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.
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.
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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
(6) R,(h) = R,(h) + B,w,(h)dR,/dX
R,(h) = R,(A) + B5w2(X)dRm/dA w2(X) = sin 2 r (X/200)
t 9) where B4 and B5 are nonlinear wavelength scale errors. Weighting function wl(X) is identical to the quadratic func- tion described in eq. (4), except scaled to wavelength. Weighting function w2(h) is a one-and-one-half-cycle sine wave. This would represent an instrument with both positive and negative wavelength errors.
Equations (4), (6), and (8) represent three different models of an instruments wavelength error. Depending on the dis- persing element in the monochromator and the scanning mechanism, different functions may be more nearly appro- priate. The dispersing elements in many spectrophotometers are nonlinear by definition. Manufacturers, as a matter of course, characterize the nonlinearity and appropriately ac- count for the nonlinearity mechanically or mathematically. In theory, wavelength errors should reduce to linear errors that are relatively simple to correct mechanically. In prac- tice, the limiting systematic error in most industrial-oriented instruments remains wavelength error, often of a nonlinear nature.
If the spectral bandwidth of a spectrophotometer varies with wavelength or if the bandwidth is excessively large, an error in the measured reflectance factor can occur. As- suming a symmetrical bandwidth function, the resulting er- ror is proportional as a first approximation to the second derivative of the measured reflectance factor with respect to wavelength. It is expressed as
where B6 is the bandwidth error and d2R,ldh2 is the second derivative of R,(X) with respect to wavelength. It is equal to
R,(X) = R,(h) + B6d2R,/dX2
(d2RldX2); (11) - R(Xi+1, + R(Xi-1) - %(A) -
(hi+l - L l > 2
Seven systematic errors considered in this article have been described by linear equations. They are summarized in Table
TABLE I . Summary of the systematic errors and their modeled equations considered in this article, including their notation and the notation of the parameters.
Svstematic error Parameter Model
Photometric zero Photometric linear
scale Photometric nonlinear
scale Wavelength linear
scale Wavelength nonlinear
scale (quadratic) Wavelength nonlinear
scale (sine wave) Bandwidth
I. For notational ease, the measured reflectance factors and their associated functional forms representing the seven sys- tematic errors can be abbreviated by variables Xo(h) through X6(h). For example, XI@) = R,(X) and X2(h) = R,(h) [lo0 - &,(A)]. Xo(X) is a special case referred to as a dummy variable and is equal to unity.
Suppose that a spectrophotometer had all these systematic errors. The difference between the measured reflectance factor and the actual reflectance factor of a material could be expressed as
M A ) - R A M = Bdr,(X) +BIXI(X) + . . . + B&&) + e(h) (12)
where Bo, B1, . . . , B6 are the weighting parameters of each systematic error and e(X) is the sum-of-squares residual error not accounted for by this model. Suppose, also, that the spectrophotometer measures from 400 through 700 nm in 10-nm increments. Thus, eq. (12) actually represents 31 equations. These equations can be written in terms of ma- trices:
Y = X B + e (13)
where Y is a (31 x 1) column vector containing R,(X) - R,(X) as its elements, X is a (31 X 7) matrix con- taining the measured functions of reflectance factor, B is a (7 x 1) column vector containing the error parameters, and e is a (3 1 X 1) column vector containing the sum-of-squares residual errors. Multiple linear regression estimates the mag- nitude of the elements in the error matrix B such that the elements in the residual error matrix e are minimized.6 The elements in matrix B are referred to as the regression coef- ficients. The magnitudes of the regression coefficients Bo- B6 indicate the magnitudes of the corresponding systematic errors for the modeled instrument. The magnitudes of the elements in matrix e indicate the model fit. A variety of statistical significance tests can be performed to evaluate the model fit.6 The most important statistic to evaluate would be the regression analysis of variance.
Once the regression coefficients are estimated, the mea- sured reflectance factor can be corrected by
&(A) = R,(h) + B&o(A) + . . . + B&j(h) (14) where Rc(h) is the corrected reflectance factor.
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The above regression model has been developed to cor- rect systematic errors. Each error described by a linear equation is a function of the measured reflectance factor. This is the inverse of diagnostic spectrophotometry in which the measured data are described as a function of the true data. For example, we describe an instrument as having a wavelength error. Implied in this statement is that the measured data are a corruption of the theoretical error-free true data. In Robertsons research, the errors described by linear equations were functions of the true data. The advantage of Robertsons method is that the elements in the coefficient matrix B directly describe systematic er- rors. When the systematic errors are functions of the measured data, eq. (14) must be inverted to yield regr...