This set of images shows the ideal low-pass filtering based on Fourier transform. The three images on top are, respectively, the original image, the ideal filter (cutoff frequency D0 is 0.15 times image size) and the low-pass filtered image, all in space domain. The three images at bottom are the corresponding Fourier transforms of the images on top. Note that the ringing effect is caused by the sharp edges of the ideal filter.



























This set of images shows the Butterworth low-pass filtering. The images are arranged in the same fashiion as the previous case. The only difference is that the Butterworth filter (first order, D0 is 0.1 times image size) is used with smooth edge compared to the ideal filter, and consequently the ringing effect is eliminated.