I read your article. Just before this:

http://www.imagemagick.org/Usage/fourier/#waves_2d

there is sentence:

But missing the example.The wave has a number of components to it.

Image example

I started to read your article because I am interested of explanation of the Fourier Transform used on images. I want to use it in my program if I will learn enough stuff. First I started to read this tutorial:

http://lodev.org/cgtutor/fourier.html

but I am stuck on the problem that I don't understand the difference between imaginary and real part displayed on those graphs (e.g. why the hell the green imaginary line is 6,6mm high on my display but the red is 3,3 mm high when I draw sinusoid).

I read that using convolution filters like gaussian blur is pretty slow (Also I have seen it in my programs) but that if I would use Fourier transform so they are much more faster. I'm curious how hard is this to learn.