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In autonomous driving, LiDAR and radar are crucial for environmental perception. LiDAR offers precise 3D spatial sensing information but struggles in adverse weather like fog. Conversely, radar signals can penetrate rain or mist due to their specific wavelength but are prone to noise disturbances. Recent state-of-the-art works reveal that the fusion of radar and LiDAR can lead to robust detection in adverse weather. The existing works adopt convolutional neural network architecture to extract features from each sensor data, then align and aggregate the two branch features to predict object detection results. However, these methods have low accuracy of predicted bounding boxes due to a simple design of label assignment and fusion strategies. In this paper, we propose a bird's-eye view fusion learning-based anchor box-free object detection system, which fuses the feature derived from the radar range-azimuth heatmap and the LiDAR point cloud to estimate possible objects. Different label assignment strategies have been designed to facilitate the consistency between the classification of foreground or background anchor points and the corresponding bounding box regressions. Furthermore, the performance of the proposed object detector is further enhanced by employing a novel interactive transformer module. The superior performance of the methods proposed in this paper has been demonstrated using the recently published Oxford Radar RobotCar dataset. Our system's average precision significantly outperforms the state-of-the-art method by 13.1% and 19.0% at Intersection of Union (IoU) of 0.8 under 'Clear+Foggy' training conditions for 'Clear' and 'Foggy' testing, respectively.