Medical image segmentation, a critical application of semantic segmentation in healthcare, has seen significant advancements through specialized computer vision techniques. While deep learning-based medical image segmentation is essential for assisting in medical diagnosis, the lack of diverse training data causes the long-tail problem. Moreover, most previous hybrid CNN-ViT architectures have limited ability to combine various attentions in different layers of the Convolutional Neural Network. To address these issues, we propose a Lagrange Duality Consistency (LDC) Loss, integrated with Boundary-Aware Contrastive Loss, as the overall training objective for semi-supervised learning to mitigate the long-tail problem. Additionally, we introduce CMAformer, a novel network that synergizes the strengths of ResUNet and Transformer. The cross-attention block in CMAformer effectively integrates spatial attention and channel attention for multi-scale feature fusion. Overall, our results indicate that CMAformer, combined with the feature fusion framework and the new consistency loss, demonstrates strong complementarity in semi-supervised learning ensembles. We achieve state-of-the-art results on multiple public medical image datasets. Example code are available at: \url{https://github.com/lzeeorno/Lagrange-Duality-and-CMAformer}.
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