## 2 Related Works
### Estimation of Robot Pose and Pipe Diameter  
The accurate estimation of robot pose and pipe diameter is important in ILI. A robot pose, describing its orientation and displacement inside a pipe, is modeled with six degrees of freedom (DoFs) consisting of roll, pitch, yaw, and X-, Y-, and Z-axis displacements. Because X-axis displacement is typically measured using an odometer in most in-pipe robots [1, 2, 4, 5], this paper focuses on a five-DoF pose excluding X-axis displacement. Modeled as an elliptic cylinder, the pipe diameter is defined by its major and minor axes. It is essential for detecting structural deformations caused by manufacturing imperfections or external pressures. These estimates are also critical for improving inspection accuracy. To tackle these challenges, various sensor-based approaches have been proposed: structured light [6, 7, 8, 9, 10, 11], 2D cameras [12, 13], and stereo cameras [14, 15]. Structured light methods project patterns onto the pipe surface and analyze their deformations in 2D camera images to estimate the robot pose and pipe diameter. In [6], the authors propose a method to estimate two-DoF poses (pitch and yaw) using a hyperbolic mirror and point light pattern by analyzing the point location in the image. A later scheme in [7] introduces a ring-shaped light pattern to estimate four-DoF poses (pitch, yaw, Y-displacement, and Z-displacement) by matching observed rings with pre-collected image databases of known poses. The schemes in [8, 9] reformulate the problem as a non-linear optimization task using ring-shaped light patterns. Additionally, a ring-shaped light is employed to estimate the pipe diameter by sampling the ring radius [10] or fitting the ring image with an ellipse [11]. However, this approach relies solely on 2D image analysis without capturing depth information, making it impossible to fully estimate five-DoF poses and pipe diameter. The 2D camera approach applies visual odometry to estimate six-DoF poses, including X-displacement. Instead of analyzing a single image, this approach utilizes sequential images to overcome the lack of depth information. [PERSON] et al. propose a feature-basedvisual odometry algorithm that extracts and matches corner features between images, enabling incremental pose updates [12]. To improve accuracy, structured light with two points is added in [13]. However, this approach depends on surface features, which can be unclear due to lighting conditions or dust, leading to matching failures. Furthermore, drift errors accumulate over time due to inaccuracies in successive frame matching. A more recent approach utilizes stereo cameras to reconstruct 3D point clouds of pipe surfaces. [PERSON] et al. employed four stereo cameras to measure the pipe surface and estimate robot pose using visual odometry applied to 3D point clouds [14]. Although this method achieved higher accuracy than 2D approaches, it still relied on surface features and demanded multi-frame processing. To address these issues, [PERSON] et al. proposed a multi-camera system that emulates four stereo cameras combined with a ring-shaped light [15]. The 3D point cloud is constructed by collecting ring light points across consecutive frames, and poses and diameters are estimated using non-linear cylinder fitting. Although this approach improves robustness, it requires rail-mounted setups for consistent pose alignment across frames and imposes high computational demands due to extensive calibration among multiple cameras. While structured light, 2D cameras, and stereo camera systems provide viable solutions for robot pose and pipe diameter estimation, they face limitations such as feature dependency, multi-frame processing, and computational complexity. In contrast, 3D ToF sensors offer a practical and scalable alternative by capturing depth data directly in a single frame without relying on features or multi-frame alignment.