Discrete Fourier Transform

1
Discrete Fourier Transform
2
FFT and Its Applications

• FFTSHIFT Shift zero-frequency component to the
  center of spectrum. For vectors, FFTSHIFT(X)
  swaps the left and right halves of X. For
  matrices, FFTSHIFT(X) swaps the first and third
  quadrants and the second and fourth quadrants.
  For N-D arrays, FFTSHIFT(X) swaps "half-spaces"
  of X along each dimension.

3
Example of 1-D Fourier Transform
4
fftBox.m Plot Fourier Spectrum

• Script file fftBox.m
• Fourier Spectrum Plot of Box function
• X1linspace(0,1,17)
• Y1ones(1,length(X1))
• X2linspace(1,16,241)
• Y2zeros(1,length(X2))
• XX1 X2 YY1 Y2
• Wabs(fftshift(fft(Y)))
• subplot(2,1,1)
• plot(X,Y,'r') axis(0 16, 0,1.2) title('Box
  function')
• subplot(2,1,2)
• plot(W,'b-')
• title('Fourier Spectrum of Box function')

5
2-D Discrete Fourier Transform
6
Example of 2-D FFT Matlab Code

• Script file fourier.m - 2D Fourier Transform
• Pictures on P.113 of Gonzalez,
  Woods, Eddins
• m128 n128
• fzeros(m,n)
• f(5671,4879)255
• F0fft2(f) S0abs(F0)
• Fcfftshift(fft2(f)) Scabs(Fc)
• Fdfft2(fftshift(f)) Sdlog(1abs(Fc))
• subplot(2,2,1)
• imshow(f,)
• subplot(2,2,2)
• imshow(S0,)
• subplot(2,2,3)
• imshow(Sc, )
• subplot(2,2,4)
• imshow(Sd, )

7
2-D FFT with CenterShift
8
2-D FFT on Texture Images