## 4 Laser-Ultrasonic Imaging of Solid Bodies
### LU Profilometry of Solids  
The LU modality of the combined PA and LU imaging system developed by [PERSON] et al. [45] was applied to the profile measurement of solid objects [52]. There are several common types of profilometers, such as contact stylus profilers, or non-contact optical profilers. However, such systems are not applicable for inspection of immersed or contaminated objects. For measurement of such objects (e.g., pipes, oil tanks, objects in cooling liquids, underwater works), the developed setup for LU profilometry provides high acoustic resolution in real time: in the longitudinal direction (along the probe pulse beam) up to 20-40 \(\upmu\)m; and in transverse, up to 150-250 \(\upmu\)m (along the direction perpendicular to the probe beam).  
2D LU images of solid samples are obtained using the back-projection algorithm (Equation (13)) described above. In Figure 6, the negative part of the images is truncated, while white boundaries denote reflections from the sample profile. Images are reconstructed under the assumption of uniform speed of sound in the whole domain of measurements. Hence, only the positions of water-solid interfaces are correctly displayed, and all the positions of the scatterers inside the object (the internal structure) are distorted.  
Let the immersion liquid (water) have speed of sound \(c_{0}\) and density \(\rho_{0}\), and let the sample have speed of sound \(c_{1}\) and density \(\rho_{1}\). As the acoustic impedance of water \(Z_{0}=\rho_{0}c_{0}\) is significantly lower than that of material of the sample (duralumin) \(Z_{1}=\rho_{1}c_{1}\), the reflection coefficient of the acoustic waves is positive, and takes the greatest value on the water-solid interface. Hence, the position of sample profile on the tomogram is defined by the first local maximum of the obtained tomograms. A set of such local maxima forms the LU image of the surface profile (Figure 7a). A Hough transform and linear least squares method are applied to the set of profile points, to find linear regions. This procedure enables calculation of the deviation of the measured sample profile from the model. Comparison of images, obtained by LU measurements and X-ray imaging has shown good agreement of the results. The discrepancy of the results was within 20-30 \(\upmu\)m in the longitudinal direction, and 150-250 \(\upmu\)m  
Figure 6: LU images \(\epsilon(r)\) of the duralumin samples reconstructed using filtered back-projection algorithm: (**a**) a cylinder 14 mm in diameter, (**b**) a cylinder 14 mm in diameter with irregular grooves 1 mm in depth, and (**c**) a cylinder 14 mm in diameter with regular grooves 2 mm in depth. Reproduced with permission from [52], OSA Publishing, 2018.  
in the transverse direction. The properties of such resolution are connected to the fundamental properties of all ultrasonic measurements: in the direction along the acoustic beam path, the resolution is significantly higher than in the perpendicular direction.  
Reconstruction of the 3D surface profiles (Figure 7b,f) was performed using step-by-step 2D imaging with sample rotation and subsequent composition of the profile lines into the surface. Figure 7e shows 3D image of the same object as in Figure 7c, but with a damaged surface. The diameters of parts, measured by X-ray tomography and LU measurements, showed agreement within 100-150 \(\upmu\)m. This discrepancy is caused mainly by misalignment of the rotational axis and the object's axis of symmetry.