## 3 Proposed Method
### Controlling the Overlay to Achieve the Target Image  
In this subsection, we explain how to compute \(\mathbf{I}_{L}\) and \(\mathbf{I}_{H}\) for a given \(\mathbf{I}_{S}\) and \(\mathbf{I}_{T}\) using Equation (2). To determine whether light subtraction was necessary and how much light should be subtracted, we calculated the differences between \(\Delta I_{T}\) and \(\mathbf{I}_{T}\), the target image computed with the Daltonization process of Equation (1) and \(\mathbf{I}_{U}\), the user-perspective image captured with the OST-HMD turned off and the LCD set to \((255,255,255)\). If \(\Delta I_{T}=\mathbf{I}_{T}-\mathbf{I}_{U}\) for some pixels is below 0, it means that light subtraction should be performed for those pixels. Because it is difficult to achieve the focus alignment between the LCD, the OST-HMD, and the monitor. we only uniformly controlled the transmittance of the LCD and then achieved the target colors by compensating individual color channels for each pixel using the OST-HMD. For the computation strategy of the transmittance, we propose two principles, the lowest transmittance principle (Principle I), and the histogram peak principle (Principle II). For both Principles I and II, each channel of all the pixels will be judged regarding the need for light subtraction. Consider the set of pixels \(P_{negative}\); arbitrary pixel \(p_{i}\) has at least one negative channel value in \(\Delta I_{Ti}^{2}\), so it requires light subtraction. Then, the light subtraction rate (LSR) and the corresponding transmittance \(t\) of the channel are calculated as  
\[LSR(C,p_{i})=\frac{\Delta C_{i}}{C_{Ui}}\]  
\[t(C,p_{i})=1-LSR(C,p_{i}).\]  
where \(C\in\{R,G,B\}\) and \(\Delta C_{i}<0\). In Principle I, the lowest transmittance of all pixels is set as the transmittance of the LCD panel; channels of pixels that are over subtracted by the transmittance \(t\) are supposed to be compensated for by the OST-HMD. In Principle II, the pixel number of each channel  
Figure 9: Response curves of the user perspective camera to the OST-HMD rendered image. and transmittance is counted as the height of the bin in the two-dimensional histogram \(H\). Each bin of the histogram, \(H(C,k)\), is combined by a chromaticity channel \(c\) with \((R,G,B)\) and \(LSR(k)\), which is in hundredths. \[H(C,k)=\sum_{p_{i}\in\{R,G,B\}}\left\{ \begin{array}{l}1,LSR(C,p_{i})=k\\ 0,otherwise\end{array} \right..\]  
Then, we sum up the histogram \(H\) by chromaticity channel and get a one-dimensional histogram \(\hat{H}\),  
\[\hat{H}(k)=\sum_{C\in\{R,G,B\}}H(C,k)\.\]  
Finally, the \(k\) corresponding to the maximal number of pixels of \(\hat{H}\) is set as the target LSR, and \(f_{subtraction}\) is set as  
\[f_{subtraction}(I_{L})=1-\underset{k}{argmax}\hat{H}(k).\]  
In theory, Principle I should be more correct. However, we adopted Principle II because of the limitations of the OST-HMD, which will be elaborated on in the discussion section. \(f_{subtraction}(I_{L})\) is computed as  
\[f_{subtraction}(I_{L})=1-\underset{c,i}{max}(\frac{\left|\Delta I_{T_{i}}^{ c}\right|}{I_{U_{i}}^{c}}),\Delta I_{T_{i}}^{c}<0,\]  
where \(t=\frac{\left|\Delta I_{T_{i}}^{c}\right|}{I_{U_{i}}^{c}}\) stands for the subtraction required by the \(c\) channel \((c=R,G,B)\) of pixel \(p_{i}\). Then, the controlling image \(I_{L}\) of the LCD is computed by the method described in Section 3.3.1, and a new user-perspective image \(I_{UI}^{\prime}\) is estimated as  
\[I_{UI}^{\prime}=f_{subtraction}(I_{L})\cdot I_{U}.\]  
If \(\Delta I_{T}\) for all pixels is greater than or equal to 0, \(I_{L}\) is set to \((255,255,255)\), no subtraction is required, and \(I_{H}\) can be calculated directly following the difference calculation. For computing \(I_{H}\), the difference between \(I_{T}\) and the estimated user-perspective image \(I_{UI}^{\prime}\) denoted as \(\Delta I_{T}^{\prime}\), is used to compute how much each channel of each pixel should be compensated for by the OST-HMD.