## 3 DIC Algorithm Configuration for Nanofiber Mat and Application Test
### Image Correlation Method by using Fast Fourier Transform (FFT)  
The conventional image correlation method uses a reference subset image as a kernel. Convolutions are operated from initial guess area for reference subset image coordinates based on image change. The strain is calculated by the coordinate change detection, which is the highest coordinate for correlation. Every coordinate in reference and all deformed images must be compared and calculated  
\begin{table}
\begin{tabular}{c c} \hline
**CC Correlation Criterion** & **Definition** \\ \hline Cross-correlation (CC) & \(C_{CC}(x_{i},y_{j},x_{i}^{\prime},y_{j}^{\prime})=\sum\limits_{i=-M}^{M}{\sum \limits_{j=-M}^{M}{\left[{f(x_{i},y_{j})g(x_{i}^{\prime},y_{j}^{\prime})} \right]}}\) \\ Normalized cross-correlation (NCC) & \(C_{NCC}(x_{i},y_{j},x_{i}^{\prime},y_{j}^{\prime})=\sum\limits_{i=-M}^{M}{ \sum\limits_{j=-M}^{M}{\left[{f(x_{i},y_{j})g(x_{i}^{\prime},y_{j}^{\prime})} \right]}}\) \\ Zero-normalized cross-correlation (ZNCC) & \(C_{ZNCC}(x_{i},y_{j},x_{i}^{\prime},y_{j}^{\prime})=\sum\limits_{i=-M}^{M}{ \sum\limits_{j=-M}^{M}{\frac{{\left[{f(x_{i},y_{j})-f_{m}}\right]}\left[{g(x_{ i}^{\prime},y_{j}^{\prime})-g_{m}}\right]}{{\Delta f\Delta g}}}}\) \\ \hline \end{tabular}
\end{table}
Table 1: Commonly used cross-correlation (CC) criteria and their definitions (M: natural number).  
\begin{table}
\begin{tabular}{c c} \hline
**SSD Correlation Criterion** & **Definition** \\ \hline Sum of squared differences (SSD) & \(C_{SSD}(x_{i},y_{j},x_{i}^{\prime},y_{j}^{\prime})=\sum\limits_{i=-M}^{M}{ \sum\limits_{j=-M}^{M}{\left[{f(x_{i},y_{j})-g(x_{i}^{\prime},y_{j}^{\prime})} \right]}^{2}}\) \\ Normalized sum of squared differences (NSSD) & \(C_{NSSD}(x_{i},y_{j},x_{i}^{\prime},y_{j}^{\prime})=\sum\limits_{i=-M}^{M}{ \sum\limits_{j=-M}^{M}{\left[{\frac{{f(x_{i},y_{j})-f_{m}}}{{\overline{f}}}- \frac{{g(x_{i}^{\prime},y_{j}^{\prime})-g_{m}}}{{\overline{g}}}}\right]}^{2}}\) \\ Zero-normalized sum of squared differences (ZNSSD) & \(C_{ZNSSD}(x_{i},y_{j},x_{i}^{\prime},y_{j}^{\prime})=\sum\limits_{i=-M}^{M}{ \sum\limits_{j=-M}^{M}{\frac{{f(x_{i},y_{j})-f_{m}}}{{\Delta f}}-\frac{{g(x_{i} ^{\prime},y_{j}^{\prime})-g_{m}}}{{\Delta g}}}}\) \\ \hline \end{tabular}
\end{table}
Table 2: Commonly used sum of squared differences correlation (SSD) criteria and their definitions.  
to obtain accurate displacement positions, so this process requires a lot of iterations. Thus, this conventional DIC method needs quite a long time to measure. However, the FFT-based DIC method compares the correlation from reference subset images to deformed subset images along the frequency range. That means that the initial guess does not need to be determined, and it also does not need iterations. It is possible to analyze DIC even faster than the conventional method. Figure 2 is the matrix calculation diagram to obtain C\({}_{\text{ZNCC}}\) correlation criteria by the FFT method.  
\[C_{\text{ZNCC}}=\frac{\sum_{i}\overline{R_{i}T_{i}}}{\sum_{i}(\overline{R_{i}} )^{2}\sum_{i}(\overline{T_{i}})^{2}} \tag{7}\]  
where \(\overline{R_{i}}=R_{i}-R_{m}\) and \(\overline{T_{i}}=T_{i}-T_{m}\), with \(R_{m}\) and \(T_{m}\) as the mean values of the intensity in the two subsets--i.e., \(R_{m}=1/N\sum_{i}R_{i}\) and \(T_{m}=1/N\sum_{i}T_{i}\). Notice that the superscript \({}^{*}\) has been omitted here, since \(R\) contains only real numbers and thus, \(R*(x)=R(x)\). \(R\otimes T=FT^{-1}\big{\{}[FT(R)]^{*}\cdot FT(T)\big{\}}\), where \(FT(\cdot)\) and \(FT^{-1}(\cdot)\) denote the Fourier transform and inverse Fourier transform, respectively.