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

.

gives a matrix formed from the

derivative of the Gaussian with respect to rows and the

derivative with respect to columns.

gives a matrix formed from the sums of the

and

derivatives.

gives an array corresponding to a Gaussian kernel with radius

in the i

index direction.

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

GaussianMatrix by default constructs discrete derivatives as finite differences.

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.

Method	"Bessel"	how to determine matrix elements
WorkingPrecision	Automatic	the precision with which to compute matrix elements
"Standardization"	True	whether to rescale and shift the matrix to account for truncation

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

For Method, derivatives of GaussianMatrix[r] are obtained by the finite difference operator. GaussianMatrix satisfies the finite difference equation .

With Method, GaussianMatrix[r] has elements proportional to the raw continuous functional form Exp.

For Method, derivatives of GaussianMatrix[r] are proportional to the partial derivatives of the functional form. The GaussianMatrix approximately satisfies the differential equation .

With , the elements of GaussianMatrix[r] will sum to 1. However, the elements of GaussianMatrix with at least one nonzero will sum to 0, and the sum of the elements, weighted in each direction by times the distance from the origin to the power of , will be 1.