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

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  Original

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  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

• 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.