An Expectation-Maximization Algorithm for the Fractal Inverse Problem

Peter Bloem Knowledge Representation and Reasoning Group VU University Amsterdam De Boelelaan 1105, 1081 HV Amsterdam, NL Steven de Rooij † Mathematical Institute University of Leiden Niels Bohrweg 1, 2333 CA Leiden, NL (Dated: February 9, 2018) 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 R H with arbitrary dimensionH . Each IFS defines a probability distribution on 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.