An Algorithm for Binary Image Segmentation Using Polygonal Markov Fields

Abstract

We present a novel algorithm for binary image segmentation based on polygonal Markov fields. We recall and adapt the dynamic representation of these fields, and formulate image segmentation as a statistical estimation problem for a Gibbsian modification of an underlying polygonal Markov field. We discuss briefly the choice of Hamiltonian, and develop Monte Carlo techniques for finding the optimal partition of the image. The approach is illustrated by a range of examples.

Mathematical Methods for Signal and Image Analysis and Representation

“…These were introduced a decade later in a series of our joint papers with M.N.M. van Lieshout and R. Kluszczynski [256,257,377,380,424] where a polygonal field optimization approach for image segmentation was advocated. Although these methods were quite succesful in global shape recognition, the problem we faced in that work was related to the lack of local parametrization tools designed to deal with intermediate scale image characteristics-even though the applied simulated annealing algorithm would eventually converge to the target polygonal segmentation, we were looking for a more efficient explicit mechanism to drive the local search.…”

“…Next, in Sects. 15.4 and 15.5 we develop a Markovian optimization dynamics for image segmentation, under which both the polygonal configuration and the underlying local activity function are subject to optimization-whereas the polygonal configuration evolves according to a simulated annealing scheme in the spirit of [256,257], the local activity function is initially chosen to reflect the image gradient information, whereupon it undergoes adaptive updates in the spirit of the celebrated Chen algorithm, see [73] and 10.2.4.c. in [333], with the activity profile reinforced along polygonal paths contributing to the improvement of the overall segmentation quality and faded along paths which deteriorate the segmentation quality.…”

An introduction to planar random tessellation models

“…The idea of using polygonal Markov field models for this purpose can be traced back to Clifford and Middleton (1989); see also Clifford and Nicholls (1994). Since the Monte Carlo methods employed at that time turned out to be rather onerous, the theme was not picked up again until the mid-2000s (Paskin and Thrun, 2005) when further theoretical results (Schreiber, 2005) motivated the development of conceptually and computationally easier algorithms (Kluszczyński et al, 2005(Kluszczyński et al, , 2007Schreiber and Lieshout, 2010;Lieshout, 2012). In the meantime, Voronoi (Green, 1995;Heikkinen and Arjas, 1998;Møller and Skare, 2001) and triangulation (Nicholls, 1998) models had also been tried.…”

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

“…To illustrate the techniques we are currently developing we shall employ the defective dynamics to establish exponential decay of correlations in the particular case of rectangular fields in Section 7 below. Another important field in which we anticipate the use of this dynamics is digital image segmentation -the defective dynamics will provide a rich and flexible class of new Monte-Carlo moves for polygonal field-based Bayesian segmentation algorithms we developed in joint work with Kluszczyński and van Lieshout [10,11,12,17].…”

“…A particularly interesting class of processes seem to be length-and area-interacting modifications of consistent fields, which not unexpectedly exhibit many features analogous to those of the two-dimensional Ising model, including the presence of first order phase transition at low enough temperatures (Nicholls,[14]; Schreiber, [15]) and low-temperature phase separation phenomenon (Schreiber,[16]). Consistent polygonal fields and their length-interacting modifications have also interesting statistical applications where they are used as priors in Bayesian image analysis (Clifford & Nicholls,[6]; Kluszczyński, van Lieshout & Schreiber, [10,11]; van Lieshout & Schreiber, [12,17]). …”