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Fourier Transform J.B. Fourier 1768-1830 PowerPoint Presentation
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Fourier Transform J.B. Fourier 1768-1830

Fourier Transform J.B. Fourier 1768-1830

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Fourier Transform J.B. Fourier 1768-1830

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  1. Fourier TransformJ.B. Fourier 1768-1830

  2. Image Enhancement in the Frequency Domain 1-D

  3. A A sin(x) 3 sin(x) B + 1 sin(3x) A+B + 0.8 sin(5x) C A+B+C + 0.4 sin(7x) D A+B+C+D A sum of sines and cosines =

  4. Fourier spectrum of step function

  5. The Continuous Fourier Transform

  6. The Continuous Fourier Transform The InverseFourier Transform The Fourier Transform 2D Continuous Fourier Transform: 1D Continuous Fourier Transform: The Inverse Transform The Transform

  7. Discrete Functions f(x) f(n) = f(x0 + nDx) f(x0+2Dx) f(x0+3Dx) f(x0+Dx) f(x0) 0 1 2 3 ... N-1 x0+2Dx x0+3Dx x0 x0+Dx The discrete function f: { f(0), f(1), f(2), … , f(N-1) }

  8. The Discrete Fourier Transform 2D Discrete Fourier Transform: (u = 0,..., N-1; v = 0,…,M-1) (x = 0,..., N-1; y = 0,…,M-1) 1D Discrete Fourier Transform: (u = 0,..., N-1) (x = 0,..., N-1)

  9. The 2D Basis Functions V u=-2, v=2 u=-1, v=2 u=0, v=2 u=1, v=2 u=2, v=2 u=-2, v=1 u=-1, v=1 u=0, v=1 u=1, v=1 u=2, v=1 U u=0, v=0 u=-2, v=0 u=-1, v=0 u=1, v=0 u=2, v=0 u=-2, v=-1 u=-1, v=-1 u=0, v=-1 u=1, v=-1 u=2, v=-1 u=-2, v=-2 u=-1, v=-2 u=0, v=-2 u=1, v=-2 u=2, v=-2 The wavelength is . The direction is u/v .

  10. The Fourier Transform Jean Baptiste Joseph Fourier

  11. Original: Real, imaginary, amplidute • F.T.; Real, imaginary, amplitude • Reconstructed

  12. Fourier spectrum |F(u,v)| The Fourier Image Fourier spectrum log(1 + |F(u,v)|) Image f

  13. FİLTERİNG

  14. Frequency Bands Image Fourier Spectrum Percentage of image power enclosed in circles (small to large) : 90%, 95%, 98%, 99%, 99.5%, 99.9%

  15. Low pass Filtering 90% 95% 98% 99% 99.5% 99.9%

  16. Noise-cleaned image Fourier Spectrum Noise Removal Noisy image

  17. Higher frequencies dueto sharp image variations (e.g., edges, noise, etc.)

  18. High Pass Filtering Original High Pass Filtered

  19. High Frequency Emphasis + Original High Pass Filtered

  20. High Frequency Emphasis Original High Frequency Emphasis High Frequency Emphasis Original

  21. High pass Filter High Frequency Emphasis High Frequency Emphasis + Histogram Equalization High Frequency Emphasis Original

  22. 2D Image - Rotated Fourier Spectrum Fourier Spectrum Rotation 2D Image

  23. Fourier Transform -- Examples Image Domain Frequency Domain

  24. Fourier Transform -- Examples Image Fourier spectrum

  25. Centered Fourier Spectum

  26. Fourier Transform of a damaged board

  27. Low Pass and High Pass Filtering

  28. Hihg Pass Filtering

  29. Gaussian Filters in Frequency and Space Domains

  30. İdeal Low pass filter

  31. İdeal Low Pass Filter

  32. Low Pass Filtering in freuencey Domain

  33. Frequency Domain and Space Domain Filters

  34. Butterworth Low Pass Filter

  35. Filtering with different Cutoff Frequencies

  36. Chapter 4 Image Enhancement in the Frequency Domain

  37. Chapter 4 Image Enhancement in the Frequency Domain

  38. Chapter 4 Image Enhancement in the Frequency Domain

  39. Chapter 4 Image Enhancement in the Frequency Domain

  40. Chapter 4 Image Enhancement in the Frequency Domain

  41. Chapter 4 Image Enhancement in the Frequency Domain

  42. Chapter 4 Image Enhancement in the Frequency Domain

  43. Chapter 4 Image Enhancement in the Frequency Domain

  44. Chapter 4 Image Enhancement in the Frequency Domain