arak

In the 1980s, Arak and Surgailis introduced a class of planar Markov fields whose realisations form a coloured tessellation of the plane.
Multi-colour random fields with polygonal realisations

The simplest and most widely studied example of a planar Markov field is the Arak process which consists of self-avoiding closed polygonal contours.
Multi-colour random fields with polygonal realisations

For the Arak process, the Hamiltonian is proportional to the total contour length by a factor two.
Multi-colour random fields with polygonal realisations

For the more general case in which both length and area terms feature in the Hamiltonian, Schreiber proposes a Metropolis–Hastings scheme based on the dynamic representation of the Arak process, which Kluszczy´nski et al. adapt and implement to solve foreground/background image segmentation problems.
Multi-colour random fields with polygonal realisations

An anisotropic Arak process can be defined through local activity functions instead of the length functional [22, 24] and allows for increased flexibility while preserving desirable basic properties including a dynamic representation.
Multi-colour random fields with polygonal realisations