Laplacian Operator with and without Diagonal Direction Elements in the Kernel

Question

This is a general question on the laplacian operator, which has two different versions. The first version is :

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

The second version includes the diagonal:

\begin{matrix} 1 & 1 & 1 \\ 1 & -8 &1 \\ 1 & 1 & 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?

Answer 1

Isotropy is uniformity in all orientations. If something is isotropic, its geometrical information is invariant from direction.

The former exhibits a clear difference between the grid directions and the directions at a 45-degree angle to the grid. This is why it is not referred as isotropic. The latter on the other hand, applies the same operation, regardless of the direction. The isotropic version is rotation invariant and no matter how the image/the filter is rotated, it generates the same response.

This becomes important when for example creating feature point detectors, where the descriptors are to be found regardless of image orientation. There are many applications in the vision literature and this is a very desired property. For more details, refer to the Wikipedia - Discrete Laplace Operator subject.

Answer 2

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.