# High-Precision Noncontact Guided Wave Tomographic Imaging of Plate Structures Using a DHB Algorithm
## 2 Tomography Theorem
A smaller correlation coefficient value means a stronger indication of the defect's appearance. This will undoubtedly make the guided wave tomographic imaging become more complex. Particularly, this adverse effect will become more serious when the air-coupled transducer is used because air-coupled transducers are insensitive to the symmetric modes of a Lamb wave. On the other hand, only one mode of the Lamb wave is needed if the DHB algorithm is used. In defect quantitative analysis, the parameter \(\beta\) determines the shape of the elliptical path. Then, the speckle of the defect shape can also be influenced. There is one optimal value for specific defects with a determined size. Other defects of different sizes cannot be imaged successfully. The influence of \(\beta\) on the imaging results is shown in Figure 1.  
The FBP algorithm is also widely used in the NDT field. Its reconstruction formula can be written as:  
\[f(\overrightarrow{x})\ =\ \int_{0}^{2\pi}\frac{D}{(D+\overrightarrow{x}\cdot \overrightarrow{e_{w}})^{2}}\int_{-\gamma_{m}}^{\gamma_{m}}(q(\lambda,\gamma) \cos\gamma)\otimes h_{H}(\gamma)d\gamma d\lambda, \tag{3}\]  
where \(\overrightarrow{x}\) is the vector of the reconstruction points, \(\lambda\) is the angle between the starting coordinate axis and the line that connects the trigger points and the origin of the starting coordinate system, \(\lambda_{m}\) is the angle range of the fan beam, \(q(\lambda,\gamma)\) is the projection value, \(\overrightarrow{e}_{w}\) is unit orthogonal vector, \(h_{H}\) is kernel of the Hilbert transform, and D is the distance between the trigger points and the intermediate point of the receiving sensors' array. Many different kinds of iterative reconstruction methods can be used for the data acquisition, as shown in Figure 2. Only the fan-beam method was used in this work to emphasize the comparison of the imaging quality of different tomographic imaging algorithms. As shown in Figure 3, it is based on the signal of a nondefective ultrasonic signals, and the nondefective signal becomes a reference signal. When the ultrasonic wave passes through the defect, the ultrasonic amplitude is attenuated, so the defect or object is imaged by utilizing the difference in amplitude between the two signals. The simulated results were made based on the attenuation law of a Lamb wave. As shown in Figure 4, results based on different detection points were made for comparison. The FBP algorithm is a technique used in radiographic inspection. It is a method of transmitting X-rays through an object and receiving it through a detector. It shoots 360 degrees around the object and displays the image. However, radiographic inspection is limited by shielding facilities or radiation regulations. In this study, ultrasonic technology using a Lamb wave was applied to compensate for such shortcomings. Figure 4 is the result of imaging by applying the simulation signal of Figure 3 to the FBP algorithm. The imaging quality can be improved by increasing the emitter and receive detection points. This method has the same characteristic as the new DHB algorithm in this respect. The number of the detection points adopted in the actual experiment was 90, 120, and 180. Figure 1: Image reconstruction results by using different values of parameter \(\beta\). Figure 2: Three common iterative reconstruction methods: (**a**) Crosshole; (**b**) Double-crosshole; (**c**) Fan-beam. The FBP algorithm has a big advantage in the aspect of imaging precision compared with RAPID. However, the problem of this algorithm is in its image artifacts. Defect imaging close to the scanning edge cannot be displayed clearly, as shown in Figure 4. These are called image artifacts. The image artifacts will undoubtedly affect the precise identification of the defects near the detection boundary. The new DHB algorithm is proposed to solve the shortcomings of the algorithms discussed above. The new DHB algorithm has good performance in the detection of plate-like structures.