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
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