# A Study on DNN-Based Practical Model for Predicting Spot Color
## Appendix A \(\Delta E_{00}^{*}\) Calculation Formula  
1. Computation of adjusted chroma \(C_{i}^{\prime}\) and hue \(h_{i}^{\prime}\). For each color, \(C_{i}^{\prime}\) is computed as \[C_{i}^{*}=\sqrt{(a_{i}^{*})^{2}+(b_{i}^{*})^{2}}\quad i=1,2\] (1) where \(a_{i}*\) and \(b_{i}*\) are the chromaticity coordinates in the CIE \(L^{*}a^{*}b^{*}\) color space. This step determines the chroma, which is a measure of color saturation or vividness. A factor \(G\) is is computed as \[G=0.5\left(1-\sqrt{\frac{\overline{C}^{\prime}}{\overline{C}^{\prime}}+25^{ \prime}}\right)\] (2) where \(\overline{C}=\frac{C_{1}^{*}+C_{2}^{*}}{2}\) is the average chroma. This adjustment factor \(G\) modifies the chroma values to reflect the viewing conditions. The adjusted chroma \(C_{i}^{\prime}\) and hue \(h_{i}^{\prime}\) are calculated as \[C_{i}^{\prime}=\sqrt{(a_{i}^{\prime})^{2}+(b_{i}^{*})^{2}}\quad i=1,2\] (11) \[h_{i}^{\prime}=\arctan(b_{i}^{*},a_{i}^{\prime})\] (12) where \(a_{i}^{\prime}=a_{i}^{*}(1+G)\). These adjusted values provide a nuanced representation of each color's properties. 2. Determination of color difference components \(\Delta L^{\prime},\Delta C^{\prime},\Delta H^{\prime}\). The difference in lightness \(\Delta L^{\prime}\) is calculated as \[\Delta L^{\prime}=L_{2}^{*}-L_{1}^{*}\] (13) where \(L_{2}^{*}\) and \(L_{1}^{*}\) are the lightness values of the two colors. The difference in chroma \(\Delta C^{\prime}\) is given by \[\Delta C^{\prime}=C_{2}^{\prime}-C_{1}^{\prime}\] (14) The difference in hue \(\Delta H^{\prime}\) is computed with special handling for angles exceeding 180 degrees, ensuring accuracy across the hue spectrum. \[\Delta H^{\prime}= \begin{cases}h_{2}^{\prime}-h_{1}^{\prime}&|h_{1}^{\prime}- h_{2}^{\prime}|\leq 180^{\circ}\\ (h_{2}^{\prime}-h_{1}^{\prime})+360^{\circ}&|h_{1}^{\prime}-h_{2}^{\prime}|>18 0^{\circ},h_{2}^{\prime}\leq h_{1}^{\prime}\\ (h_{2}^{\prime}-h_{1}^{\prime})-360^{\circ}&|h_{1}^{\prime}-h_{2}^{\prime}|>18 0^{\circ},h_{2}^{\prime}>h_{1}^{\prime}\end{cases} \] (15) \[\Delta H^{\prime}=2\sqrt{C_{1}^{\prime}C_{2}^{\prime}}\sin\left( \frac{\Delta H^{\prime}}{2}\right)\] (16) where \(\Delta H^{\prime}\) is a more refined value that incorporates both the hue difference and the chroma values of the colors, providing a measure that is more aligned with human perception of color differences. 3. Final calculation of color difference \(\Delta E_{00}^{*}\). The average values for lightness, chroma, and hue (\(\overline{L}^{\prime},\overline{C}^{\prime}\), and \(\overline{h}^{\prime}\)) are computed. \[\overline{L}^{\prime}=\frac{L_{1}^{*}+L_{2}^{*}}{2},\quad\overline{C}^{\prime }=\frac{C_{1}^{\prime}+C_{2}^{\prime}}{2}\] (17) \[\overline{h}^{\prime}= \begin{cases}\frac{h_{1}^{\prime}+h_{2}^{\prime}}{2 }&|h_{1}^{\prime}-h_{2}^{\prime}|\leq 180^{\circ}\\ \frac{h_{1}^{\prime}+h_{2}^{\prime}+360^{\circ}}{2}&|h_{1}^{\prime}-h_{2}^{ \prime}|>180^{\circ},h_{1}^{\prime}+h_{2}^{\prime}<360^{\circ}\\ \frac{h_{1}^{\prime}+h_{2}^{\prime}-360^{\circ}}{2}&|h_{1}^{\prime}-h_{2}^{ \prime}|>180^{\circ},h_{1}^{\prime}+h_{2}^{\prime}\geq 360^{\circ}\\ h_{1}^{\prime}+h_{2}^{\prime}&C_{1}^{\prime}C_{2}^{\prime}=0\end{cases} \] (18) A rotational function \(T\) is calculated to address the non-linear perception of colors, especially in the blue region. \[T=1-0.17\cos(\overline{h}^{\prime}-30^{\circ})+0.24\cos(2 \overline{h}^{\prime})+0.32\cos(3\overline{h}^{\prime}+6^{\circ})\\ -0.20\cos(4\overline{h}^{\prime}-63^{\circ})\] (19) The rotational term \(R_{T}\) is calculated to account for the interaction between chroma and hue in the blue region.