In image processing, a Guassian filter is used to blur an image. A low-pass filter is a filter that attenuates the high frequencies, preserving only smooth variations in the provided image. Is a Guassian filter a low-pass filter because it smooths the edges (or, in general, the parts of the image where there is a high variation of the intensity)? Why exactly is the Guassian filter a low-pass filter?
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Because the Fourier transform of Gaussian in the spatial/temporal domain is also a Gaussian in the frequency domain. Since a Gaussian drops off rapidly with frequency, it's a low pass filter.
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$\begingroup$ Could you please elaborate the sentence "Since a Gaussian drops off rapidly with frequency, it's a low pass filter"? $\endgroup$ – nbro Mar 28 at 16:25
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$\begingroup$ @nbro, whether it's $$ h(t) = e^{-\alpha t^2} $$ or $$ H(f) = e^{-\beta f^2} $$ as $|t|$ or $|f|$ increase, the $h(t)$ or $H(f)$ get very small. $\endgroup$ – robert bristow-johnson Mar 28 at 19:35
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$\begingroup$ @robertbristow-johnson Ok, thank you. But what is the relation between Guassian dropping as the frequency increases the the fact it is a low-pass filter? $\endgroup$ – nbro Mar 28 at 20:49
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