Can somebody give a good explanation of FFT image transform How the FFT transformed image and it's Re^2+Im^2 image can be analyzed? I just want to understand something when loiking to the image and it's frequency.
+6
A:
EDIT: There is a great introduction to the concepts here.
There's a fair bit of math behind that question. In simple terms, consider a 1-D function, such as an audio clip. The fourier transform identifies the frequencies present in that signal. Each sample in the original audio clip correlates to the amplitude of the sound wave at any given point in time. In contrast, each sample in the fourier transform identifies the amplitude of a particular frequency of oscillation. For example, a pure sine wave at 1 kHz will have a fourier transform with a single spike at the 1 kHz mark. Audio waves are combinations of many different sine waves, and the fourier transform isolates which sine waves are contributing and by how much. (Note that the real explanation requires delving into complex numbers, but the foregoing gives the essence of what's going on).
The fourier transform of an image is a simple extension of the 1-D fourier transform into two dimensions, and is achieved by simply applying the 1-D transform to each row of an image, and then transforming each column of the resulting image. It produces essentially the same thing. A picture of smooth water waves travelling in a diagonal direction will transform to a series of spikes along that same diagonal.
The fourier transform is defined over continuous functions. The FFT is an technique for efficiently evaluating the fourier transform over discrete sets of data.
Marcelo Cantos
2010-03-31 09:46:37
Good answer - it might also be worth explaining the concept of *spatial frequency* in an image, and the interpretation of the phase and magnitude of the 2D FFT.
Paul R
2010-03-31 10:33:58
Thanks for the suggestion, @Paul. Rather than bloat the answer any more, I found a good link.
Marcelo Cantos
2010-03-31 11:21:16
+1, good answer, I just like to add that FFT is an algorithm for efficiently computing the DFT. More on DFT: http://en.wikipedia.org/wiki/Discrete_Fourier_transform
frunsi
2010-03-31 11:22:07
@Marcelo: good link - plenty of practical examples rather than the usual dry mathematical treatment
Paul R
2010-03-31 12:06:16
That's a great link. I've always found fourier transforms relatively easy to understand when applied to a time-signal (like audio, or a mechanical vibration), but difficult to understand with images. It didn't occur to me that it was actually pretty simple.
notJim
2010-03-31 18:18:19
Very good link, thank you!Now reading it!
maximus
2010-04-01 05:34:35
A:
Steve Eddins of Mathworks has been discussing Fourier Transforms in general on his blog for a while now - you should check it out here.
Jacob
2010-03-31 18:16:45