An Expectation-Maximization Algorithm for the Fractal Inverse Problem
This provides a method for estimating fractal models from data, which is useful for researchers in mathematics, computer graphics, or data analysis dealing with fractal patterns, though it appears incremental as it adapts an existing algorithm to a specific problem.
The paper tackles the fractal inverse problem by developing an Expectation-Maximization algorithm to fit Iterated Function Systems (IFS) with similitudes to point cloud data in arbitrary dimensions, showing that it reconstructs known fractals with high precision parameters and approximates data outside the IFS model class.
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.