## 2 Materials and Methods
### Materials and Data Acquisition  
In order to evaluate the effectiveness of the proposed method, both simulation data and experimental data are used for image reconstructions. In simulation, five defect distributions are designed to evaluate the reconstruction results. The simulation data contains graph of defect distribution and graph of rays. The graph of defect distribution is manually designed and shows the size and shape of virtual defects, and is the expected output result of the proposed image reconstruction algorithm. The graph of rays is manually generated based on the corresponding graph of defect distribution, and is the input of the proposed method. Figure 4 illustrates these corresponding normalized defect distributions, including single circle distribution, single edge distribution, two-edge distribution, two-rectangle distribution, and three-circle distribution. In addition, three test samples are utilized in the experiments. The graph of rays in a real experiment is generated by the collected velocity data. Figure 4 also illustrates these real samples, including Chinaberry timber with hole defects, Firmiana timber with knot defects, and Pecan timber with crack defects. This study uses the self-developed portable timber tomographic imaging instrument. As shown in Figure 5, this instrument contains a stress wave signal processor box, several data wires, and 12 stress wave sensors. To detect a timber sample, sensors are equally fixed around the cross-section of a trunk section. When a sensor is knocked by the electronic hammer, all remaining sensors will receive the stress wave signal. Then, the signal processor analyzes the signals with DSP (digital signal processing) technology in order to obtain the stress wave transmission time. This paper focuses on the image reconstruction algorithm, and the details of how to collect the transmission time with the instrument have been reported in our previous work [18; 35; 36]. Figure 4: Simulation data (graph of defect distribution and graph of rays) and experimental data (real image of sample and graph of rays): (**a**) Single circle distribution; (**b**) single edge distribution; (**c**) two-edge distribution; (**d**) two-rectangle distribution; (**e**) three-circle distribution; (**f**) firmiana timber with hole defects; (**g**) chinaberry timber with knot defects; (**h**) pecan timber with crack defects. of reconstructed image is 300 \(\times\) 300 pixels. The resolution of the simulation defects image is also 300 \(\times\) 300 pixels, and the proportion of the reconstructed image size to the simulation data is 1:1. To express the appropriate visual effects for the rays graph and the reconstructed image, red is used to represent the lowest velocity values, yellow is used to represent low velocity values, and green is used to represents high velocity values. Besides, \(c_{1}\) is the control coefficient of the elliptical shape in RSEN algorithm, and \(c_{2}\) is the control coefficient of the ellipse for interpolation in the SISE algorithm. In order to show the results of the proposed method more comprehensively, the result of the RSEN algorithm and the result of the SISE algorithm are both displayed in this section. The original rays graph and segmented rays graph using the RSEN method from the simulation data with different control coefficients are shown in Figure 6. It was found that the RSEN algorithm significantly improved the original rays graph. Compared with the original graph of rays, the segmented graph of rays can reflect the spatial distribution of defects better. The approximate regions of defects are even reflected clearly in the segmented rays graph. Figure 6: Original rays graph and segmented rays graph using the RSEN method from simulation data with different control coefficients. On the other hand, all the possible values of the control coefficient \(c_{1}\) have been studied (that is, from 0 to 1). To clearly display the results, we selected three representative values of \(c_{1}\) for each sample (\(min\), \(mid\), and \(max\)).