Built-in Mathematica Symbol

GaussianMatrix

GaussianMatrix[r]
gives a matrix that corresponds to a Gaussian kernel of radius r.

}]
gives a matrix corresponding to a Gaussian kernel with radius r and standard deviation

.

GaussianMatrix[r, {n₁, n₂}]
gives a matrix formed from the n₁

discrete derivative of the Gaussian with respect to rows and n₂

discrete derivative with respect to columns.

GaussianMatrix[r, {{n₁₁, n₁₂}, {n₂₁, n₂₂}, ...}]
gives a matrix formed from the sums of the n_i1 and n_i2 derivatives.

GaussianMatrix[{{r₁, r₂, ...},

}, ...]
gives an array corresponding to a Gaussian kernel with radius r_i in the i

index direction.

GaussianMatrix[r] gives values that approximate at x index positions from the center, where =r/2.

GaussianMatrix[..., {n₁, n₂}] by default constructs discrete derivatives as finite differences.

GaussianMatrix[r, {{2, 0}, {0, 2}}] gives a matrix formed from the Laplacian of a Gaussian.

GaussianMatrix[{Automatic, , f}, ...] constructs a matrix just large enough to include at least a fraction f of the discrete integral of a Gaussian in each direction.

GaussianMatrix allows any of r, and f to be lists, specifying different values for different directions.

With the default option setting Method->"Bessel", GaussianMatrix[r] has elements proportional to Exp[-²] BesselI[x, ²], yielding a kernel with optimal discrete convolution properties.