Editing Sobel operator (section)

== Alternative operators ==

The Sobel–Feldman operator, while reducing artifacts associated with a pure central differences operator, does not exhibit a good rotational symmetry (about 1° of error).  Scharr looked into optimizing this property by producing kernels optimized for specific given numeric precision (integer, float…) and dimensionalities (1D, 2D, 3D).<ref>Scharr, Hanno, 2000, Dissertation (in German), [http://nbn-resolving.de/urn/resolver.pl?urn=urn:nbn:de:bsz:16-opus-9622 ''Optimal Operators in Digital Image Processing'' ].</ref><ref>B. Jähne, H. Scharr, and S. Körkel. Principles of filter design. In Handbook of Computer Vision and Applications. Academic Press, 1999.</ref> Optimized 3D filter kernels up to a size of 5 x 5 x 5 have been presented there, but the most frequently used, with an error of about 0.2° is:

: <math>
h_x'(:,:) =
\begin{bmatrix} 
+3 & 0 & -3 \\
+10 & 0 & -10 \\
+3 & 0 & -3 
\end{bmatrix}
\ \ \ \ \ \ \ \ \  
h_y'(:,:) =
\begin{bmatrix} 
+3 & +10 & +3 \\
 0 &   0 &  0 \\
-3 & -10 & -3 
\end{bmatrix}
</math>

This factors similarly:

* <math>\begin{bmatrix} 
3 & 10 & 3
\end{bmatrix} = \begin{bmatrix} 
3 & 1  
\end{bmatrix} * \begin{bmatrix} 
1 & 3
\end{bmatrix}</math>

{{anchor|Scharr operator}}'''Scharr operators''' result from an optimization minimizing weighted [[Mean squared error|mean squared]] angular error in the [[Fourier domain]]. This optimization is done under the condition that resulting filters are numerically consistent. Therefore they really are derivative kernels rather than merely keeping symmetry constraints. The optimal 8 bit integer valued 3x3 filter stemming from Scharr's theory is
:<math>
h_x'(:,:) =
\begin{bmatrix}
47 & 0 & -47 \\
162 & 0 & -162 \\
47 & 0 & -47 
\end{bmatrix}
\ \ \ \ \ \ \ \ \ 
h_y'(:,:) =
\begin{bmatrix} 
47 & 162 & 47 \\
0 & 0 & 0 \\
-47 & -162 & -47
\end{bmatrix}
</math>

A similar optimization strategy and resulting filters were also presented by Farid and Simoncelli.<ref>H. Farid and E. P. Simoncelli, [http://www.cs.dartmouth.edu/~farid/downloads/publications/caip97.pdf ''Optimally Rotation-Equivariant Directional Derivative Kernels''], Int'l Conf Computer Analysis of Images and Patterns, pp. 207–214, Sep 1997.</ref><ref>H. Farid and E. P. Simoncelli, [http://www.cns.nyu.edu/pub/lcv/farid03-reprint.pdf ''Differentiation of discrete multi-dimensional signals''], IEEE Trans Image Processing, vol.13(4), pp. 496–508, Apr 2004.</ref> They also investigate higher-order derivative schemes. In contrast to the work of Scharr, these filters are not enforced to be numerically consistent.

The problem of derivative filter design has been revisited e.g. by Kroon.<ref>D. Kroon, 2009, Short Paper University Twente, [http://www.k-zone.nl/Kroon_DerivativePaper.pdf ''Numerical Optimization of Kernel-Based Image Derivatives'' ].</ref>

Derivative filters based on arbitrary cubic splines were presented by Hast.<ref>A. Hast., [http://www.sciencedirect.com/science/article/pii/S0167865514000282 "Simple filter design for first and second order derivatives by a double filtering approach"], Pattern Recognition Letters, Vol. 42, no.1 June, pp. 65–71. 2014.</ref> He showed how first and second order derivatives can be computed correctly using cubic or trigonometric splines by a double filtering approach giving filters of length 7.

Another similar operator that was originally generated from the Sobel operator is the Kayyali operator,<ref>{{Cite journal|last=Dim|first=Jules R.|last2=Takamura|first2=Tamio|date=2013-12-11|title=Alternative Approach for Satellite Cloud Classification: Edge Gradient Application|journal=Advances in Meteorology|language=en|volume=2013|pages=1–8|doi=10.1155/2013/584816|issn=1687-9309|doi-access=free}}</ref> a perfect rotational symmetry based convolution filter 3x3.

Orientation-optimal derivative kernels drastically reduce systematic estimation errors in [[optical flow]] estimation. Larger schemes with even higher accuracy and optimized filter families for extended optical flow estimation have been presented in subsequent work by Scharr.<ref name="Scharr pp. 14–29">{{cite book | last=Scharr | first=Hanno | title=Complex Motion | chapter=Optimal Filters for Extended Optical Flow | series=Lecture Notes in Computer Science | year=2007 | volume=3417 | publisher=Springer Berlin Heidelberg | location=Berlin, Heidelberg | isbn=978-3-540-69864-7 | doi=10.1007/978-3-540-69866-1_2 | pages=14–29}}</ref> Second order derivative filter sets have been investigated for transparent [[motion estimation]].<ref>Scharr, Hanno, [http://www.eurasip.org/Proceedings/Eusipco/Eusipco2007/Papers/a2l-g03.pdf ''OPTIMAL SECOND ORDER DERIVATIVE FILTER FAMILIES FOR TRANSPARENT MOTION ESTIMATION'' ] 15th European Signal Processing Conference (EUSIPCO 2007), Poznan, Poland, September 3–7, 2007.</ref> It has been observed that the larger the resulting kernels are, the better they approximate derivative-of-Gaussian filters.