SYCC

The three unsigned integers ${\displaystyle Y,C_{b},C_{r}}$ of an sYCC encoded color represent fractional coordinates ${\displaystyle Y',C_{b}',C_{r}'}$ according to the formulas^[2]

${\displaystyle Y'=Y/M}$

${\displaystyle C_{b}'=(C_{b}-Z)/M}$

${\displaystyle C'_{r}=(C_{b}-Z)/M}$

where the scale factor ${\displaystyle M=2^{N}-1}$ is the maximum unsigned ${\displaystyle N}$-bit integer, and the offset ${\displaystyle Z}$ is ${\displaystyle 2^{N-1}}$ (as in the usual two's complement representation of signed integers). Conversely, the encoded integer values are given by

${\displaystyle Y={\mbox{round}}(MY')}$

${\displaystyle C_{b}={\mbox{round}}(Z+MC_{b}')}$

${\displaystyle C_{r}={\mbox{round}}(Z+MC_{r}')}$

with the resulting values clipped to the range ${\displaystyle 0..M}$.

In particular, for ${\displaystyle N=8}$ (which is the normal bit size), one gets ${\displaystyle M=255}$ and ${\displaystyle Z=128}$. Thus the fractional luma ${\displaystyle Y'}$ ranges from 0 to 1, while the fractional chroma coordinates range from ${\displaystyle -128/255\approx -0.50196...}$ to ${\displaystyle +127/255\approx +0.498039...}$.

The standard specifies that these fractional values ${\displaystyle Y',C_{b}',C_{r}'}$ are related to the non-linear fractional sRGB coordinates ${\displaystyle R',G',B'}$ by a linear transformation, described by the matrix product

${\displaystyle {\begin{bmatrix}Y'\\C_{b}'\\C_{r}'\end{bmatrix}}={\begin{bmatrix}+0.2990&+0.5870&+0.1140\\-0.1687&-0.3313&+0.5000\\+0.5000&-0.4187&-0.0813\end{bmatrix}}{\begin{bmatrix}R'\\G'\\B'\end{bmatrix}}}$

This correspondence is the same as the RGB to YCC mapping specified by the old TV standard ITU-R BT.601-5, note that the coefficients of ${\displaystyle Y'}$ here are defined still with 3 decimal places, because 4 decimal places would change legacy encoding of images (first row: Kr = 0.299 and Kg = 0.587).^[2]

The non-linear fractional sRGB coordinates ${\displaystyle R',G',B'}$ can be computed from the fractional sYCC coordinates ${\displaystyle Y',C_{b}',C_{r}'}$ by inverting the above matrix. The standard gives the approximation

${\displaystyle {\begin{bmatrix}R'\\G'\\B'\end{bmatrix}}={\begin{bmatrix}+1.0000&0.0000&+1.4020\\+1.0000&-0.3441&-0.7141\\+1.0000&+1.7720&0.0000\end{bmatrix}}{\begin{bmatrix}Y'\\C_{b}'\\C_{r}'\end{bmatrix}}}$

which is expected to be accurate enough for ${\displaystyle N=8}$ bits per component. For bit sizes greater than 8, the standard recommends using a more accurate inverse. It states that the following matrix with 6 decimal digits is accurate enough for ${\displaystyle N=16}$:

${\displaystyle {\begin{bmatrix}R'\\G'\\B'\end{bmatrix}}={\begin{bmatrix}+1.000000&+0.000037&+1.401988\\+1.000000&-0.344113&-0.714104\\+1.000000&+1.771978&+0.000135\end{bmatrix}}{\begin{bmatrix}Y'\\C_{b}'\\C_{r}'\end{bmatrix}}}$

The same standard specifies the relation between the non-linear fractional coordinates ${\displaystyle R',G',B'}$ and the CIE 1931 XYZ coordinates. The connection entails the transfer function ("gamma correction") that maps ${\displaystyle R',G',B'}$ to the linear R, G, B coordinates, and then a 3D linear transformation that relates these to the CIE ${\displaystyle X,Y,Z}$.

Since the linear transformation from sRGB to sYCC is defined in terms of non-linear (gamma-encoded) values (${\displaystyle R',G',B'}$), rather than the linear ones (${\displaystyle R,G,B}$), the ${\displaystyle Y'}$ component of sYCC is not the CIE Y coordinate, not even a function of it alone. That is, two colors with the same CIE Y value may have different sYCC ${\displaystyle Y'}$ values, and vice-versa.

The integer encoded sYCC triplet ${\displaystyle (0,0,0)}$ represents the color black whereas ${\displaystyle (255,0,0)}$ is white (more precisely, the CIE illuminant D65). More generally, triplets ${\displaystyle (Y,0,0)}$, for ${\displaystyle Y}$ in 0..255, represent shades of gray.

Note that the 8-bit integer sYCC triplet ${\displaystyle (Y,Cb,Cr)=(0,255,255)}$ has fractional coordinates ${\displaystyle (Y',Cb',Cr')\approx (0.0,0.5,0.5)}$, which, according to these matrices, has fractional non-linear sRGB coordinate ${\displaystyle G'\approx -0.5\times (0.3341+0.7141)\approx -0.528}$, and therefore has to be converted to XYZ to be realizable or perceivable. Similarly, the sYCC triplet ${\displaystyle (0,0,0)}$ has ${\displaystyle R'\approx -0.701}$ and ${\displaystyle B'\approx -0.886}$.