Delta E (CIE 2000) - Bruce Lindbloom

Delta E (CIE 2000)

The color difference, or $\Delta E$, between a sample color ($L_2$, $a_2$, $b_2$) and a reference color ($L_1$, $a_1$, $b_1$) is:

$$\Delta E = \sqrt{ \left({\Delta L'} \over {K_L S_L}\right)^2 + \left({\Delta C'} \over {K_C S_C}\right)^2 + \left({\Delta H'} \over {K_H S_H}\right)^2 + R_T \left({\Delta C'} \over {K_C S_C}\right) \left({\Delta H'} \over {K_H S_H}\right) }$$

where

$$\bar L' = (L_1 + L_2)/2$$ $$C_1 = \sqrt{a_1^2 + b_1^2}$$ $$C_2 = \sqrt{a_2^2 + b_2^2}$$ $$\bar C = (C_1 + C_2)/2$$ $$G = {1 \over 2}\left({1 - \sqrt{{{\bar C}^7} \over {{\bar C}^7 + 25^7}}}\right)$$ $$a_1' = a_1(1+G)$$ $$a_2' = a_2(1+G)$$ $$C_1' = \sqrt{{a_1'}^2 + b_1^2}$$ $$C_2' = \sqrt{{a_2'}^2 + b_2^2}$$ $$\bar C' = (C_1' + C_2')/2$$ $$h_1' = \cases{ \arctan(b_1 / a_1') & \text{if }\arctan(b_1 / a_1') \geq 0 \\ \arctan(b_1 / a_1')+360° & \text{otherwise} }$$ $$h_2' = \cases{ \arctan(b_2 / a_2') & \text{if }\arctan(b_2 / a_2') \geq 0 \\ \arctan(b_2 / a_2')+360° & \text{otherwise} }$$ $$\bar H' = \cases{ (h_1' + h_2' + 360°)/2 & \text{if }|h_1' - h_2'| \gt 180° \\ (h_1' +h_2')/2 & \text{otherwise} }$$ $$T = 1 - 0.17 \cos (\bar H' - 30°) + 0.24 \cos(2 \bar H') + 0.32\cos(3 \bar H' + 6°) - 0.20 \cos(4 \bar H' - 63°)$$ $$\Delta h' = \cases{ h_2' - h_1' & \text{if }|h_2' - h_1'| \leq 180° \\ h_2' - h_1' + 360° & \text{else if }|h_2' - h_1'| \gt 180° \text{and }h_2' \leq h_1' \\ h_2' - h_1' - 360° & otherwise }$$ $$\Delta L' = L_2 - L_1$$ $$\Delta C' = C_2' - C_1'$$ $$\Delta H' = 2 \sqrt{C_1' C_2'} \sin(\Delta h' / 2)$$ $$S_L = 1 + {{0.015 (\bar L' - 50)^2} \over {\sqrt{20 + (\bar L' - 50)^2}}}$$ $$S_C = 1 + 0.045 \bar C'$$ $$S_H = 1 + 0.015 \bar C' T$$ $$\Delta \theta = 30 \exp \left \lbrace -\left({{\bar H' - 275°} \over 25} \right)^2\right \rbrace$$ $$R_C = 2 \sqrt{{\bar C'^7} \over {\bar C'^7 + 25^7}}$$ $$R_T = -R_C \sin(2 \Delta \theta)$$ $$K_L = 1 \text{ default}$$ $$K_C = 1 \text{ default}$$ $$K_H = 1 \text{ default}$$

Implementation Notes:

1. in the formulas above, angles are expressed in degrees, not radians.
2. In computing $h_1'$ and $h_2'$', be careful with the inverse tangent since $a_1'$ and/or $a_2'$ could be zero. Instead, use special math functions to do this. In the Standard C library and most other math libraries, this function is called atan2 and is used calling atan2(b, a). In Microsoft Excel, it is called ATAN2 and its use is ATAN2(a, b). Note the argument reversal! These special functions will compute the proper inverse tangents without needing to worry about "divide by zero" conditions. Additionally, atan2 will also handle the quadrants for you.
3. Math libraries generally return inverse tangents in radians, not degrees. In Excel, use the DEGREES function to convert radians to degrees.
4. Math libraries generally require the input to the sine and cosine functions to be in radians, not degrees. In Excel, use the RADIANS function to convert degrees to radians.