Gaussian Function - Applications

Applications

Gaussian functions appear in many contexts in the natural sciences, the social sciences, mathematics, and engineering. Some examples include:

• In statistics and probability theory, Gaussian functions appear as the density function of the normal distribution, which is a limiting probability distribution of complicated sums, according to the central limit theorem.
• Gaussian functions are the Green's function for the (homogeneous and isotropic) diffusion equation (and, which is the same thing, to the heat equation), a partial differential equation that describes the time evolution of a mass-density under diffusion. Specifically, if the mass-density at time t=0 is given by a Dirac delta, which essentially means that the mass is initially concentrated in a single point, then the mass-distribution at time t will be given by a Gaussian function, with the parameter a being linearly related to 1/√t and c being linearly related to √t. More generally, if the initial mass-density is φ(x), then the mass-density at later times is obtained by taking the convolution of φ with a Gaussian function. The convolution of a function with a Gaussian is also known as a Weierstrass transform.
• A Gaussian function is the wave function of the ground state of the quantum harmonic oscillator.
• The molecular orbitals used in computational chemistry can be linear combinations of Gaussian functions called Gaussian orbitals (see also basis set (chemistry)).
• Mathematically, the derivatives of the Gaussian function can be represented using Hermite functions. The n-th derivative of the Gaussian is the Gaussian function itself multiplied by the n-th Hermite polynomial, up to scale. For example the first-derivative of the Gaussian is simply the Gaussian multiplied by x.
• Consequently, Gaussian functions are also associated with the vacuum state in quantum field theory.
• Gaussian beams are used in optical and microwave systems.
• In scale space representation, Gaussian functions are used as smoothing kernels for generating multi-scale representations in computer vision and image processing. Specifically, derivatives of Gaussians (Hermite functions) are used as a basis for defining a large number of types of visual operations.
• Gaussian functions are used to define some types of artificial neural networks.
• In fluorescence microscopy a 2D Gaussian function is used to approximate the Airy disk, describing the intensity distribution produced by a point source.
• In signal processing they serve to define Gaussian filters, such as in image processing where 2D Gaussians are used for Gaussian blurs. In digital signal processing, one uses a discrete Gaussian kernel, which may be defined by sampling a Gaussian, or in a different way.
• In geostatistics they have been used for understanding the variability between the patterns of a complex training image. They are used with kernel methods to cluster the patterns in the feature space.