## 5 Experiment of Imaging Detection
### Analysis of the Results of the Test for Imaging Detection  
The coordinates as well as the gray values of the holes for artificial defects in the left-rank arc sparse array experiment were tested. The test results are shown in Table 1. The multi-frame imaging results of the left-rank arc sparse array of ultrasonic guided waves, depicted in Figure 4, effectively capture millimeter-level defect information present in the welded structure steel plate, specifically in the region after the weld. These images reveal defects equivalent to half-wavelength sizes by maintaining a fixed transmission aperture; the rank of the receiving matrix varies accordingly. The number \(n\) of receiving array elements corresponding to the receiving aperture ranges from one to fourteen. This results in a sequence of multiple frame images corresponding to different ranks of the receiving matrix: one, four, eight, twelve, and fourteen. The full-rank imaging occurs when the rank of the receiving matrix is 14. These images, as shown in Figures 5-8, exhibit noticeable distinctions and connections in terms of gray levels, background gray levels, signal-to-noise ratios, and contrast. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Artificial Defective Holes in** & & \\
**Welded Structural Panels** & **Coordinate [X, Y]** & **Index** \\ \hline Hole A & [127; 34] & 60.71 \\ Hole B & [71; 128] & 141.7 \\ Hole C & [18; 273] & 187.2 \\ Hole D & [118; 367] & 115.4 \\ Hole E & [168; 201] & 255 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Test data on man-made defective holes in welded structural plates. Figure 4: The multi-frame imaging detection of ultrasonic guided waves in welded structural plates based on arc sparse array with left rank. (**a**) The rank of the receiving matrix is one; (**b**) The rank of the receiving matrix is four; (**c**) The rank of the receiving matrix is eight; (**d**) The rank of the receiving matrix is twelve; (**e**) The rank of the receiving matrix is fourteen. Figure 5: Multi-frame image gray level of left-rank arc sparse array. Figure 6: Background gray level of multi-frame images with left-rank arc sparse array. Figure 7: Multi-frame image gray level of left-rank arc sparse array scatterer. The coordinate data of artificially defective holes in the welded structural plate were tested to determine the coordinate values of defective holes \(A\), \(B\), \(C\), \(D\), and \(E\) in the image, to explore the relationship with the location of the defects in the actual experiments, and to draw the image, as shown in Figure 4.  
In the multi-frame imaging of the left-rank arc sparse array, the grayscale level of multi-frame images was tested data as shown in Table 2. The grayscale value of the defect at the natural focus appears to be relatively constant, indicating a consistent connection and difference between the multiple frames of images. As illustrated in Figure 5, the natural focus of the arc sparse array aligns with defect \(E\), and all of the frames of images effectively capture the scatterers within the imaging area. For defect holes \(B\) and \(C\) located outside the natural focus, the gray values show consistent changes across different frame sequence images, as seen in Figure 5. In contrast, defect holes \(A\) and \(D\), situated far away from the natural focus, exhibit consistent gray value changes in images from different frame sequences. However, these changes are more pronounced. The first to fourth frame images generally demonstrate a downward trend in gray values, with a significant change. From the fifth to tenth frames, the gray value changes remain relatively flat. The twelfth to fourteenth frame images, however, exhibit a peak value in gray values. In the fifteenth frame of the full sequence, both hole \(A\) and hole \(E\) exhibit a decreasing trend in gray values. On the other hand, holes \(B\), \(C\), and \(E\) show an increasing trend, but the magnitude of the gray value change is relatively small.