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:

Failed to parse (syntax error): {\displaystyle \mathbf{g^{(1)}} = \begin{bmatrix} +5 & +5 & +5 \\ -3 & 0 & -3 \\ -3 & -3 & -3 \end{bmatrix},\ } Failed to parse (syntax error): {\displaystyle \mathbf{g^{(2)}} = \begin{bmatrix} +5 & +5 & -3 \\ +5 & 0 & -3 \\ -3 & -3 & -3 \end{bmatrix},\ } Failed to parse (syntax error): {\displaystyle \mathbf{g^{(3)}} = \begin{bmatrix} +5 & -3 & -3 \\ +5 & 0 & -3 \\ +5 & -3 & -3 \end{bmatrix},\ } ${\mathbf{g}}^{\mathbf{(}\mathbf{4}\mathbf{)}}=\left[\begin{array}{ccc}-3& -3& -3\\ +5& 0& -3\\ +5& +5& -3\end{array}\right]$ 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

• 
  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. CiteSeerX 10.1.1.161.956. doi:10.1016/0010-4809(71)90034-6.