Topological features play an essential role in ensuring geometric plausibility and structural consistency in image analysis tasks such as segmentation and skeletonization. However, integrating topology-preserving learning based on simple points into deep learning tasks remains challenging, as existing simple point detection methods are confined to binary images and are non-differentiable, rendering them incompatible with gradient-based optimization in modern deep learning. Moreover, morphological and purely data-driven approaches often fail to guaranty topological consistency. To address these limitations, we propose a novel method that directly computes simple points on continuous-valued images, enabling differentiable topological inference. Building on this theory, we develop an efficient skeleton extraction algorithm that preserves topological structures in binary and continuous-valued images. Furthermore, we design a variational model that enforces topological constraints by preserving topologically non-removable (i.e., non-simple) points, which can be seamlessly integrated into any deep neural network segmentation with softmax or sigmoid outputs. Experimental results demonstrate that the proposed approach effectively improves topological integrity and structural accuracy across multiple benchmarks. The codes are available in https://github.com/levnsio/CSP.

翻译：拓扑特征在图像分割和骨架化等分析任务中，对于确保几何合理性和结构一致性起着关键作用。然而，将基于简单点的拓扑保持学习融入深度学习任务仍面临挑战，因为现有的简单点检测方法仅限于二值图像且不可微分，无法与现代深度学习中的梯度优化兼容。此外，形态学方法和纯数据驱动方法通常无法保证拓扑一致性。为解决这些局限，我们提出了一种新方法，可直接在连续值图像上计算简单点，从而实现可微的拓扑推理。基于该理论，我们开发了一种高效的骨架提取算法，可在二值图像和连续值图像中保持拓扑结构。此外，我们设计了一个变分模型，通过保留拓扑不可移除（即非简单）点来施加拓扑约束，该模型可无缝集成到任何具有softmax或sigmoid输出的深度神经网络分割中。实验结果表明，所提方法在多个基准上有效提升了拓扑完整性和结构精度。代码已在https://github.com/levnsio/CSP公开。