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The Fourier Transform II

Basis beeldverwerking (8D040) dr. Andrea Fuster Prof.dr . Bart ter Haar Romeny dr. Anna Vilanova Prof.dr.ir . Marcel Breeuwer. The Fourier Transform II. Contents. Fourier Transform of sine and cosine 2D Fourier Transform Properties of the Discrete Fourier Transform. Euler’s formula.

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The Fourier Transform II

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  1. Basis beeldverwerking (8D040)dr. Andrea FusterProf.dr. Bart terHaarRomenydr. Anna VilanovaProf.dr.ir. Marcel Breeuwer The Fourier Transform II

  2. Contents Fourier Transform of sine and cosine 2D Fourier Transform Properties of the Discrete Fourier Transform

  3. Euler’s formula

  4. Cosine Recall

  5. Sine

  6. Contents Fourier Transform of sine and cosine 2D Fourier Transform Properties of the Discrete Fourier Transform

  7. Discrete Fourier Transform Forward Inverse

  8. Formulation in 2D spatial coordinates f(x,y) digital image of size M x N Discrete Fourier Transform (2D) Inverse Discrete Transform (2D)

  9. Spatial and Frequency intervals Inverse proportionality Suppose function is sampled M times in x, with step , distance is covered, which is related to the lowest frequency that can be measured And similarly for y and frequency v

  10. Examples

  11. Examples

  12. Periodicity 2D Fourier Transform is periodic in both directions

  13. Periodicity 2D Inverse Fourier Transform is periodic in both directions

  14. Contents Fourier Transform of sine and cosine 2D Fourier Transform Properties of the Discrete Fourier Transform

  15. Properties of the 2D DFT

  16. Real Real Imaginary Sin (x + π/2) Sin (x)

  17. Note: translation has no effect on the magnitude of F(u,v)

  18. Symmetry: even and odd Any real or complex function w(x,y) can be expressed as the sum of an even and an odd part (either real or complex)

  19. Properties Even function (symmetric) Odd function (antisymmetric)

  20. Properties - 2

  21. FT of even and odd functions FT of even function is real FT of odd function is imaginary

  22. Even Real Imaginary Cos (x)

  23. Odd Real Imaginary Sin (x)

  24. Even Real Imaginary F(Cos(x+k)) F(Cos(x))

  25. Odd Real Imaginary Sin (x)Sin(y) Sin (x)

  26. Consequences for the Fourier Transform FT of real function is conjugate symmetric FT of imaginary function is conjugate antisymmetric

  27. Scaling property Scaling t with a

  28. Imaginary parts a

  29. Differentiation and Fourier Let be a signal with Fourier transform Differentiating both sides of inverse Fourier transform equation gives:

  30. Examples – horizontal derivative

  31. Examples – vertical derivative

  32. Examples – hor and vert derivative

  33. Thanks and see you next Wednesday!☺

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