  Accurate segmentation of tubular, network-like structures, such as vessels,
neurons, or roads, is relevant to many fields of research. For such structures,
the topology is their most important characteristic; particularly preserving
connectedness: in the case of vascular networks, missing a connected vessel
entirely alters the blood-flow dynamics. We introduce a novel similarity
measure termed centerlineDice (short clDice), which is calculated on the
intersection of the segmentation masks and their (morphological) skeleta. We
theoretically prove that clDice guarantees topology preservation up to homotopy
equivalence for binary 2D and 3D segmentation. Extending this, we propose a
computationally efficient, differentiable loss function (soft-clDice) for
training arbitrary neural segmentation networks. We benchmark the soft-clDice
loss on five public datasets, including vessels, roads and neurons (2D and 3D).
Training on soft-clDice leads to segmentation with more accurate connectivity
information, higher graph similarity, and better volumetric scores.
