Multidimensional discrete convolution: Difference between revisions

Revision as of 04:33, 11 December 2015

Revision as of 04:36, 11 December 2015

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===Problem Statement & Basics===

Similar to the one-dimensional case, an asterisk is used to represent the convolution operation. The number of dimensions in the given operation is reflected in the number of asterisks. For example, thean ''M''-dimensional convolution would be written with ''M'' asterisks. The following represents a ''M''-dimensional convolution of discrete signals:

<math>y(n_1,n_2,...,n_M)=x(n_1,n_2,...,n_M)*...*h(n_1,n_2,...,n_M)</math>

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

As with one-dimensional convolution, multidimensional convolution also produces a result that is dependent on the amount of area that overlaps between the two functions. The resulting output region of support of a discrete multidimensional convolution will be determined based on the size and regions of support of the two input signals.[[File:Picture1_wiki.png|thumb|475px|Visualization of Convolution between two Simple Two-Dimensional Signals|none]]

Listed are several properties of the two-dimensional convolution operator. Note that these can also be extended for signals of <math>N</math>-dimensions.