Existing research highlights the crucial role of topological priors in image segmentation, particularly in preserving essential structures such as connectivity and genus. Accurately capturing these topological features often requires incorporating width-related information, including the thickness and length inherent to the image structures. However, traditional mathematical definitions of topological structures lack this dimensional width information, limiting methods like persistent homology from fully addressing practical segmentation needs. To overcome this limitation, we propose a novel mathematical framework that explicitly integrates width information into the characterization of topological structures. This method leverages persistent homology, complemented by smoothing concepts from partial differential equations (PDEs), to modify local extrema of upper-level sets. This approach enables the resulting topological structures to inherently capture width properties. We incorporate this enhanced topological description into variational image segmentation models. Using some proper loss functions, we are also able to design neural networks that can segment images with the required topological and width properties. Through variational constraints on the relevant topological energies, our approach successfully preserves essential topological invariants such as connectivity and genus counts, simultaneously ensuring that segmented structures retain critical width attributes, including line thickness and length. Numerical experiments demonstrate the effectiveness of our method, showcasing its capability to maintain topological fidelity while explicitly embedding width characteristics into segmented image structures.

翻译：现有研究强调了拓扑先验在图像分割中的关键作用，特别是在保持连通性和亏格等基本结构方面。准确捕捉这些拓扑特征通常需要纳入宽度相关信息，包括图像结构固有的厚度与长度。然而，拓扑结构的传统数学定义缺乏此类维度宽度信息，使得持续同调等方法难以完全满足实际分割需求。为克服这一局限，我们提出了一种新颖的数学框架，将宽度信息显式整合到拓扑结构的表征中。该方法利用持续同调，辅以偏微分方程（PDEs）中的平滑概念，对上层集的局部极值进行修正。这一途径使得所得拓扑结构能够内在地捕捉宽度特性。我们将这种增强的拓扑描述融入变分图像分割模型。通过设计适当的损失函数，我们还能构建能够分割具有所需拓扑与宽度特性的图像的神经网络。通过对相关拓扑能量施加变分约束，我们的方法成功保持了连通性和亏格数等关键拓扑不变量，同时确保分割结构保留线宽与长度等重要宽度属性。数值实验验证了本方法的有效性，展示了其在显式嵌入宽度特征的同时保持拓扑保真度的能力。