Planar Symmetry Detection and Quantification using the Extended Persistent Homology Transform

TL;DR

This paper explores how the extended persistent homology transform can be used to detect, visualize, and quantify approximate planar symmetries in objects, providing a continuous measure rather than a binary one.

Contribution

It demonstrates a novel application of the extended persistent homology transform for symmetry analysis, highlighting its effectiveness and limitations.

Findings

Effective visualization of symmetries

Quantitative symmetry measurement achieved

Identifies limitations of the method

Abstract

Symmetry is ubiquitous throughout nature and can often give great insights into the formation, structure and stability of objects studied by mathematicians, physicists, chemists and biologists. However, perfect symmetry occurs rarely so quantitative techniques must be developed to identify approximate symmetries. To facilitate the analysis of an independent variable on the symmetry of some object, we would like this quantity to be a smoothly varying real parameter rather than a boolean one. The extended persistent homology transform is a recently developed tool which can be used to define a distance between certain kinds of objects. Here, we describe how the extended persistent homology transform can be used to visualise, detect and quantify certain kinds of symmetry and discuss the effectiveness and limitations of this method.