## 5 Discussion  
The results presented in this paper have shown that our approach obtained higher spatial resolution thickness mapping for plate-like structures using a sparse model than that of SIRT, and the visible artifacts in the tomographic reconstruction images were avoided. The computation time of our approach was also compared with that of SIRT. The computation time of the SIRT and our method used in the three experiments are as shown in Table 3. The computer, which was used for computation in our experiment, is Intel core i7-4770 quad-core desktop processor 3.4 GHz with 16 GB DDR3 memory. Figure 11: Tomographic reconstruction images for a 1.5 mm thick aluminum plate with two eccentric circular flat-bottomed holes, which have a common area. One of the holes, whose thickness loss was 30% and diameter was 30 mm, was in the center of the plate. The center coordinate of another hole was x = 260 mm, y = 260 mm, whose thickness loss was 50% and diameter was 40 mm. **(a)** The actual locations and sizes of the flaws. **(b)** The tomographic reconstruction image using the SIRT. **(c)** The tomographic reconstruction image using our method. Figure 12: Cross section of reconstruction of three circular flat-bottomed holes following the lines marked in Figure 11a along **(a)** the horizontal line and **(b)** the vertical line. From Table 3, although the computation time of our approach was greater than that of SIRT, it was enough for online corrosion monitoring. Another advantage of our approach is that the number of transducers can be decreased because of the sparsity of \(\Delta S\). Equation (7) is viewed as a sparse model. In the sparse model, \(M\) is the number of both the ultrasound transmitters and receivers respectively. The LASSO method can be used to solve the underdetermined Equation (7) as long as \(K\) is smaller than \(M^{2}\). It also implied that the number of transducers for obtaining Equation (7) can be decreased provided that \(K<M^{2}\)[41]. Figure 13 gives such an example. The monitored aluminum plate with a 20 mm diameter circular flat-bottomed hole in the center was made so that its thickness loss was 50%, as shown in Figure 4a. The monitored area was divided into \(20\times 20\) grids. In such a situation, the corrosion was in a scale close to the grid size. Figure 13a shows the result obtained by SIRT, where the number of both the ultrasound transmitters and receivers were 20. Figure 13b shows the result obtained by our approach, where the number of both the ultrasound transmitters and receivers were five. Even so, the more ultrasound transmitters and receivers were recommended for obtaining a high spatial resolution image, especially for the cases where the locations of the corrosion are unknown. The major limitation of our method was as the same as one of the LASSO method. Figure 14 gives such an example, where the monitored area is divided into \(4\times 4\) grids and eight grids have been corrode  
Figure 13: Tomographic reconstruction images for a 1.5 mm thick aluminum plate with a 20 mm diameter circular flat-bottomed hole. Thickness loss was 50% within the flaw. (**a**) The tomographic reconstruction image using the SIRT, and \(M=20\), (**b**) The tomographic reconstruction image using our method, and \(M=5\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Grid Density**} & \multicolumn{2}{c}{**Experiment I**} & \multicolumn{2}{c}{**Experiment II**} & \multicolumn{2}{c}{**Experiment III**} \\ \cline{2-7}  & **SIRT** & **Proposed** & **SIRT** & **Proposed** & **SIRT** & **Proposed** \\ \hline \(20\times 20\) & 0.01 & 0.54 & 0.01 & 0.56 & 0.01 & 0.60 \\ \(40\times 40\) & 0.02 & 0.64 & 0.02 & 0.65 & 0.02 & 0.70 \\ \(60\times 60\) & 0.11 & 0.74 & 0.11 & 0.77 & 0.11 & 0.80 \\ \(80\times 80\) & 0.14 & 0.84 & 0.14 & 0.82 & 0.14 & 0.89 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The computation time of the SIRT and our method (unit: s).