Editing Gaussian function (section)

== Discrete Gaussian ==
{{main|Discrete Gaussian kernel}}
[[File:Discrete Gaussian kernel.svg|thumb|The [[discrete Gaussian kernel]] (solid), compared with the [[sampled Gaussian kernel]] (dashed) for scales <math>t = 0.5,1,2,4.</math>]]
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]]:<ref name="lin90">[http://kth.diva-portal.org/smash/record.jsf?pid=diva2%3A472968&dswid=-3163  Lindeberg, T., "Scale-space for discrete signals," PAMI(12), No. 3, March 1990, pp. 234–254.]</ref>
<math display="block">T(n, t) = e^{-t} I_n(t)</math>
where <math>I_n(t)</math> denotes the [[modified Bessel function]]s 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.<ref name="lin90"/><ref>Campbell, J, 2007, ''[https://dx.doi.org/10.1016/j.tpb.2007.08.001 The SMM model as a boundary value problem using the discrete diffusion equation]'', Theor Popul Biol. 2007 Dec;72(4):539–46.</ref>