The Persistence Landscapes of Affine Fractals

TL;DR

This paper introduces a method to compute persistence landscapes of affine fractals via affine transformations, proving their existence and uniqueness, and illustrating the approach with examples and simulations.

Contribution

It establishes a novel connection between affine fractals and persistence landscapes, providing a way to calculate landscapes directly from transformation parameters.

Findings

Existence of a unique fixed point for the affine transformation on landscapes.

Method to compute landscapes from affine transformation parameters.

Validation of the theory through examples and simulations.

Abstract

We develop a method for calculating the persistence landscapes of affine fractals using the parameters of the corresponding transformations. Given an iterated function system of affine transformations that satisfies a certain compatibility condition, we prove that there exists an affine transformation acting on the space of persistence landscapes which intertwines the action of the iterated function system. This latter affine transformation is a strict contraction and its unique fixed point is the persistence landscape of the affine fractal. We present several examples of the theory as well as confirm the main results through simulations.