## 5 Damage Detection in Wavefield Images Based on NCA-RPCA Algorithm
### Wavefield Image Damage Identification of Single Defects Based on NCA-RPCA Algorithm
#### 5.2.4 Comparison of Algorithms  
To comprehensively demonstrate the superiority of the algorithms in this chapter, this paper conducts an in-depth comparative analysis with mainstream algorithms such as IALM, Accelerated Proximal Gradient (APG), RPCA-GD algorithm, and Fast Principal Component Pursuit (FPCP) algorithm. It will showcase the different handling effects achieved by combining various non-convex penalty functions with non-convex rank approximation functions, providing a multi-angle display of the advantages of the approximation methods proposed in this paper. First, four mainstream algorithms were used for processing, and the results showed significant limitations when handling wavefield damage images. In Figure 28a,b, the IALM and APG algorithms decompose the sparse matrix, but they excessively categorize useful information as anomalies, resulting in severe information loss and unsatisfactory processing effects. In Figure 28c, the FPCP algorithm fails to effectively retain useful information or separate outliers after decomposition. The decomposition results in Figure 28d are relatively better. Subsequently, employing the algorithm proposed in this paper for processing, the results from Figure 28 reveal that the decomposition outcomes of this chapter's algorithm are closer to the ideal. They can accurately separate the low-rank and sparse components of the wavefield image, allocating as much useful information as possible to the low-rank component while avoiding excessive redundant information in the sparse matrix. This achievement facilitates efficient restoration of the low-rank component. Therefore, through comparative analysis between Figures 28 and 29, it is evident that the algorithm proposed in this paper significantly outperforms existing traditional algorithms, better preserving the damaged information. Figure 28: _Cont._  
To further compare the advantages and disadvantages of the NCA-RPCA algorithm proposed in this paper with traditional algorithms, a comparative analysis of time efficiency in handling different types of defects was conducted. As shown in Table 3, traditional algorithms take significantly longer for damage identification, more than ten times longer than the algorithm proposed in this chapter. The proposed algorithm substantially reduces processing time, greatly enhancing SHM's real-time monitoring efficiency. The reduction in time costs provides a more efficient solution for related application areas, meeting the demands of practical application scenarios. This study conducts performance evaluations of several algorithms from various aspects, such as the rank of low-rank matrices, decomposition error, and iteration count. The analysis results from Table 4 indicate that the algorithm proposed in this chapter exhibits significant advantages across all metrics. Firstly, in the decomposed low-rank matrices, this chapter's algorithm demonstrates a higher capability to restore original data information at a lower rank level, showcasing superior information restoration ability. Secondly, the decomposition error is significantly reduced, indicating that the algorithm in this paper more effectively preserves the data's characteristics. Most importantly, there is a substantial decrease in the number of iterations, highlighting the efficiency and superiority of the algorithm proposed in this paper. These results collectively demonstrate a significant  
Figure 28: Decomposition result of traditional algorithms. (**a**) IALM decomposition result. (**b**) APG decomposition result. (**c**) FFCP decomposition result. (**d**) RPCA-GD decomposition result.