Rare event simulation for Markov random fields with application to grain growth in crystals

Stochastic models of images are very useful for applications such as segmentation, deblurring, and reconstruction. Sometimes it is important to be able to simulate, or draw samples from, a stochastic image model. For example, simulation can be used as an optimization tool for segmenting, deblurring, or reconstructing an image. Also, simulation of images that characterize a system can be helpful in understanding the system, by allowing virtual exploration of models of the system instead of expensive and time-intensive physical experimentation. There are of course many Markov chain Monte Carlo (MCMC) methods for drawing samples from the ubiquitous Markov random field (MRF) image model. However, these methods draw sample images that represent typical cases of the model. To sample images that occur with low probability, which represent rare events, a prohibitive number of Monte Carlo samples would need to be drawn using traditional MCMC. In this paper, we turn to large deviations theory and importance sampling to propose a rare-event simulation method for MRFs. We then use an impactful problem from materials science to demonstrate the application of our method. More specifically, we look at the phenomenon of abnormal grain growth in polycrystalline materials. With our proposed method, we consistently generate images containing abnormal grain growth, though this is a very challenging problem for standard Monte Carlo simulation methods. Importantly, our method can be used to simulate rare events in a broad class of imaging applications, namely those that use an MRF model.