Editing Sobel operator (section)

== Extension to other dimensions ==
The Sobel–Feldman operator consists of two separable operations:<ref>{{cite book | title= Real-time volume graphics |author= K. Engel| year=2006| pages=112–114}}</ref>
* Smoothing perpendicular to the derivative direction with a triangle filter: <math> h(-1) = 1,  h(0) = 2,  h(1) = 1 </math>
* Simple central difference in the derivative direction: <math> h'(-1) = 1,  h'(0) = 0,  h'(1) = -1 </math>

Sobel–Feldman filters for [[image derivative]]s in different dimensions with <math>x,y,z,t \in \left\{0, -1, 1\right\} </math> :

1D: <math> h_x'(x) = h'(x); </math>

2D: <math> h_x'(x,y) = h'(x)h(y) </math>

2D: <math> h_y'(x,y) = h(x)h'(y) </math>


3D: <math> h_y'(x,y,z) = h(x)h'(y)h(z)</math>

3D: <math> h_z'(x,y,z) = h(x)h(y)h'(z)</math>

4D: <math> h_x'(x,y,z,t) = h'(x)h(y)h(z)h(t)</math>

Thus as an example the 3D Sobel–Feldman kernel in ''z''-direction:

:<math>
h_z'(:,:,-1) = 
\begin{bmatrix} 
+1 & +2 & +1  \\
+2 & +4 & +2 \\
+1 & +2 & +1 
\end{bmatrix}
\quad
h_z'(:,:,0) = 
\begin{bmatrix} 
0 & 0 & 0  \\
0 & 0 & 0 \\
0 & 0 & 0 
\end{bmatrix}
\quad
h_z'(:,:,1) = 
\begin{bmatrix} 
-1 & -2 & -1  \\
-2 & -4 & -2 \\
-1 & -2 & -1 
\end{bmatrix}

</math>