Image segmentation plays a central role in computer vision. However, widely used evaluation metrics, whether pixel-wise, region-based, or boundary-focused, often struggle to capture the structural and topological coherence of a segmentation. In many practical scenarios, such as medical imaging or object delineation, small inaccuracies in boundary, holes, or fragmented predictions can result in high metric scores, despite the fact that the resulting masks fail to preserve the object global shape or connectivity. This highlights a limitation of conventional metrics: they are unable to assess whether a predicted segmentation partitions the image into meaningful interior and exterior regions. In this work, we introduce a topology-aware notion of segmentation based on the Jordan Curve Theorem, and adapted for use in digital planes. We define the concept of a \emph{Jordan-segmentatable mask}, which is a binary segmentation whose structure ensures a topological separation of the image domain into two connected components. We analyze segmentation masks through the lens of digital topology and homology theory, extracting a $4$-curve candidate from the mask, verifying its topological validity using Betti numbers. A mask is considered Jordan-segmentatable when this candidate forms a digital 4-curve with $β_0 = β_1 = 1$, or equivalently when its complement splits into exactly two $8$-connected components. This framework provides a mathematically rigorous, unsupervised criterion with which to assess the structural coherence of segmentation masks. By combining digital Jordan theory and homological invariants, our approach provides a valuable alternative to standard evaluation metrics, especially in applications where topological correctness must be preserved.

翻译：图像分割在计算机视觉中扮演着核心角色。然而，广泛使用的评估指标——无论是基于像素、区域还是边界——往往难以捕捉分割的结构与拓扑一致性。在许多实际场景中，例如医学影像或目标轮廓提取，边界上的微小误差、孔洞或碎片化预测可能导致较高的指标得分，尽管生成的掩码未能保持目标的整体形状或连通性。这凸显了传统指标的局限性：它们无法评估预测的分割是否将图像划分为有意义的内外区域。本文基于Jordan曲线定理，并针对数字平面进行调整，引入了一种拓扑感知的分割概念。我们定义了**Jordan-可分割掩码**的概念，这是一种二值分割，其结构确保图像域在拓扑上被分离为两个连通分量。我们通过数字拓扑与同调理论的视角分析分割掩码，从掩码中提取一条4-曲线候选，并利用Betti数验证其拓扑有效性。当该候选曲线形成满足β₀ = β₁ = 1的数字4-曲线，或等价地当其补集恰好分裂为两个8-连通分量时，该掩码被视为Jordan-可分割的。该框架提供了一个数学上严谨、无监督的准则，用于评估分割掩码的结构一致性。通过结合数字Jordan理论与同调不变量，我们的方法为标准评估指标提供了一个有价值的替代方案，特别是在必须保持拓扑正确性的应用场景中。