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.

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.