## 4 Numerical Results
### Estimation Results Based on Experimental Data  
In this section, we evaluate the performance of C-EPD and RU-EPD on the real data collected using a Helios2 3D ToF camera on a 20\({}^{\prime\prime}\) pipeline. Figure 9 shows the robot and pipeline used in the experiment. The Helios2 3D ToF camera was equipped in front of the robot. A PC was also installed to control and store the measurement data from the camera. The experiment pipeline consisted of 4 straight pipes, one elbow, and one T-junction. The robot's trajectory in the pipeline is marked with the red arrows in Figure 9. Due to physical constraints in the pipeline, data collection focused on two straight sections (11.9 T and 12.7 T), with 11.9 T and 12.7 T indicating that the nominal pipe thickness was 11.9 and 12.7 mm, respectively. In each pipe, we collected 50 frames from the Helios2 3D ToF camera. Figure 10 shows an example of measurement data from the Helios2 3D ToF camera. To remove the outliers, we only kept the points within \([0.5,2]\) m for estimation. Additionally, we calibrated the Helios2 3D ToF camera to remove the depth bias using the approach in [27]. In this approach, the calibration information included correction offsets which were precomputed based on multiple frames collected in pipes with known diameters. The frames were first aggregated to remove random depth errors; then, the offsets from points in the aggregated frames to the pipe surface with a known diameter were used as the correction offsets. Calibration information was computed and stored in a lookup table separately for different pose and intensity intervals to address their diversities. Since the ground-truth pose could not be measured, we only evaluated the diameter estimation.  
Figure 8: Comparison of pipe diameter estimation errors between C-EPD and RU-EPD under varying input sizes.  
Figure 9: Experimental robot and pipeline.  
Figure 11 shows the estimation results of the major axis and minor axis in the 11.9 T and 12.7 T pipes. We can see a similar trend in both pipes. There were several samples where C-EPD estimated a very high major axis compared to the nominal diameter. The minor axis estimation of C-EPD was approximately around the nominal diameter. On the other hand, the major axis with RU-EPD was much more consistent and always smaller than the one with C-EPD. Most of the minor axis estimation in RU-EPD was below the nominal diameter.  
Figure 12 shows the estimation errors of the major axis, minor axis, and mean diameter in the 11.9 T and 12.7 T pipes. Since the major axis error was positive and high with C-EPD and the minor axis error was small, it resulted in a clear estimation bias in the mean diameter as high as 4.2 mm. On the other hand, the major- and minor-axis errors with RU-EPD had opposite signs and had approximately the same value. Therefore, it led to an almost negligible estimation bias in the mean diameter of at most 0.46 mm, which was nine times smaller than that of C-EPD. We can also see that the estimation error range was very high with C-EPD. The estimation error range of the diameter was 18.16 mm in the 11.9 T pipe and 15.24 mm in the 12.7 T pipe. On the other hand, the estimation error range of the diameter with RU-EPD was at most 3.62 mm, which was at least four times smaller. The experiment results also confirm that RU-EPD had almost no estimation bias and robust estimation.  
These results demonstrate the superior performance of RU-EPD over C-EPD in both the simulated and experimental scenarios. RU-EPD effectively mitigates estimation bias and delivers robust estimations, making it a promising solution for pipeline inspection tasks.  
Figure 10: Measurement data collected by Helios2 3D ToF camera.