Laplace filter

In mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a graph or a discrete grid. For the case of a finite-dimensional graph (having a finite number of edges and vertices), the discrete Laplace operator is more commonly called the Laplacian matrix.

The discrete Laplace operator occurs in physics problems such as the Ising model and loop quantum gravity, as well as in the study of discrete dynamical systems. It is also used in numerical analysis as a stand-in for the continuous Laplace operator. Common applications include image processing, where it is known as the Laplace filter, and in machine learning for clustering and semi-supervised learning on neighborhood graphs.

There are various definitions of the discrete Laplacian for graphs, differing by sign and scale factor (sometimes one averages over the neighboring vertices, other times one just sums; this makes no difference for a regular graph). The traditional definition of the graph Laplacian, given below, corresponds to the negative continuous Laplacian on a domain with a free boundary.

Let ${\displaystyle G=(V,E)}$ be a graph with vertices ${\displaystyle \scriptstyle V}$ and edges ${\displaystyle \scriptstyle E}$. Let ${\displaystyle \phi \colon V\to R}$ be a function of the vertices taking values in a ring. Then, the discrete Laplacian ${\displaystyle \Delta }$ acting on ${\displaystyle \phi }$ is defined by

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