## 2 Model for Ultrasonic Guided Wave Ray Tomography
### Solution of the Sparse Model  
It has been understood that, when the excited frequency of a guide wave transmitted on a plate is fixed, the group velocity or slowness of the guided wave known from the dispersion curve is mainly dependent on the thickness of the plate. Namely, the slowness of the plate varies with its thickness. When a plate is corroded, the plate thickness will be changed. Suppose that the thickness of a monitored plate is fixed. It means that the upper limit of the variable thickness of the plate will be its maximum thickness. According to the maximum variable thickness of the plate, the maximum variable slowness is \(q(q\geq 0)\). Therefore, the maximum value of the difference slowness \(\Delta s_{i}\) for each grid in Equation (7) can be limited as:  
\[-q\leq\Delta s_{i}\leq q. \tag{8}\]  
When the monitored area is divided into \(N\times N\) grids, one has  
\[\sum\lvert\Delta s_{i}\rvert\leq N^{2}q, \tag{9}\]  
or, equivalently,  
\[\|\Delta S\|_{1}\leq N^{2}q. \tag{10}\]  
The \(\|\bullet\|_{p}\) denotes the \(l_{p}\)-norm of a vector. When \(p=2\), the \(l_{p}\)-norm is the Euclidean one. When \(N\) is fixed, \(N^{2}q\) will be a constant. In such a situation, it is known that the solution of Equation (7) can be formulated as the following optimization problem  
\[ \begin{array}{l}min\|\Delta T-L\Delta S\|_{2}^{2}\\ s.t.\|\Delta S\|_{1}\leq N^{2}q\end{array}  \tag{11}\]  
Figure 2: An example for the sparsity of slowness difference. Equation (11) can equivalently be rewritten as the following least absolute shrinkage and selection operator (LASSO) problem [34]:  
\[min\|\Delta T-L\Delta S\|_{2}^{2}+\alpha\|\Delta S\|_{1}, \tag{12}\]  
where \(\alpha\) is a regularization parameter. Equation (12) imposes an \(l_{1}\)-norm on the \(\Delta S\). Owing to the nature of the \(l_{1}\)-norm, it is well known that the LASSO problem can continuously be shrunk and do automatic variable selection to find the solutions. Many of the solutions, \(\Delta s_{i}\) (\(i\) = 1,2,..., \(N^{2}\)), are set to 0. It means that the solutions of Equation (12) are of a natural sparsity structure. Therefore, the sparsity of \(\Delta S\) is naturally used in the solving process. In addition, it is well known that the LASSO problem is a convex optimization problem [34]. A convex optimization problem has an optimal solution. Thus, the optimal solutions of the reconstruction image can be obtained by the LASSO approach. Certainly, based on our sparse model, both the SIRT and the regularization methods can be used to do defect imaging. Compared to the other regulation methods, the LASSO method has two main advantages as given in Reference [35]. First, it can achieve good estimation and prediction by shrinking the estimator towards 0. It means that the sparse solution of the sparse model can be obtained by the LASSO method but the other regularization methods for the \(l_{p}\)-norm with \(p>1\) cannot. For \(p<1\), their solutions are sparse but the problem is not convex and this makes the minimization very challenging computationally [36]. When the solution is not sparse, it means that many of the solutions, \(\Delta s_{i}\) (\(i\) = 1,2,..., \(N^{2}\)), are not 0. It also implies that the visible artifacts are generated. Second, there is a simple iteration way available for LASSO to converge fast, which make it very attractive computationally in terms of central processing unit and memory. Before solving Equation (12), \(\alpha\) should be known a priori. The empirical range of \(\alpha\) is from 0.0001 to 0.01. for every \(\alpha\), \(\Delta S\) can be obtained by using the CVX toolbox [37; 38] to solve Equation (12). The curve with a log-log plot of \(\|\Delta S\|_{1}\) versus \(\|\Delta T-L\Delta S\|_{2}^{2}\) as \(\alpha\) is varied, is then acquired. The curve is called the L-curve. After that, the optimal \(\alpha\) is calculated from the L-curve by Equation (11) of Reference [39]. In our experiments, the optimal \(\alpha\) calculated by this method is 0.001.