## 4 Experiments and Results
### Performance Evaluation
#### 4.2.4 Sensitivity to Parameters' Variation  
To demonstrate the robustness, as well as provide an integrated understanding, we analyzed our model's sensitivity to its three primary parameters, including contraction distance \(d_{in}\), expansion distance \(d_{out}\), and the exponent of the second term in cost function, \(Exp_{12}\). The result is displayed in  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**True Label**} & \multicolumn{8}{c}{**Recognized Labels (\%)**} & \multirow{2}{*}{**Time(s)**} \\ \cline{2-2} \cline{4-11}  & **0** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **Others** \\ \hline
**0** & **100** & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.016 \\
**1** & 0 & **100** & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.027 \\
**2** & 0 & 0 & **99.5** & 0 & 0 & 0 & 0 & 0 & 0 & **0.5** & 0.033 \\
**3** & 0 & 0 & 0 & **97.2** & 0 & 0 & **1.3** & 0 & 0 & 0 & **1.5** & 0.038 \\
**4** & 0 & 0 & 0 & 0 & **100** & 0 & 0 & 0 & 0 & 0 & 0.043 \\
**5** & 0 & 0 & 0 & 0 & **100** & 0 & 0 & 0 & 0 & 0 & 0.048 \\
**6** & 0 & 0 & 0 & **0.9** & 0 & **99.1** & 0 & 0 & 0 & 0 & 0.038 \\
**7** & 0 & 0 & 0 & 0 & 0 & 0.9 & **99.0** & 0 & 0 & **0.1** & 0.038 \\
**8** & 0 & 0 & 0 & 0 & **0.2** & 0 & 0 & **99.0** & 0 & **0.8** & 0.037 \\
**9** & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & **100** & 0 & 0.038 \\ \hline \multicolumn{11}{c}{Mean accuracy and time cost:} & \multicolumn{1}{c}{99.38} & \multicolumn{1}{c}{0.035} \\ \hline \hline \end{tabular}
\end{table}
Table 1: Confusion matrix and the time cost of the recognition. Numbers on the diagonal indicate the recognition accuracy of each gesture type.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(L_{1}\) & \(L_{2}\) & \(L_{3}\) & \(L_{4}\) & **Accuracy** \\ \hline \(\surd\) & x & x & x & 85.31\% \\ \(\surd\) & \(\surd\) & x & x & 96.87\% \\ \(\surd\) & x & \(\surd\) & x & 89.53\% \\ \(\surd\) & x & x & \(\surd\) & 88.09\% \\ \(\surd\) & \(\surd\) & \(\surd\) & x & 96.93\% \\ \(\surd\) & \(\surd\) & x & \(\surd\) & 99.35\% \\ \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) & 99.38\% \\ \hline \hline \end{tabular}
\end{table}
Table 2: Evaluation of the effectiveness of the four terms in the cost function. The “\(\surd\)” means taking the corresponding term into account, while “\(\times\)” means not.  
Figure 9, where we can see that the mean accuracy is quite stable when the parameters change. A detailed analysis is given below.  
The contraction distance \(d_{in}\) is used to eliminate fingers, meanwhile reserving the shape of the palm. It serves as the key in our two-phase DT method. As seen in Figure 9a, when it changes rapidly from \(0.3R_{0}\) to \(0.6R_{0}\), the accuracy remains almost the same--above 97%. This is a great advantage of our algorithm, since the methods based on drawing a circle depends on a carefully selected circle radius. Another parameter in our model is the expansion distance \(d_{out}\), which is used to exclude non-finger regions that are outside the palm with a too short length. With a proper \(d_{in}\) of \(0.4R_{0}\) or \(0.5R_{0}\), the varying of \(d_{out}\) from \(1/9\) to \(1/5\) of \(R_{0}\) does not affect the result much, as the accuracy always stays above 98%.  
Figure 9b shows the sensitivity of our template matching model. There are four terms in the cost function. Since the third and fourth are large penalties, we only varied the relative proportion of \(L_{1}\) and \(L_{2}\). We kept the exponent of \(L_{1}\) as 1, and then varied \(Exp_{L2}\) from 0.5 to 3. The resultant accuracies are all above 97%, and reach the peak when \(Exp_{L2}\) equals 2. It again demonstrates the effectiveness of our model, and also indicate that angle gap between two fingers serves as an important feature for this angle-based matching method.