Markov Random Fields are probabilistic models capable of describing sets of random variables with a local dependence property (the Markov property) defined on a neighborhood system. Particularly on the context of image processing, pixels can be seen as vertices of a graph defined on a finite 2-dimensional lattice, and a neighborhood system can be defined based on their relative positions to construct a MRF.

The goal of `mrf2d`

is to provide a framework for the analysis of Markov
Random Fields with pairwise interactions on 2-dimensional lattices,
including Hidden Markov Random Fields. It introduces the S4 class `mrfi`

to describe interaction structures in a very general way, being able to
adapt from very simple cases like the Ising Model to complex anisotropic
models with different types of interaction.

A complete paper describing the details of the package and examples can be found at https://arxiv.org/abs/2006.00383.

You can install the stable version of `mrf2d`

from
CRAN with:

```
install.packages("mrf2d")
```

The development version is available on the package’s Github
page. It can be installed with the
`devtools`

package by using

```
devtools::install_github("Freguglia/mrf2d")
```

`mrf2d`

introduces a programming interface for the general Markov Random
Field model in *Freguglia, Victor, Nancy L. Garcia, and Juliano L.
Bicas. “Hidden Markov random field models applied to color homogeneity
evaluation in dyed textile images.” Environmetrics (2019): e2613.* Using
specific interaction structures and parameter restrictions can lead to
important models as particular cases, such as the Potts model.

It introduces the S4 class `mrfi`

to represent interaction structures.
The `mrfi()`

function can be used to create these objects representing
interaction structures with relative positions included based on the
norm of the relative position (distance) or explicitly specified

```
interact <- mrfi(max_norm = 1, positions = list(c(4,2)))
interact
#> 3 interacting positions.
#> rx ry
#> 1 0
#> 0 1
#> 4 2
plot(interact)
```

Potentials (parameters) are represented by three-dimensional arrays, where rows and columns represent pixel label values and slices represent interacting positions.

```
potentials <- expand_array(c(-0.9, -0.9, 0.2), family = "oneeach", C = 1, mrfi = interact)
potentials
#> , , (1,0)
#>
#> 0 1
#> 0 0.0 -0.9
#> 1 -0.9 0.0
#>
#> , , (0,1)
#>
#> 0 1
#> 0 0.0 -0.9
#> 1 -0.9 0.0
#>
#> , , (4,2)
#>
#> 0 1
#> 0 0.0 0.2
#> 1 0.2 0.0
```

The negative values out of diagonal means different “colors” are less likely in that relative position.

The package has many built-in functions for sampling, potentials estimation and hidden MRF model fitting (used for image segmentation), but it also provides all the basic stack of computations used to implement algorithms for MRF models, making it suitable for development of research in Markov Random Field models.

```
set.seed(1)
img_dim <- c(200,200)
Z <- rmrf2d(img_dim, mrfi = interact, theta = potentials, cycles = 60)
dplot(Z, legend = TRUE)
```

If you’re interested in contributing or found a bug or error, please file an issue. Contributions can be done in form of code optimization, new ideas, discussing new structures, etc.

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