Discrete Laplace Operator

Theorems

If the graph is an infinite square lattice grid, then this definition of the Laplacian can be shown to correspond to the continuous Laplacian in the limit of an infinitely fine grid. Thus, for example, on a one-dimensional grid we have

$\frac{\partial^2F}{\partial x^2} = \lim_{\epsilon \rightarrow 0} \frac{+}{\epsilon^2}.$

This definition of the Laplacian is commonly used in numerical analysis and in image processing. In image processing, it is considered to be a type of digital filter, more specifically an edge filter, called the Laplace filter.

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Discrete Laplace Operator - Theorems