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| In [[colorimetry]], the '''CIE 1976 (''L''*, ''u''*, ''v''*) color space''', commonly known by its abbreviation '''CIELUV''', is a [[color space]] adopted by the [[International Commission on Illumination]] (CIE) in 1976, as a simple-to-compute transformation of the 1931 [[CIE 1931 color space|CIE XYZ color space]], but which attempted [[Color_difference#Tolerance|perceptual uniformity]]. It is extensively used for applications such as computer graphics which deal with colored lights. Although additive mixtures of different colored lights will fall on a line in CIELUV's uniform [[chromaticity diagram]] (dubbed the ''CIE 1976 UCS''), such additive mixtures will not, contrary to popular belief, fall along a line in the CIELUV color space unless the mixtures are constant in [[Lightness (color)|lightness]].
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| ==Historical background==
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| CIELUV is an [[Adams chromatic valence color space]], and is an update of the [[CIE 1964 color space|CIE 1964 (''U''*, ''V''*, ''W''*) color space]] (CIEUVW). The differences include a slightly modified [[Lightness (color)|lightness]] scale, and a modified uniform chromaticity scale in which one of the coordinates, ''v''′, is 1.5 times as large as ''v'' its [[CIE 1960 color space|1960 predecessor]]. CIELUV and [[CIELAB]] were adopted simultaneously by the CIE when no clear consensus could be formed behind only one or the other of these two color spaces.
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| CIELUV uses Judd-type (translational) [[white point]] adaptation (in contrast with CIELAB, which uses a "wrong" [[von Kries transform]]).<ref>{{cite journal|first=Deane B.|last=Judd|title=Hue saturation and lightness of surface colors with chromatic illumination|journal=[[JOSA]]|volume=30|issue=1|date=January 1940|pages=2–32| url=http://www.opticsinfobase.org/abstract.cfm?URI=josa-30-1-2|doi=10.1364/JOSA.30.000002}}</ref> This can produce useful results when working with a single illuminant, but can predict [[imaginary color]]s (i.e., outside the [[spectral locus]]) when attempting to use it as a [[chromatic adaptation transform]].<ref name="Fairchild1998">Mark D Fairchild, ''Color Appearance Models.'' Reading, MA: Addison-Wesley, 1998.</ref> The translational adaptation transform used in CIELUV has also been shown to perform poorly in predicting corresponding colors.<ref name="Alman1989">D. H. Alman, R. S. Berns, G. D. Snyder, and W. A. Larson, "Performance testing of color difference metrics using a color-tolerance dataset." ''Color Research and Application,'' '''21''':174-188 (1989).</ref>
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| == XYZ → CIELUV and CIELUV → XYZ conversions ==
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| For typical images, ''u''* and ''v''* range ±100. By definition, {{nobr|0 ≤ ''L''* ≤ 100}}, except for specular highlights{{elucidate|date=February 2013}}.{{citation needed|date=February 2013}}
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| ===The forward transformation===
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| CIELUV is based on CIEUVW and is another attempt to define an encoding with uniformity in the perceptibility of [[color difference]]s.<ref name=schanda/> The non-linear relations for ''L''*, ''u''*, and ''v''* are given below:<ref name=schanda>{{cite book|title=Colorimetry: Understanding the CIE System|first=János|last=Schanda|publisher=Wiley Interscience|year=2007|isbn=978-0-470-04904-4|quote=As 24/116 is not a simple ratio, in some publications the 6/29 ratio is used, in others the approximate value of 0.008856 (used in earlier editions of CIE 15). Similarly some authors prefer to use instead of 841/108 the expression (1/3)×(29/6)<sup>2</sup> or the approximate value of 7.787, or instead of 16/116 the ratio 4/29.|pages=61–64}}</ref>
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| :<math>
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| \begin{align}
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| L^* &= \begin{cases}
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| \left(\frac{29}{3}\right)^3 Y / Y_n,& Y / Y_n \le \left(\frac{6}{29}\right)^3 \\
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| 116 \left( Y / Y_n \right)^{1/3} - 16,& Y / Y_n > \left(\frac{6}{29}\right)^3
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| \end{cases}\\
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| u^* &= 13 L^*\cdot (u^\prime - u_n^\prime) \\
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| v^* &= 13 L^*\cdot (v^\prime - v_n^\prime)
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| \end{align}</math>
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| The quantities ''u''′<sub>''n''</sub> and ''v''′<sub>''n''</sub> are the {{nobr|(''u''′, ''v''′)}} chromaticity coordinates of a "specified white object" – which may be termed the [[white point]] – and ''Y''<sub>''n''</sub> is its luminance. In reflection mode, this is often (but not always) taken as the {{nobr|(''u''′, ''v''′)}} of the [[Diffuser (optics)|perfect reflecting diffuser]] under that illuminant. (For example, for the [[standard colorimetric observer|2° observer]] and [[standard illuminant]] C, {{nobr|1=''u''′<sub>''n''</sub> = 0.2009}}, {{nobr|1=''v''′<sub>''n''</sub> = 0.4610}}.) Equations for ''u''′ and ''v''′ are given below:<ref name="CIE15point2">''Colorimetry,'' second edition: CIE publication 15.2. Vienna: Bureau Central CIE, 1986.</ref><ref name=poynton/>
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| :<math>\begin{align}
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| u^\prime &= \frac{4 X}{X + 15 Y + 3 Z} &= \frac{4 x}{-2 x + 12 y + 3} \\
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| v^\prime &= \frac{9 Y}{X + 15 Y + 3 Z} &= \frac{9 y}{-2 x + 12 y + 3}
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| \end{align}</math>
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| ===The reverse transformation===
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| [[Image:CIE 1976 UCS.png|right|thumb|300px|{{nobr|(''u''′, ''v''′)}} chromaticity diagram, also known as the CIE 1976 UCS (uniform chromaticity scale) diagram.]]
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| The transformation from {{nobr|(''u''′, ''v''′)}} to {{nobr|(''x'', ''y'')}} is:<ref name=poynton/>
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| :<math>\begin{align}
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| x &= \frac{9u^\prime}{6u^\prime - 16v^\prime + 12}\\
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| y &= \frac{4v^\prime}{6u^\prime - 16v^\prime + 12}
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| \end{align}</math>
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| The transformation from CIELUV to XYZ is performed as follows:<ref name=poynton/>
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| :<math>\begin{align}
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| u^\prime&= \frac{u^*}{13L^*} + u^\prime_n \\
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| v^\prime&= \frac{v^*}{13L^*} + v^\prime_n \\
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| Y &= \begin{cases}
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| Y_n \cdot L^* \cdot \left(\frac{3}{29}\right)^3,& L^* \le 8 \\
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| Y_n \cdot \left(\frac{L^* + 16}{116}\right)^3,& L^* > 8
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| \end{cases}\\
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| X &= Y \cdot \frac{9u^\prime}{4v^\prime} \\
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| Z &= Y \cdot \frac{12 - 3u^\prime - 20v^\prime}{4v^\prime} \\
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| \end{align}</math>
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| ==Cylindrical representation==
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| The [[cylindrical coordinate system|cylindrical]] version of CIELUV is known as CIE LCh<sub>uv</sub>, where ''C''*<sub>''uv''</sub> is the [[colorfulness|chroma]] and ''h''<sub>''uv''</sub> is the [[hue]]:<ref name=poynton/>
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| :<math>C_{uv}^* = \sqrt{(u^*)^2 + (v^*)^2}</math>
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| :<math>h_{uv} = \operatorname{atan2}(v^*,u^*),</math>
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| where [[atan2]] function, a "two-argument arctangent", computes the [[polar coordinates|polar]] angle from a Cartesian coordinate pair.
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| Furthermore, the saturation correlate can be defined as:
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| :<math>s_{uv} = \frac{C^*}{L^*} = 13 \sqrt{(u^\prime - u^\prime_n)^2 + (v^\prime - v^\prime_n)^2}</math>
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| Similar correlates of chroma and hue, but not saturation, exist for CIELAB. See [[Colorfulness#Saturation|Colorfulness]] for more discussion on saturation.
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| ==Color and hue difference==
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| The [[color difference]] can be calculated using the [[Euclidean distance]] of the {{nobr|(''L''*, ''u''*, ''v''*)}} co-ordinates.<ref name=poynton>{{cite book|first=Charles|last=Poynton|title=Digital Video and HDTV| publisher=Morgan-Kaufmann|isbn=1-55860-792-7|year=2003|pages=226}}</ref> It follows that a chromaticity distance of <math>\sqrt{(\Delta u')^2 + (\Delta v')^2}=1/13</math> corresponds to the same Δ''E''*<sub>''uv''</sub> as a lightness difference of {{nobr|1=Δ''L''* = 1}}, in direct analogy to CIEUVW.
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| The Euclidean metric can also be used in CIELCH, with that component of Δ''E''*<sub>''uv''</sub> attributable to difference in hue as<ref name=schanda/> {{nobr|1=Δ''H''* = {{radic|''C''*<sub>1</sub>''C''*<sub>2</sub>}} 2 sin (Δ''h''/2)}}, where {{nobr|1=Δ''h'' = ''h''<sub>2</sub> – ''h''<sub>1</sub>}}.
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| ==References==
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| <references/>
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| ==External links==
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| * [http://www.efg2.com/Lab/Graphics/Colors/Chromaticity.htm Chromaticity diagrams, including the CIE 1931, CIE 1960, CIE 1976]
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| {{Color space}}
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| [[Category:Color space]]
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