## 2 Proposed Method
### Chromatic Adaptation Transform (CAT) Based on CMCCAT2000
\(R_{c}\), \(B_{c}\), and \(G_{c}\) values are the chromatically adapted cone responses obtained from the following equations, especially with the \(D\) cone response applied  
\[R_{c}=R[\alpha(R_{wr}/R_{w})+1-D], \tag{27}\]  
\[G_{c}=G[\alpha(G_{wr}/G_{w})+1-D], \tag{28}\]  
\[B_{c}=B[\alpha(B_{wr}/B_{w})+1-D], \tag{29}\]  
where  
\[\alpha=\frac{DY_{w}}{Y_{wr}}, \tag{30}\]  
where \(R_{w}\), \(B_{w}\), \(G_{w}\) and \(X_{w}\), \(Y_{w}\), \(Z_{w}\) values represent corresponding \(R\), \(G\), \(B\) and \(X\), \(Y\), \(Z\) values for the white point, respectively. \(X_{wr}\), \(Y_{wr}\), and \(Z_{wr}\) values refer to the reference white in reference illumination. From Equation (27) through Equation (29), the corresponding tristimulus value, \(X_{c}\), \(Y_{c}\), and \(Z_{c}\) are calculated as  
\[M^{-1}=\left( \begin{array}{cc}1.076450&-0.237662&0.161212\\ 0.410964&0.554342&0.034694\\ -0.010954&-0.013389&1.024343\end{array} \right), \tag{31}\]  
The forward mode converts the tristimulus values of an example under a non-daylight illumination (say, A) into those of the corresponding colors under a daylight illumination, typically D65. In estimatingcorresponding colors from a daylight to a non-daylight illumination, the reverse mode should be used, and the reverse mode of CMCCAT2000 is described as  
\[\left( \begin{array}{c}R_{w}\\ G_{w}\\ B_{w}\end{array} \right)=M\left( \begin{array}{c}X_{w}\\ Y_{w}\\ Z_{w}\end{array} \right)\left( \begin{array}{c}R_{w\tau}\\ G_{w\tau}\\ B_{w\tau}\end{array} \right)=M\left( \begin{array}{c}X_{w\tau}\\ Y_{w\tau}\\ Z_{w\tau}\end{array} \right),\left( \begin{array}{c}R^{\prime}\\ G^{\prime}\\ B^{\prime}\end{array} \right)=M\left( \begin{array}{c}X_{c}\\ Y_{c}\\ Z_{c}\end{array} \right) \tag{32}\]  
\[D=F[0.08\ log_{10}[0.5(L_{A1}+L_{A2})]+0.76-0.45(L_{A1}-L_{A2})/(L_{A1}+L_{A2})], \tag{33}\]  
where  
\[F=0.2k^{4}(5L_{A})+0.1\big{(}1-k^{4}\big{)}(5L_{A})^{\frac{1}{3}} \tag{34}\]  
where \(L_{A1}\) and \(L_{A2}\) values are the luminance of reference adapting field. \(L_{A}\) is a function of adaptation luminance and accounts for 20% of the adaptation white (\(L_{w}\)). \(D\) represents the incomplete adaptation. From Equation (32), the transformed chromatic adaptation values, \(R^{\prime}\), \(G^{\prime}\), and \(B^{\prime}\) are obtained as  
\[R^{\prime\prime}=R^{\prime}[\alpha(R_{w\tau}/R_{w})+1-D], \tag{35}\]  
\[G^{\prime}=G^{\prime}[\alpha(G_{w\tau}/G_{w})+1-D], \tag{36}\]  
\[B^{\prime\prime}=B^{\prime}[\alpha(B_{w\tau}/B_{w})+1-D], \tag{37}\]  
where  
\[\alpha=\frac{DY_{w}}{Y_{w\tau}}, \tag{38}\]  
where \(R_{w}\), \(G_{w}\), \(B_{w}\) and \(X_{w}\), \(Y_{w}\), \(Z_{w}\) values represent corresponding \(R\), \(G\), \(B\) and \(X\), \(Y\), \(Z\) values for the white point, respectively. \(X_{w\tau}\), \(Y_{w\tau}\), and \(Z_{w\tau}\) values refer to the reference white in reference illumination. From Equations (37)-(39), the resulting image is finally obtained by achieving an inversion process as  
\[\left( \begin{array}{c}X\\ Y\\ Z\end{array} \right)=M^{-1}\left( \begin{array}{c}R^{\prime\prime}\\ G^{\prime\prime}\\ B^{\prime\prime}\end{array} \right), \tag{39}\]  
If any of the values \(R^{\prime\prime}\), \(G^{\prime\prime}\), and \(B^{\prime\prime}\) are negative, then their positive equivalent must be used.