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The '''Kirsch operator''' or '''Kirsch compass kernel''' is a [[Nonlinearity|non-linear]] [[edge detection|edge detector]] that finds the maximum edge strength in a few predetermined directions. It is named after the computer scientist [[Russell A. Kirsch]].
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== Mathematical description ==
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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:
:<math>
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}
</math>
 
where z enumerates the compass direction kernels
:<math>
\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}</math> and so on.  
 
The edge direction is defined by the mask that produces the maximum edge magnitude.
 
== Example images ==
<center>
<gallery widths=200  heights=150 perrow=4>
  File:Boxfilter pavilion original.jpg|Original
  File:Kirschfilter_maximum.jpg|Maximum gradient in the 8 directions
</gallery>
<gallery widths=200 heights=150 perrow=4>
  File:Kirschfilter1.jpg|Image filtered with g1
  File:Kirschfilter2.jpg|Image filtered with g2
  File:Kirschfilter3.jpg|Image filtered with g3
  File:Kirschfilter4.jpg|Image filtered with g4
</gallery>
 
<gallery widths=200  heights=150 perrow=4>
  File:Kirschfilter5.jpg|Image filtered with g5
  File:Kirschfilter6.jpg|Image filtered with g6
  File:Kirschfilter7.jpg|Image filtered with g7
  File:Kirschfilter8.jpg|Image filtered with g8
</gallery>
</center>
 
== References ==
*{{cite journal
| last=Kirsch
| first=R.
| title=Computer determination of the constituent structure of biological images
| journal=Computers and Biomedical Research
| volume=4
| pages=315–328
| year=1971
| doi=10.1016/0010-4809(71)90034-6
| url=http://www.sciencedirect.com/science/article/pii/0010480971900346
}}
 
[[Category:Feature detection]]

Latest revision as of 08:50, 5 January 2015

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