Segmentation algorithms are prone to make topological errors on fine-scale
structures, e.g., broken connections. We propose a novel method that learns to
segment with correct topology. In particular, we design a continuous-valued
loss function that enforces a segmentation to have the same topology as the
ground truth, i.e., having the same Betti number. The proposed
topology-preserving loss function is differentiable and we incorporate it into
end-to-end training of a deep neural network. Our method achieves much better
performance on the Betti number error, which directly accounts for the
topological correctness. It also performs superiorly on other topology-relevant
metrics, e.g., the Adjusted Rand Index and the Variation of Information. We
illustrate the effectiveness of the proposed method on a broad spectrum of
natural and biomedical datasets.

我们提出了一种新的学习方法，通过设计一个连续的价值损失函数来促进分割具有与真值相同的拓扑结构，即具有相同的 Betti 数，将其并入深度神经网络的端到端训练中，以在广泛的自然和生物医学数据集上实现更好的性能。

保持拓扑结构的深度图像分割

Topology-Preserving Deep Image Segmentation

Long lived topological features are distinguished from short lived ones
(considered as topological noise) in simplicial complexes constructed from
complex networks. A new topological invariant, persistent homology, is
determined and presented as a parametrized version of a Betti number. Complex
networks with distinct degree distributions exhibit distinct persistent
topological features. Persistent toplogical attributes, shown to be related to
robust quality of networks, also reflect defficiency in certain connectivity
properites of networks. Random networks, networks with exponential conectivity
distribution and scale-free networks were considered for homological
persistency analysis.

该研究介绍了一个新的拓扑不变量 —— 持续同调，它代表了一个 Betti 数的参数化版本，可以用于区分长期和短期的拓扑特征，并且可以展示出不同网络拓扑特征之间的差异，从而反映出网络的强韧性与连通性等特性。