T-type theodolite is widely used in various test tasks due to its large measurement range, non-contact feature, high measurement accuracy, replaceable load sensor, and availability for multi-sensor sharing. The T-type theodolite has a load-carrying structure. For different test environments, different load imaging components can be installed on its side axis to reach the measurement purpose. However, after replacing the imaging detection sensor, the projection center of the optical imaging system and the position relationship between the imaging optical axis and the alignment axis of the T-type photoelectric theodolite will be changed. These factors will lead to angle deviation of the T-type photoelectric theodolite in the measurement. Therefore, the triple difference and the projection center coordinates of the T-type theodolite need to be calculated and eliminated. For the study of photoelectric theodolite, there is a lack of effective methods to simultaneously detect the projection center and triple difference. Therefore, a simple method of measuring the triple difference of T-type photoelectric theodolites is proposed in this paper. The photoelectric theodolite imaging system collects images of the near and far station poles respectively. Then, the projection center coordinates of the imaging optical system can be calculated from the image coordinates of the poles, the intrinsic parameters of the imaging system, and the encoder value of the photoelectric theodolite, thereby achieving the detection of optical axis parallelism.

Firstly, the projection equation between a point in the geodetic coordinate system and the image point on the camera image plane is derived by combining the transformation relationship of the related coordinate system [Fig. 1(a)] and the camera's pinhole imaging model [Fig. 1(b)], as well as the triple difference of the photoelectric theodolite. Secondly, the projection coordinates of the station pole vertex on the image plane are obtained according to the positive and reversed images of the photoelectric theodolite, and the imaging projection relationship combining the projection center and the triple difference is derived. The influence of the translation vector and triple difference on the miss distance of the projection point of the station pole vertex on the imaging plane is analyzed, and the projection center coordinates and triple difference of the system are calculated. Then, the optical axis parallelism of the photoelectric theodolite is detected. Finally, simulations and practical experiment results show that the proposed method is effective.

Experimental results (Table 1) show the projection center coordinate values and triple difference values obtained by linear calculation according to the proposed algorithm. Before optimization, the average of the reprojection errors is calculated, which is 0.7321 pixel in the horizontal direction and 0.7146 pixel in the vertical direction [Fig. 10 (a)]. After Levenberg-Marquadt optimization, the average of the reprojection errors in the horizontal direction is 0.2744 pixel, and that in the vertical direction is 0.2287 pixel [Fig. 10 (b)]. Therefore, it can be concluded that the accuracy after optimization has been improved by about 65%, which further verifies the effectiveness and correctness of the proposed detection method.

In this paper, the projection equation between a point in the world coordinate system and an image point on the camera image plane is derived by combining the pinhole imaging model and the triple difference of the T-type photoelectric theodolite, which can be applied to the nonlinear optimization of the photoelectric theodolite imaging system in the shooting range. Then, the calculation method of the projection center coordinate and the trilateration is derived, and a method is proposed to detect the projection center and the trilateration of the optical axis parallelism simultaneously with only two station poles, and the detailed algorithm flow chart is given. The experimental results show that the reprojection error of the system is less than 0.3 pixel, which proves that the method proposed in this paper can effectively solve the detection problem of projection center and optical axis parallelism triple difference in T-type photoelectric theodolites. However, this method is not sensitive to small-range changes of tx, so how to accurately and efficiently calculate tx to further improve the detection accuracy of the system is a problem that needs further research in the future. In addition, in the actual experiments, we only evaluate the reprojection errors and do not apply the detection results to the final theodolite measurement, especially for experimental measurements at high elevation angles (greater than 65°). This is the content that needs further experiments in the future.