## 2 Monocular Visual Digital Photography
### Cooperative Transform Shifting Parallax Method  
The cooperative transform shifting parallax method was the core algorithm of the monocular visual digital photography technique used in this study [34; 35]. It requires the photographic optical axis to be perpendicular to the reference plane. Moreover, the displacement of the deformation monitoring point was transformed into the real displacement of the deformation point, which was calculated according to the mathematical relationship between the photographic scales of the reference plane and the object surface (object plane) (Figure 3). The detailed deduction process is as follows:  
Figure 2: Distortion correction process, showing the (**a**) distorted image, (**b**) image after distortion correction, and (**c**) cut corrected image. In Equation (3), \(\Delta x^{de}\) and \(\Delta z^{de}\) of the deformation point on images on the object plane can be expressed as follows:  
\[\Delta x^{de}=\frac{\frac{S\Delta}{S\alpha}\Delta p_{x}^{de}=m\times\Delta p_{x} ^{de}}{\Delta z^{de}=\frac{S\Delta}{S\alpha}\Delta p_{z}^{de}=m\times\Delta p_{ z}^{de}} \tag{3}\]  
where \(m\) is the photographic scale of the reference plane. \(\Delta x^{de}\) and \(\Delta z^{de}\) are the horizontal and vertical displacements of the deformation points on the object plane, respectively. \(\Delta p_{x}^{de}\) and \(\Delta p_{z}^{de}\) are the horizontal and vertical parallax errors of the corresponding deformation points on images, respectively. Notably, \(\Delta p_{x}^{de}\) and \(\Delta p_{z}^{de}\) contain system errors at this time. \(\Delta p_{x}^{de}\) and \(\Delta p_{z}^{de}\) can be expressed through Equation (4):  
\[\Delta p_{x}^{de}=(\chi_{2}^{de}-\chi_{1}^{de})-(dx_{2}^{de}-dx_{1}^{de}) \tag{4}\]  
where points \(a(\chi_{1}^{de},\chi_{1}^{de})\) and \(b(\chi_{2}^{de},\chi_{2}^{de})\) are the homonymy points on the zero image and subsequent images, respectively. Meanwhile, \((dx_{1}^{de},dz_{1}^{de})\) and \((dx_{2}^{de},dz_{2}^{de})\) are system errors of the homonymy deformation points on the zero image and subsequent image, respectively. Dynamic Monitoring Test of the Standard Penetration of Large-Diameter Tubular Piles and the Relevant Construction Effect  
In this paper, a case study was carried out on the renovation and expansion project along the section of the Beijing-Taipei High-Speed Rail Corridor from Dezhou (Shandong-Hebei Border) to Qihe. This region covers the alluvial plain of the Yellow River. The field test was performed in Pingyuan County, Dezhou City. The route is shown in Figure 4. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**Calibration Method**} & \multicolumn{1}{c}{**PDMS**} & **MATLAB** \\ \hline \multirow{4}{*}{Camera internal parameters} & \(f_{x}\) & 3095.729583 & 3115.151478 \\  & \(f_{y}\) & 3081.980675 & 3101.594706 \\  & \(c_{x}\) & 1979.634456 & 1990.049731 \\  & \(c_{y}\) & 1510.257453 & 1513.00499 \\ \hline \multirow{2}{*}{Average re projection error} & Mean error in Pixel & 0.2256 & 0.3800 \\ \hline \multirow{4}{*}{Deformation parameters} & \(k_{1}\) & 0.286489 & 0.342840 \\  & \(k_{2}\) & \(-1.464689\) & \(-1.956038\) \\ \cline{1-1}  & \(p_{1}\) & \(-0.000442\) & 0.000571 \\ \cline{1-1}  & \(p_{2}\) & \(-0.000537\) & \(-0.000147\) \\ \cline{1-1}  & \(k_{3}\) & 2.365860 & 3.615712 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Internal camera parameters and distortion coefficients of the Samsung Galaxy S9 rear camera. Figure 3: Cooperative transform shift parallax method, showing the (**a**) data acquisition and processing equipment and (**b**) principle of cooperative transform shift parallax method. Through field geotechnical explorations, the strata in the study area were found to be plain fill, plain fill, silt, silty clay, aleurite, silt, aleurite, fine sand, and silt, respectively (Table 2). Among them, the seventh layer of fine sand was relatively thick, and it was one of the choices for use as the bearing stratum at the pile end.