# Three-Dimensional High-Energy Electron Radiography Method for Static Mesoscale Samples Diagnostics
## 3 The Start-to-End Simulation Studies of TDHEER Method
### Three-Dimensional Reconstruction and Results Analysis  
The three-dimensional reconstruction (TDR) method has been well developed in recent decades. Many algorithms, such as FBP (filter back projection) [15], ART (algebraic reconstruction technique), SIRT (simultaneous iterative reconstruction technique), SART (simultaneous algebraic reconstruction technique) [11], MEM (maximum entropy method) [16], have been developed and applied in many fields for solving different problems. The TDR of the mass density of an object from its two-dimensional line projection is the core of electron tomography [17; 18]. Therefore, electron tomography can be seen as a problem of inverting projection transformation to recover the distribution of the mass density of the original object. Various algorithms have been proposed to cope with the practical difficulties of this inversion problem and they differ widely in terms of their robustness with respect to noise in the data, completeness of the collected projection data set, errors in projections' orientation parameters, and the ability to efficiently handle large data sets. Our TDHEER method is similar to electron tomography but with an inverse image due to the imaging lens, so the TDR algorithm used in the electron tomography is also suitable for TDHEER. The FBP algorithm is selected in our TDHEER studies due to its computational timesaving and ease of implementation. The FBP is an analytic reconstruction algorithm designed to overcome the limitations of conventional back-projection. It applies a convolution filter to remove blurring. Here we use \"Ram-Lak\" filter in the TDHEER reconstruction [19]. The sample used for TDHEER studies is shown in Figure 3a. As in the HEER imaging process shown in Figure 1, the sample radiography images are collected of each rotation angle from 0 to 179 degree. They were used for 3D reconstruction studies. The reconstruction results with the FBP algorithm are shown in Figure 4, where (a-c) are the reconstructed slice images along the axis \(x\) from \(x-\) to \(x+\); (d-f) are the reconstructed slice images along the axis \(y\) from \(y+\) to \(y-\); (g-i) are the reconstructed slice images along the axis \(z\) from \(z-\) to \(z+\). Depending on the slices' numbers and directions, the \(x\), \(y\), and \(z\) positions of the inside ball can be attained. For each of the three axes, the thickness and the \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sample/Material** & **X (cm)** & **Y (cm)** & **Z (cm)** & **R (cm)** \\ \hline Red and big/aluminum & 0 & 0 & 0 & 0.01 \\ Blue and small/hollow & \(-\)0.004 & 0.005 & 0 & 0.003 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The spherical sample parameters used in simulation. total number of slices are 0.03 cm and 100, respectively. Therefore, the thickness of each slice is 3 \(\upmu\)m. The inner ball in each slice image along the three axes is shown from invisibility to visibility, and then to invisibility again, by which the inner ball center position of each axis is determined, shown in Table 3. The errors of the inner ball center positions are due to the slice number uncertainty of the first and last slice images with the inner ball. The reconstruction results are consistent with the original settings in the simulation, and they confirm the validation of the TDEER, from which the internal structure of the samples can be exactly determined. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Inner Ball/Material** & **X (cm)** & **Y (cm)** & **Z (cm)** \\ \hline Hollow (original settings) & \(-0.004\) & \(0.005\) & \(0\) \\ Hollow (reconstruction results) & \(0.0042\pm 0.0003\) & \(-0.0054\pm 0.0003\) & \(0\pm 0.0003\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The reconstructed inner ball \(x\), \(y\), and \(z\) positions compared with original settings.