Non-homogeneous Polygonal Markov Fields in the Plane: Graphical Representations and Geometry of Higher Order Correlations

Abstract

We consider polygonal Markov fields originally introduced by Arak and Surgailis [2]. Our attention is focused on fields with nodes of order two, which can be regarded as continuum ensembles of non-intersecting contours in the plane, sharing a number of features with the two-dimensional Ising model. We introduce non-homogeneous version of polygonal fields in anisotropic enviroment. For these fields we provide a class of new graphical constructions and random dynamics. These include a generalised dynamic representation, generalised and defective disagreement loop dynamics as well as a generalised contour birth and death dynamics. Next, we use these constructions as tools to obtain new exact results on the geometry of higher order correlations of polygonal Markov fields in their consistent regime.

Mathematical Methods for Signal and Image Analysis and Representation

“…The so-defined locally specified polygonal fields enjoy a number of striking features inherited from the previously developed polygonal models, see [11,379]. One of these is the two-dimensional germ-Markov property stating that the conditional behaviour of the field A M D inside a smooth closed curve θ depends on the outside field configuration only through the trace it leaves on θ , consisting of intersection points and the respective line directions, see [11] for details.…”

“…This allows us to define for each bounded linear segment/graph edge e in D its initial point ι ), e ∈ Edges(γ ). Observe that this construction should be regarded as a specific version of the general polygonal model given by Arak and Surgailis [11, 2.11] and an extension of the non-homogeneous polygonal fields considered in Schreiber [379] at their consistent regime (inverse temperature parameter fixed to 1). It should be also noted at this point that if the typical edge length for A M D is much smaller than the characteristic scale for oscillations of M , which is often the case in our applications below is not difficult to verify, see [379], and in fact it will be explicitly calculated in the sequel.…”

“…The present section is meant to extend the so-called generalised dynamic representation for consistent polygonal fields as developed in Schreiber [379] to cover the more general class of locally specified polygonal fields defined in Sect. 15.2 above.…”

“…This dynamics will be used in the sequel as a mechanism for update proposal generation in stochastic optimization schemes for image segmentation. We build upon [377,379] in our presentation of the dynamics based on an important concept of a disagreement loop.…”

An introduction to planar random tessellation models

“…In general, the distribution of the typical polygon in consistent polygonal Markov field models seems difficult to obtain. Partial results can be found in Schreiber (2005Schreiber ( , 2008.…”