page_title: Gaussian function
section_title: Discrete Gaussian
hierarchical_section_title: Gaussian function / Discrete Gaussian
context_page_description: In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the form
  for arbitrary real constants a, b and non zero 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 controls the width of the "bell".
  Gaussian functions are often used to represent the probability density function of a normally distributed random variable with expected value μ = b and variance σ² = c². In this case, the Gaussian is of the form:
  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.
context_section_description: One may ask for a discrete analog to the Gaussian;
  this is necessary in discrete applications, particularly digital signal processing. A simple answer is to sample the continuous Gaussian, yielding the sampled Gaussian kernel. However, this discrete function does not have the discrete analogs of the properties of the continuous function, and can lead to undesired effects, as described in the article scale space implementation.
  An alternative approach is to use the discrete Gaussian kernel:
  where  denotes the modified Bessel functions of integer order.
  This is the discrete analog of the continuous Gaussian in that it is the solution to the discrete diffusion equation (discrete space, continuous time), just as the continuous Gaussian is the solution to the continuous diffusion equation.
caption: The discrete Gaussian kernel (solid), compared with the sampled Gaussian kernel (dashed) for scales 
    
      
        
          t
          =
          0.5
          ,
          1
          ,
          2
          ,
          4.
        
      
      {\displaystyle t=0.5,1,2,4.}