The Laplace Operator
The Laplace operator is a scalar operator defined as the dot product (inner product) of two gradient vector operators:
In $N = 2$ dimensional space, we have:
When applied to a 2-D function $f (x, y)$, this operator produces a scalar function:
In discrete case, the second order differentiation becomes second order difference. In 1-D case, if the first order difference is defined as
then the second order difference is
Note that $\bigtriangleup f[n]$ is so defined that it is symmetric to the center element $f[n]$. The Laplace operation can be carried out by 1-D convolution with a kernel $[1,-2,1]$. In 2-D case, Laplace operator is the sum of two second order differences in both dimensions:
This operation can be carried out by 2-D convolution kernel with size $3\times3$:
Other Laplace kernels with size $3\times3$ can be used:
