## 5 Damage Detection in Wavefield Images Based on NCA-RPCA Algorithm
### Experimental Verification of Wavefield
#### 5.3.2 Experimental Results and Analysis  
This utilizes the full wavefield image data measured by PSV-400-3D, which is the key to the in-depth study of the real wavefield environment data. The wavefield image data in damage and non-defective cases are extracted respectively, as shown in Figure 32a,b, and processed by different algorithms for comparative analyses to reveal the advantages and disadvantages of the algorithms in dealing with real data. Through the decomposition process of the non-destructive image, it can be seen that the sparse matrix has only noise and no obvious damage points, as shown in Figure 33. From the results in Figures 34 and 35, it can be seen that compared with the processing results under lossless conditions, under lossy conditions, the sparse images decomposed by the rest of the algorithms show obvious damage points, except for the FPCP algorithm (Figure 34c), which still fails to efficiently separate out the useful information from the outliers in this processing, and all of them can accurately decompose the low-rank and sparse images. However, it can be seen in Figure 34d that the RPCA-GD algorithm, although successful in separating useful information and damaged information, is not very effective for the experimental data extracted under complex conditions. Figure 31: Experiment setup for structural plates and laser Doppler vibrometer. Figure 32: Full-wavefield image with experimental data. (**a**) non-defective conditions (**b**) destructive conditions. To compare the advantages and disadvantages of the algorithm effects in depth, this study further evaluates them comprehensively from several aspects, including the rank of the low-rank matrix, the decomposition error, the number of iterations, and the processing time, etc., which are interrelated with each other, and together constitute a comprehensive framework for the evaluation of the algorithms' performances. According to the results in Table 6, for the experimental real data, the algorithm designed in Section 4 firstly separates the low-rank matrices with a rank of seven, which possesses a much lower rank and is about three times lower than the traditional method, indicating that it can restore the original data information at a much lower rank level. Secondly, the decomposition error is also significantly reduced by an order of magnitude, and the number of iterations and processing time are also drastically reduced, which greatly improves the real-time monitoring efficiency of SHM. These comprehensive results fully demonstrate the advantages of this paper's algorithm in several aspects, highlighting the superiority of the improved algorithm in processing full-wavefield experimental data, which not only provides more accurate damage identification but also possesses high monitoring efficiency, providing a reliable solution for applications in the field of structural health monitoring. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Algorithm** & \begin{tabular}{c} **The Rank of the** \\ **Low-Rank** \\ **Matrix** \\ \end{tabular} & \begin{tabular}{c} **Error** \\ **Iterations** \\ \end{tabular} & \
\begin{tabular}{c} **The Number of** \\ **Time(s)** \\ \end{tabular} &
\begin{tabular}{c} **Processing** \\ **Time(s)** \\ \end{tabular} \\ \hline IALM & 27 & \(9.94\times 10^{-7}\) & 602 & 149.34 \\ APG & 26 & \(1.10\times 10^{-6}\) & 120 & 24.80 \\ RPCA-GD & 19 & \(8.64\times 10^{-8}\) & 58 & 16.78 \\ NCA-RPCA & 7 & \(1.06\times 10^{-10}\) & 2 & 0.83 \\ (ARCTAN) & 7 & \(1.78\times 10^{-8}\) & 2 & 0.83 \\ NCA-RPCA & 7 & \(1.24\times 10^{-9}\) & 3 & 1.53 \\ (Log) & 7 & \(6.89\times 10^{-11}\) & 2 & 0.86 \\ (SCAD) & 7 & \(1.06\times 10^{-10}\) & 2 & 0.83 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The comparative analysis of various algorithms’ performance. Figure 33: Full wavefield image decomposition results under non-destructive conditions. Figure 34: Decomposition results for traditional algorithms. (**a**) IALM decomposition results. (**b**) APG decomposition results. (**c**) FPCP decomposition results.