# Geometrical Calibration of a 2.5D Periapical Radiography System
## 3 Results  
Numerical simulations were used to verify the accuracy of this calibration method, and SolidWorks 2016 (Dassault Systemes SolidWorks Corporation, Commonwealth of Massachusetts, USA) was used to verify the simulation accuracy. The desired positions of each component and error values could be precisely set in the 3D coordinates of SolidWorks. Then, the position parameters of the phantom's projection on the digital X-ray sensor could be obtained. Finally, these position parameters were substituted into the calibration method proposed in this study to verify its accuracy and feasibility; this involved calculating whether the calculated error and the setting error are identical.  
The calibration method proposed in this study involves two steps; hence, the computer simulation was divided into two stages as well. The parameters calibrated in Step I were R, \(\psi\), and L\({}_{4}\). Table 1 lists the five setting errors for computer simulation. The calculation errors from Step I and the self-established errors were determined to be \(<0.015\%\).  
The proposed calibration method involves two steps; hence, the simulation in the computer was divided into two stages as well. The parameters calibrated in Step II were Pz, B'x, B'y, Bx, By, Px, and Py. Table 2 lists the five setting errors for computer simulation. The errors of calculation determined from Step II and the self-established errors were all smaller than \(0.01\%\).  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Test Number**} & \multicolumn{3}{c}{**L\({}_{4}\) (mm)**} & \multicolumn{3}{c}{\(\psi\) (\({}^{\circ}\))} & \multicolumn{3}{c}{**r (mm)**} \\ \cline{2-9}  & **Ideal** & **Estimate** & **Error (\%)** & **Ideal** & **Estimate** & **Error (\%)** & **Ideal** & **Estimate** & **Error (\%)** \\ \hline
1 & 5.0 & 5.00004399 & 0.00088 & 3.0 & 3.00034641 & 0.01155 & 335 & 335.000001 & 0.00000 \\
2 & 6.5 & 6.499985761 & \(-\)0.00022 & 2.8 & 2.79995015 & \(-\)0.00178 & 332 & 331.5000007 & 0.00000 \\
3 & 4.1 & 4.100159545 & 0.00389 & 3.7 & 3.70041803 & 0.01130 & 337 & 337.2999967 & 0.00000 \\
4 & 5.3 & 5.300042976 & 0.00081 & 1.3 & 1.300326732 & 0.02513 & 334 & 333.8000022 & 0.00000 \\
5 & 4.2 & 4.199978173 & \(-\)0.00052 & 5.9 & 5.899961593 & \(-\)0.00065 & 333 & 332.6999966 & 0.00000 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Calibration results of the three parameters obtained from Step I.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{4}{c}{**Group 1**} \\ \hline
**Parameter** & **Ideal** & **Estimate** & **Error (\%)** \\ \hline Pz & 350 & 350.000001 & 0.00000 \\ B’x & \(-\)12.1756613 & \(-\)12.1756622 & 0.00001 \\ B’y & 1.6572935 & 1.6572934 & \(-\)0.00001 \\ Bx & 7.8076613 & 7.8076605 & \(-\)0.00001 \\ By & 0.8407065 & 0.8407063 & \(-\)0.00002 \\ Px & \(-\)2.438 & \(-\)2.4380285 & 0.00117 \\ Py & 6.892 & 6.8919944 & \(-\)0.00008 \\ \hline \hline \multicolumn{4}{c}{**Group 2**} \\ \hline
**Parameter** & **Ideal** & **Estimate** & **Error (\%)** \\ \hline Pz & 350 & 349.999999 & 0.00000 \\ B’x & 9.25091935 & 9.25091994 & 0.00001 \\ B’y & \(-\)0.13745354 & \(-\)0.1374539 & 0.00026 \\ Bx & \(-\)10.7109194 & \(-\)10.7109188 & \(-\)0.00001 \\ By & 1.09745354 & 1.09745319 & \(-\)0.00003 \\ Px & \(-\)0.54 & \(-\)0.5399793 & \(-\)0.00383 \\ Py & 0.19 & 0.1899876 & \(-\)0.00653 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Calibration results of the parameters obtained from Step II.