This is an improved Sobel operator mainly used in edge detection. It uses 3×3 convolution matrix to compute the gradient in the x-direction (\(G_x\)) and y-direction (\(G_y\)):
\[
G_x =
\begin{bmatrix}
-1 & 0 & 1\\
-\sqrt 2 & 0 & \sqrt 2 \\
-1 & 0 & 1
\end{bmatrix} * A
~~~~~~~~~~~~
G_y =
\begin{bmatrix}
-1 & -\sqrt 2 & -1\\
0 & 0 & 0 \\
1 & \sqrt 2 & 1
\end{bmatrix} * A
\]

As with the gradient vector \( (G_x,  G_y) \), we can compute the absolute magnitude of the gradient using:

\[
G = \sqrt{G_x^2 + G_y^2}
\]

For example,  if A is the input image

\[
A =
\begin{bmatrix}
P_1 & P_2 & P_3\\
P_4 & P_5 & P_6 \\
P_7 & P_8 & P_9
\end{bmatrix}
\]

We can calculate the gradient vector components:

\[
G_x = P_3 – P_1 + \sqrt{2}(P_6 – P_4) + P_9 – P_7 \\
G_y = P_7 – P_1 + \sqrt{2}(P_8 – P_2) + P_9 – P_3
\]