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}^{1}g_{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

Original
Maximum gradient in the 8 directions

Image filtered with g⁽¹⁾
Image filtered with g⁽²⁾
Image filtered with g⁽³⁾
Image filtered with g⁽⁴⁾

Image filtered with g⁽⁵⁾
Image filtered with g⁽⁶⁾
Image filtered with g⁽⁷⁾
Image filtered with g⁽⁸⁾

References

Kirsch, R. (1971). "Computer determination of the constituent structure of biological images". Computers and Biomedical Research. 4 (3): 315–328.
doi:10.1016/0010-4809(71)90034-6
.