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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:
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| :<math>
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| 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}
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| </math>
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| where z enumerates the compass direction kernels
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| :<math>
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| \mathbf{g^{(1)}} = \begin{bmatrix}
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| +5 & +5 & +5 \\
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| -3 & 0 & -3 \\
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| -3 & -3 & -3
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| \end{bmatrix},\
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| \mathbf{g^{(2)}} = \begin{bmatrix}
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| +5 & +5 & -3 \\
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| +5 & 0 & -3 \\
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| -3 & -3 & -3
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| \end{bmatrix},\
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| \mathbf{g^{(3)}} = \begin{bmatrix}
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| +5 & -3 & -3 \\
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| +5 & 0 & -3 \\
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| +5 & -3 & -3
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| \end{bmatrix},\
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| \mathbf{g^{(4)}} = \begin{bmatrix}
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| -3 & -3 & -3 \\
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| +5 & 0 & -3 \\
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| +5 & +5 & -3
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| \end{bmatrix}</math> and so on.
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| The edge direction is defined by the mask that produces the maximum edge magnitude.
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| == Example images ==
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| <center>
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| <gallery widths=200 heights=150 perrow=4>
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| File:Boxfilter pavilion original.jpg|Original
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| File:Kirschfilter_maximum.jpg|Maximum gradient in the 8 directions
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| </gallery>
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| <gallery widths=200 heights=150 perrow=4>
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| File:Kirschfilter1.jpg|Image filtered with g1
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| File:Kirschfilter2.jpg|Image filtered with g2
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| File:Kirschfilter3.jpg|Image filtered with g3
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| File:Kirschfilter4.jpg|Image filtered with g4
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| </gallery>
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| <gallery widths=200 heights=150 perrow=4>
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| File:Kirschfilter5.jpg|Image filtered with g5
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| File:Kirschfilter6.jpg|Image filtered with g6
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| File:Kirschfilter7.jpg|Image filtered with g7
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| File:Kirschfilter8.jpg|Image filtered with g8
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| </gallery>
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| </center>
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| == References ==
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| *{{cite journal
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| | last=Kirsch
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| | first=R.
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| | title=Computer determination of the constituent structure of biological images
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| | journal=Computers and Biomedical Research
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| | volume=4
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| | pages=315–328
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| | year=1971
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| | doi=10.1016/0010-4809(71)90034-6
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| | url=http://www.sciencedirect.com/science/article/pii/0010480971900346
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| }}
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| [[Category:Feature detection]]
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