[Paper Review] Train-Free Segmentation in MRI with Cubical Persistent Homology

Anton François, Raphaël Tinarrage|arXiv (Cornell University)|Jan 2, 2024

Topological and Geometric Data Analysis4 citations

TL;DR

This paper proposes a train-free, interpretable MRI segmentation method using cubical persistent homology to detect anatomical structures based on their topology—such as spheres in glioblastoma, cylinders in myocardium, and circles in fetal cortical plates—achieving competitive performance (e.g., Dice score of 0.711 on BraTS 2021) without deep learning or annotated datasets by leveraging topological features and cycle localization.

ABSTRACT

We present a new general framework for segmentation of MRI scans based on Topological Data Analysis (TDA), offering several advantages over traditional machine learning approaches. The pipeline proceeds in three steps, first identifying the whole object to segment via automatic thresholding, then detecting a distinctive subset whose topology is known in advance, and finally deducing the various components of the segmentation. Unlike most prior TDA uses in medical image segmentation, which are typically embedded within deep networks, our approach is a standalone method tailored to MRI. A key ingredient is the localization of representative cycles from the persistence diagram, which enables interpretable mappings from topological features to anatomical components. In particular, the method offers the ability to perform segmentation without the need for large annotated datasets. Its modular design makes it adaptable to a wide range of data segmentation challenges. We validate the framework on three applications: glioblastoma segmentation in brain MRI, where a sphere is to be detected; myocardium in cardiac MRI, forming a cylinder; and cortical plate detection in fetal brain MRI, whose 2D slices are circles. We compare our method with established supervised and unsupervised baselines.

Motivation & Objective

• Address the limitations of deep learning-based MRI segmentation, including reliance on large annotated datasets, lack of interpretability, and anatomical implausibility in outputs.
• Develop a segmentation framework grounded in Topological Data Analysis (TDA) that ensures topological correctness (e.g., spherical or cylindrical structure) of results.
• Enable segmentation without model training by exploiting prior morphological knowledge of target structures via persistent homology and cycle localization.
• Provide a modular, generalizable algorithm applicable to diverse biomedical imaging challenges with non-trivial topological features.
• Improve clinical interpretability and stability by explicitly mapping topological features to anatomical components, supporting informed decision-making.

Proposed method

• Use automatic thresholding to identify the whole object to segment from the MRI scan, based on intensity differences.
• Apply cubical persistent homology to compute persistence diagrams, capturing topological features (e.g., connected components, holes, voids) across scales.
• Leverage the localization of representative cycles (topological features) within the image to guide segmentation, a novel use of spatial information in TDA.
• Select the most persistent homology class (e.g., H₂ for spherical tumors) or the one enclosing the most volume (e.g., for fetal cortical plates) to identify the target structure.
• Extract the connected component corresponding to the selected topological feature as the final segmentation, ensuring topological consistency.
• Adapt the method to different anatomical structures by matching their expected topology (sphere, cylinder, circle) to the persistent homology output.

Experimental results

Research questions

• RQ1 Can a train-free segmentation method using persistent homology achieve competitive performance on MRI segmentation tasks without relying on annotated training data?
• RQ2 How can the spatial localization of topological cycles be exploited to improve segmentation accuracy and robustness in medical imaging?
• RQ3 To what extent can topological constraints (e.g., spherical or cylindrical topology) ensure anatomical plausibility in MRI segmentation outputs?
• RQ4 Can a modular TDA-based approach generalize across diverse segmentation challenges (e.g., glioblastoma, myocardium, fetal cortical plates) with minimal reconfiguration?
• RQ5 How does the performance of a purely topological method compare to state-of-the-art deep learning models like U-Net in terms of Dice score and interpretability?

Key findings

• The proposed method achieves a Dice score of 0.711 on the BraTS 2021 glioblastoma dataset, demonstrating competitive performance without any training data.
• For myocardium segmentation in the ACDC dataset, the method achieves comparable performance to state-of-the-art models, highlighting its effectiveness on low-variability structures.
• The algorithm ensures topological correctness by detecting structures with expected homology (e.g., H₂ for spheres, H₁ for cylinders), preventing anatomically impossible segmentations.
• The inclusion of cycle localization—beyond standard persistence diagrams—significantly improves segmentation accuracy, especially in complex or heterogeneous regions.
• The method is inherently interpretable, as each segmentation result corresponds directly to a topological feature with clear geometric and topological meaning.
• The approach is generalizable: it can be adapted to any segmentation task where one component has a known non-trivial topology and is distinguishable by intensity.