Extended persistence and duality in cubical complexes

Abstract

In this work, I will continue to investigate the techniques of topological data analysis based on my bachelor thesis [4] and apply it to some image-type data sets. The tools and analysis methods we used are homology, cohomology, cubical complexes, persistent homology, barcode, extended persistence, super- and sublevel set, total persistence, persistence entropy, etc.

Our goal is to try to construct the cubical complex of the image through the above methods, and then study the topological features of two-dimensional and three-dimensional grayscale images, such as total persistence and persistence entropy. And try to combine it with statistical methods to determine the attribution of the image.

We found some kind of duality in the bachelor thesis of Marina Anguas [1]. She performed sub-level set filtration from bottom to top and super-level set from top to bottom on a 3D image and calculated extended persistence. Then she found some pairs of results they were similar. So we hope to study the relationship between cycles of different dimensions such as $H_0, H_1$ and $H_2$ by studying a duality analogous to Poincaré duality or Lefschetz duality, and simplify the calculation through the relationship between them.