Interpolation,extrapolation, and truncation in computations of CIE tristimulus values

This article addresses some general practices for the computation of CIE tristimulus values (TSVs), including different methods used for the interpolation, extrapolation, and truncation of data in the visible wavelength range. In each case, a quantitative analysis is presented on the basis of a dataset with 2365 spectral reflectances covering a wide color gamut, assuming six CIE illuminants and the two CIE standard colorimetric observers. For interpolation, it is found that among five tested methods, the (cubic) spline method is the best one when the spectral reflectance factors are uniformly sampled over the visible wavelength range. For extrapolation, it is found that the second‐order extrapolation method gives better results than the use of nearest available data points or linear extrapolation methods. With respect to truncation, considering five different usual truncated ranges, it is found that the range from 360 to 780 nm provides more accurate results than the range from 380 to 780 nm currently recommended by CIE for nonfluorescent samples, and the use of the truncated range from 400 to 700 nm in place of the CIE‐recommended range of 360 to 830 nm leads to TSVs with considerably high color differences of up to 0.9 CIELAB units. © 2016 Wiley Periodicals, Inc. Col Res Appl, 42, 10–18, 2017     

Testing the accuracy of methods for the computation of CIE tristimulus values using weighting tables

The least squares method for computing colorimetric weighting tables is presented and its connection with the optimum weights method is investigated. Each requires solving three linear systems of equations with the same coefficient matrix but three different right hand side vectors. It is shown that the two methods have nearly the same performance when the wavelength interval of the data is large. The two methods however, will perform differently when Δλ is small. Comparisons are also made between the least squares method, the optimum weights method, the zero‐ and second‐order weighting tables, and the ASTM weighting tables, both the original 1985 tables and the new 2013 Tables V and VI. The results show that the least squares method is the best for measurement intervals equal to or lower than 10 nm, and is competitive with the optimum weights method for 20 nm steps. The results presented in this article will contribute to the work of CIE Technical Committee TC1‐71 Tristimulus Integration as it seeks to make recommendations for the calculation of tristimulus values. © 2015 Wiley Periodicals, Inc. Col Res Appl, 41, 125–142, 2016     

Reevaluation of daylight spectral distributions

The spectral power distribution of colorimetric daylight illuminants was determined based on measurements1 done some 40 years ago. The data were averaged and standardized at every full 10 nm. Later, to follow colorimetric practice, these data have been interpolated (linearly) to every 5 nm, but the factors to calculate the values have not been changed. We show in this article that this is not correct and leads to some discrepancies, especially if data with even smaller step size are required. We suggest that when correcting this discrepancy one should at the same time go from the linear interpolation technique, a rather crude method for a spectral distribution with many minor minima and maxima in the spectrum, to a nonlinear one. We show that a third‐order spline interpolation can provide smooth functions resembling the original data to a high degree. © 2000 John Wiley & Sons, Inc. Col Res Appl, 25: 250–259, 2000     

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.     

A new method for computing optimum weights for calculating CIE tristimulus values

Weighing tables are widely used for calculating CIE tristimulus values. In this article, a direct method for computing optimum weighing tables for any illuminant and observer combination is developed. A comprehensive set of 1‐nm reflectance functions based upon Munsell samples is used to test various methods. Four types of weighing tables are compared. They are ASTM E308 Tables 5 and 6, ASTM E2022, and new tables proposed by this study. The results clearly show that the newly developed optimum tables outperform the other three types of tables. The new method is simple and avoids the iterative process used by Venable and adopted by ASTM for some of its tables. It may be used for calculating weighing tables for any combination of illuminant and observer. A detailed procedure and a worked example are given in the article. © 2004 Wiley Periodicals, Inc. Col Res Appl, 29, 91–103, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/col.10229     

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.     

Artwork imaging from 370 to 1630 nm using a novel multispectral system based on light‐emitting diodes

The recent use of multispectral systems as a noncontact method for analysis of artworks has already shown promising results. This study explains the application of a novel portable multispectral system based on light‐emitting diodes (LEDs) for artwork imaging. This method provides spectral information in a spectral range from 370 to 1630 nm with a 25 cm × 25 cm field of view by using two different image sensors in synchrony with 23 bands of irradiation. The spectral information for each point is estimated and validated using the pseudo‐inverse and spline interpolation methods for spectral estimation and three different evaluation metrics. The results of the metrics obtained with both estimation methods show a general good performance of the system over the whole spectral range. The experiments also showed that the selection of the training set for the pseudo‐inverse estimation has a great influence in its performance, and thus, it defines whether or not the pseudo‐inverse outperforms the spline interpolation method. The system is applied in situ to the study of Catalan art masterpieces, and the results demonstrate the potential of a cost‐effective and versatile system using various off‐the‐shelf elements to reconstruct color information and to reveal features not previously identified. © 2014 Wiley Periodicals, Inc. Col Res Appl, 40, 398–407, 2015     

Testing the behavior for metamerism of the new deviate observer (JF‐DO)

Recently,in our laboratories, a set of color‐matching functions (cmfs) has been formulated for small fields by using two groups of real observers: JAM, MM, CF and AY, JR, MR, JL, JA, FP. The measurements of these cmfs have been made using different experimental devices and methods and it has enabled us to propose a New Deviate Observer for small fields (JF‐DO). This new JF‐DO was derived from the average observer of our nine real observers, following the technique used by the CIE to establish the Standard Deviate Observer (CIE‐1989 SDO), which was established for fields of 10°, despite the CIE's assumption that it can be applied to smaller fields. In the present work, we report experimental results of the JF‐DO using metameric reflectances in comparison to the CIE‐1931 Standard Observer and to the CIE‐1989 SDO. © 2005 Wiley Periodicals, Inc. Col Res Appl, 30, 363–370, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/col.     

Correlation between visual and colorimetric scales ranging from threshold to large color difference

Psychophysical experiments of color discrimination threshold and suprathreshold color‐difference comparison were carried out with CRT‐generated stimuli using the interleaved staircase and constant stimuli methods, respectively. The experimental results ranged from small (including threshold) to large color difference at the five CIE color centers, which were satisfactorily described by chromaticity ellipses as equal color‐difference contours in the CIELAB space. The comparisons of visual and colorimetric scales in CIELAB unit and threshold unit indicated that the colorimetric magnitudes typically were linear with the visual ones, though with different proportions in individual directions or color centers. In addition, color difference was generally underestimated by the Euclidean distance in the CIELAB space, whereas colorimetric magnitude was perceptually underestimated for threshold unit, implying the present color system is not a really linear uniform space. Furthermore, visual data were used to test the CIELAB‐based color‐difference formulas. In their original forms CIEDE2000 performed a little better than CMC, followed by CIELAB, and with CIE94 showing the worst performance for the combined data set under the viewing condition in this study. © 2002 Wiley Periodicals, Inc. Col Res Appl, 27, 349–359, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/col.10081     

Smoothing spectral power distribution of daylights

CIE Division 1 entrusted TC 1–74 to investigate the possibility to develop a smooth curve version of the D illuminants. This article investigates the possibility to create such smoothed curves with minimal colorimetric error. Six different smoothing algorithms were investigated; the best method, using a locally weighted regression and smoothing algorithm, enabled a smooth function, where the worst colour difference for a 100 000 sample set was less than ΔE_ab^* = 0.5, comparing calculations using the CIE standard daylight spectrum and the smoothed function. Thus, the smoothed function can be recommended for redefining the daylight spectra for colorimetry without loosing continuity to results using the current calculation tables and for designing daylight simulator spectra. © 2012 Wiley Periodicals, Inc. Col Res Appl, 38, 316–321, 2013     

Estimating spectral reflectance from camera responses based on CIE XYZ tristimulus values under multi‐illuminants

A new spectral reflectance estimation method based on CIE XYZ values under multi‐illuminants was proposed to obtain multi‐spectral images accurately by using digital still cameras. CIE XYZ values under multi‐illuminants were initially predicted from raw RGB responses by using a polynomial model with local training samples. Then, spectral reflectance was constructed from the predicted CIE XYZ values via the pseudo‐inverse method. Experimental results indicated that the new spectral reflectance estimation method significantly outperformed the traditional colorimetric characterization method without requiring extra training samples or greatly increasing computational complexities. © 2016 Wiley Periodicals, Inc. Col Res Appl, 42, 68–77, 2017     

Chromatic luminance,colorimetric purity,and optimal aperture‐color stimuli

Chromatic luminance (i.e., luminance of a monochromatic color) is the source of all luminance, since achromatic luminance arises only from mixing colors and their chromatic luminances. The ratio of chromatic luminance to total luminance (i.e., chromatic plus achromatic luminance) is known as colorimetric purity, and its measurement has long been problematic for nonspectral hues. Colorimetric purity (p_c) is a luminance metric in contrast to excitation purity, which is a chromaticity‐diagram metric approximating saturation. The CIE definition of p_c contains a fallacy. CIE defines maximum (1.0) p_c for spectral stimuli as monochromatic (i.e., optimal) stimuli, and as the line between spectrum ends for nonspectrals. However, this line has <0.003 lm/W according to CIE colorimetric data and is therefore effectively invisible. It only represents the limit of theoretically attainable colors, and is of no practical use in color reproduction or color appearance. Required is a locus giving optimal rather than invisible nonspectral stimuli. The problem is partly semantic. CIE wisely adopted the term colorimetric purity, rather than the original spectral luminance purity, to permit an equivalent metric for spectrals and nonspectrals, but the parameter of equivalence was never clear. Since 1 p_c denotes optimal aperture‐color stimuli for spectrals, arguably 1 p_c should denote optimal stimuli consistently for all stimuli. The problem reduces to calculating optimal aperture‐color stimuli (“optimal” in energy efficiency in color‐matching) for nonspectrals, shown to comprise 442 + 613 nm in all CIE illuminants. This remedy merely requires redefinition of 1 p_c for nonspectrals as the line 442–613 nm, and gives meaningful p_c values over the hue cycle allowing new research of chromatic luminance relations with color appearance. © 2007 Wiley Periodicals, Inc. Col Res Appl, 32, 469–476, 2007     

Visual evaluation at scale of threshold to suprathreshold color difference

Visual evaluation experiments of color discrimination threshold and suprathreshold color‐difference comparison were carried out using CRT colors based on the psychophysical methods of interleaved staircase and constant stimuli, respectively. A large set of experimental data was generated ranged from threshold to large suprathreshold color difference at the five CIE color centers. The visual data were analyzed in detail for every observer at each visual scale to show the effect of color‐difference magnitude on the observer precision. The chromaticity ellipses from this study were compared with four previous published data, of CRT colors by Cui and Luo, and of surface colors by RIT‐DuPont, Cheung and Rigg, and Guan and Luo, to report the reproducibility of this kind of experiment using CRT colors and the variations between CRT and surface data, respectively. The present threshold data were also compared against the different suprathreshold data to show the effect of color‐difference scales. The visual results were further used to test the three advance color‐difference formulae, CMC, CIE94, and CIEDE2000, together with the basic CIELAB equation. In their original forms or with optimized K_L values, the CIEDE2000 outperformed others, followed by CMC, and with the CIELAB and CIE94 the poorest for predicting the combined dataset of all color centers in the present study. © 2005 Wiley Periodicals, Inc. Col Res Appl, 30, 198–208, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/col.20106     

A study of digital camera colorimetric characterization based on polynomial modeling

The digital camera is a powerful tool to capture images for use in image processing and colour communication. However, the RGB signals generated by a digital camera are device‐dependent, i.e., different digital cameras produce different RGB responses for the same scene. Furthermore, they are not colorimetric, i.e., the output RGB signals do not directly correspond to the device‐independent tristimulus values based on the CIE standard colorimetric observer. One approach for deriving a colorimetric mapping between camera RGB signals and CIE tristimulus values uses polynomial modeling and is described here. The least squares fitting technique was used to derive the coefficients of 3 × n polynomial transfer matrices, yielding a modeling accuracy typically averaging 1 ΔE units in CMC(1:1) when a 3 × 11 matrix is used. Experiments were carried out to investigate the repeatability of the digitizing system, characterization performance when different polynomials were used, modeling accuracy when 8‐bit and 12‐bit RGB data were used for characterization, and the number of reference samples needed to achieve a reasonable degree of modeling accuracy. Choice of characterization target and media and their effect on metamerism have been examined. It is demonstrated that a model is dependent upon both media and colorant, and applying a model to other media/colorants can lead to serious eye–camera metamerism problems. © 2001 John Wiley & Sons, Inc. Col Res Appl, 26, 76–84, 2001     

A review of principal component analysis and its applications to color technology

Principal component analysis, abbreviated PCA, has been an important and useful mathematical tool in color technology since the 1960s. Its uses have included defining tolerance intervals and ellipsoidal regions, estimating colorant spectral properties from mixtures, deriving CIE daylight, data reduction for large ensembles of spectra, and spectral imaging. Although PCA is a common topic in many engineering disciplines, statistics, and mathematics, many color‐technology professionals and color‐science students come from disciplines where this technique is not part of their curricula. It is from this perspective that this review publication was written. The purpose of this publication is to describe PCA and present examples in its use for colorant estimation, spectral data reduction, and defining multidimensional confidence regions for colorimetric scatter data. © 2005 Wiley Periodicals, Inc. Col Res Appl, 30, 84–98, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/col.20086     

Reduced-order distillation model using collocation method with cubic splines

A simple and compact form of reduced-order distillation model especially suitable for real-time applications is proposed. For this purpose, a modular collocation approach with the cubic spline interpolation function is developed and applied to an underlying distillation model which is constructed based on the McCabe and Thiele assumptions plus constant tray holdups. To evaluate the performance of the model, numerical simulations are carried out for the case of dynamics as well as steady states. As a consequence, it is found that the proposed reduced-order model gives better approximation than those obtained by the conventional reduced-order model with the Lagrange interpolation function.     

Color‐difference formula performance for several datasets of small color differences based on visual uncertainty

Visual uncertainty, while reported, is not used routinely when evaluating color‐difference formula performance in comparison with visual data; rather, data are analyzed assuming no uncertainty; that is, repeating the experiment would result in the identical average results. Previously, Shen and Berns developed three methods to determine whether a color‐difference formula was well‐fitting, under‐fitting, or over‐fitting visual data when visual uncertainty was considered, the method dependent on how the uncertainty was reported and the colorimetric sampling of the color‐difference stimuli. The “nonellipsoid standard error method” was used in the current analyses. Three datasets were evaluated: BFD‐P, Leeds, and Witt. For the BFD‐P data, incorporating visual uncertainty led to the same performance results as the average results, that CIEDE2000 was an improvement over CIE94, which was an improvement over CIELAB. For the Witt data, incorporating visual uncertainty led to the same performance results as the average results, that CIEDE2000 and CIE94 had equivalent performance, both an improvement over CIELAB. However, both formulas under‐fitted the visual results; thus, neither formula was optimal. For the Leeds dataset, the visual uncertainty analysis did not support the improvement of CIEDE2000 over CIE94 that occurred when evaluating the average results. Both formulas well fit the visual data. These analyses also provided insight into the tradeoffs between the number of color‐difference pairs and the number of observations when fitting a local contour of equal perceived color difference: In particular, increasing the number of observations was more important than increasing the number of color‐difference pairs. Finally, average standard error could be used to approximate visual uncertainty defined using STRESS. © 2010 Wiley Periodicals, Inc. Col Res Appl, 2011     

Comparison of real colour gamuts using a new reflectance database

A large set of data, comprising the spectral reflectances of real surface colours, has been accumulated. The data comprise 16 groups with different materials and include 85,879 measured spectra. From these data, CIELAB colorimetric coordinates were calculated under CIE illuminant D50 and the CIE 1931 standard colorimetric (2°) observer. Several published colour gamuts including those developed by Pointer and ISO reference colour gamut [ISO Graphic Technology Standard 12640‐3:2007] were compared using the present data set. It was found that the Pointer gamut is smaller than the new real data in most of the colour regions. The results also showed that the ISO reference colour gamut is larger than the new real accumulated data in most regions. The present finding indicates that there is a need to derive a new colour gamut based on the newly accumulated data for common applications. © 2013 Wiley Periodicals, Inc. Col Res Appl, 39, 442–451, 2014     

Different matrices for CIECAM02

CIECAM02 has been used to predict colour appearance under a wide range of viewing conditions, to provide a uniform colour space, and to provide a profile connection space for colour management. However, some problems have been identified with CIECAM02. CIE Technical Committee 8‐11 is currently working on the methods to resolve the problems for practical applications. Part of the CIE TC8‐11 work is to repair the lightness computational failure. This article incorporates most of the previous suggestions and makes some further tests. It is hoped that this article will aid the ongoing work of the CIE TC8‐11 and perhaps be the starting point for repairing the CIECAM02 for colour management and for the new recommendation. © 2012 Wiley Periodicals, Inc. Col Res Appl, 39, 143–153, 2014     

Inter‐observer comparison of color‐matching functions

The colorimetric difference between pairs of observers is simulated by a proper filtering of the stimulating radiation, and their comparison is made on properly defined Common Reference Frames in the tristimulus space. As examples, two comparisons are proposed: (1) Comparison between the Vos modification of the CIE 1931 Standard Colorimetric Observer and the CIE 1964 Supplementary Standard Observer: in this case, it is supposed that the difference between these two color‐vision systems is due to the macula lutea only, which with a spectral selective absorbance alters the power spectral distribution of the color stimuli. The optical density of the macular pigment is well reproduced. (2) Comparison between the Vos modification of the CIE 1931 Standard Colorimetric Observer and the CIE 1931 Standard Colorimetric Observer: in this case, the difference between these two observers could be simulated by different calibration of the photodetectors. © 1999 John Wiley & Sons, Inc. Col Res Appl, 24, 177–184, 1999