Colour conversions for DCP creation
Carl Hetherington, Dennis Couzin
1 Overview
The process of conversion from RGB to XYZ is:
1. Convert to linear RGB (i.e. apply a gamma curve, or at least an approximation to one).
2. Convert to XYZ (preserving the white point).
3. Adjust the white point.
4. Normalize values.
5. Convert to non-linear XYZ (i.e. apply a gamma curve)
2 Convert to linear RGB
This is done using either a ‘pure’ gamma function, so that for some colour value Ci the output colour Co is given by
Co = Ciγ (1)
or a modified function of the form
Ci
Ci ≤ Ct
(
Co = K ζ (2)
Ci +A
1+A Ci > Ct
where K, A, Ct and ζ are constants. This modified function approximates a ‘pure’ gamma function but changes the output
for small inputs.
1
3 Convert to XYZ
This is done by multiplying the colours by a 3 × 3 matrix. This matrix depends on the chromaticities of the RGB primaries and
the white point. Note that these are the same for BT709 and sRGB, so the corresponding matrices are the same.
The chromaticities for sRGB and BT709 are shown in Table ??. The white point is (x = 0.3127, y = 0.329); this is called
D65.
x y
Red 0.64 0.33
Green 0.3 0.6
Blue 0.15 0.06
Table 1: RGB chromaticities for sRGB and BT709
Let
Rx − Wx Wx − Bx
D= (3)
Ry − Wy Wy − By
Wx − Gx Rx − Wx
E= (4)
Wy − Gy Ry − Wy
Wx − Gx Wx − Bx
F = (5)
Wy − Gy Wy − By
D E
P = Ry + Gy + By (6)
F F
where:
• Rx , Ry : red point; Rz = 1 − Rx − Ry
• Gx , Gy : green point; Gz = 1 − Gx − Gy
• Bx , By : blue point; Bz = 1 − Bx − By
• Wx , Wy : white point
2
Then the conversion matrix C is as follows:
D E
Rx Gx
F
Bx
F
0.4123908 0.3575843 0.1804808
1 D E
C = Ry Gy By = 0.2126390 0.7151687 0.0721923 (7)
P F F
0.0193308 0.1191948 0.9505322
D E
Rz Gz Bz
F F
Then to convert RGB to XYZ we do
X R
Y = C G (8)
Z B
i.e.
X 0.4123908 0.3575843 0.1804808 R
Y = 0.2126390 0.7151687 0.0721923 G (9)
Z 0.0193308 0.1191948 0.9505322 B
Note: there is also a CIE definition of the D65 white point as (x = 0.31272, y = 0.32903), which we do not use.
4 Adjust the white point
If required we can adjust the white point of the colours from one white point S1 to another S2 . This is done by multiplication
by a Bradford matrix, B. To calculate it, we start with the Bradford chromatic adaption transform matrix M, taken from
Süsstrunk et al., Chromatic adaption performance of different RGB sensors, IS&T/SPIE Electronic Imaging, SPIE Vol. 4300
(2001).
0.8951 0.2664 −0.1614
M = −0.7502 1.7135 0.0367 (10)
0.0389 −0.0685 1.0296
3
The inverse of M is
0.9869929055 −0.1470542564 0.1599626517
M−1 = 0.4323052697 0.5183602715 0.0492912282 (11)
−0.0085286646 0.0400428217 0.9684866958
Next, compute G and H as follows
S1x
S1y
G = M 1 (12)
1 − S1x − S1y
S1y
S 2x
S2y
H = M 1 (13)
1 − S2x − S2y
S2y
Then the Bradford matrix B is given by
H1
G1 0 0
H2
B = M−1 0 0 M (14)
G2
H3
0 0
G3
(15)
5 Normalize values
Here we just multiply all colour values by the constant N where
48
N= (16)
52.37
4
6 Convert to non-linear XYZ
This is a gamma correction of 1/2.6, so that for some colour value Ci the output colour Co is given by
1/2.6
Co = Ci (17)
7 Converting from a colour matrix to chromaticities
To get back from a colour matrix C to the chromaticities:
C11
Rx = (18)
C11 + C21 + C31
C21
Ry = (19)
C11 + C21 + C31
C12
Gx = (20)
C12 + C22 + C32
C22
Gy = (21)
C12 + C22 + C32
C13
Bx = (22)
C13 + C23 + C33
C23
By = (23)
C13 + C23 + C33
C11 + C12 + C13
Wx = (24)
C11 + C12 + C13 + C21 + C22 + C23 + C31 + C32 + C33
C21 + C22 + C23
Wy = (25)
C11 + C12 + C13 + C21 + C22 + C23 + C31 + C32 + C33
