Certifiability Analysis of the Global Optimality in Camera-Based Positioning with SEC-PnP Algorithm

In GNSS degraded environments, cameras have great potential to be used as navigation sensors in various applications such as UAV landing, autonomous driving, indoor navigation, etc. Using a set of visible features with known locations on the map, the six degrees of freedom (DOF) pose (position and attitude) of the camera can be estimated. This is known as the Perspective-n-Point (PnP) problem. However, as the measurement equation is highly non-linear, the camera pose estimation result is affected by various error sources, such as an incorrect initial pose estimate or the presence of noise in the measurements (Zhu et al. (2022)). Due to the aforementioned problems, there is still a lack of methods for camera-based navigation that provide results with certified global optimality and overbounded pose estimation error. In the previous work Triolo et al. (2024), we proposed the Slack-Eigen-Convexification-PnP (SEC-PnP) algorithm to estimate the camera pose reliably even when the initial pose estimation is erroneous and the measurements are noisy. In this work, we provide a method to analytically prove the global optimality of the pose estimation results obtained by the SEC-PnP algorithm, and discuss the conditions required to fulfill the global optimality. Also, a study on the accuracy of the certified optimal solution is performed. By comparing the highest achievable accuracy of the SEC-PnP algorithm with the Cramer-Rao Lower Bound (CRLB) for certain measurement noise levels and number of the 2D-3D associations, the global optimality of the results can be validated. The proposed method with its proven global optimality has great potential to be exploited in applications with high reliability requirements.