How to calculate CIE L* a* b*

Calculation of CIE L* a* b*


The tristimulus values (X Y Z) can be calculated according to the method mentioned in CIE Y x y calculation method and the CIE L* a* b* color value can be calcu,ated using the following equation.

<Basic equation>



<Calculation>

Example] To calculate tristimulus value X Y Z at D65/2



about the calculation for item 390 nm are

1)  S(λ) ऱ(λ) = 0.0042 x 54.65 = 0.229530
2)  S(λ) ў(λ) = 0.0001 x 54.65 = 0.005465
3)  S(λ) ž(λ) = 0.0201 x 54.65 = 1.098465

The transmittance value of the measured spectrum is 86.261% at 390 nm.

1) S(λ) ऱ(λ) 𝞃(λ) = 0.229530 x 0.86261 = 0.197994
2) S(λ) ў(λ) 𝞃(λ) = 0.005465 x 0.86261 = 0.004714
3) S(λ) ž(λ) 𝞃(λ) = 1.098465 x 0.86261 = 0.947544








The sum of y^(λ) value, denominator value, in the table above is 2133.467, So, k = 100 / 2 133.467 = 0.047 315 611

Therefore, the X Y Z and the color coordinates x and y can be calculated by the above equation.

X = 1773.253 x 0.047315611 = 83.9026
Y = 1884.360 x 0.047315611 = 89.1597
Z = 2042.179 x 0.047315611 = 96.6270




by above equation (24/116)^3 = 0.008 865

1) (X/Xn)^(1/3) = (83.9026 / 95.043)^(1/3) = 0.959294 > 0.008 865
2) (Y/Yn)^(1/3) = (89.1597 / 100.00)^(1/3) = 0.962475 > 0.008 865
3) (Z/Zn)^(1/3) = (96.6270 / 108.879)^(1/3) = 0.960988 > 0.008 865 

Therefore we can use above calculated values as f(X/Xn), f(Y/Yn), f(Z/Zn) 



Using the above equation to calculate L* a* b*,

L* = 116 x 0.962 475 - 16 = 95.647 1
a* = 500 x ( 0.959 294 - 0.962 475 ) = -1.590 6
b* = 200 x ( 0.962 475 - 0.960 988) = 0.297 4

Comparison of L*, a*, b* results between calculated value with EXCEL and by using calculation Program