Bayesan analysis of deformed tessellation models

We define a class of tessellation models based on perturbing or deforming standard Tessellations such as the Voronoi tessellation. We show how distributions over this class of deformed tessellations can be used to define prior distributions for models based on tessellations, and how inference for such models can be carried out using Markov chain Monte Carlo methods; stability properties of the algorithms are investigated. Our approach applies not only to fixed dimension problems, but also to variable dimension problems, in which the number of cells in the tessellation is unknown. We illustrate our methods with two real examples. The first relates to reconstructing animal territories, represented by the individual cells of a tessellation, from observation of an inhomogeneous Poison point process. The second example involves the analysis of an image of a cross-section through a sample of metal, with the tessellation modelling the micro-crystalline structure of the metal.