1. 2D Discrete Fourier Transform (DFT)
EE 7730
2D Discrete Fourier Transform (DFT)

2. 2D Discrete Fourier Transform
2D Fourier Transform
2D Discrete Fourier Transform (DFT)
2D DFT is a sampled version of 2D FT.
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EE Image Analysis I

3. 2D Discrete Fourier Transform
2D Discrete Fourier Transform (DFT)
where and
Inverse DFT
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4. 2D Discrete Fourier Transform
It is also possible to define DFT as follows
where and
Inverse DFT
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5. 2D Discrete Fourier Transform
Or, as follows
where and
Inverse DFT
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6. 2D Discrete Fourier Transform
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7. 2D Discrete Fourier Transform
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8. 2D Discrete Fourier Transform
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9. 2D Discrete Fourier Transform
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10. Periodicity [M,N] point DFT is periodic with period [M,N] 1
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11. Periodicity [M,N] point DFT is periodic with period [M,N] 1
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12. Convolution Be careful about the convolution property!
Length=P+Q-1
Length=P
Length=Q
For the convolution property to hold, M must be greater than or equal to P+Q-1.
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13. Convolution Zero padding 4-point DFT (M=4) Bahadir K. Gunturk
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14. DFT in MATLAB
Let f be a 2D image with dimension [M,N], then its 2D DFT can be computed as follows:
Df = fft2(f,M,N);
fft2 puts the zero-frequency component at the top-left corner.
fftshift shifts the zero-frequency component to the center. (Useful for visualization.)
Example:
f = imread(‘saturn.tif’); f = double(f);
Df = fft2(f,size(f,1), size(f,2));
figure; imshow(log(abs(Df)),[ ]);
Df2 = fftshift(Df);
figure; imshow(log(abs(Df2)),[ ]);
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15. DFT in MATLAB f Df = fft2(f) After fftshift Bahadir K. Gunturk
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16. DFT in MATLAB Let’s test convolution property f = [1 1]; g = [2 2 2];
Conv_f_g = conv2(f,g); figure; plot(Conv_f_g);
Dfg = fft (Conv_f_g,4); figure; plot(abs(Dfg));
Df1 = fft (f,3);
Dg1 = fft (g,3);
Dfg1 = Df1.*Dg1; figure; plot(abs(Dfg1));
Df2 = fft (f,4);
Dg2 = fft (g,4);
Dfg2 = Df2.*Dg2; figure; plot(abs(Dfg2));
Inv_Dfg2 = ifft(Dfg2,4);
figure; plot(Inv_Dfg2);
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17. DFT in MATLAB Increasing the DFT size f = [1 1]; g = [2 2 2];
Df1 = fft (f,4);
Dg1 = fft (g,4);
Dfg1 = Df1.*Dg1; figure; plot(abs(Dfg1));
Df2 = fft (f,20);
Dg2 = fft (g,20);
Dfg2 = Df2.*Dg2; figure; plot(abs(Dfg2));
Df3 = fft (f,100);
Dg3 = fft (g,100);
Dfg3 = Df3.*Dg3; figure; plot(abs(Dfg3));
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18. DFT in MATLAB Scale axis and use fftshift f = [1 1]; g = [2 2 2];
Df1 = fft (f,100);
Dg1 = fft (g,100);
Dfg1 = Df1.*Dg1;
t = linspace(0,1,length(Dfg1));
figure; plot(t, abs(Dfg1));
Dfg1_shifted = fftshift(Dfg1);
t2 = linspace(-0.5, 0.5, length(Dfg1_shifted));
figure; plot(t, abs(Dfg1_shifted));
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19. Example
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20. Example
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21. DFT-Domain Filtering a = imread(‘cameraman.tif'); Da = fft2(a);
Da = fftshift(Da);
figure; imshow(log(abs(Da)),[]);
H = zeros(256,256);
H(128-20:128+20,128-20:128+20) = 1;
figure; imshow(H,[]);
Db = Da.*H;
Db = fftshift(Db);
b = real(ifft2(Db));
figure; imshow(b,[]); H
Frequency domain
Spatial domain
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22. Low-Pass Filtering 61x61 81x81 121x121 Bahadir K. Gunturk
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23. Low-Pass Filtering = * h DFT(h) Bahadir K. Gunturk
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24. High-Pass Filtering = * h DFT(h) Bahadir K. Gunturk
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25. High-Pass Filtering High-pass filter Bahadir K. Gunturk
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26. Anti-Aliasing a=imread(‘barbara.tif’); Bahadir K. Gunturk
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27. Anti-Aliasing a=imread(‘barbara.tif’); b=imresize(a,0.25);
c=imresize(b,4);
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28. Anti-Aliasing a=imread(‘barbara.tif’); b=imresize(a,0.25);
c=imresize(b,4);
H=zeros(512,512);
H(256-64:256+64, :256+64)=1;
Da=fft2(a);
Da=fftshift(Da);
Dd=Da.*H;
Dd=fftshift(Dd);
d=real(ifft2(Dd));
Bahadir K. Gunturk
EE Image Analysis I

29. Noise Removal
For natural images, the energy is concentrated mostly in the low-frequency components.
“Einstein”
DFT of “Einstein”
Profile along the red line
Signal vs Noise
Noise=40*rand(256,256);
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30. Noise Removal
At high-frequencies, noise power is comparable to the signal power.
Signal vs Noise
Low-pass filtering increases signal to noise ratio.
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31. Appendix
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32. Appendix: Impulse Train
The Fourier Transform of a comb function is
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33. Impulse Train (cont’d)
The Fourier Transform of a comb function is
(Fourier Trans. of 1) ?
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34. Impulse Train (cont’d)
Proof
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35. Appendix: Downsampling
Question: What is the Fourier Transform of ?
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36. Downsampling Let Using the multiplication property: Bahadir K. Gunturk
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37. Downsampling
where
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38. Example
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39. Example ?
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40. Downsampling
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41. Example
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42. Example
No aliasing if
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