Noise shows up as light gray lines or dots in the DFT. In the Imagemagick example, this noise is removed by replacing those parts with black. But you want to do the reverse, so you should add noise by adding light gray features to the DFT.

I suspect this technique is how the Imagemagick author generated the noisy twigs image: https://legacy.imagemagick.org/Usage/fourier/twigs.jpg