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2D Fourier Transform Explained with Examples
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- čas přidán 11. 12. 2020
- Explains the two dimensional (2D) Fourier Transform using examples.
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Related videos: (see: www.iaincollings.com)
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For a full list of Videos and Summary Sheets including the Matlab Code for this video, goto: www.iaincollings.com
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This guy knows what he is speaking and how to relate it graphically in the most simplest way without using lengthy jargon. Loved the detailed explanation with easy words. Thanks. 😎
I'm so glad you like the explanation and the graphical examples shown in the video. Thanks for your comment.
Thank you so much This channel needs more attention.
Glad you think so!
Thank you very much, it helped a lot especially for visualizing those concepts.
Great to hear!
Thanks for the explanation! very enlightening! 谢谢您的解释，豁然开朗！
Glad you found it useful.
It was very helpful in understand of the concepts, looking forward for the more videos. Thank you.
Glad it was helpful!
Thank you for the explanations Mr Iain, very enlightening. I'm a subscriber and I benefitted tremendously from your Convolution videos
That's great to know, thanks for your comment. Convolution was the topic that got me started with making these videos. So many students find it difficult, but I always thought that a hand drawn plot or two helped enormously. I'm glad you like this new 2D FT video. I've got a couple more 2D image processing videos in the pipeline, so keep a look out for those (probably in the new year).
Thx for the reply Mr Iain. I have in fact used 2d fourier series myself. For your upcoming series on 2d image processing, it'd be nice to see insights on how the 2d conv operator turns out to be have a 2d gaussian shape. Looking forward to seeing the series... before I forget, also thanks for the videos on the Matched Filter!
The visualization was really helpful!
Glad it helped.
Thank you very much for the clear explanation. This video really helped me understand what the FFT results of real images mean.
That's great to hear. I'm glad it was helpful!
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OMG. I love you and love the way you explain things with examples. Please keep up the good work. I can't wait to see your next videos.
Thank you! Will do!
Thank you very much for the video, the graphical instructions are very helpful !
Glad it was helpful!
Thanks for the explanation! In the future would it be possible to label the axes? It would help me in understanding the connection the time/spatial domain signal and the Fourier domain signal.
Thanks for the suggestion.
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thank you very much. i owe the understanding of the effect of phase in signal to you
Glad it helped
thanks a lot !
You're welcome!
Awesome! Thanks
I'm glad you liked it.
Great
Hey Iain, I’ve been struggling in a lot of courses applying my knowledge of signals/systems in Matlab and Simulink. I know it’s such a broad topic and can’t be covered in one or a few videos, but do you have tips on having confidence in programming abilities for signals and systems?
My main advice would be to think hard about what you expect the code to do, and what the resulting plots should look like. Don't just trust the plots. Also, study the code that others have written. I've put the code I wrote for this video on my website, so you can check it out. iaincollings.com
You're amazing. thanks so much.
So nice of you to say that. I'm glad you like the videos.
Thanks a lot!
You're welcome!
Nicely demonstrated. What sofware did you use for the 3D plots?
It's in Matlab. The complete code is available on my webpage (iaincollings.com) - just find the listing for this video, and then click on the PDF link.
Hello Professor Iain, would you mind posting the MATLAB code for this video on your website please? I can't seem to find it there, thank you very much! As always, enjoyed your explanations a lot :)
You'll find it on the following website, under the drop-down heading "Fourier Transform". It is the "Summary Sheet" link under the title "2D Fourier Transform Explained with Examples" www.iaincollings.com/signals-and-systems
I really liked this, and phase change is another important thing, special thanks for that!
I only doubt about 6:19, why deltas are diamond-shaped, I expected it to be square-shaped. Square gives you delta functions - one on the left, and one of the right when you look from each side of two dimensional model, and "diamond" gives you one on the left + one in the middle + one on the right, when looking from either side. Since this is matlab output I trust results of course, I am just puzzled.
Perhaps I should have used the “stem” plotting function in Matlab (I don’t know if they have a 2D version though). Then it would have shown them as spikes. They appear as diamond shapes because Matlab plots straight lines between function values at neighbouring grid points.
Hi professor, very insightful video! Thank you very much for the explanation!
One thing has been bugging me though. Considering the example at 9:38, what if instead of a square base I had a circular base?
Looking at the shape from any direction I'd still see a square function and therefore get a sinc in the frequency domain but the overall shape of the 2D Fourier transform results different.
What's the intuitive reason behind the difference?
Yes, great point. For the circular base, if you're looking in the x-direction, then the "slices" for particular values of y will all be square functions (as you say), but they will each have a different width. So their transforms in the y-direction will all be sinc functions, but with different widths. Then when the other part of the 2-D transform is done (in the x direction), it won't be dealing with square functions.
Professor, how can I do the 2D FFT when the domain (x,y) is not a rectangle but a circle?
That's a great question. I haven't thought about doing that before, but one obvious thing is that you could zero-pad the circular domain out to a square domain. Don't forget, the FFT is just a fast way of implementing the DFT. And the DFT implicitly assumes that the finite length of the samples (in each dimension/direction) is one period of an infinite length periodic signal. So I'm not sure a 2D FFT can be defined for a circular domain, since the lengths (and hence the periods) of each row are different. Perhaps a function could be defined with the FFT in the radial direction? I'm not sure.
Do the spaces not along the axes matter in the Fourier transform, like in a corner for example?
It depends on what you mean by "matter". The corners of the transform correspond to 2D functions that are high frequency in both axis (in the image domain).
@@iain_explains I think I understand. So if I had taken the fourier transform of a simple sine wave at a 45 degree angle from the x and y axis, the image in the frequency domain would also have points with a vector from the origin at that same angle?
@@iain_explains Forgive me, I'm fairly new to the 2d dft. Ultimately I'm just trying to figure out how to interpret a dft matrix in the frequency domain for two dimensions.
Yes. That's right. You can try it yourself to see. I provided the Matlab code at iaincollings.com and you can easily modify it yourself.
Below I've put some code that I just wrote for your 45 degree example. It shows that the Fourier transform is rotated (as you suggested it would be). Note that you also see the duplicate copies from the repeated basis functions, out at the corners, since the diagonal is longer than the side of the image.
N=5;
Step=0.1;
NN=N/Step;
x=sin(2*pi*[0:Step:N-Step]);
xx=zeros(NN,NN);
for i=1:NN
xx(i,i)=x(i);
%xx(i,NN-i+1)=x(i);
end
figure(1)
mesh(xx)
XX=fft2(xx);
figure(2)
mesh(abs(fftshift(XX)))
@@iain_explains Thank you!