## 1 Introduction  
It is well-known that corrosion detection and monitoring are vital for preserving material integrity and extending the life cycle of, for example, industrial infrastructure, aircrafts, pipelines, and oil installations [1; 2; 3; 4]. In many structure health monitoring (SHM) applications, ultrasonic guided wave tomography is used as a reliable tool for detecting and monitoring corrosion [5; 6; 7; 8; 9; 10; 11]. Sizing the corrosion region in a plate-like or a pipe-like structure is very important for corrosion detection and monitoring [12; 13; 14]. Conventional ultrasonic corrosion mapping is carried out by point-by-point scanning on the surface of the materials [15; 16] and calculating the thickness from the arrival time of the reflected waves. In contrast, in guided wave tomography, the guided wave is excited in a plate or a pipe, passing through the monitoring area, and measured by a surrounding transducer array. The corrosion area can then be obtained by reconstructing a velocity map of the guided wave and converting it to a thickness map. Ray tomography uses the guided wave time-of-flight (TOF) to reconstruct the slowness distribution [17; 18]. However, the reconstruction via this method is of a relatively low resolution, which is limited by the width of the first Fresnel zone. In Reference [19], a wave speed map is reconstructed using the curved ray tomography with simulation, even though the size of a sound speed anomaly is smaller than the resolution length of the inversion method. Diffraction tomography implements a higher resolution using the Born approximation if the scattering generated by the corrosion is very weak [20; 21]. It is clear that it is unlikely to be such a case for many corrosions of interest. The hybrid algorithm reported in Reference [22] initially uses a low-resolution ray tomography, and later gradually iterates with a diffraction tomography which can achieve a high-resolution tomography, when the scattering is not very weak. However, its performance relies on the background estimate being sufficiently accurate via ray tomography. Besides on these three tomography algorithms, the full wave-form inversion (FWI) method presents a novel nonlinear inverse scattering model to implement reconstruction images of guided wave tomography [23]. Compared with diffraction tomography, FWI allows higher order diffraction and scattering and has the same theoretical resolution limit [24]. Literature [14; 25; 26] also report that more accurate inversion results are achieved when this method is applied to monitor the plate-like materials. The major limitation of this method is its computational complexity and cost, and it can be acceptable as an offline imaging method. Because of the low computational complexity of ray tomography, it is widely used in many online SHM applications, such as the chemical industry and aviation industry [27]. Ray tomography is a promising method for corrosion monitoring, especially for online applications. In this paper, the method based on ray tomography for corrosion monitoring is studied. Currently, the simultaneous iterative reconstruction technique (SIRT) [28] is a widespread used method in the ray tomography. This method is based on an underdetermined linear model. Such a linear model known is an infinite number of solutions, which can fit measured data. It means that the solution obtained by the conventional iterative reconstruction technique cannot be guaranteed as the optimal one. In order to cope with this problem, a sparse model based on differences between the inspected and original slowness of guided waves as an imaging approach for mapping plate thickness by using ray tomography is proposed in this paper. Although our model is also underdetermined, the optimal solution of the reconstruction image can be given when the constraints on the sparsity of the slowness difference distribution are valid. Based on our model, the LASSO approach is applied to obtain the solution of the sparse-constrained minimization problem.