# High-Precision Noncontact Guided Wave Tomographic Imaging of Plate Structures Using a DHB Algorithm
## 2 Tomography Theorem
The imaging quality can be significantly improved compared with other algorithms, such as RAPID or FBP. Particularly, it can give accurate size and shape of different kinds of defects. The geometry of the corresponding fan-beam iterative reconstruction method is shown in Figure 5. Its reconstruction formula is Equation (4) [24]. The relevant parameters in Equation (4) are the same as in Equation (3). The unit orthogonal vector \(\overrightarrow{e}_{w}\) is represented by \(\overrightarrow{a}(\lambda)/\|\overrightarrow{a}(\lambda)\|\) and is substituted into Equation (3). The notation \(g_{m}(\lambda,\gamma)\) is used to denote the measured projections. Three steps are included in using this method: the difference of the projection data, the Hilbert transform of the difference data, the distance weight, and back projection. The corresponding geometry schematic can be referenced in Figure 5.  
\[P(x,y)\ =\ \sum_{i\ =\ 1}^{N-1}\mathrm{P}_{k}(x,y)\ =\ \sum_{K\ =\ 1}^{M}\frac{SDC}{\beta-1}\{\beta-R(x,y,x_{TK},y_{TK},x_{RK},y_{RK})\} \tag{2}\]0  
\[P(x,y)\ =\ \sum_{i\ =\ 1}^{N-1}\mathrm{P}_{k}(x,y)\ =\ \sum_{K\ =\ 1}^{M}\frac{SDC}{\beta-1}\{\beta-R(x,y,x_{TK},y_{TK},x_{RK},y_{RK})\} \tag{2}\]1 Simulated results from the filtered back projection (FBP) algorithm: (**a**) 180 Points; (**b**) 120 Points; (**c**) 90 Points. \[P(x,y)\ =\ \sum_{i\ =\ 1}^{N-1}\mathrm{P}_{k}(x,y)\ =\ \sum_{K\ =\ 1}^{M}\frac{SDC}{\beta-1}\{\beta-R(x,y,x_{TK},y_{TK},x_{RK},y_{RK})\} \tag{2}\]2 Guided wave signal: (**a**) No defect signal; (**b**) Defect signal. The same simulated results have been made by using the DHB algorithm. As shown in Figure 6, the problem of image artifacts can be solved perfectly. The imaging quality is also improved to a large extent. In addition, short-scan conditions can also be relaxed once the new DHB algorithm is used. It is generally believed that accurate and stable reconstruction of any region of interest (ROI) requires line integrals of the object density for all lines passing through the object. This condition leads to the notion of circular short scan. As shown in Figure 7, \(\pi+\theta\) is referred to as a short scan. \[P(x,y)\ =\ \sum_{i\ =\ 1}^{N-1}\mathrm{P}_{k}(x,y)\ =\ \sum_{K\ =\ 1}^{M}\frac{SDC}{\beta-1}\{\beta-R(x,y,x_{TK},y_{TK},x_{RK},y_{RK})\} \tag{2}\]3 Fan-beam geometry based on the equiangular rays. \[P(x,y)\ =\ \sum_{i\ =\ 1}^{N-1}\mathrm{P}_{k}(x,y)\ =\ \sum_{K\ =\ 1}^{M}\frac{SDC}{\beta-1}\{\beta-R(x,y,x_{TK},y_{TK},x_{RK},y_{RK})\} \tag{2}\]4 Simulated result from the Difference Hilbert Back Projection (DHB) algorithm: (**a**) 180 Point; (**b**) 120 Point; (**c**) 90 Point. \[P(x,y)\ =\ \sum_{i\ =\ 1}^{N-1}\mathrm{P}_{k}(x,y)\ =\ \sum_{K\ =\ 1}^{M}\frac{SDC}{\beta-1}\{\beta-R(x,y,x_{TK},y_{TK},x_{RK},y_{RK})\} \tag{2}\]5 Schematic of the short-scan problem: (**a**) A conventional short-scan of \(\pi+\theta\) allows reconstruction of the whole object; (**b**) A continuous scan of less than \(\pi+\theta\) allows reconstruction of all object points inside the convex hull of the scan; (**c**) A scan of three equally spaced segments of \(80^{\circ}\) each allows reconstruction of a triangular ROI in the center of the object. All line integral measurements through the object are required for any ROI reconstruction using the conventional FBP algorithm. However, this problem can be relaxed using the new DHB algorithm. As shown in Figure 7c, small central ROIs can be reconstructed using several short arcs if only an ROI inside the object needs to be reconstructed. Reconstruction is possible for any ROI inside the central triangular region obtained by joining the beginning point of each arc with the end point of the next arc.