Kirsch operator

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.

Example images

File:Boxfilter pavilion original.jpg|Original File:Kirschfilter_maximum.jpg|Maximum gradient in the 8 directions File:Kirschfilter3.jpg|Image filtered with g(1) File:Kirschfilter2.jpg|Image filtered with g(2) File:Kirschfilter1.jpg|Image filtered with g(3) File:Kirschfilter8.jpg|Image filtered with g(4) File:Kirschfilter7.jpg|Image filtered with g(5) File:Kirschfilter6.jpg|Image filtered with g(6) File:Kirschfilter5.jpg|Image filtered with g(7) File:Kirschfilter4.jpg|Image filtered with g(8)

References

• {{cite journal