Editing Deconvolution (section)

{{Short description|Reconstruction of a filtered signal}}{{Not to be confused with|Upsampling}}[[File:Deconvolution_of_an_astronomical_image.png|thumb|right|Before and after deconvolution of an image of the lunar crater [[Copernicus (lunar crater)|Copernicus]] using the [[Richardson–Lucy deconvolution|Richardson-Lucy]] algorithm.]]

In [[mathematics]], '''deconvolution''' is the [[Inverse function|inverse]] of [[convolution]]. Both operations are used in [[signal processing]] and [[image processing]]. For example, it may be possible to recover the original signal after a filter (convolution) by using a deconvolution method with a certain degree of accuracy.<ref>{{cite web |last=O'Haver |first=T. |title=Intro to Signal Processing - Deconvolution |url=http://www.wam.umd.edu/~toh/spectrum/Deconvolution.html |publisher=University of Maryland at College Park |access-date=2007-08-15}}</ref> Due to the measurement error of the recorded signal or image, it can be demonstrated that the worse the [[signal-to-noise ratio]] (SNR), the worse the reversing of a filter will be; hence, inverting a filter is not always a good solution as the error amplifies. Deconvolution offers a solution to this problem.

The foundations for deconvolution and [[time-series analysis]] were largely laid by [[Norbert Wiener]] of the [[Massachusetts Institute of Technology]] in his book ''Extrapolation, Interpolation, and Smoothing of Stationary Time Series'' (1949).<ref>{{cite book |last=Wiener |first=Norbert |author-link=Norbert Wiener |year=1949 |title=Extrapolation, Interpolation, and Smoothing of Stationary Time Series: With Engineering Applications |url=https://direct.mit.edu/books/oa-monograph/4361/Extrapolation-Interpolation-and-Smoothing-of |publisher=[[MIT Press]] |isbn=9780262257190}}</ref>  The book was based on work Wiener had done during [[World War II]] but that had been classified at the time.  Some of the early attempts to apply these theories were in the fields of [[weather forecasting]] and [[economics]].