# High-Precision Noncontact Guided Wave Tomographic Imaging of Plate Structures Using a DHB Algorithm
## 2 Tomography Theorem  
Tomography imaging was used to obtain the section view of the specimen under testing. Image reconstruction from the data projection is a mathematical problem. The original object can be copied from a large number of projection signals. The results of computed tomography are the distribution image of the linear attenuation coefficients. The DHB algorithm used in this work has evolved from the traditional FBP algorithm. The experiment results show that the imaging quality obtained from the DHB algorithm is much better than that obtained from other analogous algorithms such as RAPID or FBP. These two frequently used algorithms are also discussed to emphasize the advantages of the new DHB algorithm. The theoretical foundation and the corresponding experimental results will be illustrated in the following part of this paper. The RAPID is based on a damage index known as the signal difference coefficient (SDC). This SDC can be taken as a measurement of how statistically different the defect signals are from a baseline nondefect signal. Damage indexes for all transducer pairs are spatially distributed. Those damage indexes are summed to generate an image of the damage area. The first step of this algorithm is to calculate SDC between the current signal \(x_{ij}(t)\) and the reference \(y_{ij}(t)\). The index \(i\) is for the transmitter and \(j\) is for the receiver. Mathematically, the SDC can be stated as:  
\[SDC_{ij}\ =\ 1-\left|\frac{\int_{t_{0}}^{t_{0}+\Delta T}[x_{ij}(t)-\mu_{x}][y_{ ij}(t)-\mu_{y}]^{2}dt}{\sqrt{\int_{t_{0}}^{t_{0}+\Delta T}[x_{ij}(t)-\mu_{x}]^{2} dt\int_{t_{0}}^{t_{0}+\Delta T}[y_{ij}(t)-\mu_{y}]^{2}dt}}\right|, \tag{1}\]  
where \(t_{0}\) is the direct arrival time for each transducer pair, \(\mu_{x}\) and \(\mu_{y}\) are the mean values of the corresponding current signal and baseline signal, and \(\Delta T\) is the time window. The next step after the SDC values for all sensor pairs are calculated is image reconstruction. An image is generated by distributing each SDC value spatially on the image plane in an elliptical pattern. The two foci of the ellipses are the two transducers' locations. A parameter \(\beta\) controls the size of the ellipse. The amplitude linearly tapers from its maximum value along the line connecting the ellipse foci to zero on the periphery of the ellipse. The parameter \(\beta\) is a shape factor that controls the size of the elliptical distribution. Its value can be arbitrary, but greater than 1.0. Finally, the image amplitude at each pixel is a linear summation of the location probabilities from each transmitter-receiver pair (M). In Equation (2), the coordinate position of the image is displayed. \(x_{RK},y_{RK}\) is the coordinate position of the sensor with \(x_{RK},y_{RK}\) is the coordinate position of the receiving sensor. \(P_{K}(x,y)\) is the image value of image obtained from a pair of sensors and sensors. \(P(x,y)\) is the sum of the image values obtained from the sensors of all pairs (M) and the receiving sensor. \(R(x,y,x_{TK},y_{TK},x_{RK},y_{RK})\) is a formula showing the relationship between the coordinate position of the image and the position of the sensor. The larger the \(R(x,y,x_{TK},y_{TK},x_{RK},y_{RK})\), the farther away from the shortest distance of the pair of sensors. \(\beta\) is a shape factor that limits the coverage of \(R(x,y,x_{TK},y_{TK},x_{RK},y_{RK})\). \[P(x,y)\ =\ \sum_{i\ =\ 1}^{N-1}\mathrm{P}_{k}(x,y)\ =\ \sum_{K\ =\ 1}^{M}\frac{SDC}{\beta-1}\{\beta-R(x,y,x_{TK},y_{TK},x_{RK},y_{RK})\} \tag{2}\]  
The RAPID has been widely used in the field of nondestructive testing. However, some disadvantages of this algorithm are gradually being exposed with the increasing demands of imaging quality. Those disadvantages are listed as follows. In signal extraction, the time window \(\Delta T\) in Equation (1) should cover at least two modes of Lamb waves because it is necessary to know the difference between the current signal and baseline signal.