利用流形信息提升医学图像分割边界精度，减少标注依赖。

Achieving precise medical image segmentation is vital for effective treatment planning and accurate disease diagnosis. Traditional fully-supervised deep learning methods, though highly precise, are heavily reliant on large volumes of labeled data, which are often difficult to obtain due to the expertise required for medical annotations. This has led to the rise of semi-supervised learning approaches that utilize both labeled and unlabeled data to mitigate the label scarcity issue. In this paper, we introduce the Manifold-Aware Local Feature Modeling Network (MANet), which enhances the U-Net architecture by incorporating manifold supervision signals. This approach focuses on improving boundary accuracy, which is crucial for reliable medical diagnosis. To further extend the versatility of our method, we propose two variants: MA-Sobel and MA-Canny. The MA-Sobel variant employs the Sobel operator, which is effective for both 2D and 3D data, while the MA-Canny variant utilizes the Canny operator, specifically designed for 2D images, to refine boundary detection. These variants allow our method to adapt to various medical image modalities and dimensionalities, ensuring broader applicability. Our extensive experiments on datasets such as ACDC, LA, and Pancreas-NIH demonstrate that MANet consistently surpasses state-of-the-art methods in performance metrics like Dice and Jaccard scores. The proposed method also shows improved generalization across various semi-supervised segmentation networks, highlighting its robustness and effectiveness. Visual analysis of segmentation results confirms that MANet offers clearer and more accurate class boundaries, underscoring the value of manifold information in medical image segmentation.