Page 64 - Computational Colour Science Using MATLAB

CIELAB AND CIELUV COLOUR SPACE 51

1
h ab ¼ tan ðb*=a*Þð180=pÞ, ð5.3Þ
where the term 180/p is necessary to convert the output of the inverse tan
function from radians to degrees. The polar coordinates are useful since the
differences in the chroma term C* can be correlated with differences in perceived
ab
colourfulness, and differences in the hue term h can be correlated with
ab
differences in perceived hue. Equations (5.2) and (5.3) can easily be inverted
(Green, 2002a),
a* ¼ C* cosðh ab p=180Þ, ð5.4Þ

b* ¼ C* sinðh ab p=180Þ. ð5.5Þ

If the tristimulus values of the neutral are known, then it is possible to invert
Equations (5.1),

3
Y ¼ Y n fðY=Y n Þ , if fðY=Y n Þ 4 ð0:008856Þ 1=3 ,
Y ¼ Y n ðfðY=Y n Þ 16=116Þ=7.787Þ, if fðY=Y n Þ4 ð0:008856Þ 1=3 ,
3
X ¼ X n fðX=X n Þ , if fðX=X n Þ 4 ð0:008856Þ 1=3 ,
X ¼ X n ðfðX=X n Þ 16=116Þ=7.787Þ, if fðX=X n Þ4 ð0:008856Þ 1=3 , ð5.6Þ
3
Z ¼ Z n fðZ=Z n Þ , if fðZ=Z n Þ 4 ð0:008856Þ 1=3 ,
1=3
Z ¼ Z n ðfðZ=Z n Þ 16=116Þ=7.787Þ,if fðZ=Z n Þ4 ð0:008856Þ ,
where

fðY=Y n Þ¼ ðL* þ 16Þ=116,

fðX=X n Þ¼ a*=500 þ fðY=Y n Þ
and
fðZ=Z n Þ¼ fðY=Y n Þ b*=200.

The formulae for computing CIELUV coordinates are given as Equations (5.7),
1=3
L* ¼ 116ðY=Y n Þ 16, if Y=Y n 4 0:008856,
L* ¼ 903:3ðY=Y n Þ, if Y=Y n 4 0:008856,

ð5.7Þ
u* ¼ 13L*ðu u n Þ,
0
0
v* ¼ 13L*ðv v n Þ,
0
0
where u and v are the coordinates of the so-called uniform chromaticity space,
0
0
CIE 1976 UCS, which is a linear transform of the more usual xy chromaticity
space,