ESBD_397352_1740001781986_03. Attachment A.pdf

Standard Practice for

Calculation of Color Tolerances and Color Differences from Instrumentally Measured Color Coordinates1

This standard is issued under the fixed designation D2244; the number immediately following the designation indicates the year of original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A superscript epsilon (´) indicates an editorial change since the last revision or reapproval.

This standard has been approved for use by agencies of the U.S. Department of Defense.

INTRODUCTION

This practice originally resulted from the consolidation of a number of separately published methods for the instrumental evaluation of color differences. As revised in 1979, it included four color spaces in which color-scale values could be measured by instruments, many of which were obsolete, and the color differences calculated by ten equations for different color scales. The sections on apparatus, calibration standards and methods, and measurement procedures served little purpose in the light of modern color-measurement technology. The revision published in 1993 omitted these sections, and limited the color spaces and color-difference equations considered, to the three most widely used in the paint and related coatings industry. A previous revision added two new color tolerance equations and put one of the color difference equations from the 1993 version in an informative appendix for historical purposes.

1. Scope

1.1 This practice covers the calculation, from instrumentally measured color coordinates based on daylight illumination, of color tolerances and small color differences between opaque specimens such as painted panels, plastic plaques, or textile swatches. Where it is suspected that the specimens may be metameric, that is, possess different spectral curves though visually alike in color, Practice D4086 should be used to verify instrumental results. The tolerances and differences determined by these procedures are expressed in terms of approximately uniform visual color perception in CIE 1976 CIELAB opponent-color space (1),2 CMC tolerance units (2), CIE94 tolerance units (3), the DIN99o color difference formula given in DIN 6176 (4), or the CIEDE2000 color difference units (5).

1.2 For product specification, the purchaser and the seller

shall agree upon the permissible color tolerance between test specimen and reference and the procedure for calculating the color tolerance. Each material and condition of use may require specific color tolerances because other appearance factors, (for example, specimen proximity, gloss, and texture), may affect the correlation between the magnitude of a measured color difference and its commercial acceptability.

1.3 This standard does not purport to address all of the

safety concerns, if any, associated with its use. It is the responsibility of the user of this standard to establish appro-priate safety, health, and environmental practices and deter-mine the applicability of regulatory limitations prior to use.

1.4 This international standard was developed in accor-

dance with internationally recognized principles on standard-ization established in the Decision on Principles for the

Development of International Standards, Guides and Recom-mendations issued by the World Trade Organization Technical

Barriers to Trade (TBT) Committee.

2. Referenced Documents

2.1 ASTM Standards:3

D1729 Practice for Visual Appraisal of Colors and Color Differences of Diffusely-Illuminated Opaque Materials

D4086 Practice for Visual Evaluation of Metamerism E284 Terminology of Appearance E308 Practice for Computing the Colors of Objects by Using the CIE System E805 Practice for Identification of Instrumental Methods of

1 This practice is under the jurisdiction of ASTM Committee E12 on Color and Appearance and is the direct responsibility of Subcommittee E12.04 on Color and Appearance Analysis.

Current edition approved June 1, 2023. Published June 2023. Originally approved in 1964. Last previous edition approved in 2022 as D2244 – 22. DOI:

10.1520/D2244-23.

2 The boldface numbers in parentheses refer to the list of references at the end of this standard.

3 For referenced ASTM standards, visit the ASTM website, www.astm.org, or contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM

Standards volume information, refer to the standard’s Document Summary page on the ASTM website.

Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States

1� � � � � � � � � � �

D2244

Purchase of Portable Spectrophotometer for Materials and Tests Division Specification No. TxDOT 493-77-47328

Solicitation No. 601440000047328 Attachment A

Color or Color-Difference Measurement of Materials E1164 Practice for Obtaining Spectrometric Data for Object-

Color Evaluation

2.2 Other Standards:

DIN 6176 English translation of DIN 6176:2018-10 Far-bmetrische Bestimmung von Farbabstanden bei Korper-farben nach der DIN 99o-Formel, (Colorimetric determi-nation of colour differences of object colours according to the DIN 99o formula)4

3. Terminology

3.1 Terms and definitions in Terminology E284 are appli-

cable to this practice.

3.2 Definitions of Terms Specific to This Standard:

3.2.1 colorimetric spectrometer, n—spectrometer, one com-ponent of which is a dispersive element (such as a prism, grating or interference filter or wedge or tunable or discrete series of monochromatic sources), that is normally capable of producing as output colorimetric data (such as tristimulus values and derived color coordinates or indices of appearance attributes). Additionally, the colorimetric spectrometer may also be able to report the underlying spectral data from which the colorimetric data were derived.

3.2.2 color tolerance equation, n—a mathematical expression, derived from acceptability judgments, which dis-torts the metric of color space based on the coordinates in that color space, of a reference color, for the purpose of single number shade passing.

3.2.2.1 Discussion—The color tolerance equation computes

a pass/fail value based on which of the pair of specimens is assigned the designation “standard.” Thus, inter-changing the reference and test specimens will result in a change in the predicted level of acceptance between the specimens while the perceived difference is unchanged. A color difference equation quantifies distance in a color space using the metric of that space. Inter-changing the reference and test specimens does not change either the perceived or predicted color differences.

4. Summary of Practice

4.1 The differences in color between a reference and a test

specimen are determined from measurements made by use of a spectral based or filter based colorimeter. Reflectance readings from spectral instruments are converted by computations to color-scale values in accordance with Practice E308, or these color-scale values may be read directly from instruments that automatically make the computations. Color-difference units are computed, from these color-scale values, and approximate the perceived color differences between the reference and the test specimen.

5. Significance and Use

5.1 The original CIE color scales based on tristimulus

values X, Y, Z and chromaticity coordinates x, y are not uniform visually. Each subsequent color scale based on CIE values has had weighting factors applied to provide some degree of uniformity so that color differences in various regions of color space will be more nearly comparable. On the other hand, color differences obtained for the same specimens evaluated in different color-scale systems are not likely to be identical. To avoid confusion, color differences among specimens or the associated tolerances should be compared only when they are obtained for the same color-scale system. There is no simple factor that can be used to convert accurately color differences or color tolerances in one system to difference or tolerance units in another system for all colors of specimens.

5.2 Color differences calculated in ∆E00 units (6) are highly recommended for use with color-differences in the range of 0.0 to 5.0 ∆E*ab units. This color-difference equation is appropri-ate for and widely used in industrial and commercial applica-tions including, but not limited to, automobiles, coatings, cosmetics, inks, packaging, paints, plastics, printing, security, and textiles.

5.3 Users of color tolerance equations have found that, in

each system, summation of three, vector color-difference components into a single scalar value is very useful for determining whether a specimen color is within a specified tolerance from a standard. However, for control of color in production, it may be necessary to know not only the magni-tude of the departure from standard but also the direction of this departure. It is possible to include information on the direction of a small color difference by listing the three instrumentally determined components of the color difference.

5.4 Selection of color tolerances based on instrumental

values should be carefully correlated with a visual appraisal of the acceptability of differences in hue, lightness, and saturation obtained by using Practice D1729. The three tolerance equa-tions given here have been tested extensively against such data for textiles and plastics and have been shown to agree with the visual evaluations to within the experimental uncertainty of the visual judgments. That implies that the equations themselves misclassify a color difference with a frequency no greater than that of the most experienced visual color matcher.

5.5 While color difference equations and color tolerance

equations are routinely applied to a wide range of illuminants, they have been derived or optimized, or both, for use under daylight illumination. Good correlation with the visual judg-ments may not be obtained when the calculations are made with other illuminants. Use of a tolerance equation for other than daylight conditions will require visual confirmation of the level of metamerism in accordance with Practice D4086.

6. Description of Color-Difference and Color-Tolerance

Equations

6.1 CIE 1931 and 1964 Color Spaces—The daylight colors

of opaque specimens are represented by points in a space formed by three rectangular axes representing the lightness scale Y and chromaticity scales x and y, where:

x 5 X

X1Y1Z (1) y 5 Y

X1Y1Z (2)

4 Available from TechStreet.com and Beuth Verlag GmbH, 10772, Berlin, Germany, http://www.beuth.de.

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2� � � � � � � � � � � where X, Y, and Z are tristimulus values for either the 1931 CIE standard observer (2° observer) or the 1964 CIE standard observer (10° observer) and standard illuminant D65, or other phase of daylight. These scales do not provide a perceptually uniform color space. Consequently, color differences are sel-dom if ever computed directly from differences in x, y, and Y.

6.2 CIE 1976 L* a* b* Uniform Color Space and Color-

Difference Equation (1, 7)—This is an approximately uniform color space based on nonlinear expansion of the tristimulus values and taking differences to produce three opponent axes that approximate the percepts of lightness-darkness, redness-greenness and yellowness-blueness. It is produced by plotting in rectangular coordinates the quantities L*, a*, b*, calculated as follows:

L* 5 116 f ~QY! 2 16 (3) a* 5 500 @ f ~QX! 2 f ~QY! # (4) b* 5 200 @ f ~QY! 2 f ~QZ! # (5) where□□□□□□□□□□□□□□□□□□□□□□□□□□□□□

QX 5 ~X/Xn!; QY 5 ~Y/Yn!; QZ 5 ~Z/Zn!

and□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ f~Q i! 5 Q i 1/3 if Q i.~6/29!3 else□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ f~Q i! 5 ~841/108!Q i14/29 if Q i # ~6/29!3.

Here, i varies as X, Y, and Z.

The tristimulus values Xn, Yn, Zn define the color of the nominally white object-color stimulus. Usually, the white object-color stimulus is given by the spectral radiant power of one of the CIE standard illuminants, for example, C, D65 or another phase of daylight, reflected into the observer’s eye by the perfect reflecting diffuser. Under these conditions, Xn, Yn, Zn are the tristimulus values of the standard illuminant with Yn equal to 100.

6.2.1 The total color-difference ∆Eab* between two colors

each given in terms of L*, a*, b* is calculated as follows:

∆E*ab 5 =~∆L*! 21~∆a*!21~∆b*! 2 (6) NOTE 1—The color space defined above is called the CIE 1976 L* a * b* space and the color-difference equation the CIE 1976 L* a* b* color-difference formula. The abbreviation CIELAB (with all letters capitalized) is recommended.

6.2.2 The magnitude, ∆E*ab, gives no indication of the

character of the difference since it does not indicate the relative quantity and direction of hue, chroma, and lightness differ-ences.

6.2.3 The direction of the color difference is described by

the magnitude and algebraic signs of the components ∆L*, ∆a*, and ∆b*:

∆L* 5 L*B 2 L* S (7)

∆a* 5 a*B 2 a*S (8)

∆b* 5 b*B 2 b*S (9) where L*S, a*S, and b*S refer to the reference or standard, and L*B, a*B , and b*B refer to the test specimen or batch. The signs of the components ∆L*, ∆a*, and ∆b* have the following approximate meanings (8):

1∆L* 5 lighter (10)

2∆ L* 5 darker (11)

1∆a* 5 redder ~less green! (12)

2∆a* 5 greener ~less red! (13)

1∆b* 5 yellow ~less blue! (14)

2∆ b* 5 bluer ~less yellow! (15)

6.2.4 For judging the direction of the color difference

between two colors, it is useful to calculate hue angles hab and CIE 1976 metric chroma C*ab according to the following pseudocode:

if b* 5 0 then (16) hab 5 90 sign~a*! @sign~a*! 2 1# else □□□□□□□ hab 5 180 2 ~180/π!arctan~a*/b*! 2 90 sign~b*!

end if. □□□□□□

Here sign is a function that returns the sign of the argument, and arctan is the inverse tangent function returning angles in units of radians. The units of hab calculated by the above are degrees counter-clockwise from the positive a* axis. The function sign is expected to return a minus one for negative values of the argument, a zero when the argument is zero, and a positive one for positive values of the argument.

Differences in hue angle hab between the test specimen and reference can be correlated with differences in their visually perceived hue, except for very dark colors (9). Differences in chroma ∆C*ab = ([C*ab]batch − [C*ab]standard) can similarly be correlated with differences in visually perceived chroma.

C*ab 5 =~a*!21~b*!2 (17)

6.2.5 For judging the relative contributions of lightness

differences, chroma differences, and hue differences between two colors, it is useful to calculate the CIE 1976 Metric Hue Difference ∆H*ab between the colors as follows:

∆ H*ab 5 s @2~C*ab,B C*ab,S 2 a*B a*S 2 b*B b*S!#0.5 (18) where□□□□□□□□□□□□□□□□□□□□□

□if a*S b*B.a*B b*S then (19) s 5 1 else□□□□□□□□□□□□□□□□□□□□□□ s 5 21 end if.□□□□□□□□□□□□□□□□□□□□□

When ∆L*ab is calculated as in 6.2.3 and ∆C*ab is calculated as in 6.2.4, then

∆E*ab 5 @~∆L*!21~∆C*!21~∆H*!2#0.5 (20) contains terms showing the relative contributions of light-ness differences ∆L*ab, chroma differences ∆C*ab, and hue differences ∆H*ab.

6.3 CMC Color Tolerance Equation—The Colour Measure-

ment Committee of the Society of Dyers and Colourists undertook a task to improve upon the results of the JPC79 tolerance equation (2) developed at J & P Coats thread company in the United Kingdom. It was a combination of the

D2244 − 23

3� � � � � � � � � � �

CIELAB equation and local optimization based on the position of the standard used to derive the FMC-2 equation. It was based on the more intuitive perceptual variables of lightness, chroma and hue instead of the lightness, redness/greenness and yellowness/blueness of the older equation. It is intended to be used as a single-number shade-passing equation. There should not be a need to break the equation down into perceptual components—the CIELAB components of the model do that already. Fig. 1(10) shows the CIELAB chromaticness plane (a*, b*) with a large number of CMC ellipsoids plotted on that plane. The figure clearly shows the change in area of the ellipses with increases in CIELAB metric chroma C*ab and with respect to changes in CIELAB metric hue angle h*ab. The CMC components and single number tolerances are computed as follows:

∆ECMC~l:c! 5ŒS ∆L* l ·SL

D

1S ∆C* c ·Sc

D

1S ∆H*

SH

D

(21)

The most common values for the lightness to chroma ratio l:c is (2:1) for textiles and plastics that are molded to simulate a woven material, implying that lightness differences carry half the importance of chroma and hue differences (11). The values (1:1), often assumed to represent a just perceptible difference, should be applied to materials that require very critical tolerances or have glossy surfaces. For specimens that are matte, randomly rough, or mildly textured, values intermediate between (1:1) and (2:1) can be used, with the value (1.3:1) being reported most frequently.

The color dependent functions are defined as:

SL 5

0.040975·L*

~110.01765·L*!

for L* $ 16 (22)

SL 5 0.511, for L*,16

SC 5

0.0638·C*

~110.0131·C*!

10.638

SH 5 SC~T ·f11 2 f!

where□□□□□□□□□□□□□□□□□□□□□ f 5 H ~C*!4

~C*!411900J

T 5 0.561?0.2cos~h1168°!? , if 164°,h,345° else□□□□□□□□□□□□□□□□□□□□□□

T 5 0.361?0.4cos~h135°!?

All angles are given in degrees but will generally need to be converted to radians for processing on a digital computer. In Eq 22, the values of L*, C*, and h are taken to be those of the standard specimen.

The use of a commercial factor cf is no longer recom-mended. See Appendix X3.

6.4 CIE94 Color Tolerance Equation (3)—The development

of this color tolerance equation was prompted by the success of the CMC tolerance equation. It was derived primarily from visual observations of automotive paints on steel panels. Like the CMC equation, it is based on the CIELAB color metric and uses the position of the standard in CIELAB color space to derive a set of analytical functions that modify the spacing of the CIELAB space in the region around the standard. Its weighting functions are much simpler than those of the CMC equation. CIE94 tolerances are computed as follows:

∆E*94 5 F S ∆L* kLSL

D

1S ∆C* kCSC

D

1S ∆H* kHSH

D

G 0.5

Unlike many previous color difference equations, CIE94 comes with a well defined set of conditions under which the equation will provide optimum results and departures from this set of conditions will cause the agreement between the visually evaluated color-difference and the computed color-difference to be significantly poorer. Those conditions are given in Table

1. The parameters kL, kC, kH are the parametric factors that can be used to compensate for texture and other specimen presen-tation effects. These should not be used to introduce a commercial factor into the equation. For more information on the use of commercial factors in color tolerance equations, see Appendix X3. All the k values default to 1 in the absence of specific information or agreement between parties. The param-eters SL, SC, SH are used to perform the local distortion of CIELAB color space, again based on the position of the standard specimen in that space. They are computed using the following equations:

FIG. 1 CMC Ellipse Distribution in the CIELAB (a*, b*) Plane

TABLE 1 Basis Conditions for CIE94 Tolerance Equation

Attribute Requirement

Illumination D65 source

Specimen Illuminance 1000 lx

Observer Normal color vision

Background Uniform neutral gray L * = 50

Viewing Mode Object

Sample Size >4° subtended visual angle

Sample Separation Minimum possible

Size of Color Differences 0 to 5 CIELAB units

Sample Structure Visually homogenous

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4� � � � � � � � � � �

SL 5 1 (24)

SC 5 110.045·C*

SH 5 110.015·C*

In Eq 24, the value of C* is taken to be that of the standard specimen.

6.5 DIN99o Color Difference Equation—The publication in

1996 of the paper by Rohner and Rich (4) prompted the German standards institute (DIN) to further develop and standardize a modified version as a new color difference formula that globally modeled color space using logarithms of the CIELAB coordinates rather than the linear and hyperbolic functions of CMC and CIE94. The equations derived and documented in standard DIN 6176 provides an axes rotation and the logarithmic expansion of the axes of the CIELAB formula to match perceptual color distance tolerances of a large number of color data sets, that have also been used to optimize the CIE94 and CIEDE2000 formula (6). With the advantage of using Euclidean distances in the new and expanded color space, the results of describing differences of color are even better for small color distances if compared to CIEDE2000 results (6), save the direct surrounding of the grey axes. Also, as neither the tristimulus values XYZ nor the CIELAB axes a*, b* are perceptual variables while the axes L*, C* and h* are correlates of the perceptions of lightness, chroma, and hue, it seemed appropriate to scale the differences or distances in color space following the Weber-Fechner law of perception. In 2017, the coefficients of the original DIN99 were improved, resulting in the now recommended formulas DIN99o. The new coefficients resulted in a formula which is easy to use and has equivalent performance to CMC or CIE94, and computed color differences are based only on the Euclidean distance in the DIN99o space. The procedures for computing the DIN99o formula are listed as follows:

DIN reference condition parameters:

kCH 5 kE 5 1 (25)

DIN99o lightness:

L99o 5 303.67 ln~1.0 1 0.0039 L *! ⁄kE (26)

Auxiliary variable for redness:

eo 5 a*cos~26 °!1b*sin~26 °! (27)

Auxiliary variable for yellowness:

fo 5 20.83 a* sin~26 °!10.83 b* cos~26 °! (28)

Auxiliary variable for chroma:

G99o 5 ~eo 2 1 fo

2!0.5 (29)

Auxiliary variable for hue angle: (in radians) heofo 5 0 (30) for eo = 0 and fo = 0 heofo 5 arctan~fo ⁄ eo! (31) for eo > 0 and fo ≥ 0 heofo 5 π ⁄2 (32) for eo = 0 and fo > 0 heofo 5 π1arctan~fo ⁄ eo! (33) for eo < 0 heofo 5 3π ⁄2 (34) for eo = 0 and fo < 0 heofo 5 2π1arctan~fo ⁄ eo! (35) for eo > 0 and fo < 0

DIN hue angle: (in degrees) h99o 5 heofo180° ⁄π126° (36) for heofo < 334°π/180° h99o 5 ~heofo 2 2 π!180° ⁄π126° (37) for heofo ≥ 334°π/180°

DIN99o chroma:

C99o 5 @ln ~1 1 0.075Go!# ⁄~0.0435kCH kE! (38)

DIN99o redness-greenness:

a99o 5 C99ocos~h99o π ⁄ 180 °! (39)

DIN99o yellowness-blueness:

b99o 5 C99osin~h99o π ⁄ 180 °! (40)

DIN99o lightness difference:

∆L99o 5 L99oB 2 L99oS (41)

DIN99o red-green difference:

∆a99o 5 a99oB 2 a99oS (42)

DIN99o yellow-blue difference:

∆b99o 5 b99oB 2 b99oS (43)

DIN99o color difference:

∆E99o 5 @~∆L99o!2 1 ~∆a99o!2 1 ~∆b99o!2#0.5 (44) or equivalently DIN99o chroma difference:

∆C99o 5 C99oB 2 C99oS (45)

DIN99o hue difference:

∆H99o 5 2~a99oB b99oS 2 a99oS b99oB!

@0.5 ~C99oB C99oS 1 a99oB a99oS 1 b99oB b99oS!#0.5 (46)

DIN99o color difference:

∆E99o 5 @~∆L99o!2 1 ~∆C99o!2 1 ~∆H99o!2#0.5 (47) where subscript S refers to the product standard and sub-script B refers to the current product batch or test sample. Here arctan is the inverse tangent function returning the function value into angles in units of radians. The units of h99o calculated by the above are degrees counterclockwise for the range from 0 to 360 from the positive eo axis.

The values of the parameters kE and kCH for most applica-tions will be unity. These factors refer to a reference set of viewing conditions that are applicable to most color-difference industrial viewing practices and are essentially those of Table 1 of this standard. Changes in viewing conditions or changes in materials, such as textiles, can be taken into account by the appropriate selection of kE and kCH values. Values of kE and

D2244 − 23

5� � � � � � � � � � � kCH other than unity lead to changes in both color-difference and color coordinates. If it is intended to use k factors other than unity, their values shall be agreed upon and indicated in the measurement report.

6.6 CIEDE2000 Color Difference Equation (5)—The devel-

opment of this color difference equation grew out of the research being performed to try to determine which of the two color tolerances equations, CMC or CIE94, was the better formula. In the process, the researchers came to the conclusion that neither formula was truly optimum. Therefore the CIE set up a new technical committee, TC 1-47, Hue & Lightness Dependant Correction to Industrial Colour Difference Equations, to recommend a new equation that addresses the short-comings in both color tolerance equations. One of the major weaknesses of the color tolerance equations was using the position of the reference color in CIELAB color space for computing the local distortion of CIELAB color space. When the identifications of the two specimens are reversed (calling the original test specimen the reference and the original reference now the test specimen) the computation results in a different computed color difference. This is contrary to what is observed. Visually, there is no change in the magnitude of the difference between the specimens simply by switching roles.

By using the position of the arithmetic average color between the two specimens to compute the local distortions to CIELAB color space, the roles of the two specimens may be switched without changing the magnitude of the computed color-difference, in full agreement with the visual assessments. The report from CIE TC 1-47 has shown that CIEDE2000 out-performs both CMC and CIE94 across a wide array of specimens. The CIEDE2000 color differences are computed from the following equations:

L ' 5 L* a ' 5 ~11G! ·a* b ' 5 b* (48)

C ' 5 =a '21b '2 if b ' 5 0 then□□□□□□□□□□□□□□□□□□ h ' 5 90 sign~a '! @sign~a '! 2 1# else□□□□□□□□□□□□□□□□□□□□□□□ h ' 5 180 2 ~180/π!arctan~a ' /b '! 2 90 sign~b '!

end if.□□□□□□□□□□□□□□□□□□□□□

Here sign and arctan are functions that are defined in and are expected to return values as stated in 6.2.4.

G 5 0.5·S 1 2Œ C *̄ 7

C *̄71257 D where C *̄ is the arithmetric mean of the CIELAB C* values for the pair of specimens (standard and batch).

∆L ' 5 L 'B 2 L 'S

∆C ' 5 C 'B 2 C 'S

∆H ' 5 s @2 ~C 'B C 'S 2 a 'B a 'S 2 b 'B b 'S! #0.5 where□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ s 5 1 if a 'S b 'B.a 'B b 'S, else s 5 21.

∆E00 2 5 S ∆L ' kL·SL

D

1S ∆C ' kC·SC

D

1S ∆H ' kH·SH

D

1RT·S ∆C ' ·∆H '

kC·SC·kH·SH

D ∆ E00 5 =∆E00

The specimen or industry dependent parameters are kL, kC, kH (all defaulting to unity in the absence of specific information or agreement between parties). SL, SC, SH and RT. The three S terms operate on the, assumed orthogonal, CIELAB coordi-nates and the RT term computes a rotation of the color difference volume in the blue and purple-blue regions of the CIELAB diagram. The four color space terms are computed as follows:

SL 5 11

0.015·~L '̄ 2 50! 2

=201~L '̄ 2 50! 2

SC 5 110.045·C '̄

SH 5 110.015·C '̄ ·T

RT 5 2 sin ~2·∆θ! ·RC

RC 5 2·Œ C '̄7

C '̄71257

∆θ 5 30·expS2F ~h '̄ 2 275°!

G

D T 5 1 2 0.17· cos ~h '̄ 2 30°!10.24· cos ~2h '̄!10.32· cos ~3h '̄16°!

2 0.20· cos ~4h '̄ 2 63°!

The following pseudocode (12) will calculate h '̄ for substi-tution in the above equation:

p5~h 'S1h 'B!/2 q5Abs~h 'S2h 'B!

if C 'S C 'B50 then h '̄52p else if q.180 then if p,180 then h '̄5p1180 else□□□□□□□□□□□□□□□□□□□□□□□□ h '̄5p2180 end if□□□□□□□□□□□□□□□□□□□□□ else□□□□□□□□□□□□□□□□□□□□□ h '̄5p end if□□□□□□□□□□□□□□□□□□

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6� � � � � � � � � � �

Here Abs means the absolute value of the argument.

While not obvious from this listing, all displayed angles are assumed to be given in degrees, including ∆θ and thus must generally be converted into radians for trigonometric analysis on digital computers.

6.6.1 Using the arithmetic average of the CIELAB color

coordinates of the reference and test specimens to compute the local distortion of CIELAB color space introduces a new problem. Current color tolerance difference equations which base the distortion of CIELAB space on the position of the standard allows a user to predefine the acceptance volume. This is convenient for certain textile sorting applications and for graphical quality control charting. Such a predetermination is not possible with CIEDE2000. Nor is it possible or reasonable to plot groups of colors in terms of the modified space coordinates, L*, a', b* since the meaning of a' is determined uniquely for each pair of colors. Thus the equation is highly optimized for pairwise comparison of a product standard to a production test specimen but not for statistical process control.

7. Test Specimens

7.1 This practice does not cover specimen preparation

techniques. Unless otherwise specified or agreed, prepare specimens in accordance with appropriate test methods and practices.

8. Procedure

8.1 Select appropriate geometric conditions for color mea-

surement in accordance with Practice E805.

8.2 Operate the instrument in accordance with the manufac-

turer’s instructions and the procedures given in Practice E1164.

8.3 When a colorimetric spectrometer is used, obtain the

reflectance values of the reference specimen and test specimens, in turn, at a sufficient number of wavelength intervals to permit accurate calculation of CIE tristimulus values. See Practice E308.

8.4 Measure at least three portions of each specimen surface

to obtain an indication of uniformity. Record the location where these measurements were made on the specimen.

9. Calculation

9.1 Calculate color-scale values L*, a*, b*, and local

tolerance weights (SL, SC, SH) if not obtained automatically.

9.2 Calculate color differences ∆E*ab,∆ ECMC and their

components, or ∆E94, ∆E99, or ∆E00, if not obtained automatically, as described in 6.2 – 6.6, respectively.

10. Report

10.1 Report the following information:

10.1.1 Total color difference ∆ECMC, ∆E94, ∆E99, or ∆E00 of

each test specimen from its reference.

10.1.2 For CIELAB color differences, L*, a*, b* for the

reference, ∆L*, ∆a*, ∆b* and if desired ∆hab, ∆C*ab, and ∆H*ab for each specimen.

10.1.3 For other color tolerance or color difference metrics, only the CIELAB coordinates should be reported as the local distortions do not necessarily provide continuous, visually correlated parameters.

10.1.4 For non-uniform specimens, range of color-

difference magnitudes obtained for different areas of the specimens.

10.1.5 Description or identification of the method of prepar-ing the specimens.

10.1.6 Identification of the instrument used, by the manu-

facturer’s name and model number.

10.1.7 The illuminant-observer combination and the color-

difference equation used.

11. Precision and Bias

11.1 Since the precision and bias of a test method cannot be

separated from the effect of the specimens and materials and since this practice does not address the issues related to the preparation and presentation of specimens, no definitive state-ment about precision and bias can be made. The next section, uses data from a commercial collaborative testing program to illustrate precision for one material. Because of the many trigonometric functions and power functions involved in com-puting the color space parameters, all computations should be carried out in IEEE floating point format to greatest number of bits of precision available on the computational system, usually known as double precision.

11.2 The Collaborative Testing Services Color and Color

Difference Collaborative Reference Program (13) has sur-veyed the precision of color and color-difference measure-ments by sending out pairs of painted chips exhibiting small color differences on a quarterly basis since 1971. In a typical report (No. 111, February, 2000), 118 instruments were in-volved. Table 2 gives the mean color differences and their standard deviations for the groups of instruments considered separately in the intercomparison, together with the conditions of analysis and measurement.

11.2.1 Reproducibility—Based on the between-laboratory

standard deviations, two color-difference results, obtained by operators in different laboratories measuring opaque, matte

TABLE 2 Precision of Calculated Color Differences Determined for Various Conditions of Measurement and Analysis

Measurement Conditions ∆E

Equation

No. of

Instruments

Mean

∆E

Standard

Deviation R*A

Geometry Illuminant Observer

45°/0° D65 1964 CIELAB 54 1.05 0.07 0.21

45°/0° D65 1964 CMC(2:1) 54 0.55 0.03 0.09

SphereB D65 1964 CIELAB 282 1.00 0.06 0.18

SphereB D65 1964 CMC(2:1) 282 0.53 0.03 0.09

A R* is the approximate inter-laboratory precision = 3.0 × standard deviation.

B Specular component included for integrating-sphere measurements.

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7� � � � � � � � � � � paint on sealed white paper stock should be considered suspect if they differ by more than the values shown in column R* of Table 2.

11.3 Precision—Based on the within-laboratory standard

deviations, the precision of color-difference measurements, summarized in Table 2, was equivalent to the precision of measured values of color as reported in the literature (14, 15) and is thus likely to be representative of the precision obtain-able for all production materials.

12. Keywords

12.1 color; color difference; color metrics; color spaces;

color tolerances

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APPENDIXES

(Nonmandatory Information)

X1. COLOR SPACES AND COLOR DIFFERENCE METRICS NO LONGER RECOMMENDED FOR NEW USERS

X1.1 Hunter LH, aH, bH Color Space and Color-Difference

Equation—This approximately uniform color space (16) is produced by plotting in rectangular coordinates the quantities LH, aH, bH calculated as follows:

LH 5 100 S Y

Yn

D aH 5 Ka

S X

Xn

D 2 S Y

Yn

D S Y

Yn

D bH 5 Kb

S Y

Yn

D 2 S Z

Zn

D S Y

Yn

D where X, Y, and Z are CIE daylight tristimulus values obtained from a measurement or other source and Ka and Kb are coefficients that vary with the illuminant-observer combi-nation to which the tristimulus values refer. In general, Ka

5175 ~Xn/98.074!

2 and Kb570 ~Zn/118.232!

2 where Xn and Zn are the X and Z tristimulus values for the perfect reflecting diffuser in the chosen illuminant-observer combination. Examples of Ka and Kb are tabulated in Table X1.1.

X1.1.1 The total color-difference ∆EH between two colors each given in LH, aH, bH is calculated as follows:

∆EH 5 @~∆LH!21~∆aH!21~∆bH!2# 2 (X1.4) where:

∆LH 5 LH ,B 2 LH ,S (X1.5)

∆aH 5 aH ,B 2 aH ,S (X1.6)

∆bH 5 bH ,B 2 bH ,S (X1.7) where LH,S, aH,S, bH,S refer to the reference or standard and LH,B, aH,B, bH,B refer to the test specimen or batch. The signs of the components ∆ LH, ∆ aH, ∆bH have the same approximate meaning as do their counterparts in 6.2.3.

X2. EXAMPLE CALCULATIONS FOR COLOR TOLERANCE EQUATIONS

X2.1 See Table X2.1.

TABLE X1.1 Some Selected Values of Ka and Kb for Various CIE Standard Observers and CIE Standard and Recommended

Illuminants

Illuminant/Observer Ka Kb

A – 1931 2° 185.21 38.403

A –1964 10° 186.30 38.195

C – 1931 2° 175.00 70.000

C – 1964 10° 174.30 63.379

D50 – 1931 2° 173.52 58.481

D50 – 1964 10° 173.79 58.092

D55 – 1931 2° 172.85 61.798

D55 – 1964 10° 172.96 61.387

D65 – 1931 2° 172.28 67.175

D65 – 1964 10° 172.06 66.687

D75 – 1931 2° 172.21 71.292

D75 – 1964 10° 171.71 70.710

F2 –1931 2° 175.99 52.849

F2 – 1964 10° 179.58 53.486

F7 – 1931 2° 172.27 67.133

F7 – 1964 10° 172.95 66.805

F11 – 1931 2° 177.56 51.642

F11 – 1964 10° 180.09 52.144

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TABLE X2.1 Example Calculations for Color Tolerance EquationsA,B

Color Coordinate STD-1 BAT-1 STD-2 BAT-2 STD-3 BAT-3 STD-4 BAT-4 STD-5 BAT-5

X 19.4100 19.5525 22.4800 22.5833 28.9950 28.7704 4.1400 4.4129 4.9600 4.6651

Y 28.4100 28.6400 31.6000 31.3700 29.5800 29.7400 8.5400 8.5100 3.7200 3.8100

Z 11.5766 10.5791 38.4800 36.7901 35.7500 35.6045 8.0300 8.6453 19.5900 17.7848

L* 60.2574 60.4626 63.0109 62.8187 61.2901 61.4292 35.0831 35.0232 22.7233 23.0331 a* -34.0099 -34.1751 -31.0961 -29.7946 3.7196 2.2480 -44.1164 -40.0716 20.0904 14.9730 b* 36.2677 39.4387 -5.8663 -4.0864 -5.3901 -4.9620 3.7933 1.5901 -46.6940 -42.5619

C* 49.7194 52.1857 31.6447 30.0735 6.5490 5.4474 44.2792 40.1031 50.8326 45.1188 hab* 133.160 130.910 190.683 187.810 304.609 294.373 175.086 177.728 293.280 289.382

SLCMC 1.1965 1.2224 1.2064 0.8878 0.6646

SCCMC 2.5589 2.0653 1.0228 2.4259 2.5848

f 0.9998 0.9991 0.7014 0.9998 0.9999

T 0.7515 0.7599 0.6369 0.7513 0.5991

SHCMC 1.9231 1.5700 0.7623 1.8229 1.5487

∆L* 0.2052 -0.1922 0.1391 -0.0599 0.3098

∆C* 2.4663 -1.5712 -1.1016 -4.1761 -5.7138

∆H* -1.9999 -1.5472 -1.0657 1.9430 -3.2580

∆E*ab 3.1819 2.2134 1.5390 4.6063 6.5847

∆ECMC(1:1) 1.4282 1.2549 1.7684 2.0258 3.0870

∆ECMC(2:1) 1.4205 1.2474 1.7656 2.0250 3.0604

SL94 1.0000 1.00000 1.00000 1.00000 1.00000

SC94 3.23737 2.42401 1.29470 2.99256 3.28747

SH94 1.74579 1.47467 1.09824 1.66419 1.76249

∆E94 1.3910 1.2481 1.2980 1.8204 2.5561 eo -14.669 -13.428 -30.521 -28.571 0.9803 -0.1547 -37.989 -35.319 -2.4122 -5.2003 fo 39.4301 41.8558 6.9380 7.7923 -5.3744 -4.5196 18.8815 15.7662 -42.144 -37.199

Go 42.0704 43.9569 31.2993 29.6141 5.4631 4.5222 42.4223 38.6783 42.2125 37.561 heofo 1.9270 1.8812 2.9181 2.8753 4.8928 4.6782 2.6803 2.7217 4.6552 4.5735 kE 1 1 1 1 1 1 1 1 1 1 kCH 1 1 1 1 1 1 1 1 1 1

C99o 32.7444 33.5141 27.7746 26.8899 7.8942 6.7137 32.8899 31.2919 32.8033 30.7926 h99o 136.407 133.787 193.193 190.744 306.337 294.040 179.571 181.944 292.724 288.042 a99o -23.7152 -23.1909 -27.0416 -26.4186 4.6776 2.7350 -32.8890 -31.2739 12.6717 9.5368 b99o 22.5784 24.1946 -6.3391 -5.0130 -6.3591 -6.1314 0.2461 -1.0617 -30.2569 -29.2786

L99o 64.0969 64.2936 66.7259 66.5432 65.0856 65.2185 38.9422 38.8798 25.7851 26.1220

∆E99o(Lab) 1.7105 1.4765 1.9604 2.0791 3.3012

∆C99o 0.7697 -0.8848 -1.1805 -1.5980 -2.0106

∆H99o -1.5148 -1.1679 -1.5596 1.3286 -2.5965

∆L99o 0.1967 -0.1828 0.1329 -0.0624 0.3368

∆E99o(LCH) 1.7105 1.4765 1.9604 2.0791 3.3012

L*ave 60.3600 62.9148 61.3597 35.0532 22.8782

C*ave 50.9525 30.8591 5.9982 42.1911 47.9757

G 0.0017 0.0490 0.4966 0.0063 0.0026 a' -34.0678 -34.2333 -32.6195 -31.2542 5.5669 3.3643 -44.3939 -40.3237 20.1424 15.0118

C' 49.7590 52.2238 33.1428 31.5202 7.7488 5.9950 44.5557 40.3550 50.8532 45.1317 h' 133.21 130.96 190.20 187.45 315.92 304.14 175.12 177.74 293.33 289.43

C'ave 50.9914 32.3315 6.8719 42.4554 47.9924 h'ave 132.084 188.822 310.031 176.429 291.381

∆L* 0.2052 -0.1922 0.1391 -0.0599 0.3098

∆C' 2.4648 -1.6226 -1.7538 -4.2007 -5.7215

∆H’ -2.0018 -1.5490 -1.3995 1.9430 -3.2653

SL 1.1427 1.1831 1.1586 1.2148 1.4014

SC 3.2946 2.4549 1.3092 2.9105 3.1597

SH 1.9951 1.4560 1.0717 1.6476 1.2617

RC 1.9932 1.8527 0.0218 1.9759 1.9897

∆θ 0.0000 0.0002 4.2110 0.0000 19.5282

RT 0.0000 0.0000 -0.0032 0.0000 -1.2537

T 1.3010 0.9402 0.6952 1.0168 0.3636

∆E00 1.2644 1.2630 1.8731 1.8645 2.0373

Color

Coordinate

STD-6 BAT-6 STD-7 BAT-7 STD-8 BAT-8 STD-9 BAT-9 STD-10 BAT-10

X 15.6000 15.9148 73.0000 73.9351 73.9950 69.1762 0.7040 0.6139 0.2200 0.0933

Y 9.2500 9.1500 78.0500 78.8200 78.3200 73.4000 0.7500 0.6500 0.2300 0.1000

Z 5.0200 4.3872 81.8000 84.5156 85.3060 79.7130 0.9720 0.8510 0.3250 0.1452

L* 36.4612 36.2715 90.8027 91.1528 90.9257 88.6381 6.7747 5.8714 2.0776 0.9033 a* 47.8580 50.5065 -2.0831 -1.6435 -0.5406 -0.8985 -0.2909 -0.0974 0.0795 -0.0621 b* 18.3852 21.2231 1.4410 0.0447 -0.9208 -0.7239 -2.4247 -2.2282 -1.1350 -0.5515

C* 51.2680 54.7844 2.5329 1.6441 1.0677 1.1538 2.4421 2.2303 1.1378 0.5550 hab

* 21.0148 22.7924 145.326 178.441 239.583 218.857 263.160 267.469 274.004 263.419

SL 0.9090 1.4295 1.4303 0.5110 0.5110

SC 2.5947 0.7944 0.7052 0.7890 0.7095

f 0.9999 0.1456 0.0261 0.1356 0.0279

T 0.5836 0.7600 0.6949 0.6246 0.5878

SH 1.5144 0.7666 0.6996 0.7488 0.7008

∆L* -0.1897 0.3501 -2.2876 -0.9033 -1.1743

∆C* 3.5164 -0.8888 0.0861 -0.2117 -0.5828

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X3. COMMERCIAL FACTORS IN COLOR TOLERANCE AND COLOR DIFFERENCE EQUATIONS

X3.1 Scope—A commercial factor cf may be introduced into any of the above color tolerance or color difference equations for the purpose of rescaling the volume of the acceptable region to units that are convenient, or customary. It is possible, for instance, by scaling two standards that would otherwise have different tolerance values in a way that each has the same acceptable nominal value as the other, say, one unit.

X3.2 A definition of the term, commercial factor, follows:

X3.2.1 commercial factor, n—in colorimetry, a scalar factor used to scale color-difference values to convenient, or customary, units.

X3.3 Using one form of the CIELAB color difference equation as an example, a commercial factor could be imple-mented as shown in the following equation:

∆E*ab,CF5cf 5 cf=~∆L*!21~∆C*!21~∆H*!2 (X3.1)

X3.4 Commercial factors are always multiplicative, never divisive. Commercial factors less than unity make the reported units smaller and thus the tolerable volume in old units larger, and commercial factors larger than one make the reported units larger and the tolerable volume in old units smaller.

TABLE X2.1 Continued

Color

Coordinate

STD-6 BAT-6 STD-7 BAT-7 STD-8 BAT-8 STD-9 BAT-9 STD-10 BAT-10

∆H* 1.6441 1.1631 -0.3993 0.1766 -0.1444

∆E*ab 3.8864 1.5051 2.3238 0.9441 1.3189

∆ECMC(1:1) 1.7490 1.9009 1.7026 1.8034 2.4491

∆ECMC(2:1) 1.7396 1.8890 0.9901 0.9533 1.4274

SL94 1.00000 1.00000 1.00000 1.00000 1.00000

SC94 3.30706 1.11398 1.04805 1.10989 1.05120

SH94 1.76902 1.03799 1.01602 1.03663 1.01707

∆E94 1.4249 1.4194 2.3226 0.9388 1.3063 eo 51.0740 54.6985 -1.2406 -1.4576 -0.8895 -1.1249 -1.3244 -1.0643 -0.4261 -0.2976 fo -3.6977 -2.5443 1.8329 0.6313 -0.4902 -0.2131 -1.7030 -1.6268 -0.8756 -0.3888

Go 51.2077 54.7577 2.2133 1.5884 1.0157 1.1449 2.1573 1.9440 0.9738 0.4896 heofo 6.2109 6.2367 2.1658 2.7328 3.6453 3.3288 4.0514 4.1330 4.2595 4.0592 kE 1 1 1 1 1 1 1 1 1 1 kCH 1 1 1 1 1 1 1 1 1 1

C99o 36.2537 37.4846 3.5305 2.5874 1.6877 1.8938 3.4476 3.1288 1.6205 0.8291 h99o 21.8591 23.3368 150.092 182.580 234.859 216.727 258.129 262.806 270.052 258.572 a99o 33.6471 34.4180 -3.0603 -2.5848 -0.9714 -1.5179 -0.7092 -0.3918 0.0015 -0.1643 b99o 13.4981 14.8490 1.7604 -0.1165 -1.3801 -1.1325 -3.3739 -3.1042 -1.6205 -0.8126

L99o 40.3745 40.1777 92.0605 92.3665 92.1680 90.1614 7.9192 6.8752 2.4506 1.0679

∆E99o(Lab) 1.5677 1.9602 2.0944 1.1240 1.6100

∆C99o 1.2309 -0.9431 0.2061 -0.3188 -0.7914

∆H99o 0.9508 1.6910 -0.5634 0.2680 -0.2318

∆L99o -0.1968 0.3060 -2.0066 -1.0440 -1.3827

∆E99o(LCH) 1.5677 1.9602 2.0944 1.1240 1.6100

L*ave 36.3664 90.9778 89.7819 6.3231 1.4905

C*ave 53.0262 2.0885 1.1108 2.3362 0.8464

G 0.0013 0.4999 0.5000 0.4999 0.5000 a' 47.9197 50.5717 -3.1244 -2.4651 -0.8108 -1.3477 -0.4363 -0.1461 0.1192 -0.0931

C' 51.3256 54.8444 3.4407 2.4655 1.2269 1.5298 2.4637 2.2330 1.1412 0.5593 h' 20.99 22.77 155.24 178.96 228.63 208.24 259.80 266.25 275.99 260.42

C'ave 53.0850 2.9531 1.3784 2.3483 0.8503 h'ave 21.8781 167.101 218.436 263.02 268.20

∆L* -0.1897 0.3501 -2.2876 -0.9033 -1.1743

∆C' 3.5189 -0.9751 0.3029 -0.2306 -0.5819

∆H' 1.6444 1.1972 -0.4850 0.2638 -0.2165

SL 1.1943 1.6110 1.5930 1.6517 1.7246

SC 3.3888 1.1329 1.0620 1.1057 1.0383

SH 1.7357 1.0511 1.0288 1.0336 1.0099

RC 1.9949 0.0011 0.0001 0.0005 0.0000

∆θ 0.0000 0.0000 0.1794 23.848 27.865

RT 0.0000 0.0000 0.0000 -0.0004 0.0000

T 0.9239 1.1546 1.3916 0.9549 0.7787

∆E00 1.4146 1.4440 1.5381 0.6386 0.9076

A If this table is used to check a computer program, discrepancies of ±0.0001 and occasionally ±0.0002 may arise due to roundoff, and do not call into question the program’s correctness.

B The values in this table were calculated using 64-bit variables. With the use of 32-bit variables, one can expect approximately two digits less precision.

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X3.5 Commercial factors are not part of the definition of the color-difference unit resulting from that equation, and thus, reporting of the use of a commercial factor and its magnitude is essential. Examples of ways in which commercial factors might be reported follow:

∆E*ab,CF51.2 5 0.84

∆E00,CF50.8 5 1.6

∆ECMC,CF52 5 2.4

∆EHunterLAB,CF50.9 5 0.81

REFERENCES

(1) Commission Internationale de l’Éclairage, Publication CIE No. 15, Colorimetry, Central Bureau of the CIE, Vienna, 2018.

(2) Clark, F. J. J., McDonald, R., and Rigg, B., “Modification to the JPC 79 Colour-Difference Formula,” Journal of the Society of Dyers and

Colorists, Vol 100, 1984, pp. 128-132.

(3) Commission Internationale de l’Éclairage, Technical Report 116, Industrial Colour-Difference Evaluation, Central Bureau of the CIE, Vienna, 1995. (Available from TechStreet.com, or CIE.org.at.)

(4) Rohner, E., und Rich, Danny C., “Eine angenähert gleichförmige Metrik für industrielle Farbtoleranzen von Körberfarben,” Die Farbe, 42, Heft 4-6, 1996, pp. 207-220.

(5) Commission Internationale de l’Éclairage, Technical Report 142- 2001, Improvement to Industrial Colour Difference Equation, Central Bureau of the CIE, Vienna, 2000. (Available from TechStreet.com, or CIE.org.at.)

(6) ISO/CIE 11664—6:2014 Colorimetry – Part 6 CIEDE2000 Formula

(7) Robertson, A. R.,“The CIE 1976 Color-Difference Formulae,” Color

Research and Application, Vol 2, 1977, pp. 7–11.

(8) McLaren, K., and Taylor, P. F., “The Derivation of Hue-Difference

Terms from CIELAB Coordinates,” Color Research and Application, Vol 6, 1981, pp. 75–77.

(9) McLaren, K., “CIELAB Hue-Angle Anomalies at Low Tristimulus

Ratios,” Color Research and Application, Vol 5, 1980, pp. 139–143.

(10) McDonald, Roderick,“Color Communication in the 90s,” Textile

Chemist and Colorist, Vol 24, No 4, 1992, pp. 11-15.

(11) Commission Internationale de l’Éclairage, Technical Report 101, Parametric effects in colour-difference evaluation, Central Bureau of the CIE, Vienna, 1993. (Available from TechStreet.com, or CIE.or-g.at.)

(12) Sharma, G., The CIEDE2000 Color Difference Formula: Implemen-tation Notes, Supplementary Test Data, and Mathematical Observations, Color Research and Application, Vol 30, 2005, 21-30.

(13) “Color and Appearance Collaborative Reference Program for Color and Color-Difference,” Collaborative Testing Services, Inc., McLean, VA.

(14) Billmeyer, F. W., Jr., and Alessi, P. J., “Assessment of Color- Measuring Instruments,” Color Research and Application, Vol 6, 1981, pp. 195–202.

(15) Rich, D. C., “Colorimetric Repeatability and Reproducibility of CHROMA-SENSOR Spectrocolorimeters,” DIE FARBE, Vol 37, 1990, pp. 247-261.

(16) Hunter, R. S. and Harold, R. W., The Measurement of Appearance, 2nd Ed. Wiley-Interscience, New York, NY, 1987, pp. 148–152.

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