Search results
Results from the WOW.Com Content Network
Gaussian functions are widely used in statistics to describe the normal distributions, in signal processing to define Gaussian filters, in image processing where two-dimensional Gaussians are used for Gaussian blurs, and in mathematics to solve heat equations and diffusion equations and to define the Weierstrass transform.
In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional normal distribution to higher dimensions.
Common integrals in quantum field theory are all variations and generalizations of Gaussian integrals to the complex plane and to multiple dimensions. [ 1 ] : 13–15 Other integrals can be approximated by versions of the Gaussian integral.
A different technique, which goes back to Laplace (1812), [3] is the following. Let = =. Since the limits on s as y → ±∞ depend on the sign of x, it simplifies the calculation to use the fact that e −x 2 is an even function, and, therefore, the integral over all real numbers is just twice the integral from zero to infinity.
Suppose and are real random variables such that (,) is a 2-dimensional normal random vector. Then the complex random variable = + is called complex normal random variable or complex Gaussian random variable. [3]: p. 500
A function in the Schwartz space is sometimes called a Schwartz function. A two-dimensional Gaussian function is an example of a rapidly decreasing function. Schwartz space is named after French mathematician Laurent Schwartz .
Gaussian convolution can be effectively approximated via implementation of a Finite impulse response (FIR) filter. The filter will be designed with truncated versions of the Gaussian. For a two-dimensional filter, the transfer function of such a filter would be defined as the following: [17]
In two dimensions, i.e. the bivariate case, the Fréchet–Hoeffding theorem states ... to address the limitations of the Gaussian copula and of copula functions more ...