Colorimetry · pag. 1 of 2

• 1. The spectrophotometric approach to measure the color
  
  
  • 1.1 Measuring spectral data of the reflectance
  • 1.2 Calculating the tristimulus values X,Y,Z (CIE 1931, CIE 1964)
  • 1.3 Calculating the CIE L*a*b* values and CIE L*C*h* values (CIE1964)
  
  
• 2. Calculating the color data out of RGB data from a color sensor
  
  
  • 2.1 Formula
  • 2.2 Limitations of RGB based systems
  
  

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1. The spectrophotometric approach to measure the color

1.1 Measuring spectral data of the reflectance

(1.1.1) R(λ) = WhiteData(λ) * M(λ) / W(λ)

where is: R(λ) reflectance spectrum
WhiteData(λ) reflectance spectrum of the white reference
M(λ) measured data on the sample
W(λ) measured data on the white reference

That means: The measurement of the reflectance spectrum is a relative measurement; just a comparison between the data measured on the white reference and on the sample.

1.2 Calculating the tristimulus values X,Y,Z (CIE 1931, CIE 1964)

(1.2.1) X = λ S(λ) * x(λ) * R(λ)
Y = λ S(λ) * y(λ) * R(λ)
Z = λ S(λ) * z(λ) * R(λ)

where is: X,Y,Z CIE 1931 or CIE 1964 tristimulus values
R(λ) reflectance spectrum
S(λ) spectrum of the illumination *)
x,y,z(λ) spectral tristimulus values **)

*) The tables of the illumination (daylight D65 for example) are defined by the CIE.
**) The spectral tristimulus values x,y,z(λ) are representing the spectral sensitivity of the human eye. The tables are given in the CIE publications. For small samples (2°) there are the tables of the CIE 1931 publication, for large samples (10°) there are the tables of the CIE 1964 publication.

With the spectrophotometric approach, the calculations of the X,Y,Z are done numerically. With a colorimeter, these calculations are done by the hardware of the filters and of the light source. Changes of the characteristics of the spectrophotometer (light source, ...) do not change the results of X,Y,Z values. (see also 1.1).

1.3 Calculating the CIE L*a*b* values and CIE L*C*h* values (CIE1964)

The (approximately) uniform color space Lab is a non linear transformation of the XYZ color space:

(1.3.1) L* = 25*(100*Y/Yo) 1/3 - 16
a* = 500*[(*X/Xo) 1/3 - ((*Y/Yo) 1/3 ]
b* = 500*[(*Y/Yo) 1/3 - ((*Z/Zo) 1/3 ]

where is: X,Y,Z CIE 1931 or CIE 1964 tristimulus values
Xo,Yo,Zo X,Y,Z values of a perfect white sample
L* CIE Lab L - value (lightness in Lab color space)
a* CIE Lab a - value (red – green value)
b* CIE Lab b - value (yellow – blue value)

Color differences are calculated as follows:
(1.3.2) λL = L1 – L2
λa = a1 – a2
λb = b1 – b2
λE = (λL2 + λa2 + λb2 )1/2

where is: λL, λa, λb Color differences in the CIE L*a*b* color space
L1, a1, b1 L*a*b* values of sample 1
L2, a2, b2 L*a*b* values of sample 2
λE total color difference (distance between the 2 point in the color space)

CIE LCh values: polar coordinates of the Lab values

(1.3.3) L: as L in Lab
C = (a2 + b2 )1/2
h = arc tan (b/a)

where is: L, a, b L*a*b* values
C Chroma (saturation) in the L*a*b* color space
h Hue in the L*a*b* color space

Differences in the CIE LCh color space:

(1.3.4) λL = L1 – L2
λC = C1 – C2
λE = (λL2 + λa2 + λb2 )1/2
λH = (λE2 - λL2 - λC2 )1/2

where is: λL, λC, λH Color differences in the CIE L*C*h* color space
L1, C1, L2, C2 L*C* values of two samples
λL, λa, λb, λE see above

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