## 4 Laser-Ultrasonic Imaging of Solid Bodies
### LU Measurements of Geometry of Solids with Complex Surface  
The ability to segment the profile in LU images allows for the internal part of samples with complex surface to be imaged correctly [54]. In this way, such limitations of X-ray tomography for solids, as low sensitivity and requirement of access to inspected object from all sides, can be overcome for a set of applications. The refraction of ultrasonic waves that occurs at the water-solid interface can be taken into account by a two-step algorithm based on a heuristic ray tracing technique demonstrated in [54], for solids with piecewise-linear surface profiles. First, the profile is obtained in the form of the equations of linear segments analogous to the LU profilometry (Section 4.1). Second, a part of image corresponding to the internal part of the sample is recalculated, accounting for the changes in the time-of-flight and amplitude, introduced by refraction on the solid-water interface. The propagation of acoustic waves in the approximation of geometrical acoustics is shown in Figure 8. The total path of the acoustic ray can be divided into forward and backward paths. Then, the total travel time is  
\[\mathcal{T}_{ijm}=\mathcal{T}_{ij}^{forw}+\mathcal{T}_{ijm}^{back}, \tag{14}\]  
where \(\mathcal{T}_{ij}^{forw}\), \(\mathcal{T}_{ijm}^{back}\) are the travel times on forward (from PA generator to \((i,j)\) pixel of image) and backward (from \((i,j)\) pixel of image to \(m\)-th detector) paths, correspondingly. The positions of ray entry \((x_{e},z_{e})\) and exit \((x_{o},z_{o})\) from sample should be calculated to determine these times, under the assumption of homogeneous speed of sound in both media. The coordinates of the entry point are  
Figure 7: **(a,b)** Line of tomogram maxima and 3D image of a cylindrical sample. (**c,d**) Images of the cylinder with irregular grooves. (**e,f**) Images of the cylinder with irregular grooves and slots. (**c,e**) Reconstructed 3D images of cylinder surface. (**d,f**) 2D projections of the corresponding 3D maps. Bright color denotes more distant parts from the center of the sample, and dark color denotes closer parts to the center. The blue lines are auxiliary, denoting the boundaries of the relief. Adapted with permission from [52], OSA Publishing, 2018.  
calculated from the Snell's law, and the equations of line profiles know from the LU profilometry. In the first Born approximation, the inhomogeneity, located in the point \(\left(x_{i},z_{j}\right)\), scatters waves in all directions, and the coordinates of the exit point \(\left(x_{o},z_{o}\right)\) are calculated following the [PERSON]'s principle of least time for \(\mathcal{T}_{ijm}^{back}\), which is achieved at the profile lines:  
\[\left\{ \begin{array}{c}\mathcal{T}_{ijm}^{back}(x_{o},z_{o})=\frac{1}{c_{0}} \sqrt{\left(z_{o}-z_{m}\right)^{2}+\left(x_{o}-x_{m}\right)^{2}}+\frac{1}{c_{1} }\sqrt{\left(z_{o}-z_{i}\right)^{2}+\left(x_{o}-x_{j}\right)^{2}}\rightarrow \text{min},\\ z_{o}=k_{s}x_{o}+b_{s},\end{array} \right. \tag{15}\]  
where \(k_{s}\) and \(b_{s}\) are the coefficients of the \(s\)-th line of surface profile. The gradient descent method can be applied to solve this minimization problem. The expression for the recalculated part of the image corresponding to the internal part of the object can be obtained from Equation (13), with modified times-of-flight and amplitudes:  
\[\epsilon^{\prime}\left(\mathbf{r}_{ij}\right)=\sum_{m=1}^{N}p\left(\mathbf{r}_{m}, \mathcal{T}_{ijm}\right)\cdot\mathcal{D}\left(\mathbf{r}_{m},\mathbf{r}_{ij}\right) \cdot W_{ijk}^{ol}W_{ijm}^{el}, \tag{16}\]  
where \(W_{ijk}^{ol},W_{ijm}^{el}\) are the coefficients used to account for the partial transmission and mode-conversion (longitudinal waves in water into transverse waves in the solid sample) on the forward and backward paths, correspondingly. They are evaluated standard formulas from [123]. The sample (Figure 9a) is a PMMA cylinder with complex internal and external generatrices.