## What is Gaussian random field theory?

A Gaussian random field (GRF) within statistics, is a random field involving Gaussian probability density functions of the variables. A one-dimensional GRF is also called a Gaussian process. An important special case of a GRF is the Gaussian free field.

What are Markov random fields used for?

Markov random fields find application in a variety of fields, ranging from computer graphics to computer vision, machine learning or computational biology. MRFs are used in image processing to generate textures as they can be used to generate flexible and stochastic image models.

### What is MRF in image processing?

∎ In real images, regions are often homogenous; neighboring pixels usually have similar properties (intensity, color, texture, …) ∎ Markov Random Field (MRF) is a probabilistic. model which captures such contextual. constraints.

What is the random field theory?

Random field theory (RFT) is a recent body of mathematics defining theoretical results for smooth statistical maps [1]. Values in a random field are usually spatially correlated in one way or another, in its most basic form this might mean that adjacent values do not differ as much as values that are further apart [2].

#### What is the difference between Markov networks and Bayesian networks?

A Markov network or MRF is similar to a Bayesian network in its representation of dependencies; the differences being that Bayesian networks are directed and acyclic, whereas Markov networks are undirected and may be cyclic. The underlying graph of a Markov random field may be finite or infinite.

What is local Markov property?

Local Markov Property – The random variable associated with a node Xi is independent of the rest of the nodes in the graph given its immediate neighbors Ni, i.e. Xi ⊥ X\i\Ni |XNi .

## What is a clique in Markov network?

Cliques are the subset or a subgraph of an undirected graphical model such that every two distinct vertices in clique are adjacent to each other. In conclusion, A distribution factorizes over a Markov Network, H if P can be expressed as follows, where D represents complete subgraph in H.

What is texture segmentation?

Texture segmentation is the process of partitioning an image into regions with different textures containing similar group of pixels. Basically, it aims at segmenting a textured image into several regions having the similar patterns.

### How was random field theory used?

The theory has been versatile in dealing with many of the thresholding problems that we encounter in func- tional imaging. Among many other applications, it can be used to solve our problem of finding the height threshold for a smooth statistical map which gives the required family–wise error rate.

Is HMM a Bayesian network?

Simply stated, hidden Markov models are a particular kind of Bayesian network. In section 3 we will provide a short tutorial on Bayesian networks and describe how HMMs and other Markov models relate to them.

#### What is a Markov random field?

In the domain of physics and probability, a Markov random field ( MRF ), Markov network or undirected graphical model is a set of random variables having a Markov property described by an undirected graph. In other words, a random field is said to be a Markov random field if it satisfies Markov properties.

What is an example of a Gaussian random field?

Consider for example a sequence of Gaussian Markov random fields on sublattices with spacing 1/ m, m =1,2,…, which have individual spectral densities of the form

## What is piecewise linear Gaussian Markov random-field approximation?

A piecewise linear Gaussian Markov random-field approximation is constructed that globally approximates the true random field up to a given resolution. This is a particularly interesting feature in the context of spatial point process modelling.

Can we use Gaussian Markov random fields to model spatial data?

The advantages of working with Gaussian Markov random fields have been well exploited for fitting hierarchical Bayesian models for spatial data. However, as pointed out by the authors, one often wishes to model data drawn from Gaussian fields (GFs).