Fourier Transform and Frequency Filters

Fourier transform allows us to smoothen or sharpen images using filters. An image is first converted from spatial to frequency domain (high components=edges; low components=smooth regions) using Fourier transform then it is multiplied with a filter function (lowpass, highpass, bandpass filter) in a pixel-by-pixel fashion, and converted back to spacial domain.
For example, a lowpass filter will reduce high frequencies and retain low frequencies resulting to a smoother image.

original image

resulting image after applying lowpass filter