Neighbourhood Potential

A Neighbourhood Potential is the combination of a raster with a kernel.

r := Potential(d,K) implies that $r_{ij} = \sum\limits_{kl} d_{i-k, j-l} \times K_{kl}$

This is AKA Convolution. See also the Potential function and the Potential with kernel.

If we ignore the terms that are cut off at the borders, then $\sum\limits_{ij} R = \sum\limits_{ij} d \times \sum\limits_{kl} K$. The values unit of R is defined as the product of the values units of d and K.

Note the similarity of this operation with the multiplication of two polynomials in two unknowns, take:

$d(x,y) := \sum\limits_{ij} d_{ij} \times x^i \times y^j$

$K(x,y) := \sum\limits_{kl} K_{kl} \times x^k \times y^l$

$R(x,y) := \sum\limits_{ij} R_{ij} \times x^i \times y^j$

It follows that

R(x,y) = d(x,y) * K(x,y) if we ignore the terms that are cut off at the borders.