# empirical modeling of systematic spectrophotometric errors

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Ro S. Berns Kevin Y H. Petersen* Munsell Color Science Laboratory Center for Imaging Science Rochester Institute of Technology Post Office Box 9887 Rochester, New York 14623-0887

Empirical Modeling of Systematic Spectrophotometric Errors

Spectrophotometers, as electro-mechanical-optical de- vices, perform at a Jinite level of accuracy. This accuracy is limited by such factors as monochromator design, detec- tor linearity, and cost. Generally, both the diagnosis and correction of spectrophotometric errors require a number of calibrated standard reference materials and considerable effort and commitment on the part of the user. A technique using multiple linear regression has been developed, based on the measurement of several suitably chosen standard reference materials, to both diagnose and correct systematic spectrophotometric errors, including photometric zero er- rors photometric linear and reonlinear scale errors, wave- length linear and nonlinear scale errors, and bandwidth errors. The use of a single chromatic ceramic tile to correct systematic errors improved the colorimetric accuracy of a set of chromatic and neutral tiles by a factor of two for a typical industrial-oriented spectrophotometer . Greater im- provement can be achieved by increasing the number of tiles and perj+orming a separate regression at each measured wavelength., These techniques have been extremely useful in improving inter-instrument agreement for instruments with similar geometry.

Introduction

In general, spectrophotometers perform at a finite level of accuracy. As electro-mechanical-optical devices, they ex- hibit measurement errors relative to a theoretical error-free instrument that users must accept. These errors can be con- veniently divided into systematic and random errors. Sys- tematic errors include errors resulting from wavelength, bandwidth, detector linearity, nonstandard geometry, and

*Present address: The Pentagon, Washington, DC 0 1988 by John Wiley & Sons, Inc.

Volume 13, Number 4, August 1988

polarization. Random errors result from drift, electronic noise, and sample presentation. Qualitatively, systematic errop affect accuracy while random errors affect repeata- bility. The distinction between the two classifications de- pends on the situation. Thermochromism, i.e., a change in color (spectral reflectance factor) caused by a change in temperature,' would be classified as a random error if the temperature of a thermochromic material unexpectedly changed before a measurement. If a thermochromic material was calibrated at one temperature and measured at another, the error in spectral reflectance factor would be systematic. The most prevalent errors in modern instruments are as- sociated with stray light, wavelength scale, bandwidth, ref- erence white calibration, and thermochromism.

Carter and Billmeye? summarized an Inter-Society Color Council technical report that described material standards for calibrating, performance testing, and diagnostic check- ing of color-measuring instruments. This article provided an excellent overview on the proper methodologies to cal- ibrate spectrophotometers and diagnose systematic errors. Unfortunately, many industrial laboratories do not, as a matter of course, test instrument performance. In addition, many current instruments provide spectral data only at such discrete wavelength intervals as 10 or 20 nm. The difficulties in accurately diagnosing wavelength errors, in particular, for these instruments has further hampered diagnostic ef- forts.

Diagnosis, however, is only the first step to improving instrument performance. The second step is to correct an instrument's systematic errors. This involves both me- chanical and mathematical adjustments. In many indus- trial environments only mechanical adjustments tend to be made. Techniques to correct small systematic errors mathematically, performed routinely by standardizing lab- oratories, are calculation intensive and beyond the scope of most industrial laboratories. Billmeyer and Alessi3 as- sessed the colorimetric accuracy of typical color-measur-

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ing instruments in which only mechanical adjustments were Photometric Zero Error made. These instruments had an average colorimetric er- ror of about one CIELAB color-difference unit compared with results from a standardizing laboratory when rnea- suring the reflectance factor of chromatic and achromatic tiles. Based on informal investigations in our laboratory, current color-measuring instruments designed for the in- dustrial community are capable of better performance when properly maintained. Nonetheless, the lack of agreement between instruments of similar geometry but different manufacturers and the difficulties in maintaining close agreement for groups of identical instruments suggest that more sophisticated techniques to reduce systematic errors may be needed.

During the past several years, we have analyzed various mathematical techniques to diagnose and correct systematic errors inherent in our industrial-oriented spectrophotometers in order to improve their colorimetric performance and inter- instrument agree~nent .~ This article describes one method particularly suited to industrial environments.

The method is based on the use of multiple linear regres- sion in which systematic errors, modeled by a series of linear equations, are minimized in a least-squares sense between instrumental measurements and standardized measure- ments. This method was outlined by Robertson.s He dem- onstrated its utility in diagnosing photometric zero errors, linear photometric scale errors, and linear wavelength errors in a General Electric recording spectrophotometer. We have done further testing of his equations, derived equations to describe nonlinear systematic errors, and extended the sta- tistical method by including stepwise linear regression. This technique has been applied to reflectance-factor measure- ments but is also suitable to transmittance measurements. Since most industrial-oriented spectrophotometers report percent reflectance factor rather than reflectance factor, the remainder of this article and all calculations are based on units of percent reflectance factor.

Mathematical Description of Systematic Errors

The method is based on the use of multiple linear regres- sion, which requires modeling systematic errors as a se- ries of linear equations. It should be noted that a linear model implies only that the parameters are linear; the model may bc nonlinear. For example, it is well known that the relationship between luminous reflectance factor and Munsell value can be well modeled by a fifth-order po- lynomial, a nonlinear function. Multiple linear regression can be used to calculate the weightings of each order of the model, i.e., to calculate the parameters of the model. The success of the technique is limited by the accuracy of the postulated model. If the relationship between lu- minous reflectance factor and Munsell value was assumed to follow a straight line (a first-order polynomial), the model fit would be poor. With respect to modeling sys- tematic spectrophotometric errors, greater knowledge of the mechanics of the device will enable a more accurate estimate of the appropriate model.

A photometric zero error is an offset of the entire pho- tometric scale. It is often caused by stray light associated with input optics, the use of a black trap with a finite re- flectance factor, or ignoring detector dark current. It is ex- pressed as

RLA) Rrn(h) + BU (1) where Rr(X) is the true or reference reflectance factor, R,(h) is the measured reflectance factor to be evaluated, and Bo is the photometric zero error.

Photometric Linear Scale Error

A photometric linear scale error is an error that is pro- portional to the reflectance-factor measurement. It is most often caused by an improperly calibrated white standard (100%-line error) or one that has physically changed since initial calibration. It is expressed as

RLh) = R m ( h ) + BlRm(X) ( 2 ) where B1 is the photometric scale error.

Photometric Nonlinear Scale Error

A photometric nonlinear scale error is most often caused by detector nonlinearity. Since spectrophotometers are cal- ibrated such that zero and 100% reflectance factor are set, a nonlinear error can be approximately expressed as

Rr(X) = R m ( h ) + Bzl100 - Rm(A)IRrn(X) (3) where B2 represents a nonlinear photometric scale error. It is helpful to think of 100 - R,,(h) as a nonlinear weighting function of the photometric scale error. This quadratic func- tion typifies errors that are small at the ends of the photo- metric scale and larger in the middle.

Wavelength Linear Scale Errors

A wavelength scale error is an error in the measured reflectance factor resulting from a shift in the wavelength scale. The resulting error in reflectance factor is approxi- mately proportional to the first derivative of the measured reflectance factor. It is expressed as

R,(X) = R,(X) + B,dR,/dh (4) where B3 is the wavelength scale error and dR,ldh is the first derivative of R,(X) with respect to wavelength. The first derivative is equal to

where i is an index of wavelength. In this study the first derivatives of the first and last measured wavelengths were set equal to those of the second and second-to-last measured wavelengths, respectively.

244 COLOR research and application

Wavelength Nonlinear Scale Error

Many instruments, in fact, have wavelength scale errors that are nonlinear with respect to wavelength. Based on evaluating several industrial instruments using didymium filters and calculating inflection points, wavelength scales, in a sense, were weighted by functions approximately ex- pressed as

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