Multidimensional discrete convolution: Difference between revisions

Revision as of 03:06, 20 June 2016

Revision as of 11:22, 20 May 2017

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===Approximation by FIR Filter===

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:<ref name=":0">{{cite journal|last1=Getreuer|first1=Pascal|title=A Survey of Gaussian Convolution Algorithms|journal=Image Processing On Line|date=2013|pages=286–310|urldoi=http://dx.doi.org/10.5201/ipol.2013.87|accessdate=12 November 2015}}</ref>

<math>H(z_1,z_2)=\frac{1}{s(r_1,r_2)} \sum_{n_1=-r_1}^{r_1}\sum_{n_2=-r_2}^{r_2}G(n_1,n_2){z_1}^{-n_1}{z_2}^{-n_2}</math>

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<math>H(z)=\frac{1}{2r+1} \frac{z^r-z^{-r-1}}{1-z^-1}</math>

Typically, recursive passes 3, 4, or 5 times are performed in order to obtain an accurate approximation.<ref name=":0" /> A suggested method for computing ''r'' is then given as the following:<ref>{{Cite journal|url = http://dx.doi.org/10.1109/TPAMI.1986.4767776|title = Efficient synthesis of Gaussian filters by cascaded uniform filters|last = Wells|first = W.M.|date = 1986|journal = IEEE Transactions on Pattern Analysis and Machine Intelligence|doi = 10.1109/TPAMI.1986.4767776|pmid = |access-date = }}</ref>

<math>\sigma^2=\frac{1}{12}K((2r+1)^2-1)</math> where ''K'' is the number of recursive passes through the filter.