An Expectation-Maximization Algorithm for the Fractal Inverse Problem

TL;DR

This paper introduces an Expectation-Maximization algorithm to fit fractal models, specifically Iterated Function Systems, to data, enabling high-precision reconstruction and approximation of complex data sources.

Contribution

The paper develops a novel EM algorithm for the fractal inverse problem using IFS models, demonstrating effective data fitting and approximation capabilities.

Findings

Successfully reconstructs known fractals from data.

Converges to high-precision model parameters.

Approximates data sources outside the IFS class.

Abstract

We present an Expectation-Maximization algorithm for the fractal inverse problem: the problem of fitting a fractal model to data. In our setting the fractals are Iterated Function Systems (IFS), with similitudes as the family of transformations. The data is a point cloud in ${\mathbb R}^H$ with arbitrary dimension $H$. Each IFS defines a probability distribution on ${\mathbb R}^H$, so that the fractal inverse problem can be cast as a problem of parameter estimation. We show that the algorithm reconstructs well-known fractals from data, with the model converging to high precision parameters. We also show the utility of the model as an approximation for datasources outside the IFS model class.