Tracking Temporal Evolution of Topological Features in Image Data

2025-08-24

AI Summary

Existing methods lack a unified statistical modeling framework for spatiotemporal dependencies of topological features—such as connected components and holes—in image time series. This paper introduces the first topological dynamic inference method for spatiotemporal image sequences: it constructs high-dimensional topological structures by incorporating the temporal dimension, employs zigzag persistent homology to characterize the temporal evolution of low-dimensional topological features, and integrates statistical hypothesis testing to jointly identify spatial and temporal significance. The approach represents the first systematic integration of topological data analysis (TDA), zigzag persistence, and statistical inference, establishing an interpretable and verifiable topological modeling paradigm. Applied to single-cell wound-healing image sequences, it successfully captures the birth, death, and migration of topological structures. Simulation experiments demonstrate superior detection sensitivity and localization accuracy compared to existing TDA-based methods.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: Other Foundations of Computer Vision
• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

Topological Data Analysis (TDA) can be used to detect and characterize holes in an image, such as zero-dimensional holes (connected components) or one-dimensional holes (loops). However, there is currently no widely accepted statistical framework for modeling spatiotemporal dependence in the evolution of topological features, such as holes, within a time series of images. We propose a hypothesis testing framework to identify statistically significant topological features of images in space and time, simultaneously. This addition of time may induce higher-dimensional topological features which can be used to establish temporal connections between the lower-dimensional features at each point in time. The temporal evolution of these lower-dimensional features is then represented on a zigzag persistence diagram, as a topological summary statistic focused on time dynamics. We demonstrate that the method effectively captures the emergence and progression of topological features in a study of a series of images of a wounded cell as it repairs. The proposed method outperforms a current approach in a simulation study that includes features of the wound healing process. Since, the wounded cell images exhibit nonlinear, dynamic, spatial, and temporal structures during single-cell repair, they provide a good application for this method.

Problem

• Modeling spatiotemporal dependence in topological feature evolution
• Identifying statistically significant topological features across space-time
• Capturing emergence and progression of holes in time series images

Innovation

• Hypothesis testing framework for spatiotemporal topological features
• Zigzag persistence diagram for temporal evolution representation
• Captures emergence and progression of topological features