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EE 4780

EE 4780. 2D Discrete Fourier Transform (DFT). 2D Discrete Fourier Transform. 2D Fourier Transform. 2D Discrete Fourier Transform (DFT). 2D DFT is a sampled version of 2D FT. 2D Discrete Fourier Transform. Inverse DFT. 2D Discrete Fourier Transform (DFT). where and .

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EE 4780

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  1. EE 4780 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.

  3. 2D Discrete Fourier Transform • Inverse DFT • 2D Discrete Fourier Transform (DFT) where and

  4. 2D Discrete Fourier Transform • Inverse DFT • It is also possible to define DFT as follows where and

  5. 2D Discrete Fourier Transform • Inverse DFT • Or, as follows where and

  6. 2D Discrete Fourier Transform

  7. 2D Discrete Fourier Transform

  8. 2D Discrete Fourier Transform

  9. 2D Discrete Fourier Transform

  10. Periodicity • [M,N] point DFT is periodic with period [M,N] 1

  11. Periodicity • [M,N] point DFT is periodic with period [M,N] 1

  12. Convolution • Be careful about the convolution property! Length=P+Q-1 Length=Q Length=P For the convolution property to hold, M must be greater than or equal to P+Q-1.

  13. Convolution • Zero padding 4-point DFT (M=4)

  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)),[ ]);

  15. DFT in MATLAB f Df = fft2(f) After fftshift

  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);

  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));

  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));

  19. Example

  20. Example

  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

  22. Low-Pass Filtering 61x61 81x81 121x121

  23. Low-Pass Filtering h = * DFT(h)

  24. High-Pass Filtering h = * DFT(h)

  25. High-Pass Filtering High-pass filter

  26. Anti-Aliasing a=imread(‘barbara.tif’);

  27. Anti-Aliasing a=imread(‘barbara.tif’); b=imresize(a,0.25); c=imresize(b,4);

  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, 256-64:256+64)=1; Da=fft2(a); Da=fftshift(Da); Dd=Da.*H; Dd=fftshift(Dd); d=real(ifft2(Dd));

  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);

  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.

  31. Appendix

  32. Appendix: Impulse Train • The Fourier Transform of a comb function is

  33. Impulse Train (cont’d) • The Fourier Transform of a comb function is (Fourier Trans. of 1) ?

  34. Impulse Train (cont’d) • Proof

  35. Appendix: Downsampling • Question: What is the Fourier Transform of ?

  36. Downsampling • Let Using the multiplication property:

  37. Downsampling where

  38. Example

  39. Example ?

  40. Downsampling

  41. Example

  42. Example No aliasing if

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