Conversion Between RGB and HSI - Introduction to Video and Image Processing - page 196

−− WP

P r

P g + P b − = + 2 / 3 (P r + P g + P b ) + 3 / 9 ⇔

P r

−− WP P b − P g + = + 1 / 3 (D.9)

where the last reduction is possible since P r + P g + P b = 1. Inserting the above

expressions into Eq. D.5 yields

⎡ ⎤ ⎡ ⎤

⎛ ⎞

2 / 3
−

P r − 1 / 3

⎣ ⎦ • ⎣ ⎦

⎝ 1 / 3 P g − 1 / 3 ⎠

− 1 / 3 P b − 1 / 3

cos − 1

θ = P r ⇔

√ 2 / 3

+ P g + P b − · 1 / 3

cos − 1 ( 2 P r / 3

− 2 / 9 ) + ( − P g / 3 + 1 / 9 ) + ( − P b / 3 + 1 / 9 )

θ = 9 · 2 P r ⇔

P b − P g +

9 · 3 ·

+ 1 / 3

cos − 1

2 P r − P g − P b

θ = 6 P r (D.10)

6 P b −

6 P g +

+ 2

The final step is now to replace (P r ,P g ,P b ) with (P R ,P G ,P B ) . Recall that P r =
P R /J , P g = P G /J , P b = P B /J , where J = P R + P G + P B . Inserting yields

cos − 1

1 /J · ( 2 P R − P G − P B )
1 /J 2 ( 6 P R + θ = ⇔

6 P G + 6 P B ) − 2

cos − 1

2 P R − P G − P B
6 P R +

θ = ⇔

6 P G + 6 P B −

2 J 2

cos − 1 1 / 2

2 P R − P G − P B

θ = · P R + ⇔

P G + P B −

P R · P G − P R · P B − P G · P B

cos − 1 1 / 2

P B )
√ (P R − P G )(P R − P G ) + (P R − P B )(P G − P B )

(P R − P G ) + (P R −

= ·

θ (D.11)

Note that the last expression is often applied in order to optimize the conversion
from an implementation point of view. Since Eq. D.5 is only valid in the range
θ ∈[ 0° , 180° ] the final expression for hue is given below, see Appendix B for details.

θ,

if P G ≥ P B ;

H = (D.12)

360° − θ, otherwise