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 \(\theta_{0,0,r} = 0\) for all relative positions
\(r\).
Additionally, each family of restrictions may introduce other restrictions:
This family assumes the model is defined by a single parameter by adding the restriction
$$\theta_{a,b,r} = \phi * 1(a != b).$$
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.
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 \(r\):
$$\theta{a,b,r} = \phi_r * 1(a != b).$$
All parameters \(\theta_{a,b,r}\) 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).
$$\theta_{a,b,r} = \sum_d \phi_{d,r} * 1(|a-b| == d)$$
The same as 'absdif', but parameters may differ between positive and
negative differences.
$$\theta_{a,b,r} = \sum_d \phi_{d,r} * 1(a-b == d)$$
No additional restriction, all parameters other than \(\theta_{0,0,r}\) vary freely.
vignette("mrf2d-family", package = "mrf2d")
A paper with detailed description of the package can be found at doi: 10.18637/jss.v101.i08 .