Accurate tristimulus values from spectral data

Inter-instrument color measurement errors of up to 2 CIELAB units are typically introduced in the process of numerically integrating spectrophotometric data to obtain tristimulus values. These errors arise partly from inadequacies in the definition of the tristimulus values themselves, but mainly from failing to take into account the effect the instrument function has on the spectrophotometric data. Through using precise definition of the tristimulus values and through using source-observer weighting tables tailored to each specific instrument, it is practical to reduce this component of colorimetric error to less than 0.1 CIELAB unit for spectrophotometers with bandwidths up to 20 nm.     

An evaluation of some tristimulus weights

Tristimulus values were calculated for the 1964 observer and illuminant D65 for four theoretical specimens, with radiance factor data of 1-nm bandpass width, by the CIE recommended procedure and used as accurate references. Tristimulus values were then calculated for the same four theoretical specimens with six sets of published weights, with radiance factors assumed to have a bandwidth equal to the measurement interval and the bandpass profile to be triangular. The difference of the wide bandwidth calculated tristimulus values from the reference tristimulus values was considered to be an error and was expressed as CIE L*a*b* color difference.     

The influence of spectral bandpass on accuracy of tristimulus data

When calculating tristimulus values from spectral radiance factors which were measured with a spectral bandpass of 10 nm, using data from the same specimen measured with a spectral bandpass of 1 nm and calculated with a 1-nm interval as standard, and evaluating the significance of the difference in tristimulus values with the CIELAB color difference formula, it is highly probable that a 1-nm calculation interval will give no greater accuracy than a 10-nm calculation interval. For this reason it is recommended that the CIE reconsider their approval of the deletion of the 10-nm calculation interval from the new edition of Publication CIE 15.     

Spectral‐reflectance‐factor deconvolution and colorimetric calculations by local‐power expansion

The experimental data of the spectral‐reflectance factor are considered as dependent on the instrument‐spectral‐bandwidth function in order to perform their deconvolution and to compute the tristimulus values. The deconvolution is performed by local‐power expansion. In the case that the spectral‐bandpass dependence regards only the spectral transmittance of the monochromator, the goodness of this technique is evaluated by simulation (1325 reflectance factors of the Munsell samples are considered as trial functions) and compared with other usual techniques: Stearns and Stearns method for bandpass error, ASTM‐weighting function interpolation, and Venable‐ASTM weighting function. The zero order of the deconvoluted spectral‐reflectance factor can be related to the Stearns and Stearns method for bandpass error. With respect to any other technique, the second‐order deconvolution, for the CIE standard illuminants, gives color differences lower by a factor 0.1 or more for a bandpass Δλ = 10 nm, color differences lower by a factor 0.3 or more for a bandpass Δλ = 20 nm and, for the CIE fluorescent illuminants, color differences generally lower. © 2000 John Wiley & Sons, Inc. Col Res Appl, 25, 176–185, 2000     

Colorimetry and its accuracy in the measurement of fluorescent materials by the two-monochromator method

The fundamental procedure involved in the two-monochromator method for determining the spectral radiance factors of fluorescent materials is described, and some measured results are presented. The colorimetric accuracy of the method was confirmed by comparing two sets of tristimulus values: (1) those based on the two-monochromator method and calculated from the reflected and the fluorescent spectral radiance factors and the spectral power distributions of a tungsten-halogen lamp and a xenon lamp (which were determined separately); and (2) those calculated from the total radiance factors obtained from the measurement under polychromatic illumination, where the fluorescent materials were irradiated with the tungsten and the xenon lamp. The color differences between these sets of tristimulus values in CIE-LUV color space were estimated to be less than 1.0 unit, irrespective of the illuminant or the source.     

CIE Method for Calculating Tristimulus Values

Between 1981 and 1983, a working Group of USTC-1.3 of the CIE prepared new recommendations giving for the first time detailed instructions for the calculation of CIE tristimulus values, for inclusion in Publication CIE No. 15.2, a revision of the CIE document on colorimetry first published in 1971. The new recommendations state that the standard method of performing the integration forming the basic definition of tristimulus values shall be by summation at a wavelength interval of 1 nm over the wavelength ranCe 360–830 nm, but that for most colorimetric purposes the approximation of summation at a 5-nm interval over the range 380–780 nm should suffice. Recognizing that measured data fulfilling these requirements are not usually available, recommendations were also made concerning abridgement, interpolation, extrapolation, truncation, and the calculation of weighting factors. Although tables of weighting factors were not included in the recommendations to the CIE, they have since been calculated in cooperation with the Working Group and published by the ASTM. This article describes the recommendations, now accepted by the CIE and providing clear and complete guidelines for the uniform calculation of CIE tristimulus values.     

Reconstruction of reflectance spectra using weighted principal component analysis

The weighted principal component analysis technique is employed for reconstruction of reflectance spectra of surface colors from the related tristimulus values. A dynamic eigenvector subspace based on applying certain weights to reflectance data of Munsell color chips has been formed for each particular sample and the color difference value between the target, and Munsell dataset is chosen as a criterion for determination of weighting factors. Implementation of this method enables one to increase the influence of samples which are closer to target on extracted principal eigenvectors and subsequently diminish the effect of those samples which benefit from higher amount of color difference. The performance of the suggested method is evaluated in spectral reflectance reconstruction of three different collections of colored samples by the use of the first three Munsell bases. The resulting spectra show considerable improvements in terms of root mean square error between the actual and reconstructed reflectance curves as well as CIELAB color difference under illuminant A in comparison to those obtained from the standard PCA method. © 2008 Wiley Periodicals, Inc. Col Res Appl, 33, 360–371, 2008     

The CIELAB reversal in calibration and verification

A CIELAB anomaly, in which smaller spectrophotometric errors at all wavelengths lead to larger CIELAB differences, is identified. It is shown that the reversal can occur throughout tristimulus space and is colorimetrically important during calibration procedures. Three numerical examples of the reversal, using data from the BCRA tiles, are given. The reversal cannot be attributed entirely to metamerism, which itself may cause large spectrophotometric error leading to small CIELAB difference. The effect is compounded by the nonlinearity of CIELAB relative to tristimulus coordinates. A recommendation for avoiding the reversal is offered. © 2004 Wiley Periodicals, Inc. Col Res Appl, 30, 66–68, 2005; Published online in Wiley InterScience (www.interscience. wiley.com). DOI 10.1002/col.20076     

Transformation of NCS data into CIELAB colour space

The data for the Natural Colour System (NCS) were transformed to the CIE 1976 (L*, a*, b*) system (CIELAB). The transformation involved taking the CIE tristimulus values of the nominal NCS notations, i.e., the colorimetric aim points representing the NCS, converting these values to CIELAB, and plotting them in the CIELAB space. All the data have been calculated using CIE standard illuminant C and the 1931 standard observer. The data show that no simple relationship exists between the NCS and CIELAB systems.     

CIELAB Hue-Angle Anomalies at Low Tristimulus Ratios

Hue angles calculated using the CIELAB recommendation are usually dependent on the luminance factors of the stimulus if any tristimulus ratio is below the critical figure at which the Pauli linear function is applied. This can cause errors of up to 35° in the case of stimuli of low luminance lying on the spectrum locus or purple line. Errorrs are unlikely to occur in the case of surface colors, but in the case of transparent object colors their frequency and magnitude are such as to make CIELAB unsuitable as a color space or as a color-difference formula whenever a tristimulus ratio is below the critical figure. Practical considerations suggest that in such cases the best solution is to use a modified Judd polynomial function, and this course is suggested for consideration by the CIE Colorimetry Committee and the committee responsible for developing an ISO Standard for the color measurement of plastics.     

Colour management of a low‐cost four‐colour ink‐jet printing system on textiles

A low‐cost four‐colour (RBYK) dye‐based ink‐jet printing system for textiles was introduced in this study, in which red and blue inks were employed instead of the magenta and cyan inks used in half‐tone printing. The basis of a colour‐management system for this device was developed by determining the mapping between XYZ tristimulus values of output colours and the digital RBYK values using polynomial transforms. A second‐order equation was found to give the best performance with an average characterisation error of under 7 CIELAB units.     

A hybrid adaptation strategy for reconstruction reflectance based on the given tristimulus values

Recently, Cao et al proposed an adaptive weighting method for the training samples for reflectance reconstruction according to both colorimetric and spectral reflectance similarities for a given vector defined by tristimulus values. It was shown the Cao et al method outperforms the other methods including the regression estimation method in terms of multiple evaluation criteria. In this article, motivated by the work of Cao et al, a hybrid weight is introduced, which results in the size of the training samples selected is half of that used by the Cao et al method. Simulation results showed that the proposed method performs equally well as or slightly better than the Cao et al method, but uses less central processing unit time than that used by the Cao et al method. It was also found that about 100 training samples selected is good enough for the proposed method.     

The calculation of weight factors for tristimulus integration

This article defines the method of calculation of weight sets for tristimulus integration published by the ASTM in its Method E 308–85. The preparation of weight sets for any measurement interval or wavelength range by Lagrange cubic interpolation from standard tables of data at other intervals is described. In addition, methods of truncating weight sets to wavelength ranges of less than 360 nm to 830 nm are discussed. The importance of calculating and maintaining the chromaticity of the neutral point is noted. An example of calculation of a set of weights is given in an Appendix, along with an example computer program that will calculate a set of weights at 20 nm intervals from standard illuminant-observer data given at 1 nm intervals.     

Error propagation analysis in color measurement and imaging

We apply multivariate error-propagation analysis to color-signal transformations. Results are given that indicate how linear, matrix, and nonlinear transformations influence the mean, variance, and covariance of color-measurements and color-images. Since many signal processing paths include these steps, the analysis is applicable to color-measurement and imaging systems. Expressions are given that allow image noise or error propagation for a spectrophotometer, colorimeter, or digital camera. In a computed example, error statistics are propagated from tristimulus values to CIELAB coordinates. The resulting signal covariance is interpreted in terms of CIELAB error ellipsoids and the mean value of color-difference measures, and . The application of this analysis to system design is also illustrated by relating a tolerance to equivalent tristimulus-value error statistics. © 1997 John Wiley & Sons, Inc. Col Res Appl, 22, 280–289, 1997     

Dependence between apparent color and extractable color in paprika

The color properties of 96 paprika samples were evaluated by tristimulus reflectance measurements. The extractable color (ASTA units) of all these samples was also determined. The linear correlation between individual CIELAB parameters and extractable color was very poor. Several color indices used with other foods were shown to be of insufficient accuracy for predicting the extractable color in paprika. A new color index for paprika (PACI) is proposed based on the CIELAB coordinates L* (lightness), a* (red‐blue), and h (hue angle), and it is calculated as “1000a*/(L*+h)”. This new index showed a high correlation with the logarithm of extractable color (r = 0.9662) and was able to distinguish between sample groups of different ASTA units. © 1999 John Wiley & Sons, Inc. Col Res Appl, 24, 93–97, 1999     

基于镜像热源原理测定固体材料热物性的方法及应用

基于平行热线法结合镜像热源原理,提出了一种新的固体材料热物性参数测算模型,在热线法测试原理的基础上,以试样绝热边界为界线,设与真实热源对称位置处存在虚拟镜像热源,以此消除绝热边界造成的热积聚效应影响,测试时可不需再限制实验时间和试样厚度. 当相邻时刻材料的热物性参数计算结果大于判别准则时,引入镜像热源对计算温度进行修正. 为防止修正过程所用热物性参数对实验初期计算值的依赖,模型对实测温度进行两次修正. 以石棉板为研究对象,理论分析结合计算结果表明,两次修正结果不同,但差异不大,且第二次修正后各组热物性参数计算结果更稳定. 对石棉板、大理石、硼硅玻璃、硅砖等4种材料的薄板和厚板进行了热物性测定,结果与文献值较吻合,最大误差均小于5%,验证了本测定方法适用于薄板和厚板试样,有效提升了热线法测定精度,扩大了应用范围.     

Toward a more accurate and extensible colorimetry. Part VI. Improved weighting functions. Preliminary results

This Part VI is a progress report, with two motivations. (1) To publish the new method of extraction of weighting functions, and to show the demonstrated large reduction of tristimulus error in an array of ten disparate visually-matching pairs of white lights, and (2) to attempt to interest others in joining the work. The direct extraction of improved weighting functions (WFs) from an array of visually matching pairs of white lights is the subject of Part VI. This new approach is made necessary by our finding (Part I) that color-matching functions by either the Maxwell method or by the maximum saturation method lead to large errors (discrepancies) in computed chromaticities of pairs of visually-matching lights. Using spectral power distributions (SPDs) of 5 types from Part IV, eight observers make 5 strongly metameric visual matches to the same broadband reference white light, with 1.3° visual field and 70 cd/m² luminance of the reference white. Each of the resulting 5 SPDs is averaged over the 8 observers, and the 5 averaged SPDs are formed into 10 pairs (the five averaged visually-matching lights taken two at a time). Tristimulus values X, Y, and Z are computed for each member of each pair by the CIE 1931 weighting functions (color-matching functions) x , y , and z . Absolute ΔX, the tristimulus error (the difference between computed X1 and X2 of the visually-matching lights), is computed for each pair and summed over the 10 pairs, as are ΔY and ΔZ. The often-large 10-pair total tristimulus error TTE is computed for X, Y, or Z. For example TTE_X is the sum of the ten absolute ΔX's of the 10 pairs. Then x is progressively altered in spectral shape by an algorithm that on each iteration reduces TTE_X. Weighting functions y and z are altered in turn. Reduction to 1–3%, of the TTE initially associated with the CIE weighting function, is achieved in this preliminary work. The changes in shape of the resulting functions are discussed. The simpler term “weighting function” is used rather than “color-matching function” for these, and it is recognized that, when finally correct, these functions should represent the three spectral sensitivities of the normal human visual system. © 1998 John Wiley & Sons, Inc. Col Res Appl, 23, 226–233, 1998     

Uniform Chromaticity Scales — New Experimental Data

Sets of coloured paint and textile samples have been prepared, each set consisting of pairs of samples giving small chromaticity differences with a good coverage of the possible directions. For most sets the lightness differences were negligible, but for a few sets extra samples showing significant lightness differences were included. For each set one difference was taken to be standard and the other differences were assessed as a ratio of the standard difference by panels of 19 to 24 observers. Chromaticity discrimination ellipses were calculated for each set. Experiments in which pairs from different sets were inter-compared enabled the sizes of the ellipses to be correctly adjusted relative to each other. In total 42 sets (536 sample pairs) were investigated under Illuminant D65 and 39 sets (531 sample pairs) were investigated under Illuminant A. When plotted on a chromaticity diagram the ellipses formed a fairly regular pattern, but marked differences compared to other sets of ellipses were noted. The ellipses increased in size as Y decreased, particularly at very low values of Y. The agreement between ΔE values from various colour- difference equations and the experimental results was poor, although newer equations (JPC and FCM) appeared to be somewhat better than the 1976 CIELAB equation. Various checks on the repeatability and consistency of the experimental results gave confidence in the latter and suggest that the equations are seriously in error.     

CRT colorimetry. part I: Theory and practice

The relationship was derived for computer-controlled color CRT displays between spectral radiant exitance emitted and digital counts. The derivation was historical and could be traced to pioneering work in photographic sensitometry, vacuum tube physics, and broadcast television. By performing radiometric measurements relative to a display's maximum exitance, the model simplified to a two-stage model. The first stage was a nonlinear transformation relating normalized digital-to-analog converter values to device-dependent monitor tristimulus values using model parameters of gain, offset, and γ. The second stage was a linear transformation where the device-dependent monitor tristimulus values were transformed to device-independent CIE tristimulus values. By using the model, colorimetric characterization accuracy of better than 0.5 CIELAB color-difference units for 125 colors sampling the display color gamut was achieved by measuring the CIE tristimulus values of only eight colors. The model had equivalent performance to methods using extensive measurements and table lookup. © 1993 John Wiley & Sons, Inc.     

Theoretical and practical aspects of selected fiber-blend color-formulation functions

Color matching blends of precolored fiber using three different methods was studied. Best color-matching accuracy was obtained using a two-constant Kubelka-Munk (KM) procedure. First-formula color differences averaged 1.6 CIELAB units and were found to be within the experimental error of 1.6 CIELAB units. Useful approximations were obtained using the methods proposed by Friele and by Stearns. First-formula color matches averaged 2.4 CIELAB units for the Stearns and 2.7 CIELAB units for the Friele methods. The methods are mathematically compared and the merits of each are discussed. Where possible, interpretation of the empirical parameters each method employs is attempted. It is pointed out that absorption and scattering constants calculated for fibers using the KM formalism are not true KM absorption and scattering constants. It is demonstrated that too literal an interpretation of these constants leads to apparent anomalies. It is shown that the fiber KM scattering constants which are normally considered unchanged as dye is applied cannot be considered unchanged if these same fibers are subsequently to be used in blends with other colored fibers.