Physiological causes of individual variations in color-matching functions

The present study confirms that the individual variations of Stiles' 20 color-matching functions are physiologically well predicted by those of eye lens and macular optical densities. The principal-component analysis of the blue color-matching functions shows the significance of the two independent spectral components, which are expected to correspond to lens and macular optical densities. The lens and the macular densities for each of Stiles' 20 observers are estimated by using their published data physiologically measured. The estimated densities predict well each of Stiles' 20 color-matching functions. The singular-value decomposition of the estimated 20 color-matching functions provides a good estimate of the standard-deviate observer derived from the original Stiles 20 color-matching functions by using the same procedure.     

Prediction of observer variation in estimating colorimetric values

This article describes a method for computing easily the contribution of observer variation to colorimetric values. Stiles's 20 observers are used for representing color-normal observers. Deviation functions are derived by applying singular-decomposition analysis to their 20 color-matching functions. It is shown that variances and covariances of the 20 color-matching functions are estimated correctly by using the four deviation functions with high contribution factors. By using these deviation functions, a method is developed to derive the confidence ellipsoid of tristimulus values, the confidence ellipse of chromaticity coordinates, and the confidence interval of tristimulus value Y, in the colorimetric computations of a reflecting sample and an illuminant. With a slight modification, the method also can be used to predict the degree of mismatch found by color-normal observers for pairs of reflecting samples that are a metameric match with respect to the average observer and a standard illuminant.     

Comparison of methods for assessing observer metamerism

Studies on observer metamerism reported so far are classified into two series. One is studies based on the color-matching functions of Stiles's 20 observers. The other is experimental studies by using the Davidson and Hemmendinger (D-H) Color Rule and color-normal actual observers. The large discrepancy of the degree of observer metamerism between the above two series of studies was analyzed by using the color-matching functions of Stiles's 20 observers and the D–H Color Rule. The results confirmed that the discrepancy in the observer-metamerism indices was caused by different computational procedures used for deriving the indices.     

Measuring color-matching functions. Part I

A visual colorimeter was designed to implement a new model for the determination of color-matching functions developed in the Munsell Color Science Laboratory. An instrument utilizing laser primaries was originally developed. The instrument was later improved to incorporate CRT primaries and seven interference filters to make matches to simulated daylight in a 2° bipartite field. The system was designed to minimize the strain on observers. Color-matching functions of naïve observers can be measured in approximately 30 minutes. Results from an individual observer correlate well with data collected on the National Research Council of Canada's Trichromator. Part II of this article gives the color-matching functions results for 18 observers and 20 repetitions by a single observer.     

Observer variability in metameric color matches using color reproduction media

Standard color-matching functions are designed to represent the mean color-matching response of the population of human observers with normal color vision. When using these functions, two questions arise. Are they an accurate representation of the population? And what is the uncertainty in color-match predictions? To address these questions in the dual context of human visual performance and cross-media reproduction, a color-matching experiment was undertaken in which twenty observers made matches between seven different colors presented in reflective and transmissive color reproduction media and a CRT display viewed through an optical apparatus that produced a simple split-field stimulus. In addition, a single observer repeated the experiment 20 times to estimate intra-observer variability. The results were used to evaluate the accuracy of three sets of color-matching functions, to quantify the magnitude of observer variability, and to compare intra- and inter-observer variability in color-matching. These results are compared with various techniques designed to predict the range of color mismatches. The magnitude of observer variability in this experiment also provides a quantitative estimate of the limit of cross-media color reproduction accuracy that need not be exceeded. On average, the differences between matches made by two different observers was approximately 2.5 CIELAB units. © 1997 John Wiley & Sons, Inc. Col Res Appl, 22, 174–188, 1997     

Field trials for assessing the method of observer metamerism adopted by CIE

The following field trials are made for assessing the method of observer metamerism adopted by CIE. (1) The individual variation of metameric match was assessed between a fluorescent-lamp light and each of three different matching stimuli by the CIE method. The high precision of visual color match was confirmed for the 6-primary Donaldson colorimeter. The prediction was compared with experimental results for a similar fluorescent lamp. (2) The individual variation predicted by the CIE method was compared with that directly derived by using the Stiles original 20 color-matching functions. The effectiveness of the CIE method was confirmed. (3) It was clarified that the individual variation of colorimetric values on a single test stimulus corresponds to that for the metameric match between the test stimulus and a mixture of the CIE 10° r, g, b primaries. (4) The actual observer variation found by Stiles and Wyszecki in the field trial on the CIE 1964 color-matching functions was tested using the CIE method. The method is effective to assess the intrusion of other factors in actual color match, in addition to the individual variation of color-matching functions.     

New Equipment for the Measurement of Color-Matching Functions

This paper describes equipment designed to measure the color-matching functions of a human observer, together with the results obtained. Its future potentialities are discussed, as well as its limitations, accúracy, and reliability. The results obtained are in close agreement with the CIE 1964 Standard Colorimetric Observer data and will enable a statistical study of individual color-matching characteristics to be made.     

Do Tristimulus Values have Units?

It is shown that tristimulus values have been endowed with various dimensions, ranging from power to no dimensions at all. Values of the color-matching functions (which are tristimulus values by one definition) have no dimensions, but can have units in the sense of inches-per-meter. When associated with real primaries, color-matching functions can have the units of power (e.g., watts) of the indicated wavelength per unit power (watt) of the indicated primary. Color-matching functions associated with non-real primaries (such as CIE XYZ) have no units at all. If the color-matching functions are defined to comprise tristimulus weights instead of values. the dimensional inconsistency can be avoided. The indeterminacy of units, on the other hand, is intrinsic to the affine property of color-matching space, and cannot be eliminated.     

Practical transformations of CIE color-matching functions

By using the CIE color-matching functions, we describe a numerical method to derive new color-matching functions C(Λ) which satisfy the conditions (1) C(Λ) ≥ 0, (2) C(Λ) with a single peak, and (3) C(Λ)'s with the least overlaps.     

Color and color mixture: Scalar and vector fundamentals

This article explores the consequences of the Wyszecki hypothesis, that every color stimulus (radiometric function) comprises two parts, a fundamental and a residual. (Wyszecki's terms were fundamental color stimulus and metameric black.) The fundamental alone is processed by the visual system and evokes the color sensation. The residual is not processed and is without effect on color sensations. Metamers, color stimuli evoking the identical color sensation but with different radiometric functions, have the same fundamental but different residuals. Matrix R, an orthogonal projector, resolves any radiometric function into its fundamental and residual. Unlimited numbers of metamers may be generated by adding other residuals to the same fundamental. The color-matching equation has historically been written with tristimulus values as coefficients, but so written, the equation balances only psychologically. When the stinuli of the color-matching equation are replaced by the fundamentals processed by the visual system (after the residuals are ejected by matrix-R operations), the equation balances psychologically, physically, and mathematically. Wyszecki's fundamental has two representations, the scalar fundamental as conceived by Wyszecki and the vector fundamental as developed in this article. Vector fundamentals define a fixed, invariant, fundamental color space governed by Euclidean geometry. The properties of this space are explored. The article develops relationships between color-matching functions, fundamentals, and orthonormal color functions. Two new specifications for color stimuli or sensations are introduced, tricolor values and tricolor coordinates. The article assumes knowledge of ordinary algebra and geometry, and procedures for computation of tristimulus values.     

Hue uniformity and the CIELAB space and color difference formula

The hue uniformity of the CIELAB system is investigated using a hue circle of Munsell colors at value 6 and chroma 14 and experimentally determined hue coefficient data. CIELAB hue differences for equal Munsell hue increments are found to vary up to nearly a factor 4, and hue coefficients differ from the experimentally determined ones by up to 40% at certain wavelengths. Dominant wavelengths assigned by the CIELAB system to individual Munsell hues are found to vary up to 35 nm from those of the Munsell Renotations. Four other color space systems are compared with widely differing but comparable results. The CIE 2° color-matching functions are adapted to result in a set of opponent-color functions accurately representing the Munsell Hue and Chroma data. A call is made for the experimental determination of the “standard hue observer” as a step toward an improved color space/color-difference formula. © 1998 John Wiley & Sons, Inc. Col Res Appl, 23, 314–322, 1998     

Formulation of a standard deviate observer by a nonlinear optimization technique

A standard deviate observer (SDO) has been formulated that can predict the average color mismatches of metameric color pairs by 20 Stiles' observers. A nonlinear optimization technique, rather than a statistical analysis, has been employed to derive it. The colorimetric characteristics of the present SDO are compared with those of SDOs previously proposed.     

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     

A better colorimetric standard observer for color-vision studies: The stiles and burch 2° color-matching functions

The validity of the 1931 CIE Standard Observer color-matching functions (c.m.f.) rests on the assumption that V_λ is a linear combination of actually measured c.m.f. It is shown here that no combination of c.m.f. can reproduce any photometric function. For research on basic questions of color vision it is suggested that the c.m.f. of the Stiles and Burch 2° preliminary study be used. These c.m.f. are reproduced here, after correction and renormalization for a calibration error in the original study. Since photometry appears to be intrinsically more complicated and less well understood than basic colorimetry, it is also suggested the two disciplines be considered as separate aspects of the visual process.     

Abnormal Hue-Angle change of the gemstone tanzanite between CIE illuminants d65 and a in CIELAB color space

The blue-to-purple color appearance change observed in some rare specimens of the gemstone tanzanite between daylight and incandescent light is contrary to the hue-angle change calculated between CIE illuminants D65 and A in CIELAB color space. This abnormal calculated hue-angle change for tanzanite can be corrected by using the spectral sensitivity functions of the three kinds of cone photoreceptors to directly calculate color. This study suggests that the cone spectral sensitivity functions are more fundamental in color calculations than the CIE color-matching functions.     

How strong metamerism disturbs color spaces

Systems for arranging and describing color include “color spaces” and “color order systems.” In a color space, tristimulus values R, G, and B are computable for every light (every point in the space). In familiar color spaces, such computation makes use of three functions of wavelength (the color-matching functions that define one of the CIE Standard Observers), one function corresponding to each of R, G, and B. In the presence of strong metamerism (marked spectral difference between the spectral power distributions of a pair of visually matching lights), the color-matching functions may report that one light of the pair has an entirely different color from that of the other member of the visually matching pair of lights. The CIE Standard Observer embodying those color-matching functions “sees” the two visually matching lights as entirely different in color, that is, it reports entirely different sets of R, G, and B for the two visually matching lights, and, thus, an entirely different chromaticity. In an example given here, each of the CIE Standard Observers assigns a strong green color to lights that are seen by normal human observers as a visual match to a hueless reference white. On the other hand, color order systems comprising sets of real objects in a specified illuminant, and which are assembled (visually arranged) by normal observers, as are the Munsell and OSA sets, do not suffer from the type of trouble discussed here. Color spaces depending on mathematical functions of R, G, and B are at risk: both Standard Observers are shown to plot visually identical lights at widely varying points in familiar color spaces (e.g., delta E*_ab = 40–50). © 1998 John Wiley & Sons, Inc. Col Res Appl, 23: 402–407, 1998     

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     

多变量系统广义预测目标函数解耦设计

提出一种带设定值观测器的广义预测目标函数解耦控制算法,在目标函数中用控制量的增量代替控制量,使控制增量变化不至于过于剧烈,并对目标函数进行解耦设计,加入设定值观测器进一步减弱耦合,还引进了阶梯式控制策略,大大减少了计算量.最后给出了球磨机模型的仿真结果.     

Tracing a metameric match to individual variations of color vision

A metameric match made by an observer under established experimental conditions might not be the same as made by another, indicating that the color vision differs between the two observers. We have analyzed the variations of normal color vision using metameric surfaces. First, using a set of cone fundamentals, we modeled the matches of a theoretical observer with normal color vision on the Davidson & Hemmendinger (D&H) Color Rule. We also derived deviate cone fundamentals by changing the macular pigment density and the lens density, and by shifting the long-wave-sensitive photopigment along the wave-number axis. The results showed shifts in the matches for the deviate observer of no more than one sample on the D&H Color Rule, the largest shifts being due to lens density. Second, we modeled the matches made on the D&H Color Rule of 8 observers by computing their personalized cone fundamentals using their independently recorded measures of macular pigment density and lens density, and their Rayleigh matches. The results show that the use of personalized cone fundamentals provides a better prediction than the use of data from a theoretical observer. © 1998 John Wiley & Sons, Inc. Col Res Appl, 23: 379–389, 1998     

Global Color Metrics and Color-Appearance Systems

The Munsell Color System, the color-appearance system most widely used, has been constructed primarily on the basis of local perceptual uniformity. However, when it is used as a criteron for a uniform chromaticity scale, its global structure is important as well, and this structure is examined in three ways. First, the coordinates H, V, C are shown to function as a conceptual framework in identifying colors. Second, a summary of 12 studies by multidimensional scaling methods based on large color differences is given. It shows that the main part of the Munsell space is approximately Euclidean. Third, a new method based on estimation of principal hue components in each color, 120 in an experiment, yields the same result. Chromatic response functions as in opponent-color theories are obtained with object colors by this method. In all the results with Japanese observers, 5PB and 5B appear to be too close and both deviate toward 5G compared with the Munsell notation. Whether the finding is due to the observers is discussed.