Editing Sobel operator (section)

==  Technical details  ==
 
As a consequence of its definition, the Sobel operator can be implemented by simple means in both hardware and software: only eight image points around a point are needed to compute the corresponding result and only integer arithmetic is needed to compute the gradient vector approximation.  Furthermore, the two discrete filters described above are both separable:

:<math>\begin{bmatrix} 
1 & 0 & -1 \\
2 & 0 & -2 \\
1 & 0 & -1 
\end{bmatrix} = \begin{bmatrix} 
1 \\
2 \\
1  
\end{bmatrix} \begin{bmatrix} 
1 & 0 & -1
\end{bmatrix} = \begin{bmatrix} 
1 \\
1  
\end{bmatrix} * \begin{bmatrix} 
1 \\
1  
\end{bmatrix} \begin{bmatrix} 
1 & -1
\end{bmatrix} * \begin{bmatrix} 
1 & 1
\end{bmatrix}</math>
:<math>\begin{bmatrix} 
\ \ 1 & \ \ 2 & \ \ 1 \\
\ \ 0 & \ \ 0 & \ \ 0 \\
-1 & -2 & -1 
\end{bmatrix} = \begin{bmatrix} 
\ \ 1 \\
\ \ 0 \\
-1  
\end{bmatrix} \begin{bmatrix} 
1 & 2 & 1
\end{bmatrix} = \begin{bmatrix} 
1 \\
1  
\end{bmatrix} * \begin{bmatrix} 
\ \ 1 \\
-1  
\end{bmatrix} \begin{bmatrix} 
1 & 1
\end{bmatrix} * \begin{bmatrix} 
1 & 1
\end{bmatrix}
</math>

and the two derivatives '''G'''<sub>''x''</sub> and '''G'''<sub>''y''</sub> can therefore be computed as

:<math>
\mathbf{G}_x = \begin{bmatrix} 
1 \\
2 \\
1
\end{bmatrix} * \left ( \begin{bmatrix} 
1 & 0 & -1  
\end{bmatrix} * \mathbf{A} \right )
\quad \mbox{and} \quad 
\mathbf{G}_y = \begin{bmatrix} 
\ \ 1 \\
\ \ 0 \\
-1  
\end{bmatrix} * \left ( \begin{bmatrix} 
1 & 2 & 1
\end{bmatrix} * \mathbf{A} \right )
</math>

In certain implementations, this separable computation may be advantageous since it implies fewer arithmetic computations for each image point.

Applying convolution ''K'' to pixel group ''P'' can be represented in pseudocode as:

:<math>N(x,y) = \sum_{i=-1}^{1} \sum_{j=-1}^1 K(i,j)\, P(x-i,y-j)</math>

where <math>N(x,y)</math> represents the new pixel matrix resulted  after applying the convolution ''K'' to ''P''; ''P'' being the original pixel matrix.