# Geometrical Calibration of a 2.5D Periapical Radiography System
## 4 Discussion
The literature related to hardware position calibration for CT or micro-CT [14; 15; 33; 34] indicates that phantoms can be used to calibrate 5 to 11 hardware position error parameters [14], with the 7 most common presented as follows: (1) sensor offset along the x axis, (2) sensor offset along the y axis, (3) angle of sensor tilted around the x axis, (4) angle of sensor tilted around the y axis, (5) angle of sensor tilted around the z axis, (6) perpendicular distance from the digital X-ray sensor to the rotation axis, and (7) perpendicular distance from the rotation axis to the X-ray tube. Numerous scholars have used phantoms to calibrate the hardware positions of CT or micro-CT instruments [32; 33]. Calibration can be divided into two types, single- and multi-angle projection. Various scholars have proposed multiprojection calibration using 12 or 6 [35] steel balls to form two-ring phantoms. [PERSON] [34] and [PERSON] [14] have proposed single projection methods using four- and nine-point phantoms to determine component position errors. However, these calibration methods for CT or micro-CT cannot be applied to the 2.5D Periapical Radiography System. The major problem is that the system shoots using digital tomosynthesis with stationary sensors and objects, which differ from those used in CT or micro-CT. Currently, digital breast tomosynthesis is the commonest digital tomosynthesis approach used in clinical settings. In 2003, General Electric Company proposed a patent for geometrical calibration that applies to breast tomosynthesis [36]. In laboratory settings, researchers have also attempted calibration using two-ring phantoms or five-steel-ball phantoms [35], but these methods are unsuitable for the 2.5D Periapical Radiography System. This is mainly because the area of the digital X-ray sensors used in this system is considerably smaller than those of other systems. In addition, the object (tooth) is extremely close to the sensor; consequently, the magnification level is insufficient for accurately identifying the form and characteristics of the phantoms. For the calibration of digital tomosynthesis systems with small-area digital X-ray sensors, [PERSON] also proposed using a two-steel-ball phantom, but this method requires an estimated focal spot position before calibration and calculation can be conducted [19]. In this study, a new digital tomosynthesis method was proposed that involves two-step calibration using a two-steel-ball phantom with known geometrical positions. In the first step, an X-ray tube is used to shoot one 2D X-ray image at three different positions (P, Q, and R) separately. Then, the positions of the two steel balls are captured on the 2D image and substituted into equations to calculate the following three hardware position parameters: (1) the distance r between the focal spot, Point P, and the center of the X-ray tube rotation structure; (2) the included angle \(\psi\) between the straight line (\(\overline{\text{PR}}\)) connected by the focal spot, Point P, and the rotation axis and the straight line (\(\overline{\text{NR}}\)) that intersects the sensor orthogonally; and (3) the distance L\({}_{4}\) between the rotation axis, Point R, and the linking line of the two steel balls on the phantom. Next, the hardware components are adjusted in accordance with the parameters obtained in Step I before calibration is conducted in Step II. In the second step, the X-ray tube is used to shoot one 2D X-ray image at two positions (P and Q). Then, the positions of the two steel balls on the 2D image are captured and entered into equations to calculate the offsets (Px, Py, and Pz) of Point P of the X-ray tube from the sensor center on axes x, y, and z. According to the computer simulation results, the calibration method proposed in this study has extremely high accuracy. The errors of the estimation method from the ideal positions in Steps I and II were smaller than 0.015% and 0.01%, respectively. This method is applicable to digital X-ray sensors in small areas.