## 2 Proposed Methodology
### Waveform Imaging Using a Modified Recurrence Plot
However, the yellow area from the proposed RP did not show any significant change as the amplitude of the noise increased; this is depicted in Figure 6d-f. Therefore, it was verified that the proposed RP is more robust to random noise than the original RP. It is worth noting that the yellow area of the proposed RP kept the shape under noisy conditions while the original RP did not. Figure 4: RP images of simulated aperiodic signals: (**a**) original RP of \(f(x)\), (**b**) original RP of \(2f(x)\), (**c**) original RP of \(4f(x)\), (**d**) proposed RP of \(f(x)\), (**e**) proposed RP of \(2f(x)\), and (**f**) proposed RP of \(4f(x)\). Figure 5: Simulated aperiodic signals with random noise. From visual inspection, it was observed that the yellow area from the proposed RP was more sensitive to amplitude change of the waveforms, while the proposed RP was more robust to random noise. To quantitatively verify the relationship, the yellow area ratio was calculated by dividing the number of pixels of the yellow area by the total number of pixels. As shown in Figure 7 and Table 1, the increase of the yellow area's ratio between \(f(x)\) and \(4f(x)\) using the proposed RP was 56.43% (= 56.65%-0.22%); whereas, the increase using the original RP was only 24.95% (= 87.74%-62.79%). These results show that the proposed RP was twice as sensitive to the amplitude change of the simulated waveform. For the original RP with no random noise, \(w(x)\), and \(2w(x)\) random noise, the yellow area increased to 78.39%, 93.95%, and 94.64%, respectively. However, for the proposed RP with no random noise, \(w(x)\), and \(2w(x)\) random noise, the yellow area did not show such an increase; results were 29.54%, 29.70%, and 29.86%, respectively. Therefore, it was concluded that the proposed RP is sensitive to amplitude changes of the simulated waveform, while being robust to random noise. Figure 6: RP images of simulated noisy aperiodic signals: (**a**) original RP of \(2f(x)\), (**b**) original RP of \(2f(x)+\omega(x)\), (**c**) original RP of \(2f(x)+2\omega(x)\), (**d**) proposed RP of \(2f(x)\), (**e**) proposed RP of \(2f(x)+\omega(x)\), and (**f**) proposed RP of \(2f(x)+2\omega(x)\). Figure 7: \(Cont\). The performance of the proposed RP was also evaluated using another type of waveform signal that emulates the evolution of the dynamic resistance signals during resistance spot welding. We followed a procedure similar to that used for analysis of the simulated aperiodic signals. As presented in Figures 8 and 9, it was visually observed that the proposed RP is more sensitive to the amplitude change of waveforms than the original RP. As depicted in Figures 10 and 11, it was visually verified that the proposed RP is more robust to random noise than the original RP. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & \(f(x)\) & \(2f(x)\) & \(4f(x)\) & \(2f(x)+\omega(x)\) & \(2f(x)+2\omega(x)\) \\ \hline Original RP & 62.79\% & 78.39\% & 87.74\% & 93.95\% & 94.64\% \\ Proposed RP & 0.22\% & 29.54\% & 56.65\% & 29.70\% & 29.86\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of yellow area ratio for a simulated aperiodic waveform. Figure 8: Simulated dynamic resistance signals. Figure 7: Performance comparison of the original and proposed RPs using a simulated aperiodic waveform: (**a**) the change of yellow area as the waveform amplitude increases and (**b**) the change of yellow area as the random noise increases. To quantitatively verify the relationship, the yellow area ratio was calculated. As shown in Figure 12 and Table 2, the increase of the yellow area's ratio between \(f_{1}(x)\) and \(f_{3}(x)\) using the proposed RP was 21.75% (= 93.68%-71.93%); whereas, the increase using the original RP was only 45.06% (= 59.23%-14.17%). These results show that the proposed RP was three times more sensitive to the amplitude change of the simulated waveform. For the original RP with no random noise, \(w(x)\), and \(2w(x)\) random noise, the yellow area increased to 86.97%, 87.73%, and 89.51%, respectively.