In many scenarios, especially biomedical applications, the correct delineation of complex fine-scaled structures such as neurons, tissues, and vessels is critical for downstream analysis. Despite the strong predictive power of deep learning methods, they do not provide a satisfactory representation of these structures, thus creating significant barriers in scalable annotation and downstream analysis. In this dissertation, we tackle such challenges by proposing novel representations of these topological structures in a deep learning framework. We leverage the mathematical tools from topological data analysis, i.e., persistent homology and discrete Morse theory, to develop principled methods for better segmentation and uncertainty estimation, which will become powerful tools for scalable annotation.

翻译：在许多场景中，尤其是生物医学应用领域，正确描绘神经元、组织和血管等复杂精细结构对下游分析至关重要。尽管深度学习方法具有强大的预测能力，但其无法为这些结构提供令人满意的表示，从而在可扩展标注和下游分析中造成重大障碍。在本论文中，我们通过提出深度学习框架中的新型拓扑结构表示来应对这些挑战。我们利用拓扑数据分析中的数学工具（即持续同调和离散莫尔斯理论）开发出原则性方法，以实现更优的分割与不确定性估计，这些方法将成为可扩展标注的强大工具。