This set of images shows the ideal low-pass filtering using Fourier transform. The 3 images on top are the original image, the ideal filter (cutoff frequency Do is 0.15 times image size) and the low-pass filtered image respectively, all in space domain. The 3 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, Do is 0.1 times image size) is used with smooth edge compared to the ideal filter, and consequently the ringing effect is eliminated.