Extended  Absrtact
Introduction
Today, With the improvement of UAV technology as a spatial data collection platform, using the UAV photogrammetric method for mapping aims has become more popular. The advantages of this method include cost-effectiveness, speeding up the project process, high resolution of spatial data, and production of various spatial products such as orthophoto mosaic, digital surface and ground models, 3D virtual model, and 3D map. From quality point of view, in addition to the network design in UAV photogrammetry projects, the camera and its accurate calibration are essential too. Metric cameras have a strong geometry, and their calibration parameters are known and stable with the smallest possible values. In spite of high accuracy outputs of metric cameras, it is practically impossible to use them in ultra-light public drones due to their high weight, size and cost. Therefore, today, non-metric and unstable digital cameras are conventional in UAV photogrammetric systems.However, many efforts are being made to reduce this weakness by improving the geometric quality of lightweight and inexpensive non-metric cameras. Despite of these efforts, application of non-metric cameras will not yet give us acceptable products without some practical considerations such as reducing flight altitude, increasing image side lap and overlap, and using high density of ground control points, which leads to a significant increase of cost and time. The main problem with these non-metric cameras is the weak geometry of their components that makes a high instability in the camera calibration parameters. This highlights the importance of proper geometric calibration of these cameras.
 
Materials & Methods
So far, several distortion models have been used to calibrate the metric cameras such as Brown model with a maximum of 12 parameters, including principal distance, principal point coordinates, lens radial and decentering distortions and affinity. These parameters are simultaneously estimated in a bundle adjustment with self-calibration process. Therefore, it can be said that this model considers fixed physical parameters for geometric modeling of the camera by which many images acquired in a photogrammetric block. If non-metric camera geometry is not modeled by a dynamic model with local spatial and temporal distortion parameters, some local systematic errors remain in the image observations. These systematic errors cause the estimation of unknown parameters in the least square adjustment is biased. Though this solution significantly improves the result of non-metric cameras in UAV photogrammetry, some errors in the 3D reconstruction remain yet due to low strength of observation equations set which comes from dynamic nature of the camera distortion model.The dynamic image distortions lead to parallax in stereoscopic vision and horizontal/vertical steps in the boundaries of connected 3D models. This paper proposes a post-processing method to reduce dynamic image distortions after conventional self-calibration of a non-metric camera with Brown model. The proposed method is based on local modeling of the image residuals using a finite element method. The data used in this study are photogrammetric drone images taken by ILCE-7RM2T, FC6310 and FC300S cameras. The proposed algorithm has been implemented in MATLAB programming environment and Agisoft Metashapesoftware has been used for initial processing.
 
Results & Discussion
As mentioned, the proposed algorithm is a post-processing task which reduces the image residuals and increases the geometric compatibility of 3D stereovision models.One of the critical indicators in the photogrammetric mapping production line is the quality of stereoscopic vision and the study of the vertical steps between connected 3D models. Because, photogrammetric map production requires stereo vision and the amount of model steps is used as a criterion for evaluation of image geometric distortion level. It can conclude that the use of the above idea is very effective in non-metric cameras with high geometric instability. The results of our experiments performed on the UAV photogrammetry data with low camera geometric stability indicate a60% reduction in the vertical steps of the models in stereoscopic vision and a 70% reduction in image residuals. This leads to a higher geometric quality of digital-elevation, 3D model, orthophoto, and map with 3D stereoscopic vision process. On the other hand, using this algorithm for non-metric cameras with higher geometric stability has a lower effect on the results. In our experiments, it was shown the vertical steps between 3D models can be reduced by 15% to 20%. However, there are still consecutive stereo models with quick steps in this type of camera, which will improve the geometric errors in stereoscopic vision if we ignore the computational costs.
 
Conclusion
The results of our experiments performed on the UAV photogrammetry data with low camera geometric stability indicate a 70% reduction in image residuals and a 60% reduction in the vertical steps of the models in stereoscopic vision. In this paper, the behavior of image residuals, the rate of model step reduction, and processing time in different dimensions of the distortion grid were investigated, and the grid dimensions of 150 to 200 pixels were recommended to apply the proposed method. Suggestions for further research are summarized in three sections. First, various factors such as the weight of observations and the weight of constraint equations can affect the estimation of the distortion grade, which can be estimated from the VCE method. Another point to consider in completing the proposed solution is to apply the temporal dependence between distortion grids in consecutive images. Also, although the proposed method uses the idea of finite elements as post-processing, it is more accurate to estimate this grid of distortion at the same time as the bundle adjustment.