An Expectation-Maximization Algorithm for the Fractal Inverse Problem

Under submission with

see the pre-print at the arXiv

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 data sources outside the IFS model class.