# Geometrical Calibration of a 2.5D Periapical Radiography System
## 2 Materials and Methods
### Geometric Calibration Algorithm
From Step I: parameters r (distance of the focal spot of the X-ray tube to Point R), \(\psi\) (included angle between the linking line formed by Point P and Point R and the straight line orthogonal to the sensor), and L\({}_{4}\) (distance between the rotation axis position and the phantom) can be calculated. 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. From Step II: parameters Px (focal spot x-coordinate value), Py (focal spot y-coordinate value), and Pz (focal spot z-coordinate value) can then also be calculated. (See Appendix A for a more detailed derivation about step I.) Before Step II is conducted, we rotate the X-ray tube by -U\({}^{\circ}\) (calculated from Step I): specifically, we adjust the focal spot to the position in which its linking line with the rotation axis is orthogonal to the sensor (Figure 5). Figure 4: Flow chart of Step I. Step I 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 (\(\psi\)) between the linking line of the focal spot (Point P) and rotation axis (\(\overline{\rm PR}\)) and the straight line orthogonal to the sensor, and the distance (L\({}_{4}\)) between the rotation axis (Point R) and the linking line between the two steel balls of the phantom: (**a**) The actual hardware positions directly after assembly; (**b**) the X-ray tube’s focal spot rotates +\(\oplus^{\circ}\); (**c**) the X-ray tube’s focal spot rotates -\(\Theta^{\circ}\). P is the position of the focal spot, R is the position of the rotation axis, B and B’ are the positions of the two steel balls on the phantom, and D and D’ are the positions of the projections of the two steel balls on the sensor. Then, we employ Step II (Figure 6) to calculate the deviational distances (Px, Py, and Pz) on the x, y, and z axes of Point P away from the center of the sensor. Similarly, the two-steel-ball phantom is expressed to the sensor, and the X-ray tube shoots one 2D image at Point P (Figure 6a) with the projective distance of the phantom on the sensor as L1 (= \(\overline{\text{D}\,\text{D}^{\prime}}\)). Then, the X-ray tube is rotated by \(\theta^{\circ}\) (\(\theta\) can be any value less than 60\({}^{\circ}\) except 0\({}^{\circ}\), while 30\({}^{\circ}\) is recommended) and moved from Point P to Q (Figure 6b) in order to shoot another 2D image, with the projective distance of the phantom on the sensor being L2 (= \(\overline{\text{D}\,\text{D}^{\prime}}\)). We substitute the known parameters By, By\({}^{\prime}\), L1, L2, d, and r into the derived equations described as follows to calculate the three parameters Px, Py, and Pz, which are the error values between the focal spot and the ideal position. In addition, Pz + r + L4 + Bz is the distance between the focal spot and the sensor. (See Appendix A for a more detailed derivation about step II.) Figure 5: Before Step II is conducted, we rotate the X-ray tube by -\(\psi^{\circ}\) (calculated from step I): specifically, we adjust the focal spot to the position in which its linking line with the rotation axis is orthogonal to the sensor. (**a**) Original position of the X-ray tube. First, rotate the X-ray tube by -\(\psi^{\circ}\) (calculated from Step I). (**b**) Hardware positions after adjustment. The focal spot is adjusted to where its linking line with the rotation axis is orthogonal to the sensor. Figure 6: Flow chart of Step II. Step II is used to calculate the deviational distances (Px, Py, and Pz) on the x, y, and z axes of Point P away from the center of the sensor. First the X-ray tube is at Point P to shoot a 2D image, then moved from Point P to Q in order to shoot another 2D image. (**a**) Point P after the calibration in Step I. (**b**) Focal point rotates +\(\theta^{\circ}\).