Topological accuracy in medical image segmentation is a highly important
property for downstream applications such as network analysis and flow modeling
in vessels or cell counting. Recently, significant methodological advancements
have brought well-founded concepts from algebraic topology to binary
segmentation. However, these approaches have been underexplored in multi-class
segmentation scenarios, where topological errors are common. We propose a
general loss function for topologically faithful multi-class segmentation
extending the recent Betti matching concept, which is based on induced
matchings of persistence barcodes. We project the N-class segmentation problem
to N single-class segmentation tasks, which allows us to use 1-parameter
persistent homology making training of neural networks computationally
feasible. We validate our method on a comprehensive set of four medical
datasets with highly variant topological characteristics. Our loss formulation
significantly enhances topological correctness in cardiac, cell, artery-vein,
and Circle of Willis segmentation.

提出了一种通用的损失函数，用于在多类别分割中实现拓扑准确性，并通过在持久性条码上引入匹配概念进行优化，将多类别分割问题转化为单类别分割任务，从而实现了神经网络的快速训练，并在四个医学数据集上验证了该方法的有效性。