The Kirsch operator or Kirsch compass kernel is a

non-linear In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathe ...

edge detector that finds the maximum edge strength in a few predetermined directions. It is named after the computer scientist

Russell Kirsch Russell A. Kirsch (June 20, 1929 – August 11, 2020) was an American engineer at the National Bureau of Standards (now known as the National Institute of Standards and Technology). He was recognized as the developer of the first digital image s ...

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_=\max_\sum_^1\sum_^1g_^\cdot f_

where z enumerates the compass direction kernels g: :

\mathbf = \begin 
+5 & +5 & +5 \\
-3 &  0 & -3 \\
-3 & -3 & -3 
\end,\

\mathbf = \begin 
+5 & +5 & -3 \\
+5 &  0 & -3 \\
-3 & -3 & -3 
\end,\

\mathbf = \begin 
+5 & -3 & -3 \\
+5 &  0 & -3 \\
+5 & -3 & -3 
\end,\

\mathbf = \begin 
-3 & -3 & -3 \\
+5 &  0 & -3 \\
+5 & +5 & -3 
\end

and so on. The edge direction is defined by the mask that produces the maximum edge magnitude.

Example images

File:Boxfilter pavilion original.jpg, Original File:Kirschfilter_maximum.jpg, Maximum gradient in the 8 directions File:Kirschfilter3.jpg, Image filtered with g⁽¹⁾ File:Kirschfilter2.jpg, Image filtered with g⁽²⁾ File:Kirschfilter1.jpg, Image filtered with g⁽³⁾ File:Kirschfilter8.jpg, Image filtered with g⁽⁴⁾ File:Kirschfilter7.jpg, Image filtered with g⁽⁵⁾ File:Kirschfilter6.jpg, Image filtered with g⁽⁶⁾ File:Kirschfilter5.jpg, Image filtered with g⁽⁷⁾ File:Kirschfilter4.jpg, Image filtered with g⁽⁸⁾

References

*{{cite journal , last=Kirsch , first=R. , title=Computer determination of the constituent structure of biological images , journal=Computers and Biomedical Research , volume=4 , issue=3 , pages=315–328 , year=1971 , doi=10.1016/0010-4809(71)90034-6 , citeseerx=10.1.1.161.956 Edge detection