Abstract
Markov random fields on two-dimensional lattices are behind many image analysis methodologies. mrf2d provides tools for statistical inference on a class of discrete stationary Markov random field models with pairwise interaction, which includes many of the popular models such as the Potts model and texture image models. The package introduces representations of dependence structures and parameters, visualization functions and efficient (C++ based) implementations of sampling algorithms, common estimation methods and other key features of the model, providing a useful framework to implement algorithms and working with the model in general. This paper presents a description and details of the package, as well as some reproducible examples of usage.
Highlights
A Markov Random Field (MRF) is a generalization of the well-known concept of a Markov Chain where variables are indexed by vertices of a graph instead of a sequence and the notion of memory is substituted by the neighborhood of that graph
While there is a continuous development of methodologies used in MRFs in the theoretical field, implementing new algorithms is a challenge in practice, mostly due to the high-dimensionality of the problem and the complexity of the data structures required to represent the data in this type of problem
Potts Model as a particular case The Potts model (Potts 1952) is one of the most important MRF model used in image segmentation because it can assign higher probability for equal-valued pairs of nearest-neighbors, creating large regions of pixels with the same values
Summary
A Markov Random Field (MRF) is a generalization of the well-known concept of a Markov Chain where variables are indexed by vertices of a graph instead of a sequence and the notion of memory is substituted by the neighborhood (edges) of that graph. Markov Random Fields on lattices, or more generally, Gibbs distributions, have been studied in Statistical Mechanics as models for interacting particle systems They range from the very simple Ising model (or its generalization Potts model) with pairwise nearest-neighbor interaction to models with more complex interaction types, presenting long-range and/or higher-order interaction. While there is a continuous development of methodologies used in MRFs in the theoretical field, implementing new algorithms is a challenge in practice, mostly due to the high-dimensionality of the problem and the complexity of the data structures required to represent the data in this type of problem. The mrf2d package (Freguglia 2020) provides a complete framework for statistical inference on discrete-valued MRF models on 2-dimensional lattice data, where all the elements used by algorithms (such as conditional probabilities, pseudo-likelihood function, simulation, sufficient statistics and more) are available for the user, as well as many built-in model fitting functions.
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