# Nayatani Color Model

The Nayatani color apperance model is described in the paper Lightness dependency of Chroma scales of a nonlinear color-appearance model and its latest formulation by Y. Nayatani, H. Sobagaki, K. Hashimoto, and T. Yano, published by Color Res. Appl. Vol 20 pages 156-167 (1995), and is primarily geared towards modeling illumination (non-imaging) systems. Equations used are as follows:

 Adapting Luminance $$L_o = \frac{Y_o E_o}{100 \pi}$$ Normalizing Luminance $$L_o = \frac{Y_o E_{or}}{100 \pi}$$ $\xi$ $$\xi = \frac{0.48105 x_o + 0.78841 y_o - 0.08081}{y_o}$$ $\eta$ $$\eta = \frac{-0.272 x_o + 1.11962 y_o + 0.0457}{y_o}$$ $\zeta$ $$\zeta = \frac{0.91822 \left ( 1 - x_o - y_o \right )}{y_o}$$ Adapted cone response $$\begin{vmatrix} R_o\\ G_o\\ B_o \end{vmatrix} = \frac{Y_o E_o}{100 \pi} \begin{vmatrix} \xi\\ \eta\\ \zeta \end{vmatrix}$$ $\beta_1$ $$\beta_1 \left ( x \right ) = \frac{6.469 + 6.362 x^{0.4495}}{6.469 + x^{0.4495}}$$ $\beta_2$ $$\beta_2 \left ( x \right ) = \frac{8.414 + 8.091 x^{0.5128}}{8.414 + x^{0.5128}}$$ Cone Response $$\begin{vmatrix} R\\ G\\ B \end{vmatrix} = \begin{vmatrix} 0.40024 & 0.7076 & -0.08081\\ -0.2263 & 1.16532 & 0.0457\\ 0 & 0 & 0.91822 \end{vmatrix} \begin{vmatrix} X\\ Y\\ Z \end{vmatrix}$$ Scaling coefficient $e(R)$ $$e(R)= \left\{\begin{matrix} 1.758 & R \geq 20 \xi \\ 1 & R < 20 \xi \end{matrix}\right.$$ Scaling coefficient $e(G)$ $$e(G)= \left\{\begin{matrix} 1.758 & G \geq 20 \eta \\ 1 & G < 20 \eta \end{matrix}\right.$$ Achromatic Response $$\begin{eqnarray} Q = \frac{41.69}{\beta_1(L_{or})}\left ( \frac{2}{3} \beta_1(R_o) e(R) \log_{10} \frac{R+n}{20 \xi + n} + \\ \frac{1}{3} \beta_1(G_o) e(G) \log_{10} \frac{G+n}{20 \eta + n}\right ) \end{eqnarray}$$ Red-Green Chromatic Response $$\begin{eqnarray} t = \beta_1(R_o) \log_{10} \frac{R+n}{20 \xi + n} - \frac{12}{11}\beta_1(G_o) \log_{10}\frac{G+n}{20 \eta + n} + \\ \frac{1}{11} \beta_2(B_o) \log_{10}\frac{B+n}{20 \zeta + n} \end{eqnarray}$$ Yellow-Blue Chromatic Response $$\begin{eqnarray} p = \frac{1}{9}\beta_1(R_o) \log_{10} \frac{R+n}{20 \xi + n} + \frac{1}{9}\beta_1(G_o) \log_{10}\frac{G+n}{20 \eta + n} - \\ \frac{2}{9} \beta_2(B_o) \log_{10}\frac{B+n}{20 \zeta + n} \end{eqnarray}$$ Hue Angle $$\theta = \tan^{-1} \left ( \frac{p}{t} \right )$$ Brightness $$B_r = Q + \frac{50}{\beta_1(L_{or})} \left ( \frac{2}{3} \beta_1(R_o) + \frac{1}{3} \beta_1(G_o)\right )$$ Reference White Brightness $$\begin{eqnarray} B_{rw} = \frac{41.69}{\beta_1(L_{or})} \left ( \frac{2}{3} \beta_1(R_o) 1.758 \log_{10} \frac{100 \xi + n}{20 \xi + n} + \\ \frac{1}{3} \beta_1(G_o) 1.758 \log_{10} \frac{100 \eta + n}{20 \eta + n}\right ) + \frac{50}{\beta_1(L_{or})} \left ( \frac{2}{3} \beta_1(R_o) + \\ \frac{1}{3} \beta_1(G_o)\right ) \end{eqnarray}$$ Achromatic Lightness $$L_p^* = Q + 50$$ Normalized Achromatic Lightness $$L_n^* = \frac{100 B_r}{B_{rw}}$$ Red-Green Saturation $$S_{RG} = \frac{488.93 E_s(\theta) t }{\beta_1(L_{or})}$$ Yellow-Blue Saturation $$S_{YB} = \frac{488.93 E_s(\theta) p }{\beta_1(L_{or})}$$ Total Saturation $$S = \sqrt{S_{RG}^2 + S_{YB}^2}$$ Red-Green Chroma $$C_{RG} = \left ( \frac{L_p^*}{50} \right )^{0.7} S_{RG}$$ Yellow-Blue Chroma $$C_{YB} = \left ( \frac{L_p^*}{50} \right )^{0.7} S_{YB}$$ Total Chroma $$C = \left ( \frac{L_p^*}{50} \right )^{0.7} S$$ Red-Green Colorfulness $$M_{RG} = \frac{C_{RG} B_{rw}}{100}$$ Yellow-Blue Colorfulness $$M_{YB} = \frac{C_{YB} B_{rw}}{100}$$ Total Colorfulness $$M = \frac{C B_{rw}}{100}$$

## Results

 $B_r$ Brightness $L_p$ Lightness $L_n$ Normalized lightness $\theta$ Hue Angle $S$ Saturation $C$ Chroma $M$ Colorfulness