## 6 Conclusions  
This paper introduces the utilization of the left-rank arc sparse array for conducting research on the multi-frame imaging detection of ultrasonic guided waves in welded structures. It discusses the concept of the arc sparse array and its commonly used receiving aperture. By employing the wave amplitude matrix and the wave time matrix, the study analyzes the correlation between multi-frame imaging and the left-rank receiving aperture. Furthermore, it explores the fundamental principles of multi-frame imaging using the left-rank arc sparse array in the presence of weld scattering conditions within the plate. The test results of the detection demonstrate that during the application of multi-frame imaging with the left-rank arc sparse array under weld scattering conditions, an increase in the left-rank receiving aperture leads to a decline in both the gray level of the multi-frame images and the trend of the background gray level curve. Notably, when the left rank reaches approximately 64% of the full rank, the slope of the image gray level and the background gray level decreases gradually. At this point, an inflection point emerges in the curve of the signal-to-noise ratio, indicating an improved imaging effect. This outcome is advantageous for effectively capturing and characterizing the scattering information within the imaging area. The research findings indicate that the multi-frame imaging technique using the left-rank arc sparse array for ultrasonic guided waves enables an effective characterization of millimeter-level defect scatterers, which are comparable in size to the wavelength, in welded structure plates under weld scattering conditions. The imaging experiment validates the feasibility of the proposed imaging theory. This research contributes valuable insights that \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{5}{c}{**Image Signal-to-Noise Ratio (SNR)**} \\ \hline
**Transducer Frame** & **Hole \(A\)** & **Hole \(B\)** & **Hole \(C\)** & **Hole \(D\)** & **Hole \(E\)** \\
**Rate (No.)** & & & & & \\ \hline
1 & 8.06 & 12.04 & 13.23 & 11.65 & 17.67 \\
2 & 8.79 & 15.71 & 17.03 & 14.49 & 20.83 \\
3 & 9.28 & 16.74 & 17.92 & 15.69 & 21.55 \\
4 & 9.43 & 17.80 & 18.28 & 15.91 & 22.39 \\
5 & 10.52 & 19.02 & 20.51 & 17.33 & 22.99 \\
6 & 11.83 & 19.97 & 21.58 & 17.99 & 24.32 \\
7 & 12.47 & 20.15 & 22.58 & 18.39 & 24.31 \\
8 & 13.00 & 20.85 & 22.29 & 18.67 & 24.44 \\
9 & 12.70 & 20.38 & 22.21 & 17.89 & 24.05 \\
10 & 13.36 & 20.92 & 22.95 & 18.53 & 25.17 \\
11 & 11.97 & 19.46 & 21.57 & 17.32 & 24.20 \\
12 & 12.13 & 19.31 & 21.55 & 17.40 & 24.11 \\
13 & 12.98 & 19.90 & 22.13 & 18.02 & 24.71 \\
14 & 13.70 & 20.32 & 22.59 & 18.66 & 25.16 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Signal-to-noise ratio simulation data of multi-frame images of left-rank arc sparse array. can serve as a reference for further advancements and practical applications in the field of multi-frame imaging detection using irregular arrays of ultrasonic guided waves. Conceptualization, S.Y. and X.Z. ; methodology, X.Z. ; software, S.Y. ; validation, X.Z., S.Y. and M.D. ; formal analysis, S.Y. and X.Z. investigation, S.Y. ; resources, X.Z. and M.D. ; data curation, S.Y. ; writing--original draft preparation, S.Y. ; writing--review and editing, S.Y. and X.Z. ; visualization, X.Z. and M.D. ; supervision, J.Z. ; project administration, Y.G. ; funding acquisition, Y.G. All authors have read and agreed to the published version of the manuscript. This research was funded by the support of the National Natural Science Foundation of China (Nos. 12164001, 12134002), the Key Research and Development Project of Ningxia Hui Autonomous Region of China (No. 2021 BE04006), and the Ningxia Engineering Research Center for Hybrid Manufacturing System of China (No. 2023 GCJS008). Not applicable. The raw data supporting the conclusions of this article will be made available by the authors on request.