2.16/0.90 2.18/0.84 2.41/1.11 2.09/0.67 2.29/0.82 2.39/1.06 2.10/0.76 2.36/1.08 1.55/0.0014 2.10/1.03 1.77/0.55 2.16/0.89 2.15/0.87 2.39/1.10 2.09/0.65 2.27/0.83 2.39/1.04 2.09/0.75 2.37/1.08 1.47/0.0015 2.09/1.01 1.73/0.57 2.14/0.88 2.10/0.89 2.37/1.08 2.07/0.65 2.25/0.84 2.39/1.04 2.08/0.74 2.37/1.07 1.37/0.0016 2.09/0.99 1.68/0.60 2.12/0.87 2.06/0.90 2.34/1.05 2.05/0.67 2.21/0.87 2.38/1.04 2.05/0.74 2.36/1.07 1.27/0.0017 2.09/0.99 1.65/0.62 2.12/0.87 2.03/0.89 2.34/1.02 2.01/0.71 2.19/0.87 2.37/1.04 2.02/0.74 2.36/1.05 1.16/0.0018 2.10/1.00 1.59/0.64 2.11/0.89 2.00/0.89 2.32/1.01 2.00/0.78 2.17/0.89 2.36/1.06 2.00/0.76 2.36/1.05 1.08/0.0019 2.12/1.02 1.55/0.65 2.11/0.92 1.99/0.88 2.31/1.00 2.00/0.86 2.16/0.92 2.36/1.09 1.98/0.79 2.37/1.07 1.03/0.0020 2.22/1.04 1.51/0.65 2.18/0.94 1.98/0.87 2.32/0.99 2.03/0.96 2.18/0.93 2.44/1.12 2.01/0.80 2.48/1.09 1.03/0.0021 2.11/1.11 1.48/0.68 2.08/1.04 1.96/0.92 2.24/1.06 2.17/1.02 2.15/1.03 2.35/1.17 1.93/0.89 2.45/1.13 1.07/0.0022 2.13/1.20 1.48/0.67 2.10/1.11 1.98/0.92 2.26/1.08 2.19/1.07 2.17/1.06 2.35/1.23 1.95/0.94 2.49/1.19 1.15/0.0023 1.84/0.95 1.30/0.52 1.82/0.86 1.78/0.76 2.01/0.90 1.90/0.88 1.93/0.85 2.11/0.93 1.75/0.76 2.23/0.95 1.26/0.0024 1.83/1.02 1.35/0.50 1.82/0.91 1.82/0.74 2.03/0.90 1.90/0.92 1.94/0.86 2.08/0.98 1.76/0.79 2.23/1.00 1.36/0.0025 1.84/1.08 1.40/0.47 1.84/0.94 1.86/0.72 2.05/0.89 1.91/0.94 1.97/0.87 2.06/1.02 1.78/0.82 2.25/1.05 1.46/0.0026 1.86/1.12 1.46/0.45 1.86/0.97 1.90/0.71 2.07/0.89 1.93/0.95 1.99/0.87 2.06/1.05 1.81/0.84 2.28/1.09 1.55/0.0027 1.88/1.16 1.51/0.43 1.89/0.98 1.93/0.69 2.09/0.89 1.94/0.94 2.02/0.86 2.07/1.08 1.85/0.85 2.31/1.13 1.61/0.0028 1.91/1.17 1.55/0.42 1.92/0.99 1.94/0.68 2.10/0.88 1.95/0.93 2.04/0.85 2.08/1.09 1.88/0.85 2.35/1.16 1.64/0.0029 2.86/1.89 1.98/0.90 2.69/1.69 2.53/1.27 2.68/1.32 2.78/1.62 2.78/1.61 3.15/2.25 2.47/1.38 3.14/1.93 1.64/0.0030 2.86/1.89 1.99/0.93 2.69/1.71 2.53/1.30 2.68/1.35 2.77/1.63 2.78/1.64 3.14/2.26 2.47/1.39 3.15/1.94 1.61/0.0031 2.86/1.90 1.99/0.96 2.68/1.73 2.52/1.35 2.68/1.37 2.75/1.66 2.77/1.68 3.14/2.28 2.46/1.42 3.16/1.97 1.56/0.0032 2.86/1.92 1.97/1.00 2.68/1.76 2.50/1.40 2.68/1.41 2.73/1.71 2.75/1.73 3.14/2.31 2.45/1.45 3.16/2.00 1.48/0.0033 2.86/1.94 1.95/1.05 2.68/1.80 2.47/1.45 2.69/1.44 2.71/1.77 2.74/1.79 3.15/2.35 2.44/1.50 3.16/2.03 1.38/0.0034 2.86/1.97 1.91/1.09 2.68/1.85 2.45/1.51 2.70/1.48 2.68/1.85 2.72/1.85 3.16/2.40 2.43/1.55 3.16/2.07 1.27/0.0035 2.87/2.02 1.88/1.13 2.69/1.90 2.44/1.56 2.72/1.53 2.67/1.94 2.72/1.92 3.17/2.46 2.43/1.61 3.16/2.12 1.17/0.0036 2.88/2.07 1.85/1.16 2.70/1.96 2.44/1.61 2.75/1.58 2.66/2.05 2.72/1.98 3.19/2.52 2.43/1.68 3.16/2.18 1.08/0.00

Each column corresponds to an observer a. The first column provides an index to label each one of the intersection points between thelines passing through the center and the boundary of the ellipse.

TABLE II. Average values DEa,m and standard deviations sa;m of color-differences for the different colorcenters.

DEa,m/ra,mm ¼ 1

Achromaticm ¼ 2Green

m ¼ 3Red

m ¼ 4Blue

m ¼ 5Yellow

SB1 2.22/1.34 4.19/1.45 2.26/0.49 1.98/0.43 2.04/0.62SB2 1.65/0.70 3.95/1.39 2.04/0.48 1.83/0.35 1.84/0.52SB3 2.19/1.20 4.07/1.39 2.23/0.55 1.99/0.48 2.03/0.68SB4 2.11/1.00 4.05/1.42 2.29/0.59 2.05/0.48 2.06/0.73SB5 2.36/1.12 4.01/1.34 2.26/0.55 2.05/0.52 2.10/0.76SB6 2.15/1.19 3.62/1.16 2.17/0.59 1.94/0.48 1.92/0.60SB7 2.26/1.17 4.18/1.55 2.25/0.61 2.11/0.55 2.23/0.93SB8 2.47/1.50 4.10/1.45 2.21/0.61 1.90/0.45 2.03/0.80SB9 2.05/1.01 4.34/1.59 2.28/0.53 2.05/0.48 2.17/0.84SB10 2.53/1.39 4.55/1.69 2.37/0.63 2.12/0.56 2.13/0.74CIE1931 1.35/0.00 3.90/0.00 2.19/0.00 2.08/0.00 2.04/0.00

m ¼ 1, achromatic center; m ¼ 2, yellow center; m ¼ 3, blue center; m ¼ 4, red center; and m ¼ 5, green center.

Volume 34, Number 3, June 2009 197

the corresponding tristimulus values. Finally, note that the

data for the reference white used for transformation to

CIELAB are also provided in Ref. 19.

The averaged color-differences/standard deviations

hDEia;m0;n /r(DE)a;m0;n obtained along each straight line at the

achromatic color center (m ¼ 1) are listed in Table I for

each one of the observers reported in Ref. 24. We also

provide the color-differences computed for the standard

observer (a ¼ st) in the last column of this table. It

should be pointed out that these differences are not as

small as expected by taking into account that they are

computed between color stimuli that belong to the same

color-difference threshold. In the case of considering the

reference observer, stimuli (Xst;m;j1;n , Xst;m;j

2;n , Xst;m;j3;n ) and

(Xst;m;l1;0 , Xst;m;l

2;0 , Xst;m;l3;0 ) should be indistinguishable for all

value of m, l, and j. A visual inspection of Table I reveals

that for each observer the color-differences between the

considered pairs of physical stimuli qn1,j(k) and q0

1,l(k) (l, j¼ 1 , � � � , 5) present a high scattering of the data. A simi-

lar behavior is obtained for the other color centers consid-

ered in Ref. 19. For the sake of brevity, and to avoid an

excessive length of the article, we do not reproduce the

complete list of tables corresponding to all the color

centers, although the data are available from the authors

on request. Instead of reproducing the complete list of

color-differences, we resort to list in Table II the average

DEa,m given by Eq. (8) and the standard deviations ra,m

for each observer and all the color centers. The data listed

in Table II indicate that the discrepancies in evaluating

color-differences not only depend on the cmf’s of the

FIG. 2. Percentage of deviations da0,n22st,m (open circles) and da0,n

82st,m (þ) (a) m ¼ 1 achromatic center, (b) m ¼ 2 yellowcenter, (c) m ¼ 3 blue center, (d) m ¼ 4 red center, and (e) m ¼ 5 green center.

198 COLOR research and application

observer under consideration but present strong variations

depending on the considered color center.

Note that in a general case, the interval of color mis-

match for a given observer a, which is given by [DEa,m 2

ra,m, DEa,m þ ra,m], depends on the considered color

center (m). The greatest intervals are obtained in the case

of the green color center (m ¼ 2). It should be pointed

out that although the chromaticity ellipses were deter-

mined for the same confidence level,19 the color-differ-

ences for the green color centers in CIELAB units are

larger in average than those associated to the other color

centers, even for the reference observer (CIE1931 stand-

ard observer in this work).

It is desirable to get a parameter which allows the

quantitative determination of the discrepancies when eval-

uating the mean color differences for a given observer ain a color center m for each direction n. One way to do it

is to compute the relative deviations by using the follow-

ing magnitude:

daa�st;m0;n ¼

hDEist;m0;n � hDEia;m0;n

hDEist;m0;n

����������� 100 : (9)

This magnitude provides a local estimation of relative dis-

crepancy in evaluating color-differences in each direction

for a given color center. Figure 2(a) shows the magnitude

da0,na–st,1 for the achromatic color center (m ¼ 1), and for

observers SB2 (a ¼ 2) and SB8 (a ¼ 8). This figure

reveals that the magnitude da0,na–st,1 presents deviations as

high as 140% in estimating color-differences for the ach-

romatic color center. Note that observer SB8 (SB2) is that

which in average presents the high (low) values of color-

differences closest to those obtained for the standard ob-

server, according to the data listed in Table II. Note that

for observer SB2, the percentage of deviations is around

20% for several pairs of metameric samples and at certain

value of n the percentage is increased up to 40%. This

asymmetry in the percentage of deviations is an indication

that the color-difference thresholds for observer SB2 and

the reference observer differ in shape and orientation each

other. This difference is strongly appreciated for the case

of observer SB8. Figures 2(b)–2(e) present the percentage

of deviations for the same observers and the other color

centers. Note that the discrepancies in evaluating color-

differences are dramatically modified when considering

different color centers. The smallest discrepancies are

those associated to the yellow color center [Fig. 2(b)],

whereas the largest discrepancies are those associated to

the green color center [Fig. 2(e)].

The results previously found provide a further confir-

mation concerning the nonreciprocity between the colori-

metric behavior of two real observers.3 Furthermore, they

provide a convincing demonstration of how the variability

of cmf’s not only influences the absolute perception of

physical stimuli, but the evaluation of small color-differ-

ences.

Finally, if we want to evaluate the global degree of

similarity in evaluating color-differences among the set of

observers and the reference observer, we look for a statis-

tical descriptor. At first sight, we could try to use the av-

erage over all color centers of quantities DEa,m, i.e.,

DEa ¼ 1=m

P5m¼1 DE

a;m. However this is not an adequate

procedure since the quantities DEa,m present a high scat-

tering in the corresponding standard deviations (ra,m), i.e.,

the statistical data are different from each other since ra,m

6¼ ra,m0in the case that m 6¼ m0. Thus we propose the use

of an alternative statistical descriptor such as the coeffi-

cient of variation CVa,m, defined as follows:

CVa;m ¼ sa;m

DEa;m (10)

The magnitude defined in Eq. (10) incorporates the aver-

age value of the color-differences at color center m and

its corresponding standard deviation. The further averag-

ing of quantities given by Eq. (10) over all color centers

provides a global estimator of the similarity between ob-

server a and the reference observer st in evaluating color-

differences, i.e., we may determine the following:

CVa ¼ 1=mX5

m¼1

CVa;m: (11)

The average value of the coefficient of variation for the

reference observer is CVst ¼ 0.23. The results for the set

of observers are listed in Table III. The observers may be

arranged in order of likeness in the perception of color-

differences with regard to the standard observer by ana-

lyzing the similarity between CVst and CVa: the results

are listed in column 3 in Table III. It is remarkable that

the order of likeness obtained by using this method is

very similar to that obtained in Ref. 4 where the authors

considered the order of likeness between a given set of

observers and a reference observer in evaluating the abso-

lute specification of colors (see Table VII in Ref. 4).

CONCLUSIONS

In this work, we have examined how small color-differ-

ences are dependent upon the characteristic spectral

responsivities of the visual system, thus variations in the

cmf’s will produce variations in the specification of the

TABLE III. Average values of the coefficients ofvariations and order of likeness in the perception ofcolor-differences.

Observer CVa Order of likeness

SB2 0.304 1SB4 0.336 2SB5 0.336 3SB6 0.341 4SB1 0.342 5SB3 0.344 6SB9 0.346 7SB10 0.366 8SB7 0.369 9SB8 0.379 10

Volume 34, Number 3, June 2009 199

small color-differences. We have shown that a metameric

color-match perceived by one observer may appear to be

a significant mismatch to another observer, as a conse-

quence of observer metamerism. In this work, we have

provided a method to estimate the degree of metameric

mismatch by using information concerning color-differ-

ence thresholds together with a software tool for produc-

ing metameric spectral distributions.21,22

On average, the differences between matches made by

two different observers ranges in the interval from 1.3 to

5.2 CIELAB units, with an average value in the order of

2 CIELAB units. Our theoretical method predicts color-

differences, which are in the order of magnitude

experimentally, found by Alfvin and Fairchaild16 for

interobserver variability. The results obtained in this work

provide a quantitative estimate of the limit of cross media

color reproduction accuracy.

ACKNOWLEDGMENTS

The authors acknowledge the comments raised by the ref-

erees whose valuable comments helped us in improving

our work. They also thank Encarnacion Carreno for revi-

sing the grammar of the manuscript.

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