mrf2d-family: Parameter restriction families

In Freguglia/mrf2d: Markov Random Field Models for Image Analysis

Parameter restriction families

Description

Different parameter restrictions can be included in estimation processes to make sure mrf2d can successfully include a wide range of model types in its inference functions.

For model identifiability, at least one linear restriction is necessary. mrf2d always assume θ_{0,0,r} = 0 for all relative positions r.

Additionally, each family of restrictions may introduce other restrictions:

'onepar'

This family assumes the model is defined by a single parameter by adding the restriction

θ_{a,b,r} = φ * 1(a != b).

Here 1() denotes de indicator function. In words, the parameter must be the same value for any pair with different values and 0 for any equal-valued pair.

'oneeach'

Similar to 'onepar', parameters are 0 for equal-valued pairs and a constant for pairs with different values, but the constant may differ between different relative positions r:

θ{a,b,r} = φ_r * 1(a != b).

'absdif'

All parameters θ_{a,b,r} with the same absolute difference between a and b must be equal within each relative position r. (Note that 'absdif' is equal to 'oneeach' for binary images).

θ_{a,b,r} = ∑_d φ_{d,r} * 1(|a-b| == d)

'dif'

The same as 'absdif', but parameters may differ between positive and negative differences.

θ_{a,b,r} = ∑_d φ_{d,r} * 1(a-b == d)

'free'

No additional restriction, all parameters other than θ_{0,0,r} vary freely.

Author(s)

Victor Freguglia

See Also

vignette("mrf2d-family", package = "mrf2d")

A paper with detailed description of the package can be found at doi: 10.18637/jss.v101.i08.

Freguglia/mrf2d documentation built on Feb. 13, 2023, 3:29 a.m.