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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:

$h_{n,m}={\mathrm{m}\mathrm{a}\mathrm{x}}_{\mathrm{z}\mathrm{=}\mathrm{1},\mathrm{\dots },\mathrm{8}}\mathrm{\sum }_{\mathrm{i}\mathrm{=}\mathrm{-}\mathrm{1}}^{\mathrm{1}}\mathrm{\sum }_{\mathrm{j}\mathrm{=}\mathrm{-}\mathrm{1}}^{\mathrm{1}}{\mathrm{g}}_{\mathrm{i}\mathrm{j}}^{\mathrm{(}\mathrm{z}\mathrm{)}}\mathrm{\cdot }{\mathrm{f}}_{\mathrm{n}\mathrm{+}\mathrm{i},\mathrm{m}\mathrm{+}\mathrm{j}}$

where z enumerates the compass direction kernels

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

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

Example images

• 
  Original
• 
  Maximum gradient in the 8 directions

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

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

References

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