###### Abstract
(2020), and the experimental results show that the non-convex methods can have better outlier identification than the traditional methods. Then, [PERSON] et al. (2020) proposed a non-convex rank approximation method using non-convex functions to approximate the rank function and improve the efficiency and accuracy of matrix decomposition. Moreover, [PERSON] et al. proposed a non-convex robust principal component analysis algorithm [PERSON] et al. (2020), which introduced several non-convex penalty functions to approximate the rank function and sparse penalty function in the original RPCA problem. They studied defect detection in wind turbine blades under dimensionality reduction conditions, achieving significant improvements in both computational efficiency and accuracy. However, a more comprehensive analysis of the types of outliers was not conducted in the experimental study. Further, [PERSON] et al. proposed a fabric defect detection algorithm based on multi-level deep feature fusion and non-convex total variation regularized RPCA (NTV-RPCA) [PERSON] et al. (2020). This method effectively detected defects in images, improving adaptability and detection accuracy. [PERSON] and his colleagues proposed the Robust Principal Component Thermography (RPCT) method based on RPCA to detect defects in Carbon Fiber Reinforced Plastic (CFRP) samples [PERSON] et al. (2020). The accuracy of defect detection has significantly improved, but there is room for further optimization in terms of time efficiency. [PERSON] et al. introduced an unsupervised surface defect detection method based on non-convex total variation (TV) regularized RPCA and kernelization [PERSON] et al. (2020). They considered the defect-free background as the low-rank part and the defect region as the sparse part. The non-convex optimization greatly improved the solution accuracy and handled cluttered backgrounds in the low-rank subspace better. In 2022, [PERSON] and his team utilized a model that combines time-frequency analysis images with a non-convex Robust PCA method to diagnose faults in intelligent diesel engines [PERSON] et al. (2020). By incorporating the \(\ell_{p}\) norm, they transformed the optimization problem into a least-squares numerical problem with an \(\ell_{p}\) penalty term. They relaxed the original problem into a non-convex optimization problem, enabling the proposed method to adaptively extract identifying features from fault signals, thus improving recognition accuracy. However, the method does not address the influence of different types of noise effectively. In this paper, based on previous experience, a newer framework of Non-Convex Approximation Robust Principal Component Analysis (NCA-RPCA) algorithms is introduced into outlier analysis for SHM, using a rank approximation function that is closer to matching the true rank than the kernel norm and a non-convex penalty function to approximate the \(l_{0}\) norm. A sparse representation is used to elaborate the damaged part in the aluminum plate for defect detection, with a lower-rank component to restore the defect-free background. Wavefield energy variations are a powerful means of highlighting the location of defects, and the experiments prove the feasibility of introducing the algorithm into wavefield experiments, as well as greatly improving the time efficiency and identification accuracy. The rest of the paper is organized as follows. Section 2 describes the Lamb wave theory, followed by Section 3, describing the basic theory of RPCA. Next, Section 4 focuses on the deep theoretical derivation and algorithmic structure of the NCA-RPCA algorithmic framework. Section 5 demonstrates experimentally the recognition reliability of the algorithm in wavefield images with single or even multiple damages and provides a comparative analysis with the conventional IALM algorithm. Section 6 summarizes the main results of this paper and provides suggestions for future research.