[Paper Review] Learning Probabilistic Topological Representations Using Discrete Morse Theory

Xiaoling Hu, Dimitris Samaras+1 more|arXiv (Cornell University)|Jun 3, 2022

Topological and Geometric Data Analysis4 citations

TL;DR

This paper proposes the first deep learning method to learn probabilistic topological/structural representations using discrete Morse theory and persistent homology, enabling structure-preserving segmentation and structure-level uncertainty estimation. By modeling a probabilistic distribution over Morse complex structures via persistence thresholds, the method generates topologically valid segmentations and guides efficient interactive proofreading through uncertainty-aware sampling.

ABSTRACT

Accurate delineation of fine-scale structures is a very important yet challenging problem. Existing methods use topological information as an additional training loss, but are ultimately making pixel-wise predictions. In this paper, we propose the first deep learning based method to learn topological/structural representations. We use discrete Morse theory and persistent homology to construct an one-parameter family of structures as the topological/structural representation space. Furthermore, we learn a probabilistic model that can perform inference tasks in such a topological/structural representation space. Our method generates true structures rather than pixel-maps, leading to better topological integrity in automatic segmentation tasks. It also facilitates semi-automatic interactive annotation/proofreading via the sampling of structures and structure-aware uncertainty.

Motivation & Objective

• To address the limitations of pixel-wise segmentation networks that fail to preserve topological integrity during inference.
• To overcome the inadequacy of pixel-wise uncertainty maps in identifying uncertain structural components in fine-scale biomedical images.
• To develop a deep learning framework that directly models structural representations instead of pixel-level predictions.
• To enable efficient semi-automatic annotation and proofreading by providing structure-level uncertainty guidance.

Proposed method

• Construct a Morse complex from a learned likelihood map using discrete Morse theory to extract topological structures such as branches and patches.
• Parameterize the structure space via a one-parameter family of Morse complexes using a persistence threshold ε.
• Learn a Gaussian distribution over the persistence threshold ε to model a probabilistic distribution over structural hypotheses.
• Generate structural segmentations via sampling from the learned probabilistic model, ensuring topological consistency.
• Estimate structure-level uncertainty by computing variance across multiple structural samples.
• Apply post-processing to grow sampled skeletons into full segmentation maps while preserving topology.

Experimental results

Research questions

• RQ1 Can deep learning be used to directly learn topological/structural representations instead of pixel-wise predictions?
• RQ2 How can persistent homology and discrete Morse theory be integrated into a deep learning framework to model structural variability?
• RQ3 Can structure-level uncertainty maps improve the efficiency of interactive proofreading in biomedical image annotation?
• RQ4 Does learning a probabilistic model over structural representations lead to better topological integrity in segmentation?
• RQ5 How does the proposed method compare to pixel-wise uncertainty estimation in guiding human annotators?

Key findings

• The proposed method achieves a VOI score of 0.804 ± 0.047 on the DRIVE dataset when β = 10 in the KL divergence loss, outperforming other β values.
• Structure-level uncertainty maps highlight entire branches with high uncertainty, unlike pixel-wise maps that uniformly mark boundary pixels.
• Sampling from the probabilistic model generates topologically valid segmentations that preserve structural integrity at inference time.
• Semi-automatic proofreading guided by structure-level uncertainty improves segmentation accuracy faster than baseline methods.
• The method enables efficient interactive annotation by prioritizing high-uncertainty branches, reducing human effort in correcting false positives and negatives.
• Empirical results show that the proposed method significantly outperforms both deterministic and pixel-wise probabilistic baselines in proofreading efficiency.