colorimetric significance of spectrophotometric errors

8
JOURNAL OF THE OPTICAL SOCIETY OF AMERICA Colorimetric Significance of Spectrophotometric Errors A. R. ROBERTSON* Applied Optics Section, Imperial Collegeof Science and Technology, London, S. W. 7, England (Received 16 November 1966) Some recent interlaboratory surveys have demonstrated poor agreement between color measurements made with different spectrophotometers. It is suggested that instruments should have a precision at least equal to that of the human eye, and that this requires that errors be represented by a distance not greater than 0.2 units in C.I.E. (U*V*W*) space. Tolerances for several different types of error in both the photo- metric and the wavelength scales of a spectrophotometer are derived by calculating their effects on a number of samples, covering a wide range of color, and comparing this effect with the tolerance of 0.2 C.I.E. units. To fulfill this tolerance, errors which are independent of wavelength should be less than 0.4% if they are proportional to spectral reflectance, and less than 0.4% of the luminous reflectance if they are independent of spectral reflectance. If the error varies systematically with wavelength, the tolerances are reduced to half these values. Constant errors of wavelength should be less than 0.2 nm, random errors less than 0.3 nm, and slit widths less than 7 nm. INDEX HEADINGS: Spectrophotometry; Color; Colorimetry. SOME recent papers 1 - 4 have described interlabora- tory comparisons of measurements of spectral re- flectance and spectral transmittance made with a number of different spectrophotometers. These com- parisons have demonstrated that variations of results often occur between different spectrophotometers, whether or not the instruments are of similar design. The purpose of this paper is to discuss the significance of these variations and to determine the range within which spectrophotometric errors must be contained if the resultant errors of computed color specifications are to be insignificant. The discussion is limited mainly to measurements of spectral reflectance, which is de- fined here as the ratio of the spectral density of radiance of the sample considered, under specified conditions of irradiation and observation, to the spectral density of radiance of a perfect diffuser receiving the same irradiance. TOLERANCE FOR COLOR MEASUREMENT ERRORS Spectrophotometers are used for a wide variety of calorimetric problems and the accuracy required of them varies from problem to problem. In some applica- tions quite large color differences can be tolerated; in this case small errors in spectrophotometry are not important, but in other applications the tolerances may be very small and the spectrophotometer is required to have very high accuracy and precision. Moreover, in some instances, failure by the spectrophotometer to detect differences of one attribute such as lightness may be less important than failure to detect other attributes such as hue or saturation. It is therefore impossible to say with any generality how accurate a * Present address: Division of Applied Physics, National Re- search Council, Ottawa, Canada. 1 W. D. Wright, J. Opt. Soc. Am. 49, 384 (1959). 2 A. R. Robertson and W. D. Wright, J. Opt. Soc. Am. 55, 694 (1965). F. W. Billmeyer, J. Opt. Soc. Am. 55, 707 (1965). 4 J. M. Vandenbelt, J. Opt. Soc. Am. 44, 641 (1954). spectrophotometer should be, but a particularly im- portant tolerance is that set by the discrimination threshold of the human eye, and it is with this limit that the errors discussed here are compared. Thus an error is said to be too large if it can cause the instrument (or instruments) to record identical results when meas- uring two samples with perceptibly different colors, or cause it to indicate that two colors identical in their spectral composition differ in color by a perceptible amount. Instrumental errors may cause differences between results from the same instrument on different occasions (lack of precision or repeatability), or be- tween different instruments (lack of accuracy), but no distinction is made between these two cases, since the analysis is the same. The requirement is that not only the precision, but also the accuracy of a spectrophotometer should be as high as the differential precision of the human eye, because spectrophotometers are often used to compare two samples which cannot be placed side by side for visual comparison, either because they are in different geographical locations, or because of the elapse of a period of time between measurement of one sample and of the other. For example, if a paint sample is found to differ in color by a just noticeable amount from a master standard, then without the aid of a spectrophotometer as precise as the human eye, it is impossible to decide whether the sample is indeed the wrong color or whether the master has faded slightly. If for any reason the spectrophotometer used is not the instrument used for the original standardization, then both instruments must also be as accurate as the human eye is precise. In other words, spectrophotom- eters should give results with which a human observer would agree if he could see the two samples (e.g., a master standard at two different periods of time) side by side, even if this situation is not in practice attainable. Of the many methods which have been suggested for estimating the perceptual size of the difference between two colors, the one recommended by the C.I.E. 691 VOLUME 57, NUMBER 5 MAY 1967

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Page 1: Colorimetric Significance of Spectrophotometric Errors

JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Colorimetric Significance of Spectrophotometric Errors

A. R. ROBERTSON*Applied Optics Section, Imperial College of Science and Technology, London, S. W. 7, England

(Received 16 November 1966)

Some recent interlaboratory surveys have demonstrated poor agreement between color measurementsmade with different spectrophotometers. It is suggested that instruments should have a precision at leastequal to that of the human eye, and that this requires that errors be represented by a distance not greaterthan 0.2 units in C.I.E. (U*V*W*) space. Tolerances for several different types of error in both the photo-metric and the wavelength scales of a spectrophotometer are derived by calculating their effects on a numberof samples, covering a wide range of color, and comparing this effect with the tolerance of 0.2 C.I.E. units.To fulfill this tolerance, errors which are independent of wavelength should be less than 0.4% if they areproportional to spectral reflectance, and less than 0.4% of the luminous reflectance if they are independentof spectral reflectance. If the error varies systematically with wavelength, the tolerances are reduced to halfthese values. Constant errors of wavelength should be less than 0.2 nm, random errors less than 0.3 nm,and slit widths less than 7 nm.INDEX HEADINGS: Spectrophotometry; Color; Colorimetry.

SOME recent papers1-4 have described interlabora-tory comparisons of measurements of spectral re-

flectance and spectral transmittance made with anumber of different spectrophotometers. These com-parisons have demonstrated that variations of resultsoften occur between different spectrophotometers,whether or not the instruments are of similar design.The purpose of this paper is to discuss the significanceof these variations and to determine the range withinwhich spectrophotometric errors must be contained ifthe resultant errors of computed color specificationsare to be insignificant. The discussion is limited mainlyto measurements of spectral reflectance, which is de-fined here as the ratio of the spectral density of radianceof the sample considered, under specified conditions ofirradiation and observation, to the spectral density ofradiance of a perfect diffuser receiving the sameirradiance.

TOLERANCE FOR COLORMEASUREMENT ERRORS

Spectrophotometers are used for a wide variety ofcalorimetric problems and the accuracy required ofthem varies from problem to problem. In some applica-tions quite large color differences can be tolerated;in this case small errors in spectrophotometry are notimportant, but in other applications the tolerances maybe very small and the spectrophotometer is requiredto have very high accuracy and precision. Moreover,in some instances, failure by the spectrophotometer todetect differences of one attribute such as lightnessmay be less important than failure to detect otherattributes such as hue or saturation. It is thereforeimpossible to say with any generality how accurate a

* Present address: Division of Applied Physics, National Re-search Council, Ottawa, Canada.

1 W. D. Wright, J. Opt. Soc. Am. 49, 384 (1959).2 A. R. Robertson and W. D. Wright, J. Opt. Soc. Am. 55, 694

(1965).F. W. Billmeyer, J. Opt. Soc. Am. 55, 707 (1965).

4J. M. Vandenbelt, J. Opt. Soc. Am. 44, 641 (1954).

spectrophotometer should be, but a particularly im-portant tolerance is that set by the discriminationthreshold of the human eye, and it is with this limit thatthe errors discussed here are compared. Thus an erroris said to be too large if it can cause the instrument(or instruments) to record identical results when meas-uring two samples with perceptibly different colors,or cause it to indicate that two colors identical in theirspectral composition differ in color by a perceptibleamount. Instrumental errors may cause differencesbetween results from the same instrument on differentoccasions (lack of precision or repeatability), or be-tween different instruments (lack of accuracy), butno distinction is made between these two cases, sincethe analysis is the same.

The requirement is that not only the precision, butalso the accuracy of a spectrophotometer should be ashigh as the differential precision of the human eye,because spectrophotometers are often used to comparetwo samples which cannot be placed side by side forvisual comparison, either because they are in differentgeographical locations, or because of the elapse of aperiod of time between measurement of one sampleand of the other. For example, if a paint sample isfound to differ in color by a just noticeable amountfrom a master standard, then without the aid of aspectrophotometer as precise as the human eye, it isimpossible to decide whether the sample is indeed thewrong color or whether the master has faded slightly.If for any reason the spectrophotometer used is notthe instrument used for the original standardization,then both instruments must also be as accurate as thehuman eye is precise. In other words, spectrophotom-eters should give results with which a human observerwould agree if he could see the two samples (e.g., amaster standard at two different periods of time) sideby side, even if this situation is not in practice attainable.

Of the many methods which have been suggestedfor estimating the perceptual size of the differencebetween two colors, the one recommended by the C.I.E.

691

VOLUME 57, NUMBER 5 MAY 1967

Page 2: Colorimetric Significance of Spectrophotometric Errors

A. R. ROBERTSON

TABLE I. Specifications of four tiles andtwenty-eight Munsell chips.

Sample

11 57298629855

11 616R8/4Y8/4G8/4B8/4P8/4Y8/12

GY8/8R5/4Y5/4G5/4B5/4P5/4R5/10

YR5/10G5/8

PB5/10P5/10R4/14B4/8P4/1212/2Y2/2G2/2B2/2P2/2R2/6

PB2/6P2/6

in 19636 has been chosen for this investigation. Accord-ing to this method, the color difference AE is given by

AE= [(A U*) 2+ (A V*)2+ (AW*)2 1, (1)

where A U*, A V*, and AW* are the differences in U*,V*, and W*, respectively, U*, V*, and WJ* beingdefined by

U*= 13W*(u-vo0)

V*= 13W*(v-vo)

TV'*= 25Y i - 17,where

i=4x/(-2x+12y+3)

(2)

(3)

v= 6y/(-2x+1 2 y+3);

x, y, and Y are the chromaticity coordinates andluminous reflectance of the 1931 C.I.E. system of colorspecification and it0 and vo are the values of it and v forthe illuminant. Alternative, but exactly equivalent,equations for i and v are

u= 4X/ (X+ 15Y+3Z)v= 6 Y/(X+ 15Y+3Z),

(3a)

where X, Y, and Z are the C.I.E. tristimulus values.The C.I.E. did not suggest a value of AE which wouldrepresent the discrimination limit of the human eye,

5 Commission International de l'Eclairage, Compt. Rend. 1963(Report of Committee E-1.3.1).

but an estimate can be made by comparing the size ofcolor differences computed by the C.I.E. method withthe size of the same color differences computed inN.B.S. units" by the formula

AENBS 2= [221 Y' (Aa2+ A±2 )i]2+ [10A (Y)] 2 ,

where2.4266x- 1.3631v-0.3214

y86.1160.7129.475.17

57.1653.5154.5855.7656.9452.2055.5618.9618.9817.4118.0218.1720.9020.3017.0121.0917.8411.909.54

10.863.624.003.193.312.954.133.342.85

(4)

X.

0.31230.31520.31510.29830.34270.36140.29300.28020.29970.45380.37220.39540.38600.27000.23080.29660.49300.49630.24430.21490.28960.55200.19000.28530.36870.35070.27720.25460.29630.43090.19780.2757

y

0.31990.32600.32670.30300.31970.37600.34280.30760.27840.47810.46510.32050.40540.36470.27000.25890.32680.40860.40310.21730.20830.31750.23310.17920.31070.35960.34690.28910.24600.30500.19110.1858

U*

0.14-0.20-0.39-1.1422.6112.11

-21.61-18.48

8.0930.76

-11.1338.9210.85

-26.66-25.94

9.0588.5657.51

-41.75-23.60

21.6996.90

-28.3924.0712.583.30

-8.52-7.90

4.9628.95

-10.438.35

V*

1.894.323.38

-2.154.69

25.087.87

-6.40-16.84

58.7149.736.64

23.027.94

-17.00-16.19

15.5431.5913.42

-37.58-33.46

13.84-22.99-36.19

1.075.581.97

-4.28- 7.82

2.50-19.40-16.53

W*

93.4081.2660.2226.2479.3077.2077.8378.5179.1876.4378.4049.6649.6947.8048.5448.7251.8651.2047.2952.0748.3240.0736.0238.3721.3822.6919.8020.2418.8723.1020.3818.44

1.0000x+2.2633Y+ 1.1054

0.5710x+ 1.2 4 4 7 y- 0.5708B=-

(5)

1.0000x+ 2.2633Av+ 1. 1504

since it is known that color differences smaller than0.2 N.B.S. units are not normally detected by thehuman eye.6 The color changes AE which would resultfrom changes of 0.5 in Y (on a scale 0-100), and 0.001in x and y (each occurring separately), were calculatedby each method for 32 samples and are listed in TableII. The 32 samples, chosen to give as wide a range ofcolors as possible, were the four neutral tiles used inone of the interlaboratory comparisons, 2 and 28 coloredchips from the Munsell Book of Color. They are listed

TABLE II. Comparison of C.I.E. and N.B.S.color difference formulas.

Color differences AE resulting from changes of:(i) AY=0.5 (ii x=0.001 (iii) Ay=0.001

Sample C.I.E. N.B.S. C.I.E. N.B.S. C.I.E. N.B.S.

11 572 0.2 0.3 0.9 0.8 0.7 0.69862 0.3 0.3 0.8 0.7 0.6 0.59855 0.4 0.5 0.6 0.6 0.4 0.4

11 616 1.4 1.1 0.3 0.4 0.2 0.3R8/4 0.3 0.3 0.8 0.7 0.6 0.5Y8/4 0.3 0.3 0.7 0.6 0.5 0.5G8/4 0.3 0.3 0.7 0.7 0.5 0.5B8/4 0.3 0.3 0.7 0.7 0.5 0.5P8/4 0.3 0.3 0.8 0.7 0.6 0.6Y8/12 0.4 0.4 0.6 0.5 0.4 0.4

GY8/8 0.3 0.3 0.6 0.6 0.4 0.4R5/4 0.7 0.6 0.5 0.5 0.4 0.4Y5/4 0.7 0.6 0.4 0.5 0.3 0.3G5/4 0.7 0.6 0.4 0.5 0.3 0.4B5/4 0.7 0.6 0.5 0.6 0.4 0.4P5/4 0.6 0.6 0.5 0.6 0.4 0.5R5/10 1.1 0.5 0.5 0.5 0.5 0.4

YR5/10 0.9 0.6 0.5 0.4 0.4 0.4G5/8 0.9 0.6 0.4 0.5 0.2 0.3

PB5/10 0.7 0.5 0.6 0.7 0.5 0.5P5/10 0.8 0.6 0.6 0.6 0.5 0.5R4/14 2.1 0.7 0.4 0.4 0.5 0.4B4/8 1.3 0.8 0.4 0.6 0.3 0.4P4/12 1.3 0.8 0.5 0.6 0.5 0.5R2/2 2.0 1.3 0.2 0.3 0.2 0.3Y2/2 1.7 1.2 0.2 0.3 0.2 0.3G2/2 2.0 1.4 0.2 0.4 0.1 0.2B2/2 2.0 1.3 0.2 0.4 0.1 0.3P2/2 2.1 1.4 0.2 0.4 0.2 0.3R2/6 2.5 1.2 0.2 0.3 0.2 0.3

PB2/6 2.6 1.3 0.2 0.4 0.2 0.3P2/6 2.8 1.4 0.2 0.4 0.2 0.3

6 D. B. Judd and G. Wyszecki, Color in Business, Science andIndustry (John Wiley & Sons, Inc., New York, N. Y., 1963).

692 Vol. 57

Page 3: Colorimetric Significance of Spectrophotometric Errors

SIGNIFICANCE OF SPECTROPHOTOMETRIC ERRORS

in Table I together with their C.I.E. (x,y,Y andUs, V*, W*) specifications, derived from spectrophoto-metric measurements of the samples. (Differences be-tween these and published specifications for Munsellchips may be due to inaccurate spectrophotometry orto fading of the chips, but are unimportant since onlya representative series of color specifications is required.)From Table II it can be seen that, although ratios asgreat as 2 occur in individual cases, the average sizeof AE calculated by the two methods is the same,7

and it has therefore been assumed that a difference of0.2 C.I.E. units will not normally be perceptible to thehuman eye. EThe name "C.I.E. unit" is used here todenote the unit of color difference defined by Eqs. (1)to (3).]

Thus, in assessing the importance of calorimetricerrors, an error greater than 0.2 C.I.E. units has beenconsidered significant since this is a tolerance which isoften required in industrial and commercial problems." 8Smaller tolerances are unlikely to be needed in theseproblems, although they may be desirable when a spec-trophotometer is used as a research tool.

RESULTS OF INTERLABORATORYCOMPARISONS

Robertson and Wright2 have published the resultsof an interlaboratory comparison made on behalf ofCommittee E-1.3.1 of the C.I.E. in which a number oflaboratories made spectrophotometric measurements offour neutral tiles. The results were summarized intables of the means and root mean square deviationsof x, y, and Y for two groups of spectrophotometers. Infurther analysis of the results, the number of groupshas been increased to three by considering GeneralElectric instruments separately from other integrating-sphere instruments. The three groups are:

(i) instruments with directional illumination andlight collection (450 incidence and normal collection orthe inverse),

(ii) instruments using integrating spheres for eitherillumination or collection (General Electric instrumentsexcluded),

(iii) General Electric instruments.

The color difference AE between each result and themean of its group has been calculated by Eqs. (1)-(3),and the root mean square values of AE are listed inTable III for each group.

The results demonstrate that none of the groups ofinstruments have given results which agree amongthemselves within the required tolerance. Thus if twosamples, visibly just different in color, are measured,one with one spectrophotometer and the other with asecond, the two instruments, even if they are of the

I See also G. Wyszecki, J. Opt. Soc. Am. 53, 1318 (1963).8 F. W. Billmeyer, Internationale Farbiagung (Luzern, Switzer-

land, 1965).

TABLE III. Summary of the results of an interlaboratory com-parison of color measurement.2 Root mean square values of thecolor differences from the mean measurements for several groupsof spectrophotometers. The figures in brackets indicate thenumber of instruments in each group.

Root mean square values of AEIntegrating- General

Luminous Directional sphere Electricreflectance instruments instruments instruments

Sample (Y) (12) (7) (13)

11572 85 1.2 0.5 0.49862 60 1.1 0.7 0.49855 30 0.8 1.2 0.4

11 616 5 0.7 3.3 0.8

same group, will not be able to differentiate withcertainty between the colors. These results are especiallydisturbing when it is remembered that the four samplesall have spectral reflectances which are almost constantwith wavelength; it is likely that similar errors formore saturated colors would cause more serious errorsin the computed color specifications. This point isdiscussed in more detail below.

The results of another interlaboratory comparisonhave been published by Billmeyer.3 In this, coloredsamples were measured as well as gray ones, but onlyGeneral Electric spectrophotometers were used. Com-parison of the spread of results with visual discrimina-tion data8 again shows errors of measurement con-siderably greater than are acceptable.

EFFECT OF SPECTROPHOTOMETRIC ERRORS

Since the basic measurement made by a spectro-photometer is of spectral reflectance (So) as a functionof wavelength (X), it is of considerable interest toknow what tolerance is allowable for errors in fOx, andfor errors in X, if the tolerance for computed colorspecifications is to be met. Several authors have esti-mated such tolerances, but have dealt mainly withlimited groups of samples. For example, Shipley andWalker9 considered the maximum error of chromaticitycoordinates (x and y) which could occur with a spectro-photometric error within certain limits, when theselected-ordinate method of integration (with 30 ordi-nates) was used. Their calculations were restricted toa series of blue filters, but showed that an accuracy ofabout 0.4% of the luminous transmittance is necessaryto achieve a precision equal to that of the human eye.Wright' used the weighted-ordinate method of integra-tion for six colored tiles and estimated that an ac-curacy of at least 0.5% is required in measurements offix. Nimerofft 0 showed how the effect of a random error infix could be calculated for any sample, and gave graphsshowing uncertainty ellipses on an x-y chromaticitydiagram for a random error of Ox of 0.1, for samples

9 T. Shipley and G. L. Walker, J. Opt. Soc. Am. 46, 1052 (1956).10 I. Nimeroff, J. Opt. Soc. Am. 43, 531 (1953).

693May 1967

Page 4: Colorimetric Significance of Spectrophotometric Errors

A. R. ROBERTSON

with x,=10.0 (scale 0-100). MacAdam" using amethod similar to that of Nimeroff showed that forsamples with a luminous reflectance of 26.0, the un-certainty of fx should not be more than 0.25.

The rest of this paper is devoted to a discussion oftolerances for several different types of error, includingwavelength errors, for 32 samples covering a wide rangeof color. Both systematic and random errors are con-sidered, because although systematic errors of the rawdata can often be recognized and removed or mini-mized before colorimetric computations are made,'2 itis apparent from the results of the intercomparisons'-4that many significant systematic errors escape suchattention. A calculation of the effect of each indicateshow serious each type of systematic error may be, andto what extent it needs to be minimized.

Integration Methods

The first step in the calculation of a color specifica-tion from a spectrophotometric curve is the weightedintegration of the values of fiN to give the tristimulusvalues X, Y, and Z

X = kfExixtdA, (6)

with similar equations for Y and Z. Ex is the relativespectral radiance of the light source (taken throughoutthis paper to be C.I.E. standard source "C"), ;Zx>, Vx,and 2, are the color mixture functions of the observer[taken to be the 1931 C.I.E. standard (20) observer],and k is a constant, chosen to give Y= 100 for a perfectdiffuser. Since these integrals are usually evaluatedapproximately by summations over a large number ofwavelengths

X=Efx(kExtxAX), etc., (7)x

using published tables of (lExxX) etc., it is importantto know what errors, if any, can be introduced by theapproximations. De Kerf'3 has studied the variouscommonly used methods of summation by computingthe tristimulus values of a number of samples usingthe weighted-ordinate method with ordinates at 1, 5,and 10 nm intervals and the selected-ordinate methodwith 30 and 100 ordinates. He assumed that weightedordinates at 1-nm intervals give the correct result, andcalculated the differences between this and the resultsof the other methods in N.B.S. units for each sample.He found that the 10-nm weighted-ordinate methodgave results within 0.2 N.B.S. units except for a fewsamples with very rapidly varying fix curves. For thisreason, and in view of the fact that it is very widelyused, this method of integration has been chosen for

"D. L. MacAdam, J. Opt. Soc. Am. 43, 533 (1953).12 K. S. Gibson, Spectroplzo!otmetry, Natl. Bur. Std. Circular 484

(1949); H. J. Keegan, J. C. Schleter, and D. B. Judd, J. Res.Natl. Bur. Std. (U. S.) 66A, 203 (1962).

13 J. L. F. de Kerf, J. Opt. Soc. Am. 48, 334 (1958).

the calculations below. Errors of the integration methodin any case have only a second-order effect on thecalculation of the effect of measurement errors, and itis thus justifiable to use this method to calculate thesize of errors which may be of the order of 0.2 N.B.S.units.

The summations have been made over the wave-length range 400-700 nm only, using color mixturefunctions slightly adjusted in the manner suggested byMacAdamr4 which assumes that the values of fx outsidethis range can be estimated by parabolic extrapolation.

Variability of Color Mixture Functions

The color mixture functions of the 1931 C.I.E.standard observer are widely used to derive from aspectrophotometric curve a color specification which isindependent of the color vision of any particularindividual. If two metameric colors are computed tobe an exact match for the standard observer, an actualobserver may see a mismatch to a greater or lesserextent depending on how much his particular colormixture functions differ from the standard functions.The spreads of such matches can be predicted' andindicate that the highest spectrophotometric precisionis not needed in some instances since human observers,the ultimate judges, disagree quite considerably. How-ever, if two nonmetameric colors match according tothe standard observer, then all actual observers willagree that they match, and in order to obtain preciseand unambiguous color specifications the standard ob-server data must be taken to have no variability. It isin these cases that the highest precision is required ofspectrophotometers.

Errors Independent of Px

The effect of any error of spectral reflectance isobviously greatest at wavelengths where the color-mixture functions are great. This is fortunate becausethe accuracy of many spectrophotometers is least atthe extremes of the visible spectrum where these func-tions are least. However, to simplify the problem, onlyerrors with regular patterns through the spectrum havebeen studied here, and no special attention has beenpaid to any particular wavelengths.

One of the simplest forms of spectrophotometricerror is an error which, although it may vary withwavelength, is independent of fi,. Such errors may becaused, for example, by stray light, by errors of the"zero line" of the instrument, or by insensitivity of themeasuring device due to noise, to the thickness of agalvanometer needle, or to the thickness of a linedrawn on graph paper by a recording instrument. Theerror may be systematic, always having the same value,

14 D. L. MacAdam, J. Opt. Soc. Am. 43, 622 (1953).1"I. Nimeroff, J. R. Rosenblatt, and M. C. Dannemiller, J.

Opt. Soc. Am. 52, 685 (1962); 1. Nimeroff, J. Opt. Soc. Am. 56,230 (1966).

694 Vol. 57

Page 5: Colorimetric Significance of Spectrophotometric Errors

SIGNIFICANCE OF SPECTROPHOTOMETRIC ERRORS

Aox, at a given wavelength, or it may occur randomlywith a standard deviation a-vv at a given wavelength.In the first case, the error of the computed tristimulusvalues is given by

AX=E XNAOX (8)

and similar equations forAY and AZ. Xx is an abbrevia-tion for (kE-xxAX). If AX, AY, and AZ are smallcompared with X, Y, and Z, the magnitude of theerror in C.I.E. units can be found by Eq. (1) with AU*,AV*, and AW* given by

c9U* (2 U*3(u*--=) AX+ AY+(AZ

a x dYI A(9)

and similar equations for AV* and AW*, where thepartial derivatives (d U*/OX) etc. are obtained bydifferentiation of Eqs. (2) and (3a). It is apparent thatif the error Aix in the spectrophotometric curve ismultiplied by a factor e at all wavelengths, then thecolor error AE will also be multiplied by a factor eprovided that AX/X, AY/Y, and AZ/Z remain smallenough for Eq. (9) to be valid. Thus, if an error A/xcauses a color error of AE= e for a particular sample,multiplication of Aflx by 0.2/e at all wavelengths willlead to an error of 0.2 C.I.E. units.

The effects of two errors of this kind have beencalculated for each of the 32 samples listed in Table I.The first is a constant error of 1.00 independent ofwavelength ($x is given throughout on a scale of 0-100),and the second is an error which varies linearly fromits maximum positive value at 400 mu to its maximumnegative value at 700 mnw

(AOX)1 =1.00, (10)

(AO.X)2= 1.68[(550 - )/I50]. (I1l)

The factor 1.68 in Eq. (11) is introduced so that theroot mean square deviation of Ox remains equal to 1.00.There are, of course, many other ways in which asystematic error may vary with wavelength, but Eq.(11) was chosen for study because it is likely to havea particularly large effect, especially on chromaticity.For each error, and for each sample, the value, e, ofAE was calculated. Then, if e=0. 2 /e, we can deducethat the maximum sizes of the errors (Alx)1 and (?,) 2such that the tolerance of AE==0.2 C.I.E. units is notexceeded, are found by multiplying the right-hand sidesof Eqs. (10) and (11) by e, giving

(Aix)i= e,

(AO3A)2= 1.68e[(550-X)/150].

(lOa)

(Ila)

In general, e, the root mean square error, has a differentvalue for each sample and for each type of error; thesevalues are listed in Table IV for the examples con-sidered here.

TABLE IV. Tolerances for errors in 0,\ such that AE (or ao&E)<0.2 C.I.E. units. (The values given are the tolerances for ejineach of the equations.)

Type of errorSystematic Random

(550-X)USA =e UAfl=1.

6 8e - ax =eSample 1 50

11572 0.47 0.16 0.369862 0.37 0.13 0.299855 0.23 0.09 0.19

11616 0.07 0.04 0.08R8/4 0.33 0.12 0.27Y8/4 0.29 0.11 0.25G8/4 0.32 0.13 0.27B8/4 0.33 0.14 0.28P8/4 0.34 0.14 0.29Y8/12 0.17 0.09 0.22GY/8 0.22 0.10 0.24R5/4 0.12 0.06 0.14Y5/4 0.14 0.06 0.13G5/4 0.13 0.07 0.14B5/4 0.14 0.08 0.16P5/4 0.15 0.07 0.16R5/10 0.07 0.06 0.13

YR5/10 0.08 0.05 0.12G5/8 0.10 0.06 0.13

PB5/10 0.15 0.10 0.19P5/10 0.13 0.08 0.17R4/14 0.04 0.04 0.09B4/8 0.09 0.07 0.12P4/12 0.10 0.07 0.14R2/2 0.05 0.03 0.06Y2/2 0.06 0.03 0.06G2/2 0.05 0.03 0.06B2/2 0.05 0.03 0.07P2/2 0.05 0.03 0.07R2/6 0.04 0.03 0.06

PB2/6 0.05 0.05 0.08P2/6 0.05 0.04 0.08

The derivation of the effectis a little more complicated,

of a random error, oApx,since in this case it is

necessary to take into account the correlations amongthe three tristimulus values. The variances of X, Y,and Z are given by

VX=F Xx2cT-X2, etc.

and the covariances by

CYZ=F YXZXaX2\, etc.x

(12)

(13)

assuming that the values of Ox at different wavelengthsare uncorrelated and that o-x is small compared withOx at each wavelength. The variance of U* is given by

(au*\2 / U*\2 /2U*\ 2

Vu),= VX± Y+ V+zx d adY / az

[a (i au* au a*0/u*+ 2 cy+ - Cyst

oaus uu*+ -) ()CXy} (14)

695May 1967

Page 6: Colorimetric Significance of Spectrophotometric Errors

A. R. ROBERTSON

TABLE V. Tolerances for percentage errors, fx, of ,3, such thatzE (or o-&g) <0.2 C.I.E. units. (The values given are the toler-ances for c in each of the equations.)

Type of errorSystematic Random

(550-X)jf=c fx=1.68c x ac

Sample 150

11 572 0.54 0.19 0.429862 0.61 0.23 0.489855 0.78 0.31 0.65

11616 1.38 0.68 1.45R8/4 0.60 0.21 0.48Y8/4 0.60 0.25 0.52G8/4 0.61 0.27 0.51B8/4 0.61 0.24 0.49P8/4 0.61 0.20 0.47Y8/12 0.49 0.29 0.52

GY8/8 0.53 0.31 0.53R5/4 0.70 0.29 0.73Y5/4 0.80 0.43 0.82G5/4 0.80 0.50 0.83B5/4 0.77 0.40 0.76P5/4 0.85 0.30 0.75R5/10 0.44 0.23 0.61

YR5/10 0.54 0.36 0.74G5/8 0.68 0.60 0.84

PB5/10 0.66 0.33 0.69P5/10 0.71 0.25 0.71R4/14 0.40 0.26 0.68B4/8 0.79 0.55 0.98P4/12 0.72 0.28 0.85R2/2 1.35 0.68 1.66Y2/2 1.45 0.83 1.71G2/2 1.49 1.05 1.86B2/2 1.47 0.93 1.82P2/2 1.50 0.76 1.89R2/6 0.93 0.51 1.73

PB2/6 1.09 0.82 1.72P2/6 1.19 0.66 1.85

Similar equations apply for Vv* and Vw*. For a completespecification of the error distribution in (U*,V*,W*)space, the covariances Cu*v*, etc. should also be calcu-lated, but for this investigation it has been consideredsufficient to calculate the rms value of AE. This hasbeen termed the standard deviation of the computedcolor and is denoted by OAE. It is measured from themean, or true color (AE=0), and is not the standarddeviation of AE itself which would be measured fromthe mean value of AE. It is easily shown to be given by

O'AE (VU*+ Vv*+ VW.)' (15)

and is independent of the covariances.As with the systematic type of error, if a random

error characterized by aox (a function of X) causes astandard deviation of e C.I.E. units, then ox must bereduced by a factor 0.2 /E at all wavelengths if thestandard deviation is not to exceed 0.2 C.I.E. units.The only type of random erior considered here is onein which the standard deviation does not vary throughthe spectrum

ox= e. (16)

Values of e, such that oAB is equal to 0.2, were calcu-lated by the method already described for systematic

errors, and are given in Table IV for each of the 32samples.

From Table IV it is seen that the most serious of thethree types of error is the systematic error which variesthrough the spectrum; tolerances for the rms deviationof Ox are as low as 0.03 for many of the darkest samples.The tolerances for the constant and random errors areabout twice as great, and similar to each other exceptthat the constant error tends to have more effect onsamples of saturated color. As is to be expected, theleast tolerances are for the samples with low luminousreflectance, and the greatest for those with highluminous reflectance. In general, constant or randomerrors must be kept within about 0.4% of the luminousreflectance, and errors varying with wavelength, asin Eq. (lla), to within about 0.2% if all of the samplesare to be measured within 0.2 C.I.E. units, althoughthese tolerances can be relaxed a little for colors ofmedium or low saturation.

Other forms of variation of an error with wavelengthcould also be considered and might require even smallertolerances. For example, Blottiau and Bertrand' 6 haveshown that an error within certain limits will have thegreatest effect if it changes sign twice through thespectrum, although such an error is perhaps less likelyto occur than the simpler type given by Eq. (lla).

Errors Proportional to 5x

In many cases, errors occur which are proportionalto OxR. These are frequently caused by errors of the re-flectance values assumed for the white reference stan-dard, as well as by errors of the "100% line" of theinstrument. If fx is the percent error, the error in Ox is

AIx=fX3X/100. (17)

Tolerances for three forms of fx have been calculatedby a process exactly analogous to that used for errorsindependent of Ox. The three types of error consideredwere a constant error

(fx)i = c, (18)

a systematic error varying linearly through the spectrum

(fx)2= 1.68c[(550- X)/150] (19)

and a random error with standard deviation inde-pendent of wavelength

af =C. (20)

Tolerances for c are given for all three forms oferror in Table V which shows that the systematicallyvarying error is again the most important, with toler-ances down to 0.2% for some of the lighter samplesfor which errors proportional to fx are most significant.The other two types of error have tolerances as low as0.4% in some cases.

16 F. Blottiau and G. Bertrand, Rev. Opt. 39, 209 (1960).

696 Vol. 57

Page 7: Colorimetric Significance of Spectrophotometric Errors

SIGNIFICANCE OF SPECTROPHOTOMETRIC ERRORS

Other Integration Methods

For any type of systematic error, all the differentintegration methods lead to the same calorimetricerror, except for errors attributable to the methodsthemselves; but for random errors this will not be thecase. For example, if the 5-nm interval weighted-ordinate method is used instead of the 10-nm intervalmethod the errors will be reduced by a factor of ap-proximately 1/'%Q because twice as many values of fixare used. This reduction is similar to that which wouldoccur if fix were measured twice at each wavelengthfor the 10-nm interval method, thereby reducing o-3xby a factor of 1/V2. For integration by the selected-ordinate method there would be no covariance termsin Eq. (14) because with summations involving differ-ent sets of wavelengths for each tristimulus value, X,Y, and Z are uncorrelated if the errors at differentwavelengths are uncorrelated. This may cause an in-crease or a decrease of the resultant error, dependingon the signs of the partial derivatives (OU*/OX) etc.in Eq. (14).

Wavelength Errors

If the wavelength scale of a spectrophotometer hasa small systematic error, AX, the resultant error Afixof spectral reflectance will depend on the steepness ofthe spectral reflectance curve at each wavelength

Ax =h VAX, (21)

TABLE VI. Tolerances (in nanometers) for various types ofwavelength error, such that AE (or craB) <0.2 C.I.E. units.

Sample

11 57298629855

11 616R8/4Y8/4G8/4138/4P8/4Y8/12

GY8/8R5/4Y5/4G5/4B5/4P5/4R5/10

YR5/10G5/8

PB5/10P5/10R4/14134/8P4/12R2/2Y2/2G2/2132/2P2/2R2/6

PB2/6P2/6

Constanterror

6.82.72.95.90.70.50.50.60.40.20.30.50.40.40.30.40.20.20.20.30.20.20.20.20.92.70.80.80.90.40.40.4

Randomerror

>106.67.98.21.71.31.31.61.30.50.71.01.11.10.91.30.40.60.60.80.60.30.50.52.34.12.02.12.61.01.11.2

Slit-width error(Triangular waveband)

All >30

1416121713128

1018101614798

12108

16102139182624151722

where ix' is the first derivative of fx with respect to X.If fix is measured at 10-nm intervals between 400 and700 nm, fix' can be estimated by

Ax'= (1/20) (x+xo-fix-io)= 3Th (74100f400)=T-o 0700-0690)

forforfor

410 X 690,X=400,X= 700.

Thus, the color error, AE= e, resulting from a givenAX can be calculated in the usual way. This has beendone for each of the 32 samples for an error of 1 nm,independent of wavelength, and tolerances in nanom-eters for constant wavelength errors have been calcu-lated by

AX = 0.2/e, (23)

making the usual assumption that the effect of theerror is linear. This will be true if AX is small enoughfor the errors of the approximations involved in Eqs.(21) and (22) to be insignificant.

Similar calculations have also been made for thecase where the error occurs randomly, with standarddeviation ax. (This type of error might occur, forexample, if the wavelength scale of the spectropho-tometer cannot be read accurately.) The standarddeviation of fix at each wavelength was calculated by

Cpx=fix 'A, (24)

with Ox" estimated by Eq. (22); from these values aBE

was calculated in the usual way for each sample, from

which tolerances for ax were found. The tolerances forboth a constant error and a random error are listed inTable VI; they indicate that for some samples AXshould not exceed 0.2 nm and ax should not exceed 0.3rnm. In both cases, as is to be expected, the smallest

tolerances are for the most saturated samples, sincefor these, ON varies most rapidly with X.

Slit-Width Errors

For any given wavelength setting, a spectrophotom-eter transmits a finite band of wavelengths centered onthe nominal wavelength. If the width of this wavebandis increased, the spectrophotometer will usually bemade more sensitive because of the increased amountof light that is available, and it is therefore importantto be able to estimate the size of any errors which arecaused by departures from the ideal situation of negli-gibly small slit widths, since such errors will set alimit on how much the sensitivity may be increased bythis method. The shape of the transmitted wavebandmay be quite complicated, but a simple situation whichcan be considered is that of a triangular waveband. Inthis case, if the photocurrent in the receiver is 1 x perunit wavelength interval, then the photocurrent 1 x*for a waveband of halfwidth w nm is

Ix*= I I+i(1-)dk.-co h coA u

(25)

AXlay 1967 697

Page 8: Colorimetric Significance of Spectrophotometric Errors

A. R. ROBERTSON

Expanding I by means of Taylor's theorem, and inte-grating gives

I X EIA+irw'Ix"], (26)

where Ix" is the second derivative of IA with respectto X. Some higher-order terms'7 have been omittedsince higher powers of w are assumed to be negligiblysmall. Thus, since the reflectance, Oix, of a sample isthe ratio of the photocurrents for the sample and fora standard (usually magnesium oxide), the apparentreflectance measured with a waveband of halfwidth conm is

-X* = (figJ+&o'I2CfxI1AQ")/(Ix+ ±'2Ac x"') ' (27)

where DfxIx]j" is the second derivative of EflxIx] withrespect to X. If co is small, this reduces to

flx* 2fl+rrco'Lfx'+2 fxlx/Ix]. (28)

The last term is equivalent to a wavelength error of6W'Ix'/IX and can be treated by the procedure outlinedin the previous section. It depends on the rate ofchange with wavelength of the energy of the sourceand the sensitivity of the photocell; for many spec-trophotometers it is negligible, leaving an error of

AfX = 112C,'Xi". (29)

fix" has been estimated by

flx"= X+ flx-1i- 2fl), (30)

and the error, AE= e, resulting from the use of a 1-nmwaveband has been calculated in the usual way foreach of the 32 samples. In this case, however, the sizeof the error depends on the square of co and the toler-ances are found by

X = (0.2/e)t (31)

These tolerances, which are listed in the last column ofTable VI, show that the halfwidth should not exceedabout 7 rm if all of the most saturated colors are to bemeasured satisfactorily.

Combination of Errors

If more than one type of error should occur in agiven measurement, the total error is found by vectoraddition of the individual errors in (U*,V*,W*) space.Thus the magnitude of the total error could be eitherless than or greater than the magnitude of any of the

17 See, for example, E. P. Hyde, Astrophys. J. 35, 237 (1912).

component errors. The worst situation would occur ifall the component errors caused shifts in the samedirection in (U*,V*,W*) space, since the total errorwould then be equal to the scalar sum of the compo-nents. Thus, for example, if two errors of El and 62

C.I.E. units, respectively, were to occur together, it ispossible that the total error would be as large as (e,+ 62)

C.I.E. units, although on average it would be only(.E2+ 622)2.

CONCLUSION

Although the methods used to derive the effects ofthe various types of error involve a number of approxi-mations, they are sufficiently accurate to give goodestimates of the tolerances which should not be ex-ceeded if accurate measurements are to be made. Thesamples used for the calculations were selected to givea wide variety of colors; few opaque samples are likelyto be more difficult to measure than the most difficultof these, although the measurement of certain filterswith spectral transmittance varying very rapidly withwavelength may require even smaller tolerances. Thus,we can say that if the majority of colors are to bemeasured to within 0.2 C.I.E. units of their true colors,systematic or random errors in fx which do not varywith wavelength and are not dependent on fx, shouldnot exceed about 0.4% of the luminous reflectance ofthe sample; errors which are proportional to Ox, butindependent of wavelength should not exceed about0.4% of fix. If the error varies linearly from one end ofthe visible spectrum to the other, changing sign in themiddle, the tolerances are about half these values, andif the error changes sign twice in the visible spectrum,the tolerances may be even less. Constant wavelengtherrors should not exceed 0.2 nm; random wavelengtherrors should not exceed 0.3 nm, and the halfwidth ofthe waveband used (if it is triangular in shape) shouldnot exceed 7 nm, except for colors of low saturation.If more than one type of error occur together, toler-ances are even lower, since it is possible that the totalerror will be equal to the scalar sum of the componenterrors.

ACKNOWLEDGMENTS

I am indebted to Professor W. D. Wright for hisguidance, and for suggesting the topic of accuratecolor measurement, and also to the Science ResearchCouncil for the award of a Research Studentship forthe period of this work.

Vol. 57