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# Fourier Transform J.B. Fourier 1768-1830

Fourier Transform J.B. Fourier 1768-1830. Image Enhancement in the Frequency Domain 1-D. 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. =. Fourier spectrum of step function.

## 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

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