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Jordan-Segmentable Masks: A Topology-Aware definition for characterizing Binary Image Segmentation

Serena Grazia De Benedictis, Amedeo Altavilla, Nicoletta Del Buono

TL;DR

This work tackles the limitation of conventional segmentation metrics in capturing topological coherence by introducing Jordan-segmentable masks, a topology-aware criterion grounded in the digital Jordan Curve Theorem and homology. It identifies a 4-curve candidate within a binary segmentation and validates it using Betti numbers: $β_0=β_1=1$ for the curve and $β_0(I\setminus S)=2$ for the complement, ensuring a robust interior/exterior separation. The approach provides an unsupervised, structure-preserving check that complements pixel-wise measures, particularly in domains where connectivity matters. Experiments on the FSS-1000 dataset with standard segmentation methods illustrate the practical applicability and reveal cases where topological correctness diverges from traditional scores, underscoring the method's complementary value.

Abstract

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.

Jordan-Segmentable Masks: A Topology-Aware definition for characterizing Binary Image Segmentation

TL;DR

This work tackles the limitation of conventional segmentation metrics in capturing topological coherence by introducing Jordan-segmentable masks, a topology-aware criterion grounded in the digital Jordan Curve Theorem and homology. It identifies a 4-curve candidate within a binary segmentation and validates it using Betti numbers: for the curve and for the complement, ensuring a robust interior/exterior separation. The approach provides an unsupervised, structure-preserving check that complements pixel-wise measures, particularly in domains where connectivity matters. Experiments on the FSS-1000 dataset with standard segmentation methods illustrate the practical applicability and reveal cases where topological correctness diverges from traditional scores, underscoring the method's complementary value.

Abstract

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 -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 , or equivalently when its complement splits into exactly two -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.
Paper Structure (8 sections, 3 theorems, 1 equation, 18 figures)

This paper contains 8 sections, 3 theorems, 1 equation, 18 figures.

Table of Contents

  1. Introduction
  2. Background and basic topological notions
  3. Digital Topology
  4. A Digital Analog of the Jordan Curve Theorem
  5. Homology Groups and Betti Numbers
  6. Jordan-Segmentatable mask
  7. Jordan-segmentable mask in practice
  8. Conclusion

Key Result

Theorem 1

(Jordan Curve Theorem on $\mathbb{R}^2$) Let $C$ be a Jordan curve in the plane $\mathbb{R}^2$. Then its complement, $\mathbb{R}^2 \setminus C$, consists of exactly two connected components. One of these components is bounded (referred to as the interior), while the other is unbounded (referred to a

Figures (18)

  • Figure 1: Illustration of the various types of neighbors for a point, along with the graph structure defined by their adjacency relations.
  • Figure 2: Illustration of the minimal number of points required to define a digital curve.
  • Figure 3: Examples illustrating connectivity paradoxes in the digital plane, for 8- and 4- adjacency over $S$ and its complementary.
  • Figure 4: Construction of the 4-curve candidate $S$.
  • Figure 5: Conceptual workflow illustrating the proposed method for an homological validation of a Jordan-segmentable mask from binary segmentation.
  • ...and 13 more figures

Theorems & Definitions (7)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 2
  • Definition 3
  • Definition 4