## 3 Formulation of the Coloration Model
### Calculation of Observables Seen from the Observation Point by the Reverse Ray Tracing Method
#### 3.2.3 Digital Camera Sensitivity Coefficient and Scaling Factor  
The scaling factor 1/B has a dimension of \(\rm[radiance]^{-1}\) in this study, although in [PERSON] et al. [17] this factor was discussed based on the irradiance. Here, usage of the gray standard paper and its Lambertian reflecting property enabled us to determine the sensitivity factor of the digital camera in terms of the radiance, whereas the property could not be taken into consideration in our previous physical model described in terms of irradiance. Here, flux, radiance and irradiance are the radiometric terms, and corresponding photometric terms are luminous flux, luminance and illuminance, respectively. Let the aperture of the digital camera at the time of shooting be \(\rm f/X_{E}\), the ISO sensitivity be \(\rm I_{E}\), and the exposure time be \(\rm T_{E}\). If the linear RGB values obtained from the image are (R, G, B), the output value corresponding to the luminance is expressed by Equation (14) [17; 28]:  
\[\rm Y_{out}=0.2126R+0.7152G+0.0722B, \tag{14}\]  
where the linear RGB values (R, G, B) denoted by \(255t_{0}\) are connected to sRGB values \(255t_{1}\) by Equation (15):  
\[\rm t_{1}=\gamma(t_{0})=\left\{ \begin{array}{rl}&4.50\times t_{0}\quad\rm if \ t_{0}<0.018,\\ &1.099\times t_{0}^{0.45}-0.099\quad\rm otherwise.\end{array} \right. \tag{15}\]  
In order to convert the above output value \(\rm Y_{out}\) to an output \(\rm O_{E}\) of equivalent luminance, we use the following relation:  
\[\rm O_{E}=\frac{Y_{out}}{255}. \tag{16}\]  
Next, the illuminance at a certain position from the light source is assumed to be \(\rm E_{E}\), and this light is assumed to be reflected by the gray standard paper at this position. If this standard paper has a Lambertian surface with the albedo of \(\rm R_{E}\), the luminance \(\rm L_{E}\) of the reflected light is written by:  
\[\rm L_{E}=\frac{R_{E}}{\pi}\rm E_{E}. \tag{17}\]  
Under these conditions, the photograph of the gray standard paper will be taken. On the other hand, for the standard conditions of the digital camera, we assume that the aperture of the digital camera is \(\rm f/X_{S}\), the ISO sensitivity \(\rm I_{S}\), and the exposure time \(\rm T_{S}\). Since the output value of the digital camera should be settled as 118 in terms of sRGB [28], we obtain the standard linear output as \(\rm O_{S}=\gamma^{-1}(118/255)=0.225\) by using inverse transformation of \(\rm t_{0}=\gamma^{-1}(t_{1})\) defined by Equation (15) using the relation \(\rm t_{1}=118/255\). Assuming that the incident light flux from the source is apparently regarded as a parallel light and its illuminance is \(\rm E_{s}\), the standard output value \(\rm L_{S}\) in terms luminance is expressed by \(\rm L_{S}=E_{S}/\pi\). Therefore, the digital camera sensitivity coefficient \(\beta^{\prime}\)1 is expressed by Equation (18) [28]:  
\[ \begin{array}{ll}\beta^{\prime}_{1}&=\frac{\mathrm{O}_{\mathrm{E}}}{\mathrm{O}_ {\mathrm{S}}}\frac{\mathrm{L}_{\mathrm{E}}}{\mathrm{L}_{\mathrm{E}}}\left(\frac{ \mathrm{X}_{\mathrm{E}}}{\mathrm{X}_{\mathrm{S}}}\right)^{2}\cdot\frac{\mathrm{ I}_{\mathrm{E}}}{\mathrm{I}_{\mathrm{S}}}\cdot\frac{\mathrm{T}_{\mathrm{S}}}{ \mathrm{T}_{\mathrm{E}}},\\ &=\frac{\mathrm{O}_{\mathrm{E}}}{\mathrm{O}_{\mathrm{S}}}\frac{\mathrm{E}_{ \mathrm{E}}}{\mathrm{E}_{\mathrm{E}}\mathrm{R}_{\mathrm{E}}}\cdot\left(\frac{ \mathrm{X}_{\mathrm{E}}}{\mathrm{X}_{\mathrm{S}}}\right)^{2}\cdot\frac{\mathrm{ I}_{\mathrm{E}}}{\mathrm{I}_{\mathrm{S}}}\cdot\frac{\mathrm{T}_{\mathrm{S}}}{ \mathrm{T}_{\mathrm{E}}}.\end{array}