Visual impression of whiteness and its calorimetric definition. 11.R. Thielert and G. Schliemann

Unilever Forschungsgesellschaft mbH, Hamburg 50, Germany(Received 26 July 1979)

Some years ago a formula W = Y - c (lOOp) for calculating the whiteness W of object colors wassuggested. In this formula Y is the luminous reflectance of the sample, c is a constant, and p = <OF>/<OE> is the ratio of two distances in the 1931 CIE chromaticity diagram. <OF> is the distancebetween the chromaticity point F of the sample and a point 0 with the chromaticity coordinates x0,yo. Point 0 is the center of an ellipse with a semimajor axis a and a semiminor axis b. The angle ofinclination of a with respect to the positive direction of the x axis of the chromaticity diagram is 0.<OE> is the distance between point 0 and the intersection E of the ellipse with the straight linethrough the points 0 and F. Originally the numerical values of the parameters x0, y0, e, a, b, and cwere chosen in such a way that a good correlation between Wand visually determined scale values ofwhiteness D was obtained for all samples that were investigated. In the meantime, however, it be-came apparent that there was some discrepancy between the calculated whiteness W and the visualimpression of whiteness with very light bluish samples. Therefore, a systematic gradual optimizationof the parameters x0 , y0, e, a, and b was carried out. This resulted in a statistically significantimprovement of the correlation between D and Weven with very light bluish samples. The numericalvalues of the optimized parameters are x0opt = 0.3060, Yoop, = 0.3130, opt = 54'. The numerical val-ues of a, b, and the constant c remained unchanged, i.e., a = 0.030, b = 0.009, and c = 1/3.

Some years ago two papers were published by Lanter' andby Rduchle and Schramm 2 containing descriptions of whitescales made of sets of different bleached and opticallybrightened textile samples. The papers also contained scalevalues 6 of whiteness that were determined by visual assess-

ment.3 A white scale that is made in a similar manner and isvisually assessed in the same way is currently known as aCiba-Geigy textile white scale.4 When whiteness values Wwere calculated for this white scale by use of a formula pre-viously published by US,

5 it became apparent that no sufficient

411 J. Opt. Soc. Am., Vol. 70, No. 4, April 1980 0030-3941/80/040411-07$00.50 (0 1980 Optical Society of America 411

TABLE I. Tristimulus values, chromaticity coordinate z, and visually de-termined scale values of whiteness.

Sample X Y Z z D

CGT 130 89.16 92.58 99.09 0.3528 0CGT 140 89.77 93.05 100.88 0.3556 0CGT 150 90.13 93.24 102.78 0.3592 0.19CGT 160 90.20 93.11 103.80 0.3615 0.28CGT 170 90.20 93.01 105.05 0.3644 0.05CGT 180 90.81 93.41 107.19 0.3678 0.80CGT 190 91.14 93.66 109.14 0.3713 1.00CGT 200 91.44 93.75 110.14 0.3729 1.58CGT 210 91.65 93.96 111.21 0.3747 1.46CGT 220 91.95 94.25 112.42 0.3765 2.20CGT 230 92.12 94.35 113.79 0.3790 2.14CGT 240 92.38 94.71 114.92 0.3805 2.28

RG 11 83.50 82.69 96.10 0.3664 0RG 12 82.42 87.67 91.74 0.3504 0.36RG 13 85.03 86.14 99.13 0.3667 1.36RG 14 76.35 79.31 92.72 0.3733 1.53RG 15 76.07 79.15 91.20 0.3701 1.62RG 16 87.75 91.31 97.35 0.3522 1.73RG 17 94.02 95.54 113.10 0.3737 2.09RG 18 95.17 98.69 114.63 0.3716 2.84

H 21 75.72 78.19 87.76 0.3632 0.04H 22 86.27 89.11 95.29 0.3520 0H 23 76.71 78.79 93.90 0.3765 0.79H 24 77.83 80.42 98.07 0.3826 0.89H 25 82.79 85.95 101.14 0.3748 1.05H 26 86.57 89.47 102.74 0.3685 1.35H 27 95.17 98.69 114.63 0.3716 2.00H 28 93.04 96.01 112.55 0.3732 2.19H 29 92.51 94.97 114.87 0.3799 2.84H 30 94.21 97.13 115.58 0.3766 2.91

correlation could be obtained6 between the calculatedwhiteness values W and the visually determined scale values6 of whiteness that were published by the manufacturer of thewhite scale. We, therefore, tried to find out the reason for thisinsufficient correlation. To this end we compared the tris-timulus values of 12 samples (CGT 130-CGT 240) of a Ciba-Geigy textile white scale with scale values D of whiteness thatwere obtained by us using the method of paired comparisons.Both the color measurement and the visual scaling were madeby us using measuring devices and procedures of visual as-sessment that have already been described in connection withour earlier findings.5 The samples of the textile white scalewere, in their original condition, slightly translucent.Therefore, the color of the background influenced the colorof the samples. In order to eliminate this undesired influencewe folded the originally 8 X 11 cm rectangular samples, thusobtaining samples of 5.5 X 8.0 cm that were sufficientlyopaque. The insufficient correlation between W and 6 waspresumed to be caused by those samples of the textile whitescale whose luminous reflectances were the highest. There-fore, we investigated two additional sets of samples which, inopposition to the sets that we used in our earlier investigations,contained samples exhibiting luminous reflectances Y > 93.These samples were of the same kind as those used in ourearlier investigations, i.e., they consisted of stacks of 8 X 12cm rectangular sheets of paper. 5 One of these two additionalsets consisted of eight samples some of these being eithergreenish-white or reddish-white (RG 11-RG 18). This set was

similar to the set of samples RG 1-RG 10 of our earlier in-vestigations. However, it contained samples with luminousreflectances Y > 93. The other set consisted of 10 mainlyneutral white or bluish-white samples (H 21-H 30). However,in opposition to the bluish-white samples of the Ciba-Geigytextile white scale, the set of samples H 21-H 30 was composedin such a way that the tristimulus values Y of the samples werenot correlated with their chromaticity coordinates z (see TableI). The set of samples H 21-H 30 was, with respect to itscomposition, similar to the sets of samples H 1-H 10 and H11-H 20 of our earlier investigations. However, between thesamples H 21-H 30 there were some with luminous reflec-tances Y > 93.

COLOR MEASUREMENTS

Color measurements were made by use of a spectropho-tometer with an integrating-sphere attachment and a lumi-naire containing a xenon arc lamp. The samples were illu-minated diffusely by light whose relative spectral-energydistribution corresponded approximately to CIE illuminantD60. Because all samples contained fluorescing opticalbrighteners, the light reflected from the samples was com-bined with fluorescent light at certain wavelengths. Thus thetotal relative spectral radiance Px of the light diffusely re-flected and emitted from the samples at 00 was measured,where P\ is the sum of the spectral reflectance pA, of thesample and the spectral radiance fx of the fluorescent radia-tion emitted at the same wavelength X relative to the spectralradiance of a nonfluorescent nonselective white standard inthe same illumination,

PX = PX + fA. (1)

From the PA, tristimulus values X, Y, Z of the 1931 CIEcolorimetric system were calculated on the basis of illumina-tion7 with CIE illuminant D60. Further details concerning

0.34-

0.33-

y

0.32-

0.30

+ D60

00

00

0

0

0OoUd7

000

0.31 0.32

FIG. 1. Chromaticity coordinates of samples CGT 130-CGT 240.

412 J. Opt. Soc. Am., Vol. 70, No. 4, April 1980

I IlA 1.

R. Thielert and G. Schliemann 412

reflectance of the sample, and c is a constant. The quantityp is intended to characterize the saturation of the sample. Itis specified by the equation

p = (OF)/(OE),

0.33-

y

0.32-

.

0.30

00

0

@ 0

00~00 +

O *

0

0.37

.

0

0.32x

FIG. 2. Chromaticity coordinates of samples RG 11-RG 18(S) and H 21-H30 (0). The samples RG 18 and H 27 are identical. Therefore, theirchromaticity coordinates are marked by the symbol O.

these measurements have been published elsewhere.5 Thetristimulus values of the samples are listed in Table I. Thechromaticity coordinates of the samples are shown in Figs. 1and 2. These figures also include the chromaticity point ofCIE illuminant D60 and the center 0 of an ellipse used by usfor whiteness calculations in connection with our earlier in-vestigations. 5

VISUAL ASSESSMENT OF WHITENESS

Whiteness was visually assessed by the method of pairedcomparisons. The samples were assessed by 40 observers whowere mostly the same as in the earlier investigations. Fromthe judgements of these 40 observers scale values D ofwhiteness were determined. 5 The samples were illuminatedby a luminaire whose spectral-energy distribution corre-sponded approximately to that of CIE illuminant D60. Thatmeans, the color measurements and the visual assessmentswere made using light sources whose relative spectral-energydistributions were nearly identical.

In Fig. 3 the relationship is shown that exists between thescale values 6 of the samples CGT 130-CGT 240 determined 4

by the manufacturer of these samples, and the scale values Dthat were determined by us. Obviously there exists a goodcorrelation between 6 and D.

CORRELATION BETWEEN VISUALLYDETERMINED SCALE VALUES D OF WHITENESSAND COLORIMETRICALLY DETERMINEDWHITENESS VALUES W

For colorimetric determination of whiteness, the formula

W= Y- c (lOOp) (2)

was suggested by us. 5 W is the whiteness, Y is the luminous

(3)

where (OF) 'and (OE) are distances in the 1931 CIE chro-maticity diagram. (OF) is the distance between the chro-maticity point F of the sample and a point 0 with coordinatesx0 and yo. Point 0 is the center of an ellipse whose semimajorand semiminor axes are a and b, respectively. The angle ofinclination of the major axis 2a with respect to the positivedirection of the x axis of the chromaticity diagram is 0. (OE)is the distance between point 0 and the intersection E of theellipse with the straight line through the points 0 and F.

In our earlier investigations we chose x0 = 0.3090, yo =0.3170, 0 = 480, a = 0.030, b = 0.009, and c = 1/3. When wecalculated whiteness values W by use of these numericalvalues and Eqs. (2) and (3), we obtained, with those three ofthe samples CGT 130-CGT 240 whose scale values D and lu-minous reflectances were the greatest, whiteness values Wthat were obviously too small with respect to the scale values7 of these samples (Fig. 4). With the samples RG 11-RG 18we obtained a good correlation between D and W, though thesamples RG 17 and RG 18 exhibited luminous reflectances Y> 93. Among the samples H 21-H 30, there were three whosewhiteness values W were too small with respect to their scalevalues D of whiteness. These were the samples H 24, H 29,and H 30. Two of them, H 29 and H 30, exhibited luminousreflectances Y > 93. However, this also held for the samplesH 27 and H 28 and, as already mentioned, held also for thesamples RG 17 and RG 18, whose whiteness values were of the

2.0-

00

/

720 150 200 250a

FIG. 3. Relationship between visually determined Ciba-Geigy scale valuesof whiteness 6 and visually determined scale values of whiteness D ofsamples CGT 130-CGT 240.

413 J. Opt. Soc. Am., Vol. 70, No. 4, April 1980

0.34-

+ D60

U1 J-1 , ,

R. Thielert and G. Schliemann

Samples CGT 130- CGT 240

270

.190 2300.240

180

100-

90-

w80-

90-

80-

70-

60-

-n. t11b1.'0 2.0 3.0

D

Samples RG 11-RG 18

* 170 8 100-

90-

80-

* 13016

014*15

704 22

,.12

1.0 2.0 3.0 0D

Samples H21-H30

* 28 0 30027 029

026

230 *25

a24

.o 2.0 3.0D

FIG. 4. Relationship between visually determined scale values D of whiteness and calorimetrically evaluated whiteness values Wof samples CGT 130-CGT

240, RG 11-RG 18, and H 21-H 30. Whiteness values Wwere calculated with x0 = 0.3090, yo = 0.3170, 0 = 48°, a = 0.030, b = 0.009, and c

= 1/3.

expected magnitude. In contrast with these samples, thesample H 24, whose whiteness value W was also too small,

exhibited a rather small luminous reflectance (Y = 80.42).However, this sample had the greatest chromaticity coordi-nate z of all samples (Table I). All other samples with too

small whiteness values W (i.e., sapiples CGT 220, CGT 230,

CGT 240, H 29, and H 30) also had great chromaticity coor-dinates z.

Among all samples with correct whiteness values W only

the sample H 23 had a chromaticity coordinate z that was not

smaller than those of the samples with too small whitenessvalues W. The sample H 23, however, exhibited a luminousreflectance Y < 80 and presumably, therefore, had a whiteness

value W of the expected magnitude. Therefore, one obtainsobviously too small whiteness values W by use of Eqs. (2) and

(3) and the numerical values xo = 0.3090, yo = 0.3170, 0 = 480,

a = 0,030, b = 0.009, and c = 1/3 if samples are both distinctlybluish and very light.

Originally the numerical values of xo, yo, 0, a, b, and c were

simply chosen in order to obtain a good correlation betweenD and W with all samples that were investigated by us.

Moreover, we tried to obtain a difference as small as possiblebetween the ellipse that we used in our calculation and theellipse that Honjyo and Nonaka8 derived from their investi-gations. We, therefore, continued our investigations with a

systematic optimization of the correlation between D andW.

OPTIMIZATION OF THE CORRELATION BETWEEND AND W

In order to optimize the correlation between D and W, we

systematically varied the parameters xo, yo, 0, a, b and in-vestigated the relationship between these parameters and thecorrelation between D and W. In this way we evaluated thenumerical values of xo, yo, 0, a, and b, by use of which an op-timal correlation between D and W could be achieved.

The variation of the parameters was made in three succes-

sive steps:

(i) Variation of the location of the ellipse by use of which

p of Eq. (3) is specified, i.e., variation of xo, yo, and0;

(ii) variation of the size of this ellipse, i.e., variation of aand b with the ratio a/b remaining unchanged; and,

(iii) variation of the shape of the ellipse, i.e., variation ofb.

With all sets of parameters that were obtained in this way,whiteness values W were calculated using Eqs. (2) and (3).From these W and the appropriate scale values D we calcu-lated correlation coefficients

(4)

n

E7 (Di -D)(Wi -W)rL(D, W) = i= 1

n - n - 1/2E(Di-) _ (W, -W)2

i=l i=l

(L = 1, 2,. . ., N) for N = 10 sets of samples, i.e., seven setsfrom our earlier investigations and three new sets. D is themean of all n scale values Di and W is the mean of all nwhiteness values W1 of each set of samples. Each of thesecorrelation coefficients rL (D, W) is an estimate of a true butunknown correlation coefficient RL (D, W).

Next we tried to formulate the concept of optimal correla-tion between D and W more precisely. To this end we as-sumed that an optimal correlation between D and W isachieved if the mean correlation coefficient 7 (D, W) that hadto be calculated from the rL (D, W) exhibits a maximum. Amean correlation coefficient F(D, W), however, can be definedonly if all N correlation coefficients rL (D, W) are estimatesof acommon correlation coefficient R (D, W) of the populationof D and W, or, in other words, if

R1 (D, W) = R2(D, W) = *- = RL(D, W) =

= RN(D, W) = R(D, W). (5)

We, therefore, tested with which sets of parameters x 0 , yo,0, and a and b, respectively, correlation coefficients rL (D, W)were obtained that satisfied the condition expressed by Eq.(5). To this end we assumed that Eq. (5) was valid. The

justification of this assumption was tested by use of the x2

test.9 If the assumption expressed by Eq. (5) was justified at

414 J. Opt. Soc. Am., Vol. 70, No. 4, April 1980

R 170* 160

*150

140

1 130

70-

thU 1- I0Ue- . , .

R. Thielert and G. Schliemann 414

TABLE II. Optimal parameter values, correlation coefficients, and 95% confidence limits of maximum correlation coefficients of the population of Dand W

xo 0.3090 0.3070 0.3060 0.3070 0.3060 0.3070 0.3060 0.3070 0.3060 0.3070 0.3070 0.3060Yo 0.3170 0.3150 0.3140 0.3150 0.3140 0.3150 0.3130 0.3140 0.3130 0.3140 0.3140 0.31200 480 480 500 500 52° 520 540 540 560 560 580 600

rnax(D, W) - 0.965 0.970 0.969 0.971 0.970 0.970 0.970 0.971 0.970 0.969 0.967Rt - 0.946 0.952 0.951 0.954 0.953 0.953 0.953 0.954 0.953 0.951 0.949Ru - 0.978 0.981 0.980 0.981 0.981 0.981 0.981 0.982 0.981 0.980 0.979

rL (D, W) of samples

B 1-B 10 0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.997B 11-B 20 0.988 0.989 0.988 0.988 0.986 0.987 0.987 0.988 0.985 0.986 0.985 0.984CGT 130-CGT 240 0.814 0.924 0.942 0.927 0.943 0.930 0.948 0.936 0.949 0.935 0.936 0.944V 1-V 10 0.891 0.965 0.977 0.971 0.979 0.975 0.986 0.985 0.987 0.986 0.987 0.989RG 1-RG 10 0.957 0.963 0.975 0.971 0.976 0.973 0.972 0.963 0.976 0.970 0.969 0.968RG 11-RG 18 0.945 0.948 0.960 0.959 0.967 0.965 0.948 0.940 0.954 0.946 0.949 0.940F 1-F 12 0.930 0.929 0.945 0.950 0.953 0.959 0.957 0.961 0.956 0.962 0.956 0.948H 1-H 10 0.946 0.922 0.897 0.942 0.907 0.950 0.923 0.957 0.923 0.958 0.952 0.928H 11-H 20 0.982 0.978 0.965 0.975 0.962 0.972 0.953 0.963 0.952 0.961 0.959 0.942H 21-H 30 0.944- 0.980 0.987 0.978 0.987 0.975 0.983 0.972 0.982 0.969 0.967 0.977

a 95% confidence level, mean correlation coefficients F(D, W)were calculated. After that we determined the maximum 7rmax(D, W) of the mean correlation coefficients r(D, W). Thismaximum mean correlation coefficient Fmax(D, W) can beinterpreted as an estimate of the maximum correlation coef-ficient Rmax(D, W) of the population of the D and W. The95% confidence limits of Rmax(D, W) were also calculated.The x2 test, the calculation of the mean correlation coeffi-cients r-(D, W), and the calculation of the 95% confidencelimits of Rmax(D, W) were carried out following currentmathematical procedures. 9 In order to calculate the r(D, W)we started with the transformation of the rL (D, W) by use ofthe equation

± + rL(D, W)UL(D, W) = 1 In (t - rL (D,W) (6)

The use of the UL (D, W) instead of the rL (D, W) facilitatesthe calculations because the UL(D, W) are approximatelynormally distributed and the rL (D, W) are not. From the UL(D, W) we calculated estimated mean values

NZ_ (nL - 3) UL (D, W)

U(D, W) =1NY2 (nL - 3)

L=1

(7)

The nL represent the number of the pairs Di, Wi that wereused for the calculation of the rL (D, W). Each nL, therefore,is identical with the number of samples that formed togethera set.

From the U(D, W), mean correlation coefficients 7(D, W)were calculated using the equation

1 (t + (D, W)U(D, W) = hI I 7!(D, W))

NY. (nL - 3)/(nL - 1)

+ r(D, W) L=1 L (8)2 N

Y' (nL - 3)L=1

Because it is not possible to solve Eq. (8) with respect to r

(D, W), these mean correlation coefficients had to be calcu-lated by use of a gradual approximative procedure. We,therefore, first calculated from each U(D, W) an estimater*(D, W) of r(D, W) by neglecting the second term of theright-hand side of Eq. (8). We then substituted this estimater* (D, W) into the second term of the right-hand side of Eq.(8). Thus we obtained a new estimate r**(D, W) of N(D, W)using the equation

1 (1 +r**(D, W)- In - r2 I1 - F**(D, iW_)j

NY1 (nL - 3 )/(nL - l)

*(D W) L=1= U(D W) F* W) L-1

Z_ (nL - 3)L=1

(9)

This procedure was repeated until the magnitude of the lastestimate of r(D, W) did not differ from the preceding estimatein the third decimal place.

RESULTS OF OPTIMIZATION

The determination of the optimal location of the ellipse wascarried out by varying its center (xo, yo) between 0.3000 < xo< 0.3120 and 0.3050 ' yo ' 0.3200 in steps of Ax0 = 0.0010and A yo = 0.0010. The angle 0 was varied between 480 < 0' 60° in steps of AO = 20. All other parameters remainedunchanged, i.e., a = 0.030, b = 0.009, and c = 1/3. With thesenumerical values of the parameters, whiteness values W of allsamples were calculated by use of Eqs. (2) and (3). For eachset of samples and each triplet of numerical values of xo, yo,and 0, correlation coefficients rL (D, W) were calculated.From the rL(D, W), mean correlation coefficients r(D, W)were calculated using Eqs. (8) and (9) if they were defined.The results of this part of the optimization are listed in TableII. These are:

(i) The numerical values of x0 and yo with which we ob-tained maximum mean correlation coefficients 7 max(D, W) if xo and yo were varied as described above and

415 J. Opt. Soc. Am., Vol. 70, No. 4, April 1980 R. Thielert and G. Schliematin 415

loon

w

50

0 2 4D

TI r 16 8 10

Samples B11-B20100-

50-

4 l

Samples CGT 130-CG T 240

S

p

100-

I I I1 2

D

10i

Samples V 1- V 10

I I I I I V0 2 4 6

D

-?2 0 2 0 2 4 0 2D D

Samples H 1 - H 10 Samples

*0 .4

50 2

0 2 4 0 2D

H 11- H20100-

50-

D

Samples H 21- H 30

0 2 4D

FIG. 5. Relationship between visually determined scale values D of whiteness and colorimetrically evaluated whiteness values Wof all samples. Whitenessvalues Wwere calculated with xopt = 0.3060, Yoopt = 0.3130, 0 opt = 540, a = 0.030, b = 0.009, and c = 1/3.

416 J. Opt. Soc. Am., Vol. 70, No. 4, April 1980

Samples 81- 8 1O

100-

I

50-

I I I I

0 2D

w

D

100

W

50

L L . w w . - . . .

50-

R. Thielert and G. Schliemann 416

if 0 was 480, 500, 520, 540, 560, 580, and 600, respec-tively;

(ii) the numerical values of the maximum mean correlationcoefficients rmax (D, W);

(iii) the lower and upper 95% confidence limits R1 and R0of Rmax(D, W); and

(iv) the correlation coefficients rL (D, W) of all sets ofsamples from which the maximum mean correlationcoefficients were calculated.

Moreover, Table II contains the correlation coefficients rL (D,W) that we obtained with the numerical values of x 0, yo, 0,a, b, and c that we used in our earlier investigations, 5' 10 i.e.,xo = 0.3090, yo = 0.3170, 0 = 48°, a = 0.030, b = 0.009, and c= 1/3.

All correlation coefficients rL (D, W) of each set of sampleslisted in Table II do not differ significantly from each other.However, no mean correlation coefficient r(D, W) could becalculated from the rL(D, W) that were obtained with thenumerical values of xo, yo, and 0 that were used in our earlierinvestigations. This was because the assumption expressedby Eq. (5) was not justified for these rL (D, W). From all othersets of correlation coefficients rL (D, W) listed in Table II,mean correlation coefficients F(D, W) could be calculated.This can be interpreted as a statistically significant im-provement of the correlation between D and W effected bya variation of the location of the ellipse. The optimal centerof the ellipse, (xoopt, Yoopt), is obviously between 0.3060 < xo< 0.3070 and 0.3130 < yo • 0.3150, and the optimum of theangle 0 is between 520 < 0 < 560. A more precise determi-nation of (xo0 pt, Yoopt) and of 0 opt was not feasible, neither byuse of the mean correlation coefficients Fmax(D, W) nor by useof the 95% confidence limits of Rmax(D, W). The optimumof 0 we defined by choosing 0opt = 54° because this is the"middle" of the "optimal interval" of 0. The optimal centerof the ellipse was also more or less arbitrarily defined bychoosing xoopt = 0.3060 and Yoopt = 0.3130.

The determination of the optimal size of the ellipse wascarried out by introducing new semiaxes a* = ga and b* = gb.The factor g was varied between 0.90 < g < 1.40 in steps of Ag= 0.05. The numerical values of the other parameters werexoopt = 0.3060, Yo~pt = 0.3130, 0opt = 540, a = 0.030, b = 0.009,and c = 1/3. No significant improvement of the correlationbetween D and W could be obtained by the variation of thesize of the ellipse.

The determination of the optimal shape of the ellipse was

made by varying the semiminor axis b between 0.0040 ' b '0.0120 in steps of Ab = 0.0005. The numerical values of theother parameters were x~opt = 0.3060, Yoopt = 0.3150, 0 opt =540, a = 0.030, c = 1/3. The variation of the shape also didnot result in a statistically significant improvement of thecorrelation between D and W. Therefore, we did not continuethe gradual optimization.

In Fig. 5 the relationship between whiteness values W thatwere calculated with xoopt 0.3060, Yopt =,0.3130, Oopt = 540,a = 0.030, b = 0.009, and c = 1/3 by use of Eqs. (2) and (3) andvisually determined scale values D is shown. As can be seenfrom Fig. 5 and the appropriate correlation coefficients rL (D,W) listed in Table II, good correlation between D and W wasobtained for all sets of samples. From Fig. 4 it could bedemonstrated that calculated whiteness values W of very lightbluish samples CGT 220, CGT 230, CGT 240, H 24, H 29, andH 30 were too small with respect to their visual impression ofwhiteness if those numerical values of x0 , yo, and 0 werechosen that were used in our earlier investigations. From Fig.5 it can be seen that all calculated whiteness values W of thesesamples were of the expected magnitude if the calculationswere made using the optimized parameters x0 opt = 0.3060,Y~opt = 0.3130, and 0 opt = 540.

'J. Lanter, "Pruifung und Bewertung von optischen Aufhellern," SVFFachorgan Textilveredl. 19, 469-480 (1964).

2A. Rauchle and W. Schramm, "Allgemeine Betrachtungen zurvisuellen und farbmetrischen Beurteilung von gebleichten und mitAufhellern behandelten textilen Geweben," Textilveredlung 2,719-729 (1967).

3 J. Lanter, "Die Bewertung des Aufhelleffektes und die Bestimmungder Echtheiten optisch aufgehellter Textilien," Ciba Rundsch. 152,26-32 (1960).

4 Anonymous, "Visuelle Weissbewertung," Ciba-Geigy Rundsch.1973/1, 10-13 (1973).

5R. Thielert and G. Schliemann, "Visual impression of whiteness andits colorimetric definition," J. Opt. Soc. Am. 63, 1607-1612(1973).

6 R. Griesser (private communication).7G. Wyszecki and W. S. Stiles, Color Science (Wiley, New York, 1967),

p. 289.8K. Honjyo and M. Nonaka, "Perception of white in a 100 field," J.

Opt. Soc. Am. 60, 1690-1694 (1970).9A. Hald, Statistical Theory with Engineering Applications (Wiley,

New York, 1967), p. 608.10The correlation coefficients of the samples F 1-F 12 and H 1-H 10

that are listed in Table II of our previous paper 5 deviate slightlyfrom those listed in Table II of this paper. This is because thewhiteness values W calculated previously 5 for these samples (seeTable I of our previous paper) were slightly incorrect, presumablydue to computational errors. The correlation coefficients F (D, W)listed in Table II of this paper are the correct ones.

417 J. Opt. Soc. Am., Vol. 70, No. 4, April 1980 R. Thielert and G. Schliemarm 417