Monocular stereo vision features low cost and compactness compared to binocular stereo vision and has a broader application prospect in space-constrained environments. Stereo vision systems based on dual-biprism are widely employed in engineering measurement due to their adjustable field of view (FoV). Compared to other types of monocular vision systems, this method is compact and easy to adjust. Topography reconstruction and deformation measurement are the main application purposes of monocular vision systems. The error factors existing in the imaging system should be considered and evaluated to obtain high-precision measurement results. The acquisition and reconstruction of depth information are crucial for accuracy. The depth equation derived from the optical geometry can be adopted to analyze factors affecting the reconstruction accuracy. Analyzing the influence of object distances and angles on image depth information and disparity in depth equations can provide references for system layout and optimization. Additionally, the artificially placed dual-biprism has offset and rotation, and the errors caused by postures will change the imaging model which is derived in the ideal state. Therefore, model correction considering posture errors is important for high-precision imaging. Meanwhile, the dual-biprism posture will lead to the FoV difference. The quantitative study of the FoV caused by the posture can be helpful for the reasonable arrangement of system layout and object positions. Based on the previous studies, to make the monocular stereo system composed of dual-biprism more applicable to high-precision topographic reconstruction and deformation measurement, we will conduct an in-depth study on the influences of systematic errors and prismatic postures on the FoV.

The depth equation of the monocular vision system is expressed by geometrical optics and the ray tracing method. By making a small angle assumption and ignoring the distance between the dual-biprism and the camera, a depth equation with parameters such as disparity, included angle, and object distance can be obtained, as demonstrated in Eq. (8). By solving the partial derivative of the depth equation, the relationship among object distance, included angle, and disparity is obtained, as illustrated in Eqs. (9)-(10). The classification of prism postures is discussed, including rotation around the base point and offset along the x- or z-direction, as shown in Fig. 3. According to the systematic error introduced by the prism postures, the imaging model is further modified. Furthermore, the modified model is utilized to analyze the influence of prism postures on the FoV, as described in Eqs. (12)-(14). The experiments include verifying the validity of the derivation of the depth equation by leveraging the DIC results as true values, proving the model correctness by calculating the coordinates of the corner points, and investigating the FoV changes caused by the prismatic postures by matching the coordinates of the corresponding points. First, the experiment of object distance change is carried out. After keeping the object distance unchanged, the included angle of the prism is changed to evaluate the influence of the object distance and included angle on the disparity respectively. The DIC results are compared with the results of Eq. (8) to verify the derivation correctness. The dual-biprism is offset according to the classifications, the image is collected before and after posture changes, and the pixel coordinates of the corners of the whole field are extracted by the corner recognition method. The angular coordinates and offset distance before posture change are substituted into Eqs. (12)-(14). The calculated pixel coordinates are compared with the pixel coordinates identified above to verify the equation derivation correctness. Finally, the influence of postures on the FoV is determined by tracking the pixel coordinates of specific corners in the calibration plate before and after the prism posture changes.

The depth equation for the monocular stereo vision system can be described as Eq. (8). The influence of the parameters on the disparity can be obtained by solving the derivatives regarding the object distance and included angle for the depth equation respectively. The derived equations can be expressed as Eqs. (9) and (10). The depth equation description shows that the disparity decreases with increasing distance and shows a nonlinear change. As shown in Fig. 3, all three posture classifications cause a change in the standard virtual point model. The camera is calibrated after the device is placed to verify the derivation validity. As shown in Fig. 8, the FoV changes introduced by the postures can be obtained by tracking the corner points extracted in the calibration board before and after the posture variations. When the prism group rotates 1° clockwise around the base point, the FoV in each channel will shift anticlockwise, which will also cause the FoV in the overall overlapping area to move in this direction. If only the right prism rotates 1° clockwise, it will make the pixel coordinates of the virtual point in the right channel shift 57 pixels to the right, and the FoV offset of the side channel will reduce the overlapping FoV of the system. When the prism group is offset to the right by 1 mm along the x-direction, the same trend will be introduced. If only the right prism is offset, the virtual point will be offset by 49 pixels to the right. Meanwhile, when the prism group moves 1.4 mm along the positive half-axis of the z-direction, there is no significant FoV change in Fig. 8(c). The speckle images before and after the object distance and angle changes are captured, with the disparity map obtained. Then, the depth map can be computed using Eq. (8), in which the depth information of each point in the overlapping FoV can be obtained. The profile of the measured object can be obtained using the coordinate transformation method. The derivation correctness of the equation can be verified by selecting three cross-sections on the object and comparing the profiles of the object obtained by the two methods, as shown in Figs. 9-11. The corrected models considering the prismatic postures are illustrated in Eqs. (12)-(14). The pixel coordinates of the corner points obtained before the posture change are calculated by substituting them into Eqs. (12)-(14) to obtain the offset coordinates, which can be compared with the pixel coordinates of the corner points extracted after the offset to verify the correctness of the corrected model, as shown in Fig. 12.

The relationship between depth equation and disparity in prism-splitting type monocular stereo vision systems is studied, with the system error introduced by the dual-biprism postures considered. The depth equation of the system is derived by combining the virtual point model and ray tracing method. By solving the derivative of the depth equation, the influence of object distance and included angle on disparity is studied. The results show that the disparity of the image increases with the reducing object distance and rising included angle. The imaging model is modified to address the system errors introduced by postures. The experimental results show that the pixel coordinates of virtual points can be accurately calculated using the modified model with known offset distances of the dual-biprism and world coordinates of spatial points, which can determine the mapping relationship of spatial points for different prism postures. Finally, the rotation of the dual-biprism or the offset along the direction perpendicular to the optical axis of the camera will cause the FoV of the system to change, while the posture change along the optical axis of the camera will only reduce the imaging range. Finally, we can provide references for high-precision reconstruction and deformation measurement of monocular stereo vision systems composed of optical elements.