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| The '''difference''' or '''distance''' between two colors is a [[metric (mathematics)|metric]] of interest in [[color science]]. It allows people to quantify a notion that would otherwise be described with adjectives, to the detriment of anyone whose work is color critical. Common definitions make use of the [[Euclidean distance]] in a [[device independent]] [[color space]].
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| ==Delta E==
| |
| The [[International Commission on Illumination]] (CIE) calls their distance metric Δ''E''<sup>*</sup><sub>''ab''</sub> (also called Δ''E*'', dE*, dE, or "Delta E") where [[delta (letter)|delta]] is a [[Greek letter]] often used to denote difference, and '''E''' stands for ''Empfindung''; German for "sensation". Use of this term can be traced back to the influential [[Hermann von Helmholtz]] and [[Ewald Hering]].<ref>http://books.google.com/books?id=DrduOSrOFegC&pg=PA188&lpg=PA188&dq=(grundempfindung%7Cempfindung)+helmholtz+color&source=web&ots=g_T3sFQ7eG&sig=4HBhygBAc3zW-XfGVPlaykl72bA</ref><ref>http://books.google.com/books?id=OoESifAi9ZsC&pg=PA278&lpg=PA278&dq=empfindung+color&source=web&ots=-z6JbcN54V&sig=-Bww3AcNWpnDQp1Lu7yqGBFyUOw</ref>
| |
| | |
| Different studies have proposed different Δ''E'' values that have a [[JND]] (just noticeable difference). Unempirically, a value of '1.0' is often mentioned, but in a recent study, Mahy et al. (1994) assessed a JND of 2.3 Δ''E''. However, perceptual non-uniformities in the underlying [[CIELAB]] color space prevent this and have led to the CIE's refining their definition over the years, leading to the superior (as recommended by the CIE) 1994 and 2000 formulas.<ref>Real World Color Management, Second Edition (Bruce Fraser)</ref> These non-uniformities are important because [[Color vision#Physiology of color perception|the human eye is more sensitive to certain colors than others]]. A good metric should take this into account in order for the notion of a "[[just noticeable difference]]" to have meaning. Otherwise, a certain Δ''E'' that may be insignificant between two colors that the eye is insensitive to may be conspicuous in another part of the spectrum.<ref>[http://www.aim-dtp.net/aim/evaluation/cie_de/index.htm Evaluation of the CIE Color Difference Formulas]{{dead link|date=April 2009}}</ref>
| |
| The values L, a, and b (like used in the equations below) are usually double precision values scaled between (-1,1). When a Lab image is saved, L is used as an unsigned 8bit integer, and a and b as signed 8bit integers, in order to keep the image normal in file size.
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| | |
| ===CIE76===
| |
| The 1976 formula is the first color-difference formula that related a measured to a known Lab value. This formula has been succeeded by the 1994 and 2000 formulas because the Lab space turned out to be not as perceptually uniform as intended, especially in the saturated regions. This means that this formula rates these colors too highly as opposed to other colors. | |
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| Using <math>({L^*_1},{a^*_1},{b^*_1})</math> and <math>({L^*_2},{a^*_2},{b^*_2})</math>, two colors in [[L*a*b*]]:
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| :<math>\Delta E_{ab}^* = \sqrt{ (L^*_2-L^*_1)^2+(a^*_2-a^*_1)^2 + (b^*_2-b^*_1)^2 }</math>
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| <math>\Delta E_{ab}^* \approx 2.3</math> corresponds to a [[JND]] (just noticeable difference).<ref>{{cite book|publisher=[[CRC Press]]|title=Digital Color Imaging Handbook|year=2003|author=Gaurav Sharma|isbn=0-8493-0900-X|url=http://books.google.com/?id=OxlBqY67rl0C&pg=PA31&vq=1.42&dq=jnd+gaurav+sharma|edition=1.7.2}}</ref>
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| | |
| ===CIE94===
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| The 1976 definition was extended to address perceptual non-uniformities, while retaining the L*a*b* color space, by the introduction of application-specific weights derived from an automotive paint test's tolerance data.<ref>{{cite web|url=http://www.colorwiki.com/wiki/Delta_E:_The_Color_Difference |title=Delta E: The Color Difference |publisher=Colorwiki.com |date= |accessdate=2009-04-16}}</ref>
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| Δ''E'' (1994) is defined in the [[colorfulness#Chroma in CIE 1976 L*a*b* and L*u*v* color spaces|L*C*h* color space]] with differences in lightness, chroma and hue calculated from [[CIELAB|L*a*b* coordinates]]. Given a reference color<ref>Called such because the operator is not [[commutative]]. This makes it a [[quasimetric]].</ref> <math>(L^*_1,a^*_1,b^*_1)</math> and another color <math>(L^*_2,a^*_2,b^*_2)</math>, the difference is:<ref>{{cite web|author=Bruce Justin Lindbloom |url=http://www.brucelindbloom.com/Eqn_DeltaE_CIE94.html |title=Delta E (CIE 1994) |publisher=Brucelindbloom.com |date= |accessdate=2011-03-23}}</ref><ref>{{cite web|url=http://www.colorpro.com/info/software/heggie.html |title=Colour Difference Software by David Heggie |publisher=Colorpro.com |date=1995-12-19 |accessdate=2009-04-16}}</ref><ref>{{cite web|url=http://www.physics.kee.hu/cie/newcie/nc/DS014-4_3.pdf|title=CIE 1976 L*a*b* Colour space draft standard|accessdate=2011-03-23}}</ref>
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| | |
| :<math>\Delta E_{94}^* = \sqrt{ \left(\frac{\Delta L^*}{k_L S_L}\right)^2 + \left(\frac{\Delta C^*_{ab}}{k_C S_C}\right)^2 + \left(\frac{\Delta H^*_{ab}}{k_H S_H}\right)^2 }</math>
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| where:
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| :<math>\Delta L^* = L^*_1 - L^*_2</math>
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| :<math>C^*_1 = \sqrt{ {a^*_1}^2 + {b^*_1}^2 }</math>
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| :<math>C^*_2 = \sqrt{ {a^*_2}^2 + {b^*_2}^2 }</math>
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| :<math>\Delta C^*_{ab} = C^*_1 - C^*_2</math>
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| :<math>\Delta H^*_{ab} = \sqrt{ {\Delta E^*_{ab}}^2 - {\Delta L^*}^2 - {\Delta C^*_{ab}}^2 } = \sqrt{ {\Delta a^*}^2 + {\Delta b^*}^2 - {\Delta C^*_{ab}}^2 }</math>
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| :<math>\Delta a^* = a^*_1 - a^*_2</math>
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| :<math>\Delta b^* = b^*_1 - b^*_2</math>
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| :<math>S_L = 1</math>
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| :<math>S_C = 1+K_1 C^*_1</math>
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| :<math>S_H = 1+K_2 C^*_1</math>
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| | |
| and where ''k<sub>C</sub>'' and ''k<sub>H</sub>'' are usually both unity and the weighting factors ''k<sub>L</sub>'', ''K''<sub>1</sub> and ''K''<sub>2</sub> depend on the application:
| |
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| {| class="wikitable" border="1"
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| |-
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| !
| |
| ! graphic arts
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| ! textiles
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| |-
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| | <math>k_L</math>
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| | 1
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| | 2
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| |-
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| | <math>K_1</math>
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| | 0.045
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| | 0.048
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| |-
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| | <math>K_2</math>
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| | 0.015
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| | 0.014
| |
| |}
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| Geometrically, the quantity <math>\Delta H^*_{ab}</math> corresponds to the arithmetic mean of the chord lengths of the equal chroma circles of the two colors.
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| <ref>Georg A. Klein: Industrial Color Physics: P.147 - ISBN 978-1-4419-1196-4</ref>
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| | |
| ===CIEDE2000===
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| Since the 1994 definition did not adequately resolve the [[perceptual uniformity]] issue, the CIE refined their definition, adding five corrections:<ref>{{cite journal|title=The CIEDE2000 color-difference formula: Implementation notes, supplementary test data, and mathematical observations |journal=Color Research & Applications |publisher=[[Wiley Interscience]] |first=Gaurav |last=Sharma |coauthors=Wencheng Wu, Edul N. Dalal |volume=30 |issue=1 |pages=21–30 |doi=10.1002/col.20070 |url=http://www.ece.rochester.edu/~gsharma/ciede2000/ciede2000noteCRNA.pdf |year=2005 |ref=CITEREFSharma2005 }}</ref><ref>{{cite web |author=Bruce Justin Lindbloom |url=http://www.brucelindbloom.com/Eqn_DeltaE_CIE2000.html |title=Delta E (CIE 2000) |publisher=Brucelindbloom.com |date= |accessdate=2009-04-16}}</ref>
| |
| * A hue rotation term (R<sub>T</sub>), to deal with the problematic blue region (hue angles in the neighborhood of 275°):<ref>[http://www.brucelindbloom.com/MunsellCalcHelp.html#BluePurple The "Blue Turns Purple" Problem], Bruce Lindbloom</ref>
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| * Compensation for neutral colors (the primed values in the L*C*h differences)
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| * Compensation for lightness (S<sub>L</sub>)
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| * Compensation for chroma (S<sub>C</sub>)
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| * Compensation for hue (S<sub>H</sub>)
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| | |
| :<math>\Delta E_{00}^* = \sqrt{ \left(\frac{\Delta L'}{k_L S_L}\right)^2 + \left(\frac{\Delta C'}{k_C S_C}\right)^2 + \left(\frac{\Delta H'}{k_H S_H}\right)^2 + R_T \frac{\Delta C'}{k_C S_C}\frac{\Delta H'}{k_H S_H} }</math>
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| ::<small>'''Note:''' The formulae below should use degrees rather than radians; the issue is significant for ''R<sub>T</sub>''.</small>
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| ::<small>The ''k<sub>L</sub>'', ''k<sub>C</sub>'', and ''k<sub>H</sub>'' are usually unity.</small>
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| :<math>\Delta L^\prime = L^*_2 - L^*_1</math>
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| :<math>\bar{L} = \frac{L^*_1 + L^*_2}{2} \quad \bar{C} = \frac{C^*_1 + C^*_2}{2}</math>
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| | |
| :<math>
| |
| a_1^\prime = a_1^* + \frac{a_1^*}{2} \left( 1 - \sqrt{\frac{\bar{C}^7}{\bar{C}^7 + 25^7}} \right) \quad
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| a_2^\prime = a_2^* + \frac{a_2^*}{2} \left( 1 - \sqrt{\frac{\bar{C}^7}{\bar{C}^7 + 25^7}} \right)
| |
| </math>
| |
| | |
| :<math>
| |
| \bar{C}^\prime = \frac{C_1^\prime + C_2^\prime}{2} \mbox{ and }
| |
| \Delta{C'}=C'_2-C'_1 \quad
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| \mbox{where }
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| C_1^\prime = \sqrt{a_1^{'^2} + b_1^{*^2}} \quad
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| C_2^\prime = \sqrt{a_2^{'^2} + b_2^{*^2}} \quad
| |
| </math>
| |
| | |
| :<math>
| |
| h_1^\prime=\text{atan2} (b_1^*, a_1^\prime) \mod 360^\circ, \quad
| |
| h_2^\prime=\text{atan2} (b_2^*, a_2^\prime) \mod 360^\circ
| |
| </math>
| |
| :: <small>'''Note:''' The inverse tangent (tan<sup>−1</sup>) can be computed using a common library routine <code>atan2(b, a′)</code> which usually has a range from −π to π radians; color specifications are given in 0 to 360 degrees, so some adjustment is needed. The inverse tangent is indeterminate if both ''a′'' and ''b'' are zero (which also means that the corresponding ''C′'' is zero); in that case, set the hue angle to zero. See {{harvnb|Sharma|2005|loc=eqn. 7}}.</small>
| |
| | |
| :<math>
| |
| \Delta h' = \begin{cases}
| |
| h_2^\prime - h_1^\prime & \left| h_1^\prime - h_2^\prime \right| \leq 180^\circ \\
| |
| h_2^\prime - h_1^\prime + 360^\circ & \left| h_1^\prime - h_2^\prime \right| > 180^\circ, h_2^\prime \leq h_1^\prime \\
| |
| h_2^\prime - h_1^\prime - 360^\circ & \left| h_1^\prime - h_2^\prime \right| > 180^\circ, h_2^\prime > h_1^\prime
| |
| \end{cases}
| |
| </math>
| |
| :: <small>'''Note:''' When either ''C′''<sub>1</sub> or ''C′''<sub>2</sub> is zero, then Δh′ is irrelevant and may be set to zero. See {{harvnb|Sharma|2005|loc=eqn. 10}}.</small>
| |
| | |
| :<math>
| |
| \Delta H^\prime = 2 \sqrt{C_1^\prime C_2^\prime} \sin (\Delta h^\prime/2), \quad \bar{H}^\prime=\begin{cases}
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| (h_1^\prime + h_2^\prime + 360^\circ)/2 & \left| h_1^\prime - h_2^\prime \right| > 180^\circ \\
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| (h_1^\prime + h_2^\prime)/2 & \left| h_1^\prime - h_2^\prime \right| \leq 180^\circ
| |
| \end{cases}
| |
| </math>
| |
| :: <small>'''Note:''' When either ''C′''<sub>1</sub> or ''C′''<sub>2</sub> is zero, then {{overbar|H}}′ is ''h′''<sub>1</sub>+''h′''<sub>2</sub> (no divide by 2; essentially, if one angle is indeterminate, then use the other angle as the average; relies on indeterminate angle being set to zero). See {{harvnb|Sharma|2005|loc=eqn. 7 and p. 23}} stating most implementations on the internet at the time had "an error in the computation of average hue".</small>
| |
| | |
| :<math>
| |
| T = 1 - 0.17 \cos ( \bar{H}^\prime - 30^\circ )
| |
| + 0.24 \cos (2\bar{H}^\prime)
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| + 0.32 \cos (3\bar{H}^\prime + 6^\circ )
| |
| - 0.20 \cos (4\bar{H}^\prime - 63^\circ)
| |
| </math> | |
| | |
| :<math>
| |
| S_L = 1 + \frac{0.015 \left( \bar{L} - 50 \right)^2}{\sqrt{20 + {\left(\bar{L} - 50 \right)}^2} } \quad
| |
| S_C = 1+0.045 \bar{C}^\prime \quad
| |
| S_H = 1+0.015 \bar{C}^\prime T
| |
| </math>
| |
| | |
| :<math>R_T = -2 \sqrt{\frac{\bar{C}'^7}{\bar{C}'^7+25^7}} \sin \left[ 60^\circ \cdot \exp \left( -\left[ \frac{\bar{H}'-275^\circ}{25^\circ} \right]^2 \right) \right]</math>
| |
| | |
| ===CMC l:c (1984)===
| |
| In 1984, the Colour Measurement Committee of the [[Society of Dyers and Colourists]] defined a difference measure, also based on the L*C*h color model. Named after the developing committee, their metric is called '''CMC l:c'''. The [[quasimetric]] has two parameters: lightness (l) and chroma (c), allowing the users to weight the difference based on the ratio of l:c that is deemed appropriate for the application. Commonly used values are 2:1<ref>Meaning that the lightness contributes ''half'' as much to the difference (or, identically, is allowed ''twice'' the tolerance) as the chroma</ref> for acceptability and 1:1 for the threshold of imperceptibility.
| |
| | |
| The distance of a color <math>(L^*_2,C^*_2,h_2)</math> to a reference <math>(L^*_1,C^*_1,h_1)</math> is:<ref>{{cite web|author=Bruce Justin Lindbloom |url=http://www.brucelindbloom.com/Eqn_DeltaE_CMC.html |title=Delta E (CMC) |publisher=Brucelindbloom.com |date= |accessdate=2009-04-16}}</ref>
| |
| | |
| :<math>\Delta E^*_{CMC} = \sqrt{ \left( \frac{L^*_2-L^*_1}{l S_L} \right)^2 + \left( \frac{C^*_2-C^*_1}{c S_C} \right)^2 + \left( \frac{\Delta H^*_{ab}}{S_H} \right)^2 }</math>
| |
| | |
| <math>S_L=\begin{cases} 0.511 & L^*_1 < 16 \\ \frac{0.040975 L^*_1}{1+0.01765 L^*_1} & L^*_1 \geq 16 \end{cases} \quad S_C=\frac{0.0638 C^*_1}{1+0.0131 C^*_1} + 0.638 \quad S_H=S_C (FT+1-F)</math>
| |
| | |
| <math>F = \sqrt{\frac{C^{*^4}_1}{C^{*^4}_1+1900}} \quad T=\begin{cases} 0.56 + |0.2 \cos (h_1+168^\circ)| & 164^\circ \leq h_1 \leq 345^\circ \\ 0.36 + |0.4 \cos (h_1+35^\circ) | & \mbox{otherwise} \end{cases}</math> | |
| | |
| CMC l:c is designed to be used with [[CIE Standard Illuminant D65|D65]] and the [[CIE 1931 color space#CIE standard observer|CIE Supplementary Observer]].<ref>[http://www.hunterlab.com/appnotes/an10_96ar.pdf CMC<!-- Bot generated title -->]</ref>
| |
| | |
| ==Tolerance==
| |
| [[Image:CIExy1931 MacAdam.png|250px|right|thumb|A MacAdam diagram in the [[CIE 1931 color space]]]]
| |
| | |
| '''Tolerancing''' concerns the question "What is a set of colors that are imperceptibly/acceptably close to a given reference?" If the distance measure is perceptually uniform, then the answer is simply "the set of points whose distance to the reference is less than the just-noticeable-difference (JND) threshold." This requires a perceptually uniform metric in order for the threshold to be constant throughout the [[gamut]] (range of colors). Otherwise, the threshold will be a function of the reference color—useless as an objective, practical guide.
| |
| | |
| In the [[CIE 1931 color space]], for example, the tolerance contours are defined by the [[MacAdam ellipse]], which holds L* ([[lightness (color)|lightness]]) fixed. As can be observed on the diagram on the right, the [[ellipse]]s denoting the tolerance contours vary in size. It is partly due to this non-uniformity that lead to the creation of [[CIELUV]] and [[CIELAB]].
| |
| | |
| More generally, if the lightness is allowed to vary, then we find the tolerance set to be [[ellipsoid]]al. Increasing the weighting factor in the aforementioned distance expressions has the effect of increasing the size of the ellipsoid along the respective axis.<ref>http://www.xrite.com/documents/literature/en/L10-024_Color_Tolerance_en.pdf</ref>
| |
| | |
| <!-- A diagram would handily illustrate the last paragraph. I am not going to post a {{reqdiagram}} tag here in order not to make the article ugly, but you can contribute if you like. -->
| |
| | |
| {{clear}}
| |
| | |
| == See also ==
| |
| * [[CIELAB]]
| |
| | |
| ==Footnotes==
| |
| {{reflist|30em}}
| |
| | |
| ==Further reading==
| |
| * {{cite journal|title=Historical development of CIE recommended color difference equations|first=Alan R.|last=Robertson|journal=Color Research & Application|year=1990|volume=15|issue=3|pages=167–170|doi=10.1002/col.5080150308|url=http://www3.interscience.wiley.com/cgi-bin/fulltext/114184816/PDFSTART}}
| |
| * {{cite journal|title=Uniformity of some recent color metrics tested with an accurate color-difference tolerance dataset|first=M.|last=Melgosa|coauthors=Quesada, J. J. and Hita, E.| url=http://www.opticsinfobase.org/abstract.cfm?URI=ao-33-34-8069|date=December 1994|journal=[[Applied Optics]]|volume=33|issue=34|pages=8069–8077|doi=10.1364/AO.33.008069|pmid=20963027}}
| |
| * {{cite book|title=Colour Physics for Industry|last=McDonald|first=Roderick|coauthors=Hill, MacDonald, Nobbs, Rigg, Sinclair, Smith|editor=Roderick McDonald|isbn=0-901956-70-8|publisher=[[Society of Dyers and Colourists]]|year=1997|edition=2E}}
| |
| | |
| ==External links==
| |
| * [http://www.brucelindbloom.com/ColorDifferenceCalc.html Bruce Lindbloom's color difference calculator]. Uses all metrics defined herein.
| |
| * [http://www.ece.rochester.edu/~gsharma/ciede2000/ The CIEDE2000 Color-Difference Formula], by Gaurav Sharma. Implementations in MATLAB and Excel.
| |
| * [http://kodisha.net/color-names/ Color Similarity Tool], by Dragan Bajcic, Implementations in PHP and Javascript.
| |
| * [http://code.google.com/p/python-colormath/ python-colormath]. Implementation in Python.
| |
| * [http://www2.konicaminolta.eu/eu/Measuring/pcc/en/index.html Precise Color Communication] Konica Minolta Sensing.
| |
| | |
| [[Category:Color space]]
| |
| [[Category:Visual perception]]
| |
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