# A Fast and Reliable Luma Control Scheme for High-Quality HDR/WCG Video
## 2 Luma Control Problem  
To define the luma control problem, let \(R,G,B\) denote the original pixel values in the linear light domain, which are to be transformed to NCL Y'CbCr 4:2:0 pixel values. For HDR-10 video [3], this transformation employs the inverse ST.2084 [2] Perceptual Quantizer (PQ), the Y'CbCr color-space conversion, the narrow-band 10-bit quantization, and chroma subsampling in this order, as described in [12] and depicted in Figure 1.  
After the processing steps, such as video encoding, transmission, reception, and decoding, the reconstructed NCL Y'CbCr 4:2:0 video is supposed to be transformed back to the RGB display signal in the linear light domain. The postprocessing for this transformation shall comprise the stages, which are exactly the inverses of the corresponding preprocessing blocks. Hence, this processing involves chroma upsampling, inverse 10-bit quantization, RGB color-space conversion, and then ST.2084 EOTF. For now, in order to consider the artifacts caused by chroma subsampling, let us leave the processing blocks from video encoding to decoding behind. If we denote by \(\hat{R},\hat{G},\hat{B}\) the reconstructed output pixel values of the postprocessing, we can define the luminance error by  
\[E=\big{\{}w_{R}\big{(}\hat{R}-R\big{)}+w_{G}\big{(}\hat{G}-G\big{)}+w_{B} \big{(}\hat{B}-B\big{)}\big{\}}^{2}, \tag{1}\]  
where \((w_{R},w_{G},w_{B})\) represents the contribution of each linear light component to luminance and is given by (0.2126, 0.7152, 0.0722) for BT.709 [13] and (0.2627, 0.6780, 0.0593) for BT.2020 [14] color gamut. Translation of this error to the one with nonlinear reconstructed signals provides  
\[E=\big{\{}w_{R}\big{(}L\big{(}\hat{R}^{\prime}\big{)}-L\big{(}R^{\prime}\big{)} \big{)}+w_{G}\big{(}L\big{(}\hat{G}^{\prime}\big{)}-L\big{(}G^{\prime}\big{)} \big{)}+w_{B}\big{(}L\big{(}\hat{B}^{\prime}\big{)}-L\big{(}B^{\prime}\big{)} \big{)}\big{\}}^{2}, \tag{2}\]  
where the \"prime\" notation, as a well-known convention, illustrates that the signal is in the \"nonlinear\" domain, and \(L(\cdot)\) denotes the ST 2084 EOTF, which is defined by  
\[L(X)=\left[\frac{\max\Big{(}\Big{(}X^{1/m}-c_{1}\Big{)},0\Big{)}}{c_{2}-c_{3} \cdot X^{1/m}}\right]^{1/n}, \tag{3}\]  
where \(m=78.84375\), \(n=0.1593017578\), \(c_{1}=0.8359375\), \(c_{2}=18.8515625\), and \(c_{3}=18.6875\). For further investigation of the reconstructed nonlinear signal, \(\big{(}\hat{R}^{\prime},\hat{G}^{\prime},\hat{B}^{\prime}\big{)}\), let us denote the chroma subsampling errors by \(\Delta C_{b}\) and \(\Delta C_{r}\), such that  
\[\hat{C}_{b}=C_{b}+\Delta C_{b},\hat{C}_{r}=C_{r}+\Delta C_{r}, \tag{4}\]  
where \(\big{(}\hat{C}_{b},\hat{C}_{r}\big{)}\) and \((C_{b},C_{r})\) represent the reconstructed and the original chroma signal pairs in nonlinear 4:4:4 format, respectively. In order to compensate for these subsampling errors, if we assume that the original luma value, \(Y^{\prime}\), is adjusted to a new one, \(\hat{Y}^{\prime}=Y^{\prime}+\Delta Y^{\prime}\), then the reconstructed signal, \(\big{(}\hat{R}^{\prime},\hat{G}^{\prime},\hat{B}^{\prime}\big{)}\), or equivalently, the reconstruction difference, \((\Delta R,\Delta G,\Delta B)=\big{(}\hat{R}^{\prime},\hat{G}^{\prime},\hat{B }^{\prime}\big{)}-(R^{\prime},G^{\prime},B^{\prime})\), can be represented by  
\[\left[ \begin{array}{c}\hat{R}^{\prime}\\ \hat{G}^{\prime}\\ \hat{B}^{\prime}\end{array} \right]=\left[ \begin{array}{cc}1&0&a_{RCr}\\ 1&a_{GCb}&a_{GCr}\\ 1&a_{BCb}&0\end{array} \right]\left[ \begin{array}{c}\hat{Y}^{\prime}\\ \hat{C}_{b}\\ \hat{C}_{r}\end{array} \right]or\left[ \begin{array}{c}\Delta R\\ \Delta G\\ \Delta B\end{array} \right]=\left[ \begin{array}{cc}1&0&a_{RCr}\\ 1&a_{GCb}&a_{GCr}\\ 1&a_{BCb}&0\end{array} \right]\left[ \begin{array}{c}\Delta Y^{\prime}\\ \Delta C_{b}\\ \Delta C_{r}\end{array}