To Boldly Split: Partitioning Space Filling Curves by Markov Chain Monte Carlo Simulation

Space filling curves are a class of fractals that are important mathematical descriptions of the appearance and shape of natural objects. There is growing interest in the modelling of such curves to measure pathology in medicine and biology. This work presents a method of modelling fractal curves, such as the boundary of brain white matter, and partitioning such curves in to segments having equal fractal dimension. Since the solution space, for a given number of contour points and a required set of partitions is very large, we employ a Bayesian framework of reversible-jump Markov chain Monte Carlo (MCMC) and a sampler based on the Metropolis-Hastings test. We detail the algorithm and present results on both simple contours (animal silhouettes) and space-filling brain contours and show the convergence characteristics of the method. We discuss its use for building compact local statistical shape models.