## 2 Unsaturated Subgrade Filler Triaxial Testing System
### Three-Dimensional Volume Measurement of Samples Using Photogrammetry  
Deformation data of the samples during the experiment are acquired using photogrammetry technology in this system. Overlapping images are captured with a digital camera, as shown in Figure 2a, and measurement markers are positioned on the loading frame and the surface of the triaxial chamber. The camera's position and orientation are inferred based on photogrammetry principles, as demonstrated by the perspective center \(S_{1}\) in Figure 2. The 3D coordinates of the measurement targets on the triaxial chamber wall are calculated from known camera parameters, allowing for the precise determination of the shape and orientation of the chamber's outer surface. By capturing images of the sample from various angles and applying image processing techniques, multiple 2D images are reconstructed into a 3D model, which precisely calculates the volume changes of unsaturated samples. This enables further analysis of shear strength characterization and constitutive model behavior during the 3D settlement consolidation process. This research proposes and refines a multiple ray-tracing technique to reconstruct the 3D model, as demonstrated in Figure 2. In Figure 2, \(S_{1}\), \(S_{2}\), and \(S_{\mathrm{n}}\) represent the positions of the camera's perspective centers, and subscripts 1, 2, and n represent the images captured at these locations. \(B_{1}\), \(B_{2}\), and \(B_{\mathrm{n}}\) represent the positions of point \(P\) in each image, and \(O_{1}S_{1}\), \(O_{2}S_{2}\), and \(O_{\mathrm{n}}S_{\mathrm{n}}\) represent the camera's shooting directions, with their magnitudes equal to the lens's focal length. For example, by using the direction obtained from the photogrammetry analysis shown in Figure 2c for image 1, the 3D coordinates of the points in the image can be calculated, thus determining the 3D coordinates of ray \(B_{1}S_{1}\). The shape and position of the triaxial chamber have been determined from the previous analysis, enabling the calculation of the intersection point \(C_{1}\) of ray \(B_{1}S_{1}\) with the chamber's outer surface, as shown in Figure 2a. By considering the known refractive indices of air (na) and the chamber wall (nc), along with the shape and position of the chamber's outer surface, the direction of the incident ray \(B_{1}S_{1}\), and the location of intersection \(C_{1}\), Snell's Law is applied to calculate the direction of the refracted ray \(C_{1}D_{1}\), as depicted in Figure 2b. A similar ray-tracing process can be applied to points B2 and Bn in images 2 and n, analyzing the same object point \(P\). If there are no errors, all traced rays \(D_{1}P\), \(D_{2}P\), and \(D_{n}P\) will converge at point \(P\) in three-dimensional space, as illustrated in Figure 2a. However, errors are inevitable in actual measurements, so the rays may not converge in three-dimensional space, as illustrated in Figure 2c. To address this issue, the least-squares optimization method proposed by [PERSON] et al. (2015) [34] may be referenced. By minimizing the sum of squared distances from a point to all traced rays, the source point of the rays is determined, allowing for the calculation of point \(P\)'s 3D coordinates, which are then used to calculate the 3D coordinates of multiple points on the sample's surface, constructing a complete 3D model to determine the total volume change of the sample. This method is subsequently employed to obtain the 3D coordinates of many points on the surface of the sample, forming an overall 3D model that yields the ability to calculate the total change in volume  
Figure 3c displays a photograph of the tensiometer. In this study, the tensiometer was saturated in a triaxial test chamber through multiple cycles of pressurization at 600 kPa to ensure complete saturation. Subsequently, the tensiometer was calibrated within the positive pressure range to ensure measurement accuracy and reliability.  
Reliability Evaluation of the 3D Settlement and Consolidation Testing System for Unsaturated Subgrade Fill