# A Fast and Reliable Luma Control Scheme for High-Quality HDR/WCG Video
## 3 Linear Approximation of EOTF
### Limitations of Fast Luma Control  
By inserting (6) into (2) and then combining (5) with (8) for \(\Delta X,X\in\{R,G,B\}\), the luminance perception error can be represented by  
\[E =\left[\sum\limits_{X\in\{R,G,B\}}w_{X}(L(X^{\prime}+\Delta X)-L(X^ {\prime}))\right]^{2} \tag{13}\] \[\qquad=\left[\sum\limits_{X\in\{R,G,B\}}w_{X}(L^{\prime}(X^{ \prime})\Delta X)\right]^{2}\] \[=\left[\sum\limits_{X\in\{R,G,B\}}w_{X}\Big{(}L^{\prime}(X^{ \prime})\Big{(}\hat{Y}^{\prime}-e_{X}\Big{)}\Big{)}\right]^{2}\]  
The luma value, \(\hat{Y}^{\prime}_{F}\), in (7) is the solution minimizing (13) and we can easily identify that this minimum error value equals zero, which was attained by the approximation of the EOTF values \(L(X^{\prime}+\Delta X_{F})\) using (6), with \(\Delta X_{F}\in\{\Delta R_{F},\Delta G_{F},\Delta B_{F}\}\) given in (10). Now, let us denote this approximated quantity by \(\hat{X}_{F}\) and its corresponding nonlinear quantity by \(\hat{X}^{\prime}_{F}\), specifically,  
\[\hat{X}_{F}=L\big{(}X^{\prime}\big{)}+L^{\prime}\big{(}X^{\prime}\big{)}\! \cdot\!\Delta X_{F}\text{ and }\hat{X}^{\prime}_{F}=L^{-1}\big{(}\hat{X}_{F}\big{)}, \tag{14}\]  
where \(L^{-1}(\cdot)\) is the inverse of the EOTF given in (3) and the position of each quantity is depicted in Figure 2. Since the zero minimum, achieved by the \(\Delta X_{F}\), is the lowest possible error of the luminance perception in (2), if we can find a luma value \(\hat{Y}^{\prime}\) producing the quantity \(\hat{X}^{\prime}_{F}\) (i.e., via (5)) for all the color components at the same time, then this value shall be the optimal one and be the same as that of the iterative solution. However, \(\Delta X\) defined in (5) and \(\hat{Y}^{\prime}\) are with equal spacing (which means that if \(\hat{Y}^{\prime}\) is changed by an amount, then \(\Delta X_{F}\) for all the color components are also changed by the same amount at the same time), but the distances from \(X^{\prime}\) to \(\hat{X}^{\prime}_{F}\) for each color component are not guaranteed to be the same, hence, the existence of such \(\hat{Y}^{\prime}\) is not generally possible. Instead, let us now consider the luma value \(\hat{Y}^{\prime}_{X(F)}\) producing the quantity \(\hat{X}^{\prime}_{F}\) for each color component, and the minimum and maximum values among \(\left\{\hat{Y}^{\prime}_{X(F)}\right\}_{X=R,G,B}\). From \(\Delta X=\hat{Y}^{\prime}-e_{X},X\in\{R,G,B\}\), we can get such \(\hat{Y}^{\prime}_{X(F)}\) as  
\[\hat{X}^{\prime}_{F}-X^{\prime}=\hat{Y}^{\prime}_{X(F)}-e_{X}\text{ or }\hat{Y}^{\prime}_{X(F)}=\hat{X}^{\prime}_{F}-X^{\prime}+e_{X},X\in\{R,G,B\}, \tag{15}\]  
and  
\[\hat{Y}^{\prime}_{min}=\min\Big{\{}\hat{Y}^{\prime}_{X(F)}\Big{\}},\hat{Y}^{ \prime}_{max}=\max\Big{\{}\hat{Y}^{\prime}_{X(F)}\Big{\}},X\in\{R,G,B\} \tag{16}\]  
Figure 2: A linear approximation of the electro-optical transfer function (EOTF) and the quantities in (14). Then, if we further consider a luma value, \(\hat{Y}^{\prime}_{a}\), which is larger than the above \(\hat{Y}^{\prime}_{max}\) (i.e., \(\hat{Y}^{\prime}_{a}\geq\hat{Y}^{\prime}_{max}\)), the reconstructed RGB signal, \(\hat{X}^{\prime}_{a}\), via (5) can be represented by  
\[\hat{X}^{\prime}_{a}=\hat{X}^{\prime}_{max}+\delta_{X(a)}=\hat{X}^{\prime}_{F} +\delta_{X(max)}+\delta_{X(a)},X\in\{R,G,B\}, \tag{17}\]  
where \(\hat{X}^{\prime}_{max}\) denotes the reconstructed RGB values from the corresponding \(\hat{Y}^{\prime}_{max}\), and \(\delta_{X(max)},\delta_{X(a)}\geq 0\) for all \(X\), because \(\Delta X=\hat{Y}^{\prime}-e_{X},X\in\{R,G,B\}\) and \(\hat{Y}^{\prime}_{a}\geq\hat{Y}^{\prime}_{max}\). Hence, the luminance error introduced by this luma value can be represented by  
\[