Multidimensional discrete convolution: Difference between revisions

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Convolution in one dimension was a powerful discovery that allowed the input and output of a linear shift-invariant (LSI) system (see [[LTI system theory]]). to be easily compared so long as the impulse response of the filter system was known. This notion carries over to multidimensional convolution as well, as simply knowing the impulse response of a multidimensional filter too allows for a direct comparison to be made between the input and output of a system. This is profound since several of the signals that are transferred in the digital world today are of multiple dimensions including images and videos. Similar to the one-dimensional convolution, the multidimensional convolution allows the computation of the output of an LSI system for a given input signal.

For example, consider an image that is sent over some wireless network subject to electroopticalelectro-optical noise. Possible noise sources include errors in channel transmission, the analog to digital converter, and the image sensor. Usually noise caused by the channel or sensor creates spatially-independent, high-frequency signal components that translates to arbitrary light and dark spots on the actual image. In order to rid the image data of the high-frequency spectral content, it can be multiplied by the frequency response of a low-pass filter, which based on the convolution theorem, is equivalent to convolving the signal in the time/spatial ___domain by the impulse response of the low-pass filter. Several impulse responses that do so are shown below.<ref>{{Cite web|title = MARBLE: Interactive Vision|url = http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/MARBLE/|website = homepages.inf.ed.ac.uk|accessdate = 2015-11-12}}</ref>

[[File:Screen Shot 2015-11-11 at 11.18.23 PM.png|none|thumb|311x311px|Impulse Responses of Typical Multidimensional Low Pass Filters]]