# Improved Acoustic Emission Tomography Algorithm Based on Lasso Regression
## 3 LASSO Algorithm Combined with AE Tomography  
In 1995, [PERSON] proposed Lasso (least absolute shrinkage and selection operator), a regression model that uses L1 regularization techniques, also referred to as Lasso regression. Regularization is a general term for a class of methods that introduces additional information to the original loss function in order to prevent overfitting and improve model generalization performance. There are generally two commonly used extra items called L1 regularization (\(\ell 1-\)norm) and L2 regularization (\(\ell 2-\)norm). L1 regularization and L2 regularization can be regarded as penalty terms of the loss function, and penalty refers to some restrictions on some parameters in the loss function [15]. The algorithm also performs to solve convex-cardinality problems as an essential reconstruction algorithm (also called the L1 heuristic) rebuilt from the L1-norm [16; 17; 18]. It is an efficient approach to reconstructing sparest solutions to specific underdetermined systems of linear equations. More specifically, there are an unknown vector \(\mathrm{x}\)\(\in\)\(\mathrm{R}^{\mathrm{N}}\) and an underdetermined matrix \(\mathrm{A}\)\(\in\)\(\mathrm{R}^{\mathrm{M}\times\mathrm{N}}\) and \(\mathrm{y}\)\(=\)\(\mathrm{A}\cdot\mathrm{x}\). The problem can be solved as follows:  
\[\min\parallel\mathrm{x}\parallel_{1}\mathrm{subject\ to}\parallel\mathrm{A} \cdot\mathrm{x}\,-\,\mathrm{y}\parallel_{2}\,<\,\varepsilon, \tag{6}\]  
where \(\parallel\cdot\parallel_{1}\) and \(\parallel\cdot\parallel_{2}\) represent the \(\ell 1-\)norm and the \(\ell 2-\)norm, respectively, and \(\varepsilon\) represents the noise boundary [19]. According to the feature of this algorithm, LASSO  
Figure 2: Potential emission time computation. regularization can make parameters sparse, and obtained parameters are a sparse matrix, which can be provided for feature selection. The characteristic of this theory can contribute to shrinking the number of coefficient matrices and obtaining local optimal solutions simultaneously. In words, the LASSO algorithm requires a fewer number of measurement paths than the other algorithm, such as the SIRT, for velocity distribution according to the number of coefficients depending on the number of paths in elastic velocity tomography. The following demonstrates the theory of AE tomography combined with the LASSO algorithm:  
\[\mathrm{T}=\mathrm{A}\cdot(\mathrm{s}_{0}+\mathrm{\Delta s}), \tag{7}\]  
where \(\mathrm{T}\) is the vector of the measured travel time, \(\mathrm{S}_{0}\) is the standard slowness in a completely healthy state, and \(\mathrm{\Delta s}\) is the slowness difference vector generated by subtracting the actual slowness from the standard slowness. All of the ray-path matrices are used to construct \(\mathrm{A}\), a coefficient matrix that is generated from all of the travel lengths. A vector representing the results of the deviation between the measured travel time and the travel time in totally healthy conditions, \(y\), is defined as:  
\[\mathrm{y}=\mathrm{T}-\mathrm{T}_{0}=\mathrm{A}\cdot\mathrm{\Delta s},\ \mathrm{where}\ \mathrm{T}_{0}=\mathrm{A}\cdot\mathrm{s}0, \tag{8}\]  
where \(\mathrm{T}_{0}\) is the travel time vector for a healthy specimen. The velocity distribution imaging of AE tomography depends on reconstructed \(\mathrm{\Delta s}\) in Equation (9), which is defined by the LASSO algorithm after the sampling phase from Equation (6). It is noted that solution the slowness vector \(\mathrm{\Delta s}\) is a defect detection problem in physical understanding:  
\[\mathrm{min}||\mathrm{\Delta s}||_{1}\ \mathrm{subject\ to}\ ||\mathrm{A} \cdot\mathrm{\Delta s}\ -\ \mathrm{y}||_{2}<\ \varepsilon, \tag{9}\]  
where the \(||\cdot||_{1}\) and \(||\cdot||_{2}\) are the \(\ell 1-\)norm and \(\ell 2-\)norm, respectively.