The Kirsch operator is a non-linear edge detector that delivers the most pronounced gradient direction as the edge strength of a pixel. Only eight discrete directions are considered, starting from 0 °, in 45 ° steps.
Analytical description An analytical description is possible as follows:
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{\ displaystyle 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 the component denotes in the (i + 2) row and the (j + 2) column of the matrix .
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{\ displaystyle {\ rm {g_ {ij} ^ {(z)}}}}
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{\ displaystyle {\ rm {g ^ {(z)}}}}
The matrices are the direction
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{\ displaystyle {\ rm {g ^ {(z)}}}}
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{\ displaystyle \ 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}}}
example Pictures
Maximum gradient of the 8 directions
<img src="https://de.wikipedia.org//de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">
