Gaussian Function Family
A Gaussian Function Family is an exponential function family of the form [math]\displaystyle{ f(x,a,b,c) = ae^{-k/m} }[/math], such that [math]\displaystyle{ k=(x-b)^2 }[/math] and [math]\displaystyle{ m=2c^2 }[/math].
- See: Gaussian Function Instance, Gaussian Probability Distribution, Weierstrass Transform, Standard Deviation, Gaussian Filter.
References
2016
- (Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/Gaussian_function Retrieved:2016-8-14.
- In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the form: : [math]\displaystyle{ f\left(x\right) = a e^{- { \frac{(x-b)^2 }{ 2 c^2} } } }[/math] for arbitrary real constants a, b and c. It is named after the mathematician Carl Friedrich Gauss.
The graph of a Gaussian is a characteristic symmetric “bell curve" shape. The parameter a is the height of the curve's peak, b is the position of the center of the peak and c (the standard deviation, sometimes called the Gaussian RMS width) controls the width of the "bell".
Gaussian functions are widely used in statistics where they describe the normal distributions, in signal processing where they serve to define Gaussian filters, in image processing where two-dimensional Gaussians are used for Gaussian blurs, and in mathematics where they are used to solve heat equations and diffusion equations and to define the Weierstrass transform.
- In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the form: : [math]\displaystyle{ f\left(x\right) = a e^{- { \frac{(x-b)^2 }{ 2 c^2} } } }[/math] for arbitrary real constants a, b and c. It is named after the mathematician Carl Friedrich Gauss.