BUILT-IN MATHEMATICA SYMBOL

# GaussianMatrix

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

GaussianMatrix[{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 derivative of the Gaussian with respect to rows and the derivative with respect to columns.

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

GaussianMatrix[{{r1, r2, ...}, }, ...]
gives an array corresponding to a Gaussian kernel with radius in the i index direction.

## Details and OptionsDetails and Options

• gives values that approximate at index position from the center, where .
• By default, the elements of sum to 1.
• 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.
• For integer r, GaussianMatrix[r, ...] yields a × matrix.
• For noninteger r, the value of r is effectively rounded to an integer.
• Options for GaussianMatrix include:
•  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
• Possible settings for the Method option are and .
• With the default option setting Method->"Bessel", has elements proportional to Exp[-2] BesselI[x, 2], yielding a kernel with optimal discrete convolution properties.
• For Method->"Bessel", derivatives of GaussianMatrix[r] are obtained by the finite difference operator. GaussianMatrix[{r, }] satisfies the finite difference equation .
• With Method->"Gaussian", has elements proportional to the raw continuous functional form Exp[-xi2/(2 2)]/().
• For Method->"Gaussian", derivatives of GaussianMatrix[r] are proportional to the partial derivatives of the functional form. The GaussianMatrix[{r, }] approximately satisfies the differential equation .
• With , the elements of will sum to 1. However, the elements of GaussianMatrix[r, {n1, n2, ...}] 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.
• With "Standardization"->True, the proportionality factor ensures that the elements of sum to 1. However, the elements of GaussianMatrix[r, {n1, n2, ...}] 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.
• With "Standardization"->False, no proportionality factor is used.

## ExamplesExamplesopen allclose all

### Basic Examples (3)Basic Examples (3)

Compute and plot a Gaussian matrix:

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Compute and plot a Gaussian vector:

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First derivative of a Gaussian matrix in the vertical dimension:

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