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Accurate airway extraction from computed tomography (CT) images is a critical step for planning navigation bronchoscopy and quantitative assessment of airway-related chronic obstructive pulmonary disease (COPD). The existing methods are challenging to sufficiently segment the airway, especially the high-generation airway, with the constraint of the limited label and cannot meet the clinical use in COPD. We propose a novel two-stage 3D contextual transformer-based U-Net for airway segmentation using CT images. The method consists of two stages, performing initial and refined airway segmentation. The two-stage model shares the same subnetwork with different airway masks as input. Contextual transformer block is performed both in the encoder and decoder path of the subnetwork to finish high-quality airway segmentation effectively. In the first stage, the total airway mask and CT images are provided to the subnetwork, and the intrapulmonary airway mask and corresponding CT scans to the subnetwork in the second stage. Then the predictions of the two-stage method are merged as the final prediction. Extensive experiments were performed on in-house and multiple public datasets. Quantitative and qualitative analysis demonstrate that our proposed method extracted much more branches and lengths of the tree while accomplishing state-of-the-art airway segmentation performance. The code is available at https://github.com/zhaozsq/airway_segmentation.