Editing Edge detection (section)

=== Other first-order methods ===
Different gradient operators can be applied to estimate image gradients from the input image or a smoothed version of it. The simplest approach is to use central differences:

:<math>
\begin{align}
L_x(x, y) & = -\frac 1 2 L(x-1, y) + 0 \cdot L(x, y) + \frac 1 2 \cdot L(x+1, y) \\[8pt]
L_y(x, y) & = -\frac1 2 L(x, y-1) + 0 \cdot L(x, y) + \frac 1 2 \cdot L(x, y+1),
\end{align}
</math>

corresponding to the application of the following filter masks to the image data:

:<math>
L_y = \begin{bmatrix}
+1/2 & 0 & -1/2
\end{bmatrix} L
\quad \text{and} \quad
L_x = \begin{bmatrix}
+1/2 \\
0 \\
-1/2
\end{bmatrix} L.
</math>

The well-known and earlier [[Sobel operator]] is based on the following filters:

:<math>
L_y = \begin{bmatrix}
+1 & 0 & -1 \\
+2 & 0 & -2 \\
+1 & 0 & -1
\end{bmatrix} L
\quad \text{and} \quad
L_x = \begin{bmatrix}
+1 & +2 & +1  \\
0 & 0 & 0 \\
-1 & -2 & -1
\end{bmatrix} L.
</math>

Given such estimates of first-order [[image derivative]]s, the gradient magnitude is then computed as:

:<math>|\nabla L| = \sqrt{ L_x^2 + L_y^2}</math>

while the gradient orientation can be estimated as

:<math>\theta = \operatorname{atan2}(L_y, L_x).</math>

Other first-order difference operators for estimating image gradient have been proposed in the [[Prewitt operator]], [[Roberts cross]], Kayyali<ref>{{Cite journal|last1=Dim|first1=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|issue=1 |pages=1–8|doi=10.1155/2013/584816|issn=1687-9309|doi-access=free|bibcode=2013AdMet201384816D }}</ref> operator and [[Frei–Chen operator]].

It is possible to extend filters dimension to avoid the issue of recognizing edge in low [[Signal-to-noise ratio|SNR]] image. The cost of this operation is loss in terms of resolution. Examples are Extended Prewitt 7×7.