Markov Chain Monte Carlo Shape Sampling Using Level Sets

Abstract

In this paper, we show how the Metropolis-Hastings algorithm can be used to sample shapes from a distribution defined over the space of signed distance functions. We extend the basic random walk Metropolis-Hastings method to high-dimensional curves using a proposal distribution that can simultaneously maintain the signed distance function property and the ergodic requirement. We show that detailed balance is approximately satisfied and that the Markov chain will asymptotically converge. A key advantage of our approach is that the shape representation is implicit throughout the process, as compared to existing work where explicit curve parameterization is required. Furthermore, our framework can be carried over to 3D situations easily. We show several applications of the framework to shape sampling from multimodal distributions and medical image segmentation.