In this paper, we investigate an innovative car-following framework where a self-driving vehicle, identified as the follower, autonomously follows another leading vehicle. We propose to design the car-following strategy based only on the environmental LiDAR data captured by the follower and the GNSS input. The proposed framework is composed of several modules, including the detection module using the PointNet++ neural network, the continuous calculation of the leader's driving trajectory, and the trajectory following control module using the BP-PID method. Comparison experiments and analysis have been performed on the Carla simulator. Results show that our proposed framework can work effectively and efficiently in the defined car-following tasks, and its success rate exceeds that of the Yolo-v5-based method by more than 13% under night conditions or rainy weather settings.