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:

$h_{n,m}=\underset{z=1,\dots ,8}{\max }\sum _{i=-1}^1\sum _{j=-1}^1{g}_{ij}^{(z)}\cdot f_{n+i,m+j}$

where z enumerates the compass direction kernels g:

${𝐠}^{\mathbf{(}\mathrm{𝟏}\mathbf{)}}=\left[\begin{array}{ccc}+5& +5& +5\\ -3& 0& -3\\ -3& -3& -3\end{array}\right],$${𝐠}^{\mathbf{(}\mathrm{𝟐}\mathbf{)}}=\left[\begin{array}{ccc}+5& +5& -3\\ +5& 0& -3\\ -3& -3& -3\end{array}\right],$${𝐠}^{\mathbf{(}\mathrm{𝟑}\mathbf{)}}=\left[\begin{array}{ccc}+5& -3& -3\\ +5& 0& -3\\ +5& -3& -3\end{array}\right],$${𝐠}^{\mathbf{(}\mathrm{𝟒}\mathbf{)}}=\left[\begin{array}{ccc}-3& -3& -3\\ +5& 0& -3\\ +5& +5& -3\end{array}\right]$ and so on.

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

Example images

• Original
  Original
• Maximum gradient in the 8 directions
  Maximum gradient in the 8 directions

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

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