The Influence of Luminance onColor-Difference Thresholds

Fernando Carreno,* Jesus Manuel ZoidoEscuela Universitaria de Optica, Universidad Complutense de Madrid, C/Arcos de Jalo´n s/n 28037 Madrid, Spain

Received 31 January 2000; accepted 30 May 2000

Abstract: We present a discussion about the influence ofluminance on the geometry of color-difference thresholds.By assuming that color matchings are normally distributedin XYZ space, we have shown [Appl Opt 38:208–219(1999)] that the probability density function of colormatches in (x, y, Y) space is not a normal distributionfunction and it is an asymmetric one. Due to this asymmetry,the curves of equally noticeable chromaticity differencesshow asymmetries around the color center. The influence ofluminance on the shape of these curves is analyzed, and weconclude that the lower the luminance, the larger the asym-metry.© 2001 John Wiley & Sons, Inc. Col Res Appl, 26, 362–368, 2001

INTRODUCTION

The size and geometry of regions of equally noticeable chro-maticity differences have been studied for a long time. In theoriginal work of MacAdam,1 he established that these regionscould be considered as ellipses. In some recent works,2–13 theresults presented in Ref. 1 are revised, and the conclusionsderived by MacAdam are called into question. In any case, theelliptic geometry is currently used in color science.14–20

It is commonly assumed21,22 that frequencies of equaliza-tions in a color-matching experiment obey a normal distribu-tion function in the representation space,RRGB, associated withthe stimuli corresponding to the instrumental primaries. In thiscase, the exponent in the distribution function is given by thecorrespondingx2 Pearson distribution function, which can beconsidered as the distance between the vector of independentnormal variables (Q1, Q2, Q3) and the vector of mean values(Q1

0, Q20, Q3

0). By associating distance with probability, theexponent in the normal distribution function provides a mea-sure of the distance between different color stimuli. The metrictensorcij obtained from the matrix of covariances,mij, providesfor a fixed value of the distance,Ds, the ellipsoid containing

inside a certain percentage of the stimuli that are indistinguish-able from the reference stimulus. When projecting the ellip-soids onto other representation spaces, such that (x, y, Y), theobtained regions are ellipses. When the fitting of the experi-mental data to an ellipsoid is carried out in tristimulus values,the projection onto (x, y, Y) space provides an ellipse. If thefitting of the experimental data is directly carried out in the (x,y, Y) space, the chromaticity threshold cannot be consideredapriori to be an ellipse, because in this space the probabilitydensity function of the color matchings does not conform to anormal distribution function. In fact Brown22 explicitly pointedout this question without further analysis.

The effects of the level of luminance on the increase of thechromaticity components of all radii of the discriminationellipsoid was noted by the authors of Ref. 21 for color centers32 and 33 (see p. 832). The influence of luminance was alsoanalyzed in Ref. 21 in detail for three: blue (19a), green (31a),and white (34a). In this work, some indications that the corre-lation coefficients change with luminance were obtained. Theauthors decided to publish only part of their results. In a laterarticle by Brown,23 he presented the final data in the (x, y, 0.2log Y) space for some color centers, including those reported inRef. 21. Brown analyzed the influence of luminance on the sizeand orientation of the discrimination ellipses.

In a recent article24 it has been demonstrated that theprobability density function in the (x, y, Y) space does notobey a normal distribution function. In that work, a methodto determine the shape of color-difference thresholds wasdeveloped and the experimental data provided in Ref. 21were analyzed. In this article, evidence of the asymmetry ofthe regions of equally noticeable chromaticity differences isalso obtained by computing the kurtosis and asymmetrycoefficients for the transformed probability density func-tion. In our previous work,24 we obtained some evidencesindicating that luminance is an important parameter with astrong influence on the shape of the regions of equallynoticeable chromaticity differences. We show that the lowerthe luminance, the larger the asymmetry. In the secondsection, we present a brief description of the theoretical

* Correspondence to: Dr. Fernando Carren˜o, Escuela Universitaria deOptica, Universidad Domplutense 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.

362 COLOR research and application

background of our analysis. An extended description can befound in Ref. 1. The results are then presented for the colorcenters studied in Ref. 23. Finally, we present the mainconclusions.

THEORETICAL BACKGROUND

Let us consider an experiment in which the color-matchingsare carried out with an instrument whose primary stimuli arelinearly related with a new set of stimuliC1, C2, andC3. Acertain color stimulus can be specified in the representationspaceRX generated by the new set of primary stimuli by itscorresponding tristimulus valuesXi (i 5 1, 2, 3). Theobserver makes color matches between a fixed referencefield and a variable field. For a certain reference stimulus(X1

0, X20, X3

0) in the fixed field of observation, he carries outn equalizations around the reference stimulus,X j 5 (X1

j ,X2

j , X3j ) ( j 5 1, . . . , n) being the tristimulus values

associated with thej -th equalization. From the set of exper-imental data, we can compute the mean values,Xi

0, thestandard deviation,s i

X, and the matrix of covariances, {m ijX}

(see Ref. 24). It is assumed that tristimulus valuesXij are

normally distributed around the mean valueXi0, thus the

probability density function for the random vectorX 5 (X1,X2, X3) obeys to a normal distribution (see Ref. 21).

Let us consider a transformation of the representationspaceRX into the (x1, x2, x3) 5 ( x, y, Y) space, commonlyused in the literature (see for instance, Refs. 21, 23, 25).When considering a plane of constant luminance (x3 5 x3

0),it has been shown that in the (x1, x2, x3

0) space the prob-ability density function is given by1

fdia~ x1, x2! 5 Ax~ x3

0!2

x23 expH2

1

2q~x1, x2!J, (1)

where

q~ x1, x2! 5 c11X Sx1x3

0

x22

x10x3

0

x20 D 2

1 2c13X Sx1x3

0

x22

x10x3

0

x20 D

3 Sx30~1 2 x1 2 x2!

x22

x30~1 2 x1

0 2 x20!

x20 D

1 c33X Sx3

0~1 2 x1 2 x2!

x22

x30~1 2 x1

0 2 x20!

x20 D2

,

Ax 5 F ~ x30!2 EE

D~ x1,x2!

1

x23 expH2

1

2q~x1, x2!dx1dx2JG21

,

and$cijX% 5 $mij

X%21.

To characterize the properties of this function, we com-pute some characteristic parameters and we compare themwith those of a normal distribution. Thus, for this purposewe evaluate the principal asymmetry coefficients,Asi, andthe principal kurtosis,Ki, given, respectively, by

Asi 5^~ xi 2 ^xi&!

3&

^~ xi 2 ^xi&!2&3/ 2 , i 5 1, 2,

(2)

Ki 5^~ xi 2 ^xi&!

4&

^~ xi 2 ^xi&!2&2 , i 5 1, 2.

For a normal distribution these parameters take the valuesAsi 5 0 and Ki 5 3. Deviations from this values forfdia( x1, x2) provide us with an estimation of the validity ofthe assumption of elliptic geometry for representing thethreshold of chromaticity discrimination around a givencolor center. For a fixed level of luminance, this threshold isobtained by imposing condition

EEDth

fdia~ x1, x2!dx1dx2 5 1 2 a, (3)

for the confidence levela. Integration process in this ex-pression must be carried out to obtain the chromaticitythresholdDth.

In Ref. 24, a numerical method is used to evaluate theintegral (3). This method proceeds as follows: let us con-sider the chromaticity coordinatesx0 5 ( x1

0, x20) of a certain

reference stimulus. Now we assume that color-matches areguided, i.e., the chromaticities of the color stimuli in thevariable field are located in a straight line containing thereference pointx0 and making an angleu with the x1 axis.The equation of this line is

x2~ x1, u ! 5 ~ x1 2 x10!tan$u% 1 x2

0. (4)

In this case, for a fixed value ofu, the probability densityfunction fdia( x1, x2) reduces to a one-dimensional distribu-tion function of the form

fu~ x1! 5 Au~ x3

0!2

~~ x1 2 x10!tan$u% 1 x2

0!3

3 expH21

2q~x1, x2~x1, u !J. (5)

Taking into account that any color-matching carried out bythe observer lies in the line given by Eq. (4) and within thelimits of the chromaticity diagram, it is clear that the chro-maticity coordinatex1 belongs to the interval [x1

(1), x1(2)]

defined by the intersection of Eq. (4) with the limits of thechromaticity diagram (see Fig. 2 in Ref. 24). We finallyconsider that the one-dimensional region of chromatic in-distinguishability over the line (4) contains 100(12 a)% ofthe color-matches carried out by the observer. The colorstimuli over the line distinguishable from the referencestimulus are given byxs 5 ( x1

s, x2s) andxi 5 ( x1

i , x2i ), with

Ex1

0

x1s

fu~ x1!dx1 51 2 a

2, (6)

Volume 26, Number 5, October 2001 363

Ex1

i

x10

fu~ x1!dx1 51 2 a

2. (7)

By using numerical methods, we can solve Eqs. (6) and (7)and, from Eq. (4), we can determine the pointsxi andxs. If werepeat this procedure for different values ofu, we can deter-mine the threshold of chromaticity discrimination for any ref-erence stimulusx0. Note that in Eqs. (6) and (7) we have notassumeda priori that the region of chromatic indistinguish-ability is symmetric with regard to the reference stimulus.

COMPARISON WITH EXPERIMENTAL DATA

When applying the method proposed in the previous sec-tion, we use the data supplied in Ref. 23: in Table II (p. 686)Brown provides the coefficients of discrimination,gij in the( x, y, 0.2 log Y) 5 ( x91, x92, x93) space. To perform ouranalysis, we require the discrimination coefficients,cij

X, inthe RX space. The relation between both spaces is known,and we can determine coefficientscij

X by means of thewell-known covariant transformation

cijX 5 O

l51

3 Om51

3 dx9ldXi

dx9mdXj

glm. (8)

All the thresholds of chromatic discrimination provided byBrown in Ref. 23 are given for a confidence level of 95%(a 5 0.05).

We have pointed out that the main criticism of the pro-cedure followed by Brown lies in the fact that the relationbetweenRX and (x91, x92, x93) spaces is nonlinear, and, inthis way, the normality hypothesis is not preserved.24

All the calculations involving integrals in the chromatic-ity diagram have been carried out using the integrationprocedure described in Ref. 26: this integration procedure isbased on the Newton rule. The convergence of the integra-tion procedure has properly been tested.

First of all, we have calculated the asymmetry coeffi-cients and the kurtosis parameters associated with probabil-ity distribution (1) for the data of observer WRJB providedin Ref. 23. The obtained results are shown in Table I.Notation in this table, and in the entire article, is coinciden-tal with that used by Brown in Ref. 23. It should be notedthat the asymmetry of distribution function (1) becomesevident. It cannot be assumeda priori that chromaticitydifference thresholds are symmetrically distributed. FromTable I, we see that deviations with regard to the normaldistribution function for the computed quantities,Asi

andKi, strongly depend on the considered color center. It isparticularly interesting to note that for color centers 19a,31a, and 34a, the asymmetry coefficients and kurtosis pa-rameters are high-deviant from those of a normal distribu-tion. This deviation increases with decreasing the lumi-nance. These quantities serve as indicators of the severity ofdeviation with regard to the normal distribution function forthe data associated with the matches carried out around agiven color center. We have found that the larger the devi-

ations of these parameters with regard to those of a normaldistribution, the larger the asymmetries. These deviationsare greater than those found in our previous study.24

By using the procedure described in Section II, thechromaticity-difference thresholds have been determinedfor all the color centers and both observers analyzed inRef. 23. The results fora 5 0.05 are illustrated in Fig. 1for the color centers 1, 2, 3, 4, and 5, and observer WRJB(note that we have determined the chromaticity thresh-olds for the same confidence level used in Ref. 23).Deviations from the elliptic geometry shown in this fig-ure are evident. The most relevant results are obtained forthe series of experiments with fixed chromaticities andvariable luminances carried out for color centers 19a,31a, and 34a. These results are shown in Fig. 2 for colorcenter 34a and observer WRJB. This figure reveals veryhigh asymmetries, some of these are due to the followingfact: for certain directionsu in the chromaticity diagramwe have found that one of the Eqs. (6) or (7) is not satisfied,

i.e.,Ex1

0

x12

fu~x1!dx1 5 bs ,12a

2, orE

x11

x10

fu~x1!dx1 5 bi ,12a

2.

In the first case, the upper limit over the line isx12 (deter-

mined by the upper intersection of the line with the chro-maticity diagram) and the lower limit,x1

i , in Eq. (7) isdetermined according to the following equation:

Ex1

i

x10

fu~ x1!dx1 5 S1 2 a

2 D 1 S1 2 a

22 bsD . (9)

In the second case, the lower limit isx11 (determined by the

lower intersection of the line with the chromaticity diagram)and the upper limit,x1

s, in Eq. (6) is determined accordingto condition

TABLE I. Asymmetry coefficients and kurtosis of dis-tribution (1) for the color centers analyzed for WRJB. Inthe first column we provide the number of the stimulusaccording to the order given in Ref. 23.

N0 As1 As2 K1 K2

1 0.1295 0.1395 3.0248 3.02982 0.2691 0.2752 3.1417 3.14733 20.0697 20.0652 2.6368 2.63634 0.0153 0.0591 3.0015 3.00714 0.3068 0.3215 3.1931 3.20885 0.3438 0.3745 3.1682 3.1881

19a 0.0838 0.1372 3.0181 3.037719a 0.0675 0.1027 3.0110 3.021119a 0.0951 0.1762 3.0255 3.062319a 0.1446 0.2738 3.0600 3.151031a 0.0058 0.0488 3.0009 3.004831a 0.0028 0.0502 3.0004 3.005131a 0.0759 0.1260 3.0149 3.031831a 0.1822 0.2175 3.0714 3.095034a 0.0159 0.0311 3.0008 3.001734a 0.0398 0.5554 3.0036 3.006234a 0.0543 0.0681 3.0065 3.009334a 0.0479 0.0807 3.0059 3.013034a 0.1784 0.1946 3.0659 3.076034a 0.2046 0.2817 3.0964 3.1593

364 COLOR research and application

Ex1

0

x1s

fu~ x1!dx1 5 S1 2 a

2 D 1 S1 2 a

22 b iD . (10)

When these particular situations are produced, the last equa-tions are necessary to guarantee that discrimination thres-

holds have been obtained for a confidence level of 100(12a). Equations (9) and (10) clearly point out the fact thatprobability of color-matching is not symmetric around thecolor center for the considered direction. This situation isillustrated in Fig. 3(a) for the color center 5 studied in Ref.

FIG. 1. Regions of chromatic indistinguishability computed from Eqs. (6)–(7) for a 5 0.05 and observer WRJB for the color centers1, 2, 3, 4, and 5 studied in Ref. 23. The boldface cross indicates the chromaticity coordinates of the reference stimulus.

Volume 26, Number 5, October 2001 365

23: the one-dimensional probability function of Eq. (5)shows an asymmetry for a certain direction in the chroma-ticity diagram for the color center under consideration. InFig. 3(b) we show the values of the integralAs5

Ex1

0

x12

fu~ x1!dx1 for some directionsu. The value ofAs is a

measure of the area under functionfu( x1) between thereference pointx1

0 and the chromaticity coordinatex12 ob-

tained from the intersection of linex2( x1, u ) with thechromaticity diagram. We see that, for certain values ofu,this integral takes values less than (12 a)/2. In these cases,the required percentage of equalizations prescribed in Eq.(6) cannot be achieved and asymmetry in the considereddirection is clearly pointed out.

These results were expected, because the asymmetry ofthe distribution function of Eq. (5) increases for values of( x1, x2, x3) far from the reference stimulus (x1

0, x20, x3

0).These results are coincidental with the experimental resultsobtained by other authors5–10 and they confirm the resultspreviously reported by us.24

We can evaluate if deviations with regard to the ellipseare severe by computing the following distances24:

ds~u ! 5 Î~ x10 2 x1

s!2 1 ~ y10 2 y1

s!2,(11)

di~u ! 5 Î~ x10 2 x1

i !2 1 ~ y10 2 y1

i !2,

in each directionu for all the color centers. In the case thatthe data fit the equation of an ellipse, these distances arecoincidental in all the considered directions. The percentageof deviation

qa~u ! 5uds 2 diu

max$ds, di%100, (12)

provides us with an estimation of the local asymmetry ineach direction for a given color center.

Figure 4 shows the magnitudeqa for color center 34a forthe different levels of luminance. From Table I, we canobserve that for this color center both the kurtosis andasymmetry coefficients take extreme values with regard toAsi 5 0 andKi 5 3. In this case, the asymmetries found byus are so marked as Parra6 indicated. In Table II, we list themaximum values ofqa for observer WRJB and each one of

FIG. 3. (a) Illustration of the asymmetry of one-dimensionalprobability density function fu(x). (b) Value of As as a functionof the angle u for color center 5. Continuous line indicatesthe value (1 2 a)/2.

FIG. 2. Regions of chromatic indistinguishability computedfrom Eqs. (6)–(7) for a 5 0.05, and observer WRJB for thedifferent levels of luminance of color center 34a. (a) (E) Y 57.00, (p) Y 5 3.52, (h) Y 5 1.16, (1) Y 5 0.556. (b) (3)Y 5 0.197, ({) Y 5 0.0557.

366 COLOR research and application

the color centers considered in Ref. 23. Note the high valuesof qa when decreasing the luminance in the series of exper-iments around color centers 19a, 31a, and 34a. These resultsshow that the lower the luminance, the larger the maximumvalues ofqa. This behavior seems to confirm that the lowerthe luminance, the higher the asymmetry.

CONCLUSIONS

Brown noted in Ref. 23 the change of orientation and size ofthe ellipses at different levels of luminance. Although thechange in shape was expected, the reorientation was sur-prising. This reorientation is reproduced when a rigorousstatistical analysis is applied to the experimental data in the( x, y, Y) space. As shown in Fig. 2, this reorientation isespecially remarkable for color center 34a (see Fig. 3 in Ref.23 for comparison purposes). These results are in agreementwith those recently reported in Ref. 25 (see Table II on p.40), although these authors carried out the fittings in (x1, x2,x3) space. We cannot carry out further analysis of the dataof Ref. 25, because the authors of this article provide onlythe final results.

As shown in Ref. 24, when assuming the normalityhypothesis in the representation space associated with theinstrumental primaries, or any linear transformation ofthem, the normality is not preserved when transforming in anonlinear fashion to other representation space, such that( x1, x2, x3). The probability density function in this spaceis not symmetric around the considered color center and,thus, the curves of equally noticeable chromaticity differ-ences are asymmetric.

The asymmetry and kurtosis parameters given by Eq. (2)seem to be good estimators of the deviation of the normalityhypothesis.

From the study presented in this work it can be concludedthat the treatment of the experimental data must be carriedout in the original representation space associated with theinstrumental primaries, or in any representation space lin-

early related with the previous one. In the case when thetreatment of the experimental data is carried out in otherspace and a nonlinear transformation is used, the probabilitydensity function must be adequately transformed, as it hasbeen suggested in both Ref. 24 and in this article.

The rigorous methods developed in Ref. 24 have provento be a powerful technique that should be used to predict thecolor-difference thresholds, if the set of color-matchingexperimental data is available for any color center of inter-est. The magnitudeqa is also a good estimator of the degreeof local asymmetry.

Finally, the effect of luminance on the asymmetry isimportant and must be considered: the results obtained forcolor centers 19a, 31a, and 34a are of great interest andclearly point out the influence of luminance on the geometryof color-difference threshold. The increase of the asymme-try at low luminance levels obviously indicates a deterio-rated chromatic discrimination. It should be desirable tocarry out an experimental study for analyzing the influenceof luminance on color-difference thresholds, to determinethe limits of validity of the elliptic geometry for represent-ing just-noticeable differences. This is of particular impor-tance when proposing and testing color-difference formulas(see Ref. 25): some authors have suggested the need of alightness-dependence correction.27–29

ACKNOWLEDGMENTS

The authors are particularly endebted to E. Bernabeu for hiscontinual advice.

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1 0.130 352 0.0603 733 0.146 154 0.0621 134 0.0342 915 0.0976 79

FIG. 4. Estimation of the local asymmetry by means of themagnitude qa(u ) for observer WRJB and color center 34a atthe different levels of luminance. Symbols as in Fig. 2.

Volume 26, Number 5, October 2001 367

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