Results and Discussions First, the experimental results of the camera with the typical ordinary focal length (f=20 mm) show that the error of the calibration method employing the perspective projection is relatively lower (Fig. 4 and Fig. 5), which indicates that the perspective projection better describes the imaging process under the current configuration. In this case, the approximation error of the affine projection model cannot be negligible. Furthermore, the results demonstrate that the proposed calibration method is effective and reliable, and can be applied to ordinary cameras within a certain threshold. Additionally, the experimental results of the camera with typical telephoto lenses (f=100 mm) show that the calibration method of the telephoto camera based on the approximation projection model has better accuracy and robustness (Fig. 6 and Fig. 7), and the affine approximation projection model can be better adapted to the imaging process of telephoto cameras.As for actual experimental results, under the condition of ordinary focal length (f=35 mm) in Table 1, despite a small deviation, the calibration results of the Weak and Para methods under the affine approximation projection are still close to those of RML and RNML methods under the perspective projection model. This indicates that the calibration method under the perspective projection model can obtain higher precision results, and the approximation error of the affine projection model cannot be ignored. Further comparison of the calibration results of the Weak and Para methods shows that the two approximation models are basically equivalent under the current configuration. In the case of the telephoto camera (f=75 mm), the calibration parameters and image pairs acquired via the stereovision are applied to reconstruct the 3D planar template (Fig. 10). The out-of-plane error of the reconstructed plane is about 0.015 mm, which is significantly lower than the reconstruction error of methods under perspective projection, thus demonstrating the correctness and effectiveness of the proposed method.

Featuring unique advantages of high spatial resolution and more detailed information, telephoto cameras have been widely employed in military fields such as aerial reconnaissance, damage assessment, and patrol monitoring, as well as civil fields including industrial precision processing, aerial photogrammetry, deformation monitoring, and traffic surveillance. However, there are practical obstacles to the adoption of telephoto lenses on small-format cameras, with potential difficulties in self-calibration. A telephoto lens will inevitably narrow the field of view (FOV), exerting detrimental effects on the performance of the central perspective projection model, which will result in over-parameterization, ill-conditioning, and subsequent numerical instability in normal equations of the bundle adjustment. Linear dependencies between the intrinsic and extrinsic parameters make it rather difficult to recover satisfactory calibration parameters in such weak geometry conditions. Therefore, an alternative affine approximation of the perspective projection model is adopted to accommodate high-accuracy and close-range photogrammetry with telephoto lenses, which makes full use of the advantages of the model, such as high linearity, simplicity, and high robustness. Additionally, a calibration method of telephoto cameras based on the affine approximation projection model is proposed. By employing the proposed model, this paper aims to obtain better calibration performance of telephoto cameras, thereby laying a foundation for the application of telephoto cameras.

As the most common affine approximation of the perspective projection model, the characteristics of weak perspective and paraperspective projection are thoroughly elaborated, according to which a calibration method of telephoto camera based on the affine approximation projection model is built. In this paper, the formation mechanism of reversal ambiguity is thoroughly analyzed, and the estimation method of calibration parameters based on the planar template under the affine approximate projection is deduced in detail. First, combined with the normalized method on line-based homography estimation method and the partitioned regularization estimation algorithm, the homography between the image plane and the cooperative planar template can be obtained. Then, the initial values of the intrinsic and extrinsic parameters of the camera are calculated under the affine approximation projection model. With the minimum sum of the residual square of the re-projection image points under the perspective projection model as the cost function, a nonlinear optimization algorithm is adopted to refine the calibration parameters for minimizing the approximation error of the affine projection model. In addition, an additional stage with a control point is attached to the planar template to address pose ambiguity.

A novel method of telephoto camera calibration based on the affine approximation projection model is proposed to address the over-parameterization of the perspective projection model in telephoto camera calibration. By the affine approximation projection model, the initial values of the calibration parameters can be obtained. Additionally, the method with the affine approximation projection model is more robust than that with the perspective projection model. With the minimum sum of the residual square of the re-projection image points under the perspective projection model as the cost function, a nonlinear optimization algorithm is adopted to refine the calibration parameters for minimizing the approximation error of the affine projection model. Besides, an additional stage with a control point is attached to the planar template to solve the pose ambiguity. Under the typical telephoto conditions where the angle of view is lower than 10? and the variation in the depth of the target along the line of sight is smaller than its average depth from the camera, the proposed method only requires the freely moving camera to observe a planar pattern shown at several different orientations, which is simple and flexible. Simulation and actual experimental results show that the proposed calibration method of the telephoto camera is effective, and the out-of-plane error of the reconstructed plane is better than 0.02 mm in the laboratory environment.