# The Reproduction and Evaluation of Star Fields with the Milky Way in a Planetarium
## 3 Modeling
### Estimation Model for Physical Factors  
The physical factors of transmittance, representation size and star density of the standard pattern were normalized as (\(C_{tra}\), \(C_{size}\), \(C_{density})=(1,1,1)\) to integrate the changes for each factor. According to their ratios to another pattern, the transmittance component was calculated by using the total area of holes on the star plate because one of the standard patterns had a value of 1.0. The representation size and star density components were derived from the changes in the physical values. Figure 6 shows a summary of each component for each projection pattern. For example, in the case of pattern S1-L1, we obtained three components (\(C_{tra}\), \(C_{size}\), \(C_{density}\)) = (0.5, 0.25, 1) because the pattern S1-L1 had half transmittance, a 1/4 area ratio, and the same density as the standard pattern. By using these physical components to represent the characteristics of each projection pattern, we constructed a common model to estimate the psychometric evaluation values for faithfulness, preference, and depth feeling. In our modeling process, we had three hypotheses regarding the evaluation results as follows:  
Figure 5: Evaluation results for all observers for each projection pattern in luminance order. (**a**) Faithfulness; (**b**) Preference; (**c**) Depth feeling. Figure 6: Component of each projection pattern. 1. Psychometric evaluations can be described using a logarithmic scale that conforms to the Weber-Fechner law. 2. Each psychometric evaluation follows a normal distribution in the logarithmic domain because the average rating value for each evaluation is unimodal, as depicted in Figure 5.
3. Humans evaluate faithfulness, preference, and depth feeling using all physical factors comprehensively, and this process can be described as addition in a numerical formula. Most psychophysical experiments using direct scaling followed a lognormal distribution [18], and a multi-layered perceptual model is often expressed by a linear model [19]. Considering these hypotheses, we proposed a common estimation model \(f\) to represent the evaluation for the evaluation index \(\Phi\) by a log-normal distribution for the Milky Way as follows:  
\[f(x^{(\Phi)})=w\frac{1}{\sqrt{2\pi\sigma}x}exp\left(-\frac{(\ln x-\mu)^{2}}{2 \sigma^{2}}\right)-2,\ \Phi\in\{\text{faithfulness, preference, depth feeling}\}, \tag{1}\]  
\[x^{(\Phi)}\Big{(}C_{tra},\ C_{size},\ C_{density}\Big{)}=p_{1}^{(\Phi)}C_{tra} +p_{2}^{(\Phi)}C_{size}+p_{3}^{(\Phi)}C_{density},\quad p_{1}^{(\Phi)}+p_{2}^{ (\Phi)}+p_{3}^{(\Phi)}=1 \tag{2}\]  
where \(\mu\) and \(\sigma\) represent the mean and standard deviation of the natural logarithm of the variable, respectively. \(p_{i}^{(\Phi)}(i=1,2,3)\) represents the weights of (\(C_{tra},\ C_{size},\ C_{density}\)), where \(w\) represents the scaling factor. By least mean-square fitting, as shown in Figure 7, they were calculated as \(\mu=4.02\) and \(\sigma=0.38\), where the fittest weights (\(p_{1}\), \(p_{2}\), \(p_{3}\)) and \(w\) were (0.25, 0.25, 0.50) and 134.15, respectively. Comparison of the differences in weights across the evaluation indices indicates that the change in density was the strongest factor in evaluating the Milky Way. The correlation coefficient for all evaluation indices was 0.92, and the correlation coefficients for each evaluation index were as high as 0.97, 0.97, and 0.86. This result indicates that our hypotheses are appropriate for estimating the evaluation of the Milky Way using the common estimation model in Equations (1) and (2). This model indicates that we can reproduce the Milky Way with an appearance well-balanced in faithfulness, preference, and depth feeling if a planetarium can be made with the estimated appropriate parameters. Figure 7: Fitting result by proposed common estimation model.