# Some equivalence relation between persistent homology and morphological dynamics

arXiv: https://arxiv.org/abs/2205.12546 · HTML: https://mlsanity.com/paper/2205.12546/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 61 references that are in the map.
- First posted on arXiv 25 May 2022.

## Builds on (most cited first, 4 of 4)

- [The Topology ToolKit](https://mlsanity.com/paper/1805.09110/) (2018)
- [Task-based Augmented Contour Trees with Fibonacci Heaps](https://mlsanity.com/paper/1902.04805/) (2019)
- [Localized Topological Simplification of Scalar Data](https://mlsanity.com/paper/2009.00083/) (2020)
- [VOIDD: automatic vessel of intervention dynamic detection in PCI procedures](https://mlsanity.com/paper/1710.04476/) (2017)

## Abstract (arXiv)

In Mathematical Morphology (MM), connected filters based on dynamics are used to filter the extrema of an image. Similarly, persistence is a concept coming from Persistent Homology (PH) and Morse Theory (MT) that represents the stability of the extrema of a Morse function. Since these two concepts seem to be closely related, in this paper we examine their relationship, and we prove that they are equal on n-D Morse functions, n ≥ 1. More exactly, pairing a minimum with a 1-saddle by dynamics or pairing the same 1-saddle with a minimum by persistence leads exactly to the same pairing, assuming that the critical values of the studied Morse function are unique. This result is a step further to show how much topological data analysis and mathematical morphology are related, paving the way for a more in-depth study of the relations between these two research fields.