An Interactive Guide To The Fourier Transform

Hey!
Very nice material. Thank you.
I have noticed a small typo in the Appendix: Projecting Onto Cycles section. You have mixed up the Fourier transform with the inverse Fourier transform :wink:

You didn’t emphasize the basic idea that the Fourier transform is a special case of a phenomenon that takes place in any real vector space with an inner product. Once you get that it’s just not that fancy and it’s only scary because we write it in a scary fashion, life becomes easier.

As a mathematician, I find it much easier to think of L^2 as being orthogonally spanned by the e(nx) functions and taking the Fourier transform is the analog of writing out an unknown vector in coordinates. What’s so hard about that?

This abstract perspective helps you practically too. You become less reliant on formulas since you have a good global understanding of the notion of a Fourier transform. In some sense, you’re rephrasing this when you talk about smoothies but I don’t know how effective the metaphor is.

Huge caveat: I am a mathematician and “math that anyone can understand” is often math that I have a hard time understanding because of the way it’s presented.

Fantastic!

I found this article incredibly helpful as a high school student in need of college-level mathematics concepts. This article helped me understand the basic concept of what the Fourier transform does, and for anyone who needs to know why it works with more math (but still only high school level), I would recommend Stephen’s link to his essay: http://linuxbio.med.buffalo.edu/Fourier/AC_Signal_Processing.html I found it incredibly easy to understand as well as very helpful. One thing about it, though: make sure you keep reading if you don’t understand something, because it is probably fully explained in the next paragraph or two.

Fourier Transform cannot get anymore explicit… And… you made it free… You’ve got a very large heart - never forget that. God bless you!!

Fantastic guide, one of the best I’ve read! Thanks much for sharing this.

Signal,

I suspect that there are several reasons you’ve not received any replies. First thing is that as soon as you move the FFT into 2-dimensional space it moves very quickly away from the core ideas of this thread. Secondly, I think that it is not clear exactly what you are asking. As far as I can tell, what you are doing is 1) taking a time series of images and running a 2D FFT on each. Then 2) throwing away the amplitude data and 3)Inverse transforming the images back into the spatial domain.

The real problem is in the statement of the next part of the problem - to wit: “I have found that moving part pixel intensity values becomes dominant”. I suspect that you are referring to the reverse transformed images. What’s more I suspect that you are also referring to an image by image comparison when specifying increase or decrease.

Anyway, keep in mind that the FFT tells you about periodicity. In 2 dimensions, this means how bits of the image are spaced (and oriented). Think about a picket fence - this would have a very strong “DOT” at a point corresponding to the picket fence’s spacing (i.e., distance from the center in the frequency domain) and orientation (i.e., angle of the radial line connecting the “DOT” to the center) in the original image. Incidentally, there should also be a “DOT” corresponding to the width of each individual picket as well as "DOT"s representing pixel spacing and size.

Now, in the next step you throw away all the data about fixed spacings in the image (the stuff that’s not moving) and keep only the data about stuff whose periodic relationship to the scene has changed (that is, the phase has changed) from image to image (The “moving” stuff), and the wonder why moving abjects are enhanced.

Does this help

In this section on Fourier transforms:
"I could say “2-inch radius, start at 45 degrees, 1 circle per second, go!”. After half a second we should be at the same spot: starting point + amount traveled = 45 + 180 = 225 degrees (on a 2-inch circle

I might be missing something basic, or there is a mistake.
If it is 1 circle per second, then I would say after 1 second we would have completed a circle and be back to the same place or starting point. How, on a circle, would you be back at the same spot after 225 degrees? In half a second at one circle per second I would think we would be 1/2 way around a circle, at a spot opposite where we started.

Thanks for writing this- hope it’s part of a future book. This link might be helpful for those who want to understand it from the frequency perspective:
http://www.dspguru.com/sites/dspguru/files/conv-dsp-tutorial.pdf

@Bob: Ah, that phrase wasn’t clear. I meant that by following the instructions, we’d each be at 225 degrees on our own circles (we’re at the corresponding positions on our own circle).

I’ve clarified the wording, thanks!

@Carl: Thanks!

Too good! Fourier transform is more clear to me now than it was ever before QQ

Hey!
Super helpful.
So how would you create a linear trajectory with a sum of sine and cosine graphs, any amplitude?

OOPS - That was supposed to be an infinite number of Odd harmonics.

Jenn,

As a follow-on, you might be interested to know about the old Maxim that a Square Wave is made up of an infinite number of even harmonics while a Triangle wave is made up of an even number of Odd harmonics. In either case the amplitude of each term decreases as a function of frequency. If you don’t believe it, try graphing the first 10 (even or odd) harmonics. If you get the phase and amplitudes right, you will see a really good approximation of a square wave or a triangle wave emerge as you add harmonics. Even Sin(t)+0.5Sin(2t) will already have a square wave character.

Jenn Ng

Sorry it took so long to reply but here is how I would answer your question. First, you are proposing to transform a non-periodic function of the form f(t)=At. By definition you can’t do this with the Fourier series being discussed here. You need to use the full integral definition instead. That is, multiply your function by the complex exponential e^-(jwt) and integrate. (That is, integrate Ate^(-jwt) dt). You should get some function of w that is a complex exponential. (w is the radial frequency 2pif). Once you get this, use euler’s formula to re-write the complex exponential as an infinite sum of sines and cosines. This will give you the indefinate integral. Depending on what you want, you will probably need to handle infinities in the integral limits when finding the definate integral - a real pain but doable. Admittedly this is all very tedious but it can be made to work.

you might enjoy this website

aha!

Hello everyone,i have a doubt…
I know that for a given signal, the sampling frequency Fs must be twice or more than maximum frequency of the signal Fm. It is easy to understand the concept for a 1D signal. But I don’t know how to calculate sampling frequency or Nyquist rate for a multidimensional signal like 2D image.So can anyone help me regarding it.???

Ugh how is everyone else understanding this? I got so confused after the applet was introduced…What do the the different paths of the different “motor cars” mean? Are you saying that every wave is a sum of infinitely smaller waves?

someone help

@Jake

“Are you saying that every wave is a sum of infinitely smaller waves?”

Instead of worrying about what one analogy or another means, look to the fundamental nature of a periodic signal and the ideas embodied in Fourier’s conjecture - i.e., that any periodic waveform can be represented as a sum of sine and or cosine waves. So, take some complex periodic waveform. By definition, it must have a period and it must repeat identically in each of these periods. This period is the fundamental frequency. Well if we remove the fundamental what are we left with? The answer is simply all the other frequencies that make up the waveform. Fourier’s methods allow us to pick any one of those frequencies and remove it. Each time we do this the waveform becomes simpler until at last there is only one frequency left. Fourier does not constrain the sizes of those frequency components - only their frequencies.