# Why exactly is the Guassian filter a low-pass filter?

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?

• @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. – robert bristow-johnson Mar 28 at 19:35