# Geometrical Calibration of a 2.5D Periapical Radiography System
## 1 Introduction  
The application of X-ray radiography in clinical dentistry is prevalent. Currently, dental radiographic images can be divided into 2D and 3D radiographic images, and 2D radiographs can be further divided into intraoral and extraoral. Equipment for capturing intraoral radiographs includes periapical, bitewing, and occlusal films, whereas that for extraoral radiographs includes panoramic and cephalometric films. The most substantial problem related to 2D radiographs is the superposition of tissue images, which leads to dentists being unable to accurately determine the solid forms of teeth or bone structures [1, 2]. The commonest form of 3D radiography is cone beam computed tomography (CBCT). This technique has the advantages of enabling 3D image developments in diagnosis, giving doctors more information related to the structures and forms of interior tissues at different depths [3]. However, CBCT image resolution can only reach a maximum of 70 \(\upmu\)m [4, 5]. This resolution is beneficial for examining patients with mandible fracture, nasal cavity, and tooth impaction, as well as for conducting preoperative planning of dental implant surgery. However, lesions such as root microfractures and root fractures are difficult to identify. In recent years, the development of digital periapical radiography has enabled an image resolution of at most 20 \(\upmu\)m [6]. This resolution enables clear observation of such lesions. However, the clinical applicability of this approach is still limited in that only 2D images can be captured [7]. To overcome the disadvantage of only providing 2D intraoral periapical film images, the present research team proposed a new piece of equipment, the 2.5D Periapical Radiography System, in 2018 [8]. This device is similar to a CBCT machine, with the most substantial difference being that a digital X-ray sensor is placed inside the patient's oral cavity. The device has a supporting frame for hanging the X-ray tube. During image capture, the tissue is observed on a rotation axis, and the X-ray tube has a limited angle of rotation (e.g., \(\pm\) 30\({}^{\circ}\)). Continuous 2D images of different angles are imported to a computer for image reconstruction using digital tomosynthesis, allowing the reconstructed images of different depths to be observed. Although the prototype of this proposed system can obtain reconstructed images in different depths, the shooting time is over 10 min because current commercially available digital X-ray sensors are unable to continuously shoot at high frame rates, rendering them inapplicable to clinical dentistry [9; 10; 11]. Therefore, our team developed a high-frame-rate sensor in 2019 [12]. This sensor can achieve a frame rate of up to 15 Hz, which substantially reduces the shooting time to under 10 s, thus increasing its clinical feasibility. However, artefacts and blurring were observed in images reconstructed using experimental in vitro images of teeth because hardware component positioning was not established through accurate geometrical calibration. Scholars have proposed calibration methods related to computed tomography (CT) and micro-CT [13; 14; 15; 16], most of which involve the use of a phantom with known interior geometrical positions for geometrical calibration of the hardware components. Phantoms for calibration are mostly items such as steel balls [15], ruby balls [14], and steel wires [13] covered in resin of a precise size. First, a single- or multi-angle projection is applied to the phantom, and feature points are captured from the obtained 2D images. Next, hardware errors are calculated, and the calculated error values are used to adjust the hardware component positions. Alternatively, the image deviation method is used to remove images that are blurred because hardware components are not installed in ideal positions [17]. However, methods proposed by previous scholars cannot be effectively applied to the 2.5D Periapical Radiography System because of the system's small sensor size and the sensor being too close to the object.