## 5 Case Studies
### Study Setup
Considering the running time and recognition performance of the experimental confirmation, some RCMIs of proper orders are selected to construct the feature vector. Tables 1 and 2 represent the corresponding average values of region and contour moment invariants of different classes respectively. Figure 4: Results of pre-processing: (**a–d**) original images; (**e–h**) pre-processed images; (**i–l**) edge images. Coefficient of variation (CV) is defined as the ratio of the standard deviation to the absolute value of mean, which is used to measure the difference between different classes. The larger the value, the easier recognition is to achieve. For different features, it is easy to find that the CV values of region moment invariants are different from those of contour moment invariants. In the case of some features, the CV values of region moment invariants are relatively small, whereas the corresponding values of contour moment invariants are quite large, forming a mutually complementary relationship. Based on the CV values, the multi-feature vector \(\mathbf{F}_{i}\) used in this paper is expressed as follows:  
\[\mathbf{F}_{i}=\left[\widetilde{C}_{13}^{\prime\prime},\ \widetilde{C}_{21}^{ \prime\prime},\ \widetilde{C}_{23}^{\prime},\ \widetilde{C}_{25}^{\prime},\ \widetilde{C}_{26}^{ \prime},\ \widetilde{C}_{32}^{\prime},\ \widetilde{C}_{44}^{\prime},\ \widetilde{C}_{55}^{ \prime}\right] \tag{31}\]  
In the optimization phase, the initialization of IFA was set as follows: the number of fireflies \(K=10\), initial and final randomization parameters \(\alpha_{0}=0.5\) and \(\alpha_{\infty}=10^{-4}\), maximum iterations \(t_{\text{max}}=50\), attractiveness \(\beta_{0}=1\), basic attractiveness \(\beta_{b}=0.2\) and light absorption coefficient \(\gamma=1\). The WSVM parameters of \(C\) and \(\tau\) for the wavelet kernel function were varied in the fixed range of \([0.01,\ 100]\). The above initialization parameters are given in Table 3. All the case studies were developed under the MATLAB 2016b programming environment, and performed on a personal computer with Intel(R) Core(TM) i7 CPU 4 GHz, 16G memory, whose operation system is Microsoft Windows 7 Ultimate. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \(\widetilde{C}_{\mathbf{13}}^{{}^{\prime}}\) & \(\widetilde{C}_{\mathbf{21}}^{{}^{\prime}}\) & \(\widetilde{C}_{\mathbf{23}}^{{}^{\prime}}\) & \(\widetilde{C}_{\mathbf{25}}^{{}^{\prime}}\) & \(\widetilde{C}_{\mathbf{26}}^{{}^{\prime}}\) & \(\widetilde{C}_{\mathbf{32}}^{{}^{\prime}}\) & \(\widetilde{C}_{\mathbf{44}}^{{}^{\prime}}\) & \(\widetilde{C}_{\mathbf{55}}^{{}^{\prime}}\) \\ \hline Stable burning state & 0.05764 & 0.1434 & 0.06644 & 0.07413 & 0.05933 & 0.1205 & 0.07037 & 0.05952 \\ Extinguishing state & 0.09043 & 0.1052 & 0.07653 & 0.06037 & 0.04384 & 0.1627 & 0.09681 & 0.04934 \\ Final extinguished state & 0.1105 & 0.1751 & 0.05140 & 0.04476 & 0.03403 & 0.06471 & 0.05474 & 0.1346 \\ Non-flame state & 0.1022 & 0.1882 & 0.2518 & 0.1948 & 0.2879 & 0.3698 & 0.2118 & 0.2461 \\ Coefficient of variation & 25.73\% & 24.18\% & 84.35\% & 73.34\% & 114.36\% & 74.18\% & 65.54\% & 74.19\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Average values of the region moment invariants. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \(\widetilde{C}_{\mathbf{13}}^{{}^{\prime}}\) & \(\widetilde{C}_{\mathbf{21}}^{{}^{\prime}}\) & \(\widetilde{C}_{\mathbf{23}}^{{}^{\prime}}\) & \(\widetilde{C}_{\mathbf{25}}^{{}^{\prime}}\) & \(\widetilde{C}_{\mathbf{26}}^{{}^{\prime}}\) & \(\widetilde{C}_{\mathbf{32}}^{{}^{\prime}}\) & \(\widetilde{C}_{\mathbf{44}}^{{}^{\prime}}\) &