ASTM D2244-23

This document is not an ASTM standard and is intended only to provide the user of an ASTM standard an indication of what changes have been made to the previous version. Because
it may not be technically possible to adequately depict all changes accurately, ASTM recommends that users consult prior editions as appropriate. In all cases only the current version
of the standard as published by ASTM is to be considered the official document.
Designation: D2244 − 22 D2244 − 23
Standard Practice for
Calculation of Color Tolerances and Color Differences from
Instrumentally Measured Color Coordinates
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), 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 appropriate safety, health, and environmental practices and determine the applicability of
regulatory limitations prior to use.
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 Oct. 1, 2022June 1, 2023. Published October 2022June 2023. Originally approved in 1964. Last previous edition approved in 20212022 as
D2244 – 21.D2244 – 22. DOI: 10.1520/D2244-22.10.1520/D2244-23.
The boldface numbers in parentheses refer to the list of references at the end of this standard.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
D2244 − 23
1.4 This international standard was developed in accordance with internationally recognized principles on standardization
established in the Decision on Principles for the Development of International Standards, Guides and Recommendations issued
by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
2. Referenced Documents
2.1 ASTM Standards:
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 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 Farbmetrische Bestimmung von Farbabstanden bei Korperfarben nach der
DIN 99o-Formel, (Colorimetric determination of colour differences of object colours according to the DIN 99o formula)
3. Terminology
3.1 Terms and definitions in Terminology E284 are applicable to this practice.
3.2 Definitions of Terms Specific to This Standard:
3.2.1 colorimetric spectrometer, n—spectrometer, one component 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 distorts 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.
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.
Available from TechStreet.com and Beuth Verlag GmbH, 10772, Berlin, Germany, http://www.beuth.de.
D2244 − 23
5.2 Color differences calculated in ∆E units (6) are highly recommended for use with color-differences in the range of 0.0 to
5.0 ∆E* units. This color-difference equation is appropriate for and widely used in industrial and commercial applications
ab
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 magnitude 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 equations 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 judgments 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
x 5 (1)
X1Y1Z
Y
y 5 (2)
X1Y1Z
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 D , or other phase of daylight. These scales do not provide a perceptually uniform
color space. Consequently, color differences are seldom 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 Q 2 16 (3)
~ !
Y
a*5 500 @ f ~Q ! 2 f ~Q ! # (4)
X Y
b*5 200 f Q 2 f Q (5)
@ ~ ! ~ ! #
Y Z
where□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
Q 5 X/X ; Q 5 Y/Y ; Q 5 Z/Z
~ ! ~ ! ~ !
X n Y n Z n
and□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
1/3 3
f Q 5 Q if Q . 6/29
~ ! ~ !
i i i
else□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
f Q 5 841/108 Q 14/29 if Q # 6/29 .
~ ! ~ ! ~ !
i i i
Here, i varies as X, Y, and Z.
The tristimulus values X ,Y ,Z define the color of the nominally white object-color stimulus. Usually, the white object-color
n n n
stimulus is given by the spectral radiant power of one of the CIE standard illuminants, for example, C,D or another phase of
daylight, reflected into the observer’s eye by the perfect reflecting diffuser. Under these conditions, X ,Y ,Z are the tristimulus
n n n
values of the standard illuminant with Y equal to 100.
n
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6.2.1 The total color-difference ∆E * between two colors each given in terms of L*, a*, b* is calculated as follows:
ab
2 2 2
∆E* 5= ∆L* 1 ∆a* 1 ∆b* (6)
~ ! ~ ! ~ !
ab
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* , gives no indication of the character of the difference since it does not indicate the relative quantity
ab
and direction of hue, chroma, and lightness differences.
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* 2 L* (7)
B S
∆a*5 a* 2 a* (8)
B S
∆b*5 b* 2 b* (9)
B S
where L* , a* , and b* refer to the reference or standard, and L* , a* , and b* refer to the test specimen or batch. The signs
S S S B B B
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 h and CIE 1976
ab
metric chroma C* according to the following pseudocode:
ab
if b* 5 0 then (16)
h 5 90 sign~a*!@sign~a*!21#
ab
else □□□□□□□
h 5 180 2 180/π arctan a*/b* 290 sign b*
~ ! ~ ! ~ !
ab
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 h calculated by the above are degrees counter-clockwise from the positive a* axis. The function sign is
ab
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 h between the test specimen and reference can be correlated with differences in their visually
ab
perceived hue, except for very dark colors (9). Differences in chroma ∆C* = ([C* ] − [C* ] ) can similarly be
ab ab batch ab standard
correlated with differences in visually perceived chroma.
2 2
C* 5= a* 1 b* (17)
~ ! ~ !
ab
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 ΔH* between the colors as follows:
ab
0.5
∆ H* 5 s @2~C* C* 2 a* a* 2 b* b* !# (18)
ab ab,B ab,S B S B S
where□□□□□□□□□□□□□□□□□□□□□
□if a* b* .a* b* then (19)
S B B S
s 5 1
else□□□□□□□□□□□□□□□□□□□□□□
s 521
end if.□□□□□□□□□□□□□□□□□□□□□
D2244 − 23
When ΔL* is calculated as in 6.2.3 and ΔC* is calculated as in 6.2.4, then
ab ab
2 2 2 0.5
∆E* 5 @~∆L*! 1~∆C*! 1~∆H*! # (20)
ab
contains terms showing the relative contributions of lightness differences ΔL* , chroma differences ΔC* , and hue differences
ab ab
ΔH* .
ab
6.3 CMC Color Tolerance Equation—The Colour Measurement 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 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* and with
ab
respect to changes in CIELAB metric hue angle h* . The CMC components and single number tolerances are computed as
ab
follows:
2 2 2
∆L* ∆C* ∆H*
∆E l:c 5 1 1 (21)
~ ! Œ
S D S D S D
CMC
l·S c·S S
L c H
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:
0.040975·L*
S 5 for L*$ 16 (22)
L
~110.01765·L*!
S 5 0.511, for L*,16
L
0.0638·C*
S 5 10.638
C
~110.0131·C*!
S 5 S T·f112 f
~ !
H C
where□□□□□□□□□□□□□□□□□□□□□
FIG. 1 CMC Ellipse Distribution in the CIELAB (a*, b*) Plane
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~C*! 2
f 5
H 4 J
~C*! 11900
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 recommended. 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:
2 2 2 0.5
∆L* ∆C* ∆H*
∆E* 5 1 1
FS D S D S D G
k S k S k S
L L C C H H
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 k ,k ,k are the parametric factors that can be used to compensate for texture and other specimen presentation effects.
L C H
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 parameters S ,S ,S are used to perform the local distortion of CIELAB color space, again based
L C H
on the position of the standard specimen in that space. They are computed using the following equations:
S 5 1 (24)
L
S 5 110.045·C*
C
S 5 110.015·C*
H
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
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
D2244 − 23
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:
k 5 k 5 1 (25)
CH E
DIN99o lightness:
L 5 303.67ln~1.0 1 0.0039 L *!⁄k (26)
99o E
Auxiliary variable for redness:
* *
e 5 a cos 26 ° 1b sin 26 ° (27)
~ ! ~ !
o
Auxiliary variable for yellowness:
* *
f 520.83a sin 26 ° 10.83b cos 26 ° (28)
~ ! ~ !
o
Auxiliary variable for chroma:
2 2 0.5
G 5 ~e 1 f ! (29)
99o o o
Auxiliary variable for hue angle: (in radians)
h 5 0 (30)
eofo
for e = 0 and f = 0
o o
h 5 arctan f ⁄ e (31)
~ !
eofo o o
for e > 0 and f ≥ 0
o o
h 5 π⁄2 (32)
eofo
for e = 0 and f > 0
o o
h 5 π1arctan~f ⁄ e ! (33)
eofo o o
for e < 0
o
h 5 3π⁄2 (34)
eofo
for e = 0 and f < 0
o o
h 5 2π1arctan f ⁄ e (35)
~ !
eofo o o
for e > 0 and f < 0
o o
DIN hue angle: (in degrees)
h 5 h 180°⁄π126° (36)
99o eofo
for h < 334°π/180°
eofo
h 5 h 2 2 π 180°⁄π126° (37)
~ !
99o eofo
for h ≥ 334°π/180°
eofo
DIN99o chroma:
C 5 @ln ~1 1 0.075G !#⁄~0.0435k k ! (38)
99o o CH E
DIN99o redness-greenness:
a 5 C cos h π ⁄ 180 ° (39)
~ !
99o 99o 99o
DIN99o yellowness-blueness:
b 5 C sin~h π ⁄ 180 °! (40)
99o 99o 99o
DIN99o lightness difference:
∆L 5 L 2 L (41)
99o 99oB 99oS
DIN99o red-green difference:
∆a 5 a 2 a (42)
99o 99oB 99oS
DIN99o yellow-blue difference:
∆b 5 b 2 b (43)
99o 99oB 99oS
DIN99o color difference:
2 2 2 0.5
∆E 5 ∆L 1 ∆a 1 ∆b (44)
@~ ! ~ ! ~ ! #
99o 99o 99o 99o
D2244 − 23
or equivalently
DIN99o chroma difference:
∆C 5 C 2 C (45)
99o 99oB 99oS
DIN99o hue difference:
2~a b 2 a b !
99oB 99oS 99oS 99oB
∆H 5 (46)
0.5
99o
0.5 C C 1 a a 1 b b
@ ~ !#
99oB 99oS 99oB 99oS 99oB 99oS
DIN99o color difference:
2 2 2 0.5
∆E 5 @~∆L ! 1 ~∆C ! 1 ~∆H ! # (47)
99o 99o 99o 99o
where subscript S refers to the product standard and subscript 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 h calculated by the above
99o
are degrees counterclockwise for the range from 0 to 360 from the positive e axis.
o
The values of the parameters k and k for most applications will be unity. These factors refer to a reference set of viewing
E CH
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 k and k values. Values of k and k other than unity lead to changes in both color-difference and color coordinates.
E CH E CH
If it is intended to use k factors other than unity, their valu
...