The Weber Fraction andAsymmetries in LuminanceThresholds

Fernando Carreno,* Jesus Manuel ZoidoEscuela Universitaria de Optica, Universidad Complutense de Madrid, Calle Arcos de Jalon, E-28037, Madrid, Spain

Received 9 July 2001; accepted 22 October 2001

Abstract: Presented in this article is a method for thedetermination of the Weber fraction by analyzing data ob-tained from color-matching experiments. The method isbased on a rigorous analysis of the probability densityfunction derived from discrimination judgments made in thexyY space. Analysis shows the thresholds for luminanceincrements and decrements are shown to be different andasymmetric and the Weber fraction to depend on the tar-get’s level of luminance and chromaticity. © 2002 Wiley

Periodicals, Inc. Col Res Appl, 27, 330–334, 2002; Published online in

Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/col.

10079

Key words: colorimetry; color matching; Weber Fraction

INTRODUCTION

It has been suggested that sensitivity to luminance change isindependent of chromaticity.1 Nevertheless, in research intocolor-difference models the question of a lightness-depen-dent correction has been considered, 2,3 although the use-fulness of this correction can be questioned. 3,4

In his original work, Brown 1 reported his analysis of thechanges in the dimensions and orientations of discrimina-tion ellipsoids determining the visual sensitivity to colordifferences. To display his experimental results, Brownopted to show the cross section of each ellipsoid in a planeof constant luminance. The radius of each ellipsoid perpen-dicular to the cross section of a plane of constant luminancewas taken as a quantity nearly equivalent to the Weberfraction, �Y/Y. Brown proposed (in the third table of thatwork)1 what he called the “fractional error of luminance,”determined by

�Y

Y�

11.53

�g33

, (1)

where �Y is the standard deviation of luminance matchesand g33 is the coefficient of the discrimination ellipsoid usedfor fitting the experimental data in the xyY space corre-sponding to the term k2 (�Y/Y)2 (see also the first equationin Brown’s work, p. 684). Note that the coefficient g33 inEq. (1) is the only surviving term in the discriminationellipsoid when considering a fixed chromaticity. Similarmethods can be used to obtain color ellipsoids that containa specified percentage of color matches in the CIE xyYspace, identical to those in Melgosa et al.3,5

This procedure of fitting experimental data directly in thexyY space has proved inadequate.6,7. We have shown that,assuming that the data in the XYZ space obey a normaldistribution function, f (X, Y, Z), the corresponding proba-bility density function in the xyY space, fr(x, y, Y), does notyield a normal distribution. This behavior is result of to thenonlinear character of the transformation from the XYZ tothe xyY space. In this article we present a robust method toachieve such a transform along with a rigorous derivation ofthe Weber fraction based on such data.6,7 We also analyzethe influence of chromaticity on luminance thresholds. Al-though we present the theoretical background of our anal-ysis is described only briefly here a fuller description can befound in an article by Carreno and Zoido.6

THEORETICAL BACKGROUND

Consider an experiment in which an observer performscolor matching between a reference field and a variablefield. For a given reference stimulus (X0, Y0, Z0), the ob-server carries out n equalizations around this referencestimulus, with (Xj, Yj, Zj; j � 1, . . . , n) the tristimulusvalues associated with the jth equalization. From the set ofexperimental data such typical statistical descriptors as the

* Correspondence to: Fernando Carreno: (email: ferpo@fis.ucm.es)© 2002 Wiley Periodicals, Inc.

330 COLOR research and application

mean match, standard deviation, and matrix of covariancescan be obtained.1,6 It is assumed that the tristimulus values(Xj, Yj, Zj) are normally distributed around the referencestimulus with the probability density function for the ran-dom vector (X, Y, Z) thus having a normal distribution.8

Consider further the transformation of the stimulus spaceXYZ into (x1, x2, x3) � (x, y, Y) space, commonly used in theliterature.5,8 For a fixed chromaticity (x1

0, x20) � (x0, y0), the

probability density function for color matching is a unidi-mensional function with luminance the independent vari-able. By following the steps we outlined in a previousarticle,6 it can be shown that in the (x1

0, x20, x3) space this

probability distribution for luminance matches is given by

fY� x10, x2

0, x3� � AY � x3�2e�1/ 2H� x3�, (2)

where H(x3) � C11X (x1

0/x20)(x3 � x3

0) � C22X (x3 � x3

0) � C33X

(1 � x10 � x2

0)/x20 (x3 � x3

0) � 2C12X (x1

0/x20 )(x3 � x3

0)2 � 2C13X

(x10/x2

0)2 (1 � x10 � x2

0) (x3 � x30)2 � 2C23

X (1 � x10 � x2

0)/x20

(x3 � x30)2. The matrix {Cij

X} is the inverse of the matrix ofcovariances.

In Eq. (2) the constant AY is determined so that thefollowing relation [Eq. (3)] is satisfied

�0

�

fY � x10, x2

0, x3�dx3 � 1, (3)

which is the well-known closing condition applied to anyprobability density function.

The factor (x3)2 in Eq. (2) means the function fY (x10, x2

0, x3)does not obey a normal distribution. Thus, special care isrequired when deriving sensitivity-to-luminance differencesat a fixed chromaticity in the xyY space, as these are likelyto be asymmetric because of their non-Gaussian character-istic.

Information can be gained by investigating the propertiesof the probability density function derived from Eq. (2). Todo this, we computed the asymmetry coefficient, As, and thekurtosis, K, as

As �� x3 � � x3 � �3

� x3 � � x3 � �23/ 2 , (4)

K �� x3 � � x3 � �4

� x3 � � x3 � �22 .

where those values enclosed in angle brackets are expectedvalues. The values of As and K return to 0 and 3, respec-tively, for a Gaussian distribution; thus, deviations fromthese values describe the deviation of Eq. (2) from normal-ity.

We can specify threshold as that region that encompasses100(1 � �)% of the color matchings accepted by an ob-server. The limits for this threshold, x3

l and x3u, are given by6

�x3

0

x3u

fY� x10, x2

0, x3�dx3 �1 � �

2, (5)

�x3

1

x10

fY� x10, x2

0, x3�dx3 �1 � �

2. (6)

Solving Eq. (5) and (6) returns the values of x3u and x3

l . Notethat in Eq. (5) and (6) we did not assume a priori that thediscrimination boundary is symmetric with the referencestimulus ( x1

0, x20, x3

0). Figure 1 shows the unidimensionalprobability function for color center 19 in Brown,1 indicat-ing the parameters of interest. If Eqs. (5) and (6) are takeninto account, the luminance threshold limit can be given by

�Y � x3u � x3

l , (7)

and, finally, the Weber fraction becomes

WF ��Y

Y�

x3u � x3

l

x30 . (8)

COMPARISON WITH EXPERIMENTAL DATA

The method developed for determination of the Weberfraction will now be applied to data from Brown1 andBrown and MacAdam.8 The calculations involving integralswere performed using the Simpson rule, so convergencewas tested properly. For the data in Brown and MacAdam8

the coefficients of discrimination, gij, specified in the (x, y,0.2 log Y) space, were transformed to obtain the correspond-ing coefficients in the XYZ space, using the method weproposed in a previous article.7

For observer WRJB Brown and MacAdam8, we com-puted the asymmetry coefficient and the kurtosis parametersassociated with the unidimensional probability function[Eq. (2)]. The results are listed in Table I. The calculationsfor kurtosis showed that its deviation differed from a Gauss-ian [Eq (3)] function in the fourth decimal number. How-ever, the asymmetry coefficients differed substantially from

FIG. 1. Unidimensional probability density function of Eq.(2) for color stimulus 19 (x3

0 � Y � 0.0259) from Brown1; x3l

and x3u determine the lower and upper limits, respectively.

Volume 27, Number 5, October 2002 331

0, indicating that the probability density function [fY(x10, x2

0,x3)] cannot be considered as a Gaussian function. From thisthe intervals x3

u � x30 and x3

0 � x31 would be expected to be

different.For all color centers and observers given in Brown and

MacAdam’s article 8 we computed the luminance thresholdsfor � � 0.16 using the procedure described previously. Ingeneral, the upper, Yu � x3

u � x30, and lower, Yl � x3

0 � x3l ,

threshold limits showed asymmetric magnitudes. A measureof this asymmetry can be obtained by analyzing the quotient

qY �Yu

Yl. (9)

Note that the higher the value of �, the lower the asymmetryfound (see a previous article6 for details of the influence of� on the shape of chromaticity thresholds). The value � �0.16 is intermediate between � � 0.32 and � � 0.05. Theformer is a very restrictive criterion, whereas the later is arelaxed one.

The magnitude qY was computed for all color centers anddrawn in Fig. 2(a) for observer WRJB; typical deviationsfrom symmetry are about 10%. For � � 0.05, deviationsfrom symmetry were the range of 35%. The magnitude qY asdefined in Eq. (9) less than unity, indicating that the detec-tion of increments of luminance was always better than thedetection of decrements of luminance. This result contrastswith that reported by Krauskopf 10 and may reflect thedifferent methodologies. Brown and MacAdam 8 used asteady-state condition for color matchings, whereas Kraus-kopf allowed a temporal variation in luminance. Our resultsare in agreement with those reported by Vingrys and Ma-hon11 (see Table II), who found higher thresholds for dec-rements.

For each color we computed the Weber fraction (WF), asdetailed previously. For observer WRJB the values of WFare shown in Figure 2, it also shows the values of 3�Y/Ywhich, according to Brown8 would provide an alternate

estimate of the Weber fraction. Deviations between thevalues obtained by us and those estimated by the methodsuggested by Brown were in the range of 12%. The averagevalue of the Weber’ fraction determined according toBrown’s proposed procedure was 0.046, whereas the aver-age value obtained using our procedure was 0.047 � 0.013.Brown and MacAdam8 recognized that their value (0.046)was high in comparison with the 0.02 usually accepted.9

Note that we found no apparent relationship between theWeber fraction (WF) and luminance (Y) over the range ofluminance studied. The values of qY are listed in Table I.

Figure 3a,b shows the values of the Weber fraction as afunction of chromaticity coordinates (x, y). These plotsreveal that the threshold of luminance is not independent onchromaticity, as first suggested by Brown.8 The high degree

TABLE I. Coefficients of asymmetry and kurtosis for observer WRJB in Brown and McAdam.8 the parameter qyis given in columns 4 and 8.

Color center As K qY Color center As K qY

1 0.125 2.9991 0.9035 21 0.094 2.9999 0.92722 0.104 2.9999 0.9198 22 0.077 2.9999 0.94003 0.108 2.9999 0.9166 23 0.095 2.9999 0.92674 0.157 2.9714 0.8786 24 0.087 2.9999 0.93255 0.101 2.9999 0.9222 25 0.099 2.9999 0.92316 0.093 2.9999 0.9279 26 0.070 2.9999 0.94587 0.106 2.9999 0.9178 27 0.127 2.9989 0.90268 0.086 2.9999 0.9331 28 0.110 2.9999 0.91529 0.159 2.9664 0.8768 29 0.077 2.9999 0.9399

11 0.133 2.9974 0.8975 30 0.090 2.9999 0.929912 0.102 2.9999 0.9209 31 0.065 2.9999 0.948913 0.103 2.9999 0.9204 32 0.063 2.9999 0.950914 0.130 2.9982 0.8997 33 0.083 2.9999 0.936215 0.144 2.9915 0.8897 34 0.070 2.9999 0.945716 0.053 2.9999 0.9587 35 0.111 2.9999 0.914717 0.114 2.9999 0.9124 36 0.066 2.9999 0.948818 0.140 2.9939 0.8922 37 0.059 2.9999 0.953719 0.125 2.9992 0.9036 38 0.087 2.9999 0.932220 0.099 2.9999 0.9236 39 0.078 2.9999 0.9395

FIG. 2. The Weber fraction as a function of luminance(semilog scale) determined using our procedure developed(�), with quantity 3�Y/Y (0) for the data in Brown and Mac-Adam.8

332 COLOR research and application

of scatter in the data does not allow a quantitative predictionof this dependence in a easy way.

We computed the asymmetry coefficient and the kurtosisparameters associated with the unidimensional probabilityfunction given in Eq. (2) for observer WRJB.1 The resultsare listed in Table II. In this case deviations of As from 0 arelower than those obtained from an analysis of color centersin Brown and MacAdam.8 For kurtosis the deviation from aGaussian [Eq. (3)] differs in the fourth decimal number forthe data from Brown.1 Because of this we expected thedeviations from normality for these color centers to be lesspronounced than for those from Brown and MacAdam.8 Forcolor centers 19, 31, and 34 note that the lower the lumi-nance, the higher is the value of As. In other words, theasymmetry in the thresholds for increments and decrementsstrongly depends on the level of luminance, particularly forlow luminance levels. This becomes evident with the im-portance of factor x3

2 in Eq. (2). This behavior is concordantwith results reported by Carreno and Zoido.7 when consid-ering chromaticity thresholds.

By following the method proposed previously we derivedqY and the Weber fraction for � � 0.16 with the data forobserver WRJB in Brown.1 The values of qY are listed inTable II. Typical deviations from symmetry are less than8%. The values of WF for this observer are plotted in Figure4 over a large luminance range together with the valuesderived from Brown’s method. In this case the deviations ofthe data obtained using both procedures are higher than forthe colors in the Brown and McAdam study.8 The averagevalue of the Weber fraction determined according to theprocedure used by Brown is 0.06 � 0.03, whereas theaverage value obtained using the procedure proposed by usis 0.025 � 0.013. Note that the values obtained by ourprocedure are close to those reported by Hecht et al.9 Alsonote that in this case the Weber fraction decreases whenluminance increases, in contrast with the data shown in Fig.2, which might reflect the larger range of luminance.

To compare the results obtained from our procedure

TABLE II. Coefficients of asymmetry and kurtosis forobserver WRJB in Brown.1. The parameter qy is givenin column 4, the luminance in column 5.

Color center As K qY Y

1 0.0913 2.9999 0.9292 0.1302 0.0847 2.9999 0.9342 0.06033 0.0565 2.9999 0.9557 0.1464 0.0454 2.9999 0.9644 0.06214 0.1018 2.9999 0.9213 0.03425 0.0581 2.9999 0.9545 0.0976

19 0.0368 2.9999 0.9710 0.92019 0.0352 3.0000 0.9723 0.46119 0.0697 2.9999 0.9456 0.092019 0.1033 2.9999 0.9202 0.025931 0.0277 2.9999 0.9781 4.8131 0.0359 3.0000 0.9717 2.4131 0.0391 3.0000 0.9692 0.48131 0.0447 2.9999 0.9649 0.13634 0.0188 3.0000 0.9851 7.0034 0.0201 3.0000 0.9840 3.5234 0.0294 3.0000 0.9768 1.1634 0.0343 3.0000 0.9729 0.55634 0.0533 2.9999 0.9582 0.19734 0.0775 2.9999 0.9397 0.0557

FIG. 3. The Weber fraction as a function of the chromaticitycoordinates (x1

0 � x0 and x10 � y0) for the data in Brown and

McAdam.8

FIG. 4. The Weber fraction as a function of luminance(semilog scale), determined using our procedure developed(�), with quantity 3�Y/Y (E) for the data in Brown.1

Volume 27, Number 5, October 2002 333

reported in this article with those given in Hecht et al.9, wefitted the derived values of WF to the formula

�Y

Y� c�1 � �kY��1/ 2 2, (10)

obtaining the values c � 0.012 and k � 30.3 for observerWRJB. These values are substantially different from thosereported by Brown1 for observer WRJB. This discrepancyarises from the method used by Brown not taking intoaccount the non-Gaussian character of the transformedprobability density function, in contrast with our method.The differences between the predictions of both methods arehigh when low levels of luminance are considered.

CONCLUSIONS

When a set of color-matching data is available, the rigorousmethod developed by Carreno and Zoido6 provides a pow-erful technique to predict luminance thresholds. The param-eter qY provides an unambiguous index of threshold asym-metry. For the data provided in Brown1 and Brown andMacAdam8 we have shown that thresholds of decrementsare higher than thresholds of increments. This conclusion isin agreement with results reported by Vingrys and Mahon.11

Our analysis indicates that the Weber fraction depends onluminance (Fig. 4). This dependence was not apparent whenanalyzing the data in Brown and MacAdam8 carried out bythe same observer but with different colors and a morelimited luminance range.

From the results obtained in this work, it also seems thatthe Weber fraction depends on chromaticity (see Fig. 3). Toquantify this dependence precisely, it would be desirable to

repeat the experimental studies to determine the influencethat chromaticity has on luminance thresholds.

The Weber fraction obtained using the methods describedin this article, differs in general with that obtained using therelation WF � 3�Y/Y, as proposed in Brown1 and Brownand MacAdam.8 We believe a rigorous analysis of theexperimental results about any sensitivity threshold shouldtake into account the method described by Carreno andZoido6 when being carried out.

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5. Melgosa M, Perez MM, Hita E. Chromaticity-discrimination thresh-olds with aperture and object colors: experimental results and predic-tions of some recent color-differences formulas. Appl Opt 1996;35:176–187.

6. Carreno F, Zoido JM. Statistics of color matching experimental data.Appl Opt 1999;38:208–218.

7. Carreno F, Zoido JM. The influence of luminance on color-differencethresholds. Color Res Appl 2001;26:362–368.

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