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 A. 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}}={\rm {{max}_{{z=1,\ldots ,8}}\sum _{{i=-1}}^{{1}}\sum _{{j=-1}}^{{1}}g_{{ij}}^{{(z)}}\cdot f_{{n+i,m+j}}}}

where z enumerates the compass direction kernels

{\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

Original
Maximum gradient in the 8 directions

Image filtered with g1
Image filtered with g2
Image filtered with g3
Image filtered with g4

Image filtered with g5
Image filtered with g6
Image filtered with g7
Image filtered with g8

References

Kirsch, R. (1971). "Computer determination of the constituent structure of biological images". Computers and Biomedical Research. 4: 315–328. doi:10.1016/0010-4809(71)90034-6.

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