# Geometrical Calibration of a 2.5D Periapical Radiography System
## 2 Materials and Methods
### Geometric Calibration Algorithm  
Ideally, for hardware structures related to digital tomosynthesis, the focal spot of the X-ray tube (Point P) should be perpendicular to and intersect orthogonally with the sensor's center (Figure 2a). In addition, the positions of the focal spot (Point P), rotation axis (Point R), and central axis of the sensor should form a straight line (Figure 2b). The reconstructed images captured using this structure did not contain artefacts resulting from hardware being uncalibrated. However, in real-world settings, the focal spot (Point P) in an uncalibrated system may cause the linked line between points P and R to become non-perpendicular to the sensor, and the linked line between points P and R may not align with the center of the sensor. Such conditions cause artefacts and blurs in the reconstructed images and influence image quality (Figure 2c). Figure 1: The 2.5D Periapical Radiography System was developed in 2018 [(8)]. (**a**) Prototype of the intraoral digital tomosynthesis system. (**b**) Reconstructed images obtained from the 2.5D prototype of the intraoral digital tomosynthesis system at different distances from the sensor surface. Figure 2: The difference between the ideal hardware position and the actual hardware component position. (**a**) Major components of the 2.5D Periapical Radiography System. (**b**) Ideal tomosynthesis geometry (focal spot, Point P; location of the two-steel-ball phantom, B and B’; projection points D and D’ of B and B’; rotation axis point R). (**c**) Real tomosynthesis geometry. The calibration method proposed in this study is conducted in two steps (Figure 3). In Step I, the focal spot of the X-ray tube is rotated and adjusted so that the linking line between the focal spot and the rotation axis is orthogonal to the sensor. In Step II, the linking line of the focal spot is adjusted so that the rotation axis is orthogonal to the sensor at its center. Step I (Figure 4) calculates the distance (r) between the focal spot of the X-ray tube (Point P) and the rotation axis of the X-ray tube's rotation structure, the angle (\(\Downarrow\)) between the linking line of the focal spot (Point P) and rotation axis (Point R) and the straight line orthogonal to the sensor (\(\overline{\text{NR}}\)), and the distance (L\({}_{4}\)) between the rotation axis (Point R) and the linking line between the two steel balls of the phantom. This calibration method requires a two-steel-ball phantom of known precise sizes, and the distance between the two steel balls (d) must be known. The two-steel-ball phantom is expressed to the sensor (Bz and Bz' are known), and the X-ray tube's focal spot shoots one 2D image at Point P (Figure 4a) with the projective distance of the phantom to the sensor calculated as L\({}_{1}\) (= \(\overline{\text{D}\text{D}^{\prime}}\)). Then, we rotate for angle \(\pm\theta\) (\(\theta\) can be any value less than 60\({}^{\circ}\) except 0\({}^{\circ}\), while 30\({}^{\circ}\) is recommended--if \(\theta\) is larger than 60\({}^{\circ}\), the projection of the steel ball will fall outside the sensor range) and move the X-ray tube to Point Q (Figure 4b) and Point R (Figure 4c) to shoot one 2D image each with the projective distance of the phantom on the sensor as L\({}_{2}\) (= \(\overline{\text{D}\text{D}^{\prime}}\)) and L\({}_{3}\) (= \(\overline{\text{D}\text{D}^{\prime}}\)). L\({}_{1}\), L\({}_{2}\), and L\({}_{3}\) can be calculated from the 2D images and the actual resolution size of the sensor. Finally, we substitute the known parameters Bz, Bz', \(\theta\), d, L\({}_{1}\), L\({}_{2}\), and L\({}_{3}\) into the derived equations as follows. Parameters r, \(\psi\), and L\({}_{4}\) can then be calculated. Figure 3: Two-step calibration process. The calibration method proposed in this study is conducted in two steps. In Step I, the focal spot of the X-ray tube is rotated and adjusted so that the linking line between the focal spot and the rotation axis is orthogonal to the sensor.