  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.
