# Comparison of Deep Transfer Learning Models for the Quantification of Photoelastic Images
## 2 Materials and Method  
### Theoretical Background  
Photoelastic materials exhibit the characteristic of changing from isotropic properties to anisotropic properties when subjected to external forces. When light passes through photoelastic materials, the refractive index changes, causing birefringence, or double refraction, even if the incident light has the same wavelength. This phenomenon is known as birefringence, or double refraction. Due to birefringence, the split beams of light have different propagation velocities, resulting in a relative time difference, often referred to as relative retardation. Relative retardation is expressed by the following equation:  
\[\delta=2\pi f\bigg{(}\frac{t}{v_{1}}-\frac{t}{v_{2}}\bigg{)}=2\pi h\frac{c}{ \lambda}\bigg{(}\frac{1}{v_{1}}-\frac{1}{v_{2}}\bigg{)}=\frac{2\pi h}{\lambda} (n_{1}-n_{2}) \tag{1}\]  
where:  
\(f=\) the frequency of incident light.  
\(t=\) the thickness of the photoelastic material.  
\(v_{1}\), \(v_{2}\) = the velocities indices of refracted light waves.  
\(c=\) the velocity of light.  
\(\lambda=\) the wavelength of incident light.  
\(n_{1}\), \(n_{2}=\) the refractive indices of two light waves.  
[PERSON] [43] established the following definition for the relationship between refractive index and stress:  
\[(n_{1}-n_{2})=C(\sigma_{1}-\sigma_{2}) \tag{2}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Abbreviations** & **Definition** \\ \hline RGB & Red, green, and blue \\ R2 & Coefficient of determination \\ MSE & Mean square error \\ MAE & Mean absolute error \\ RMSE & Root mean square error \\ \hline \hline \end{tabular}
\end{table}
Table 1: List of abbreviations used in the paper.  
where:  
\(\sigma_{1}-\sigma_{2}\) = principal stress difference.  
\(C\) = the relative stress-optic coefficient.  
The two equations above can be summarized as follows:  
\[\delta=\frac{2\pi h}{\lambda}C(\sigma_{1}-\sigma_{2}) \tag{3}\]  
On the other hand, light intensity, \(I\), can be calculated using the following formula:  
\[I=I_{a}sin^{2}\frac{\delta}{2} \tag{4}\]  
where:  
\(I_{a}\) = the brightness of incident light.  
The light intensity, \(I\), varies in a pattern of brightness and darkness that repeats according to the sine function. \"Fringes\" refers to the arrangement of the stripes that appear during this repetition, and the \"fringe order\" defines the number of fringes. For the light intensity, \(I\), to become 0, the relative delay, \(\delta\), must be a multiple of half wavelength. The fringe order, \(N\), expresses this relationship in the following manner:  
\[I=0;\ \ \delta=2N\pi(N=0,1,2\cdots) \tag{5}\]  
Ultimately, when Equation (5) is substituted into the previously mentioned Equation (3), we can summarize the relationship between the principal stress difference and the fringe order as follows:  
\[\sigma_{1}-\sigma_{2}=\frac{N\lambda}{hC} \tag{6}\]  
In other words, to calculate the changes in stress within photoelastic materials, it is necessary to evaluate the fringe order. Previously, observers quantified the photoelastic response by directly calculating the fringe order through visual inspection of the number of fringes. However, with the advancement of digital image analysis technology, it has become possible to acquire data at the pixel level of photoelastic images and quantify them. The following section introduces [PERSON]'s method and [PERSON] and [PERSON]'s method.