颜色差异

1976年的这个色彩差异公式是首个用较为均匀的Lab空间计算色彩差异值的公式。Lab比RGB等空间在感官上均匀一点，所以得到的结果会更好。不过后来人们发现Lab色彩空间，尤其是在饱和度较高的区域里，并没有设计时预想的那么“感官上均匀”，所以几次更新了ΔE公式。CIE76公式会高估较为饱和的颜色之间的差异。

应用${\displaystyle ({L_{1}^{*}},{a_{1}^{*}},{b_{1}^{*}})}$和${\displaystyle ({L_{2}^{*}},{a_{2}^{*}},{b_{2}^{*}})}$两个L*a*b*色彩空间的颜色：

${\displaystyle \Delta E_{ab}^{*}={\sqrt {(L_{2}^{*}-L_{1}^{*})^{2}+(a_{2}^{*}-a_{1}^{*})^{2}+(b_{2}^{*}-b_{1}^{*})^{2}}}}$

本公式对应的最小可觉差：${\displaystyle \Delta E_{ab}^{*}\approx 2.3}$ [6]

在汽车喷漆行业里，人们测量了大量有关人眼容差情况的数据。随着这些“应用特定”的数据的发布，人们将1976年的定义进行了进一步的发展，以更好地应对感知非均匀特性。该公式仍然采用的是Lab色彩空间。[7]

1994版ΔE是在L*C*h*色彩空间下定义的。它的亮度、色品（或称“浓度”）、色调是从L*a*b*坐标系里计算来的。对于一个基准色${\displaystyle (L_{1}^{*},a_{1}^{*},b_{1}^{*})}$和另一个颜色${\displaystyle (L_{2}^{*},a_{2}^{*},b_{2}^{*})}$，它们的差异是：[8][9][10]

${\displaystyle \Delta E_{94}^{*}={\sqrt {\left({\frac {\Delta L^{*}}{k_{L}S_{L}}}\right)^{2}+\left({\frac {\Delta C_{ab}^{*}}{k_{C}S_{C}}}\right)^{2}+\left({\frac {\Delta H_{ab}^{*}}{k_{H}S_{H}}}\right)^{2}}}}$

其中：

${\displaystyle \Delta L^{*}=L_{1}^{*}-L_{2}^{*}}$

${\displaystyle C_{1}^{*}={\sqrt {{a_{1}^{*}}^{2}+{b_{1}^{*}}^{2}}}}$

${\displaystyle C_{2}^{*}={\sqrt {{a_{2}^{*}}^{2}+{b_{2}^{*}}^{2}}}}$

${\displaystyle \Delta C_{ab}^{*}=C_{1}^{*}-C_{2}^{*}}$

${\displaystyle \Delta H_{ab}^{*}={\sqrt {{\Delta E_{ab}^{*}}^{2}-{\Delta L^{*}}^{2}-{\Delta C_{ab}^{*}}^{2}}}={\sqrt {{\Delta a^{*}}^{2}+{\Delta b^{*}}^{2}-{\Delta C_{ab}^{*}}^{2}}}}$

${\displaystyle \Delta a^{*}=a_{1}^{*}-a_{2}^{*}}$

${\displaystyle \Delta b^{*}=b_{1}^{*}-b_{2}^{*}}$

${\displaystyle S_{L}=1}$

${\displaystyle S_{C}=1+K_{1}C_{1}^{*}}$

${\displaystyle S_{H}=1+K_{2}C_{1}^{*}}$

另外，其中的k_C和k_H一般都取1，系数（权值）k_L、K₁和K₂则依应用场合不同而取不同的值。如：

	图形艺术	纹理、纺织品
${\displaystyle k_{L}}$	1	2
${\displaystyle K_{1}}$	0.045	0.048
${\displaystyle K_{2}}$	0.015	0.014

从几何学上讲，${\displaystyle \Delta H_{ab}^{*}}$这个数量即是这两种颜色的相同色度圈的弦长的算术平均值。 [11]

鉴于1994年的公式并没有充分解决感知非均匀特性的问题，CIE再次修缮了定义，并加入了5个修订系数：[12][13]

• 色调旋转项(R_T)，用来应对常出问题的蓝色区域（色相角度275°左右）；[14]
• 中性色调补偿（对应L*C*h差异）
• 亮度补偿（S_L）
• 色度补偿（S_C）
• 色调补偿（S_H）

${\displaystyle \Delta E_{00}^{*}={\sqrt {\left({\frac {\Delta L'}{k_{L}S_{L}}}\right)^{2}+\left({\frac {\Delta C'}{k_{C}S_{C}}}\right)^{2}+\left({\frac {\Delta H'}{k_{H}S_{H}}}\right)^{2}+R_{T}{\frac {\Delta C'}{k_{C}S_{C}}}{\frac {\Delta H'}{k_{H}S_{H}}}}}}$

注：下面的公式请使用角度而非弧度，尤其在R_T上需要注意。

k_L、k_C、k_H一般取1。

${\displaystyle \Delta L^{\prime }=L_{2}^{*}-L_{1}^{*}}$

${\displaystyle {\bar {L}}={\frac {L_{1}^{*}+L_{2}^{*}}{2}}\quad {\bar {C}}={\frac {C_{1}^{*}+C_{2}^{*}}{2}}}$

${\displaystyle a_{1}^{\prime }=a_{1}^{*}+{\frac {a_{1}^{*}}{2}}\left(1-{\sqrt {\frac {{\bar {C}}^{7}}{{\bar {C}}^{7}+25^{7}}}}\right)\quad a_{2}^{\prime }=a_{2}^{*}+{\frac {a_{2}^{*}}{2}}\left(1-{\sqrt {\frac {{\bar {C}}^{7}}{{\bar {C}}^{7}+25^{7}}}}\right)}$

${\displaystyle {\bar {C}}^{\prime }={\frac {C_{1}^{\prime }+C_{2}^{\prime }}{2}}{\mbox{ and }}\Delta {C'}=C'_{2}-C'_{1}\quad {\mbox{where }}C_{1}^{\prime }={\sqrt {a_{1}^{'^{2}}+b_{1}^{*^{2}}}}\quad C_{2}^{\prime }={\sqrt {a_{2}^{'^{2}}+b_{2}^{*^{2}}}}\quad }$

${\displaystyle h_{1}^{\prime }={\text{atan2}}(b_{1}^{*},a_{1}^{\prime })\mod 360^{\circ },\quad h_{2}^{\prime }={\text{atan2}}(b_{2}^{*},a_{2}^{\prime })\mod 360^{\circ }}$

Note: The inverse tangent (tan⁻¹) can be computed using a common library routine atan2(b, a′) 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 Sharma 2005，eqn. 7.

${\displaystyle \Delta h'={\begin{cases}h_{2}^{\prime }-h_{1}^{\prime }&\left|h_{1}^{\prime }-h_{2}^{\prime }\right|\leq 180^{\circ }\\h_{2}^{\prime }-h_{1}^{\prime }+360^{\circ }&\left|h_{1}^{\prime }-h_{2}^{\prime }\right|>180^{\circ },h_{2}^{\prime }\leq h_{1}^{\prime }\\h_{2}^{\prime }-h_{1}^{\prime }-360^{\circ }&\left|h_{1}^{\prime }-h_{2}^{\prime }\right|>180^{\circ },h_{2}^{\prime }>h_{1}^{\prime }\end{cases}}}$

Note: When either C′₁ or C′₂ is zero, then Δh′ is irrelevant and may be set to zero. See Sharma 2005，eqn. 10.

${\displaystyle \Delta H^{\prime }=2{\sqrt {C_{1}^{\prime }C_{2}^{\prime }}}\sin(\Delta h^{\prime }/2),\quad {\bar {H}}^{\prime }={\begin{cases}(h_{1}^{\prime }+h_{2}^{\prime }+360^{\circ })/2&\left|h_{1}^{\prime }-h_{2}^{\prime }\right|>180^{\circ }\\(h_{1}^{\prime }+h_{2}^{\prime })/2&\left|h_{1}^{\prime }-h_{2}^{\prime }\right|\leq 180^{\circ }\end{cases}}}$

Note: When either C′₁ or C′₂ is zero, then H′ is h′₁+h′₂ (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 Sharma 2005，eqn. 7 and p. 23 stating most implementations on the internet at the time had "an error in the computation of average hue".

${\displaystyle T=1-0.17\cos({\bar {H}}^{\prime }-30^{\circ })+0.24\cos(2{\bar {H}}^{\prime })+0.32\cos(3{\bar {H}}^{\prime }+6^{\circ })-0.20\cos(4{\bar {H}}^{\prime }-63^{\circ })}$

${\displaystyle S_{L}=1+{\frac {0.015\left({\bar {L}}-50\right)^{2}}{\sqrt {20+{\left({\bar {L}}-50\right)}^{2}}}}\quad S_{C}=1+0.045{\bar {C}}^{\prime }\quad S_{H}=1+0.015{\bar {C}}^{\prime }T}$

${\displaystyle R_{T}=-2{\sqrt {\frac {{\bar {C}}'^{7}}{{\bar {C}}'^{7}+25^{7}}}}\sin \left[60^{\circ }\cdot \exp \left(-\left[{\frac {{\bar {H}}'-275^{\circ }}{25^{\circ }}}\right]^{2}\right)\right]}$

在讨论人类感知时，一般使用ΔE计算色彩差异。如前所述，一般认为在ΔE系统中，1.0是刚好注意得到的量。取决于产品的受众，有时候会改用2.5作为可以接受的限制。[15]例如一般情况下，人们会说“叶子和草地是绿的”，但仔细看就能发现叶子和草地都有从黄绿到蓝绿等不同的差异。

这类差异在很多产品的品质控制中非常重要：在生产和运输中经常会有一些来自内部或外部的随机扰动，造成实际上显示的颜色和之前校准出的结果有一些差异。在最理想的情况下，要做到“一般的观察者”不能察觉这类变化。胶印过程中通常允许最高2到4个单位的ΔE色彩差异；而家用打印机造成的色差可能更大。[16][17]