Generating Topologically and Geometrically Diverse Manifold Data in Dimensions Four and Below

TL;DR

This paper explores combining algebraic topology and image processing to generate diverse, topologically rich 2D, 3D, and 4D image data, aiding the training of neural networks to recognize complex topological features.

Contribution

It introduces a novel approach that integrates algebraic topology with image processing to generate diverse topological data in multiple dimensions, including 4D.

Findings

Generated topologically rich 2D, 3D, and 4D data sets

Demonstrated the use of topology-based labels in simulation

Provided a roadmap for future 4D data generation

Abstract

Understanding the topological characteristics of data is important to many areas of research. Recent work has demonstrated that synthetic 4D image-type data can be useful to train 4D convolutional neural network models to see topological features in these data. These models also appear to tolerate the use of image preprocessing techniques where existing topological data analysis techniques such as persistent homology do not. This paper investigates how methods from algebraic topology, combined with image processing techniques such as morphology, can be used to generate topologically sophisticated and diverse-looking 2-, 3-, and 4D image-type data with topological labels in simulation. These approaches are illustrated in 2D and 3D with the aim of providing a roadmap towards achieving this in 4D.