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

TL;DR

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.

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.

Figures (12)

• Figure 1: The interval under the equivalence relation that identifies its boundary points is homeomorphic to a circle.
• Figure 2: The matching arrows of the square can be identified to construct a torus.
• Figure 3: The 3-manifolds that are considered in this project can be defined by quotient spaces of the cube $I^3$.
• Figure 4: Visualising $S^1 \times I^2$. With an appropriate labelling of $I^2$, this donut can be used to understand $S^1 \times S^2$ and $S^1 \times S^1 \times S^1$.
• Figure 5: Visualising $S^2 \times S^1$. The BACK face is spherically wrapped around the FRONT face to give $S^2 \times I$. The inner and outer boundary spheres are then identified with one another.