PSI - Issue 80

Haomiao Fang et al. / Procedia Structural Integrity 80 (2026) 53–64 H. Fang et al./ Structural Integrity Procedia 00 (2025) 000 – 000 ≜ 1 2 ‖ − −1 ‖ 22 + ‖ ‖ 1 (6) Then, sequentially update each atom in 0 via SVD to minimize the error. After the dictionary 1 being updated, fix it, and update 2 . Subsequently, fix 2 , and update 2 in the same procedure. In each iteration, the sparse representation is updated based on the dictionary −1 from the previous iteration, and the new dictionary is refined, using −1 as a warm restart. After sufficient iterations, converges, representing the most fundamental and distinctive shapelets extracted from candidates. Further details of the K-SVD dictionary learning process are provided in [18]. 3.2.3 Dictionary compilation To ensure the dictionary atoms match the dimensions of test signals, a shift invariant technique [27] is implemented to compile the dictionary based on the learnt shapelets in Section 3.2.3. Each shapelet can be extended to generate a series of dictionary atoms, all matching the length of the test signal, by embedding the shapelet information at different phase shifts while setting the remaining entries to zero. The dictionary compilation process is illustrated in Figure 4.

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Figure 4: The illustration of dictionary compilation.

3.2.4 Sparse coding Sparse coding is performed to efficiently represent the test signal as a linear combination of a small number of atoms from the dictionary. This process is defined by (7): {‖ − ‖ 2 + ‖ ‖ 1 } (7) Where y , D , x represents test signal, learnt dictionary and sparse representation matrix, respectively. λ is a regularization parameter controlling the trade-off between reconstruction accuracy and sparsity. In this study, the empirical parameter λ is set to 0.01. During the sparse coding stage, the learnt dictionary D is fixed while the sparse representation x is determined by minimizing the reconstruction error under a sparsity constraint. The sparse coding procedure is illustrated in Figure 5.

Figure 5: The illustration of sparse coding.

3.2.5 Damage index The reconstructed signal, based on the analytical dictionary, serves as a virtual baseline for subsequent damage identification on composite coupon panels. Discrepancies between the measured signal and its reconstructed baseline