Laplacian Operator with and without Diagonal Direction Elements in the Kernel This is a general question on the laplacian operator, which has two different versions. The first version is :

\begin{matrix}
  0 &amp; 1 &amp; 0 \\
  1 &amp; -4 &amp; 1 \\
  0 &amp; 1 &amp; 0
 \end{matrix}

The second version includes the diagonal:

\begin{matrix}
  1 &amp; 1 &amp; 1 \\
  1 &amp; -8 &amp;1 \\
  1 &amp; 1 &amp; 1
 \end{matrix}
Including the diagonal direction will make the laplacian isotropic. But what exactly is isotropic? How would that affect the resulting image if the diagonal components are included? What happen if I don't include the diagonal components?
All of those are Discrete Approximation of the operator - Laplace Operator.
You can discretize it in any logical manner.
In the cases above, Istotropic means if you rotate it it looks the same.
The diagonal elements means the filter would be sensitive to changes which are in 45, 135, 225 and 315 degrees, it it means it will also amplify more noise.