# A Fast and Reliable Luma Control Scheme for High-Quality HDR/WCG Video
## 2 Luma Control Problem
\right], \tag{5}\]  
where \((a_{RCr},a_{GCb},a_{GCr},a_{BCb})\) means the contribution of a chroma component for each color and is given by (1.5748, \(-0.1873\), \(-0.4681\), 1.8556) for BT.709 and (1.4746, \(-0.1646\), \(-0.5714\), 1.8814) for BT.2020. This shows the adjusted luma value, \(\hat{Y}^{\prime}\), controls the nonlinear domain reconstruction, \(\big{(}\hat{R}^{\prime},\hat{G}^{\prime},\hat{B}^{\prime}\big{)}\), and thus determines the luminance error given in (2). Hence, the luma control problem is to find the best luma value that produces the minimum luminance error of (2). Figure 1: Conventional preprocessing stages for high dynamic range and wide color gamut (HDR/WCG) video in [12]. The iterative luma control scheme in [8] searches for the best compensation using the bisection method with their proposed bounds on the optimal luma value. The iterative nature of the scheme comes from the nonlinearity of \(L\big{(}\hat{X}^{\prime}\big{)},\hat{X}^{\prime}\in\big{\{}\hat{R}^{\prime},\hat {G}^{\prime},\hat{B}^{\prime}\big{\}}\) in (2), resulting in its repeated and complex computations for each candidate luma value. To get rid of the iterative nature of the luma control scheme, [9] proposed an approximation of \(L\big{(}\hat{X}^{\prime}\big{)}\) based on a truncated Taylor series, such that  
\[L\big{(}\hat{X}^{\prime}\big{)}=L\big{(}X^{\prime}+\Delta X\big{)}=L\big{(}X^{ \prime}\big{)}+L^{\prime}\big{(}X^{\prime}\big{)}\cdot\Delta X,X^{\prime}\in \big{\{}R^{\prime},G^{\prime},B^{\prime}\big{\}}, \tag{6}\]  
where \(R^{\prime},G^{\prime},B^{\prime}\), are the original pixel values in nonlinear RGB color space and \(L^{\prime}(\cdot)\) denotes the derivative of the EOTF, \(L(\cdot)\). This approximation, after combined with (5) and (2), provides an optimal luma value, \(\hat{Y}^{\prime}_{F}\), as a closed form solution of  
\[\hat{Y}^{\prime}_{F}\!\!=\frac{w_{R}\!\cdot\!L^{\prime}(R^{\prime})\!\cdot\!e_ {R}+w_{G}\!\cdot\!L^{\prime}(G^{\prime})\!\cdot\!e_{G}+w_{B}\!\cdot\!L^{\prime} (B^{\prime})\!\cdot\!e_{B}}{w_{R}\!\cdot\!L^{\prime}(R^{\prime})+w_{G}\!\cdot \!L^{\prime}(G^{\prime})+w_{B}\!\cdot\!L^{\prime}(B^{\prime})}, \tag{7}\]  
where  
\[ \begin{array}{c}e_{R}=Y^{\prime}-a_{RC}\Delta C_{r}\\ e_{G}=Y^{\prime}-a_{GCb}\Delta C_{b}-a_{GCr}\Delta C_{r}\\ e_{B}=Y^{\prime}-a_{BCb}\Delta C_{b}\end{array}  \tag{8}\]  
This fast scheme is very simple and no longer iterative but shows limited performance for some videos having highly saturated colors. As a reason for this performance limitation, [10] pointed out that the approximation of (6) can be severely limited when \(L(X^{\prime})\) has a high curvature at the point \(X^{\prime}\) and when \(\Delta X\) is not small. To fill the performance gap between the above two luma control schemes, a modified linear approximation was proposed in [10], such that  
\[L\big{(}\hat{X}^{\prime}\big{)}=L\big{(}X^{\prime}+\Delta X\big{)}=\big{[}L \big{(}X^{\prime}\big{)}+f_{X}\big{(}X^{\prime},\Delta X_{F}\big{)}\big{]}+s \big{(}X^{\prime},\Delta X_{F}\big{)}\!\cdot\!\Delta X,\ \ X^{\prime}\in\big{\{}R^{\prime},G^{ \prime},B^{\prime}\big{\}}, \tag{9}\]  
where \(\Delta X_{F}\) is the resulting value from (5) with (7), specifically,  
\[\left[ \begin{array}{c}\Delta R_{F}\\ \Delta G_{F}\\ \Delta B_{F}\end{array} \right]=\left[ \begin{array}{ccc}1&0&a_{RCr}\\ 1&a_{GCb}&a_{GCr}\\ 1&a_{BCb}&0\end{array} \right]\left[ \begin{array}{c}\hat{Y}^{\prime}_{F}-Y^{ \prime}\\ \hat{C}_{b}-C_{b}\\ \hat{C}_{r}-C_{r}\end{array}