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We address a tracking system of 360° field of view using Light Detection And Ranging (LiDAR) sensors, which are a key module for detecting surrounding traffic and infrastructure for vehicular safety applications. We propose a joint algorithm for estimating motion and a nonparametric contour model whose complexity is O(N), with N being the number of scan points. Particularly, the proposed nonparametric model has the potential to track objects with arbitrary shape (e.g., car, pedestrian, and bicycle). A comparison of the proposed approach with other methods using parametric models shows the robustness and the quality of the proposed system. Quantitative performance evaluation against ground truth is presented in tracking a car, a pedestrian, and a bicycle. Experiments demonstrate that the proposed system substantially improves performance. Road test results show the effectiveness and efficiency of the implemented system. A lane-change alert (LCA) demonstration system is built based on the detection system, and the LCA performance is analyzed using ground truth.