  Segmentation of curvilinear structures such as vasculature and road networks
is challenging due to relatively weak signals and complex geometry/topology. To
facilitate and accelerate large scale annotation, one has to adopt
semi-automatic approaches such as proofreading by experts. In this work, we
focus on uncertainty estimation for such tasks, so that highly uncertain, and
thus error-prone structures can be identified for human annotators to verify.
Unlike most existing works, which provide pixel-wise uncertainty maps, we
stipulate it is crucial to estimate uncertainty in the units of topological
structures, e.g., small pieces of connections and branches. To achieve this, we
leverage tools from topological data analysis, specifically discrete Morse
theory (DMT), to first capture the structures, and then reason about their
uncertainties. To model the uncertainty, we (1) propose a joint prediction
model that estimates the uncertainty of a structure while taking the
neighboring structures into consideration (inter-structural uncertainty); (2)
propose a novel Probabilistic DMT to model the inherent uncertainty within each
structure (intra-structural uncertainty) by sampling its representations via a
perturb-and-walk scheme. On various 2D and 3D datasets, our method produces
better structure-wise uncertainty maps compared to existing works. Code
available at https://github.com/Saumya-Gupta-26/struct-uncertainty
