Kirsch operator

The Kirsch operator or Kirsch compass kernel is a non-linear edge detector that finds the maximum edge strength in a few predetermined directions. It is named after the computer scientist Russell Kirsch.

Mathematical description
The operator takes a single kernel mask and rotates it in 45 degree increments through all 8 compass directions: N, NW, W, SW, S, SE, E, and NE. The edge magnitude of the Kirsch operator is calculated as the maximum magnitude across all directions:
 * $$h_{n,m}=\max_{z=1,\dots,8}\sum_{i=-1}^1\sum_{j=-1}^1g_{ij}^{(z)}\cdot f_{n+i,m+j}$$

where z enumerates the compass direction kernels g:

\mathbf{g^{(1)}} = \begin{bmatrix} +5 & +5 & +5 \\ -3 & 0 & -3 \\ -3 & -3 & -3 \end{bmatrix},\ $$$$ \mathbf{g^{(2)}} = \begin{bmatrix} +5 & +5 & -3 \\ +5 & 0 & -3 \\ -3 & -3 & -3 \end{bmatrix},\ $$$$ \mathbf{g^{(3)}} = \begin{bmatrix} +5 & -3 & -3 \\ +5 & 0 & -3 \\ +5 & -3 & -3 \end{bmatrix},\ $$$$ \mathbf{g^{(4)}} = \begin{bmatrix} -3 & -3 & -3 \\ +5 & 0 & -3 \\ +5 & +5 & -3 \end{bmatrix}$$ and so on.

The edge direction is defined by the mask that produces the maximum edge magnitude.