爱华网In mathematics, a Gaussian function is a function of theform:
The graph of a Gaussian is a characteristic symmetric "bell curve"shape that quickly falls off towards plus/minus infinity. Theparameter a is the height of the curve's peak, b is the position ofthe centre of the peak, and c controls the width of the"bell".
Gaussian functions are widely used in statistics where theydescribe the normal distributions, in signal processing where theyserve to define Gaussian filters, in image processing wheretwo-dimensional Gaussians are used for Gaussian blurs, and inmathematics where they are used to solve heat equations anddiffusion equations.
Properties
Gaussian functions arise by applying the exponential function to ageneral quadratic function. The Gaussian functions are thus thosefunctions whose logarithm is a quadratic function.
The parameter c is relatedto the full width at half maximum (FWHM) of the peak accordingto
Gaussian functions are analytic, and their limit as x → ∞ is0.
Gaussian functions are among those functions that are elementarybut lack elementary antiderivatives; the integral of the Gaussian functionis the error function. Nonetheless their improper integralsover the whole real line can be evaluated exactly, using theGaussian integral
The product of two Gaussianfunctions is a Gaussian, and the convolution of two Gaussianfunctions is again a Gaussian, with .
Two-dimensionalGaussian function
Gaussian curve with a 2-dimensional domain
In two dimensions, the power to which e is raised in the Gaussianfunction is any negative-definite quadratic form. Consequently,the level sets of theGaussian will always be ellipses.
A particular example of a two-dimensional Gaussian functionis
Here the coefficient A is the amplitude,xo,yo is the center andσx, σy are the x andy spreads of the blob. The figure on the right was createdusing A = 1, xo = 0, yo= 0, σx = σy = 1.
In general, a two-dimensional elliptical Gaussian function isexpressed as
where the matrix
Using this formulation, the figure on the right can be createdusing A = 1, (xo, yo) = (0, 0), a = c = 1/2, b = 0.
