Laplacian Filter
Laplacian of $f$ can be represented as follows, where 'div' denotes divergence and 'grad' denotes gradient.
$\nabla{f}=\mathrm{div}(\mathrm{grad}(f)) = \frac{\partial^{2}f}{\partial{x^2}} + \frac{\partial^{2}f}{\partial{y^2}} + \frac{\partial^{2}f}{\partial{z^2}} $
The detailed explanation of the laplacian operation, divergence, and gradient are presented in the following urls:
The Laplacian operation w.r.t $x$ axis can be rewritten in a discrete manner:
$ \frac{\partial^2 f}{\partial{x^2}} = [f(x+1, y) - f(x,y)] - [f(x, y) - f(x-1, y)] = f(x+1, y) + f(x-1, y) - 2f(x, y) $
$ \frac{\partial{f}}{\partial{x}} = f(x+1, y) - f(x, y) $
Then, the Laplacian operation w.r.t $x$ and $y$ axes can be written as follows, It should be noted that the Laplacian operation is equal to zero (need to read more info about it).
$ \frac{\partial^2 f}{\partial{x^2}} + \frac{\partial^2 f}{\partial{y^2}} = [f(x+1, y) + f(x-1, y) + f(x, y+1) + f(x, y-1)] - 4f(x,y)=0 $
$s.t. \: \nabla^2{f}=0$