Multidimensional discrete convolution: Difference between revisions

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Revision as of 17:45, 5 September 2018

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===Row-Column Decomposition===

The row-column method can be applied when one of the signals in the convolution is separable. The method exploits the properties of separability in order to achieve a method of calculating the convolution of two multidimensional signals that is more computationally efficient than direct computation of each sample (given that one of the signals are separable).<ref>{{cite journal|last1=Sihvo|first1=Tero|last2=Niittylahti|first2=Jarkko|title=Row-Column Decomposition Based 2D Transform Optimization on Subword Parallel Processors|date=5 June 2005|accessdate=8 November 2015}}</ref> The following shows the mathematical reasoning behind the row-column decomposition approach (typically <math>h(n_1,n_2)</math> is the separable signal):

<math>y(n_1,n_2)=\sum_{k_1=-\infty}^{\infty} \sum_{k_2=-\infty}^{\infty} h(k_1,k_2)x(n_1-k_1,n_2-k_2)</math>

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===Applications===

Gaussian convolutions are used extensively in signal and image processing. For example, image-blurring can be accomplished with Gaussian convolution where the <math>\sigma</math> parameter will control the strength of the blurring. Higher values would thus correspond to a more blurry end result.<ref>{{Cite web|title = Gaussian Blur - Image processing for scientists and engineers, Part 4|url = http://patrick-fuller.com/gaussian-blur-image-processing-for-scientists-and-engineers-part-4/|website = patrick-fuller.com|accessdate = 2015-11-12}}</ref> It is also commonly used in [[Computer vision]] applications such as [[Scale-invariant feature transform]] (SIFT) feature detection.<ref>{{cite journal|last1=Lowe|first1=D.G.|title=Object recognition from local scale-invariant features|journal=Proceedings of the International Conference on Computer Vision|date=1999|volume=2|pages=1150–1157|accessdate=12 November 2015}}</ref>

==See also==