1. 7- 1 Chapter 7: Fourier Analysis Fourier analysis = Series + Transform ◎ Fourier Series -- A periodic (T) function f(x) can be written as the sum of sines and cosines of varying amplitudes and frequencies

2. 7- 2 ○ Some function is formed by a finite number of sinuous functions

3. 7- 3 Some function requires an infinite number of sinuous functions to compose

4. 7- 4 Spectrum The spectrum of a periodic function is discrete, consisting of components at dc, 1/T, and its multiples, e.g., For non-periodic functions, i.e., The spectrum of the function is continuous

5. 7- 5 ○ In complex form: Compare with

6. 7- 6 Euler ’ s formula:

7. 7- 7

8. 7- 8

9. 7- 9 Continuous case

10. 7- 10 Discrete case: ◎ Fourier Transform

11. 7- 11 Matrix form Let

12. 7- 12 。 Example: f = {1,2,3,4}. Then, N = 4,

13. 7- 13

14. 7- 14 ○ Inverse DFT Let

15. 7- 15 。 Example:

16. 7- 16 ◎ Properties ○ Linearity: e.g., Noise removal f’ = f + n, n: additive noise,

17. 8-17 Fourier spectrum noise Corresponding spatial noise

18. ○ Scaling : Show:

19. 7- 19 ○ Periodicity:

20. 7- 20 ○ Shifting:

21. 7- 21 。 Example:

22. 7- 22 ◎ Convolution: Convolution theorem: Correlation theorem ◎ Correlation

23. 7- 23 。 Discrete Case: A = 4, B = 5, A + B – 1 = 8, e.g.,

24. 7- 24 ＊ Convolution can be defined in terms of polynomial product Extend f, g to if f, g have different numbers of sample points Let Compute The coefficients of to form the result of convolution

25. 7- 25 。 Example: Let The coefficients of form the convolution

26. 7- 26

27. 7- 27 ○ Fast Fourier Transform (FFT) -- Successive doubling method

28. 7- 28

29. 7- 29 。 Time complexity : the length of the input sequence FT: FFT: Times of speed increasing: N FT FFT Ratio 4 16 8 2.0 8 84 24 2.67 16 256 64 4.0 32 1024 160 6.4 64 4096 384 10.67 128 16384 896 18.3 256 65536 2048 32.0 512 262144 4608 56.9 1024 1048576 10240 102.4

30. 7- 30 ○ Inverse FFT ← Given ← compute i. Input into FFT. The output is ii. Taking the complex conjugate and multiplying by N, yields the f(x)

31. 7- 31 ◎ 2D Fourier Transform ○ FT: IFT:

32. 7- 32 ◎ Properties ○ Filtering: every F(u,v) is obtained by multiplying every f(x,y) by a fixed value and adding up the results. DFT can be considered as a linear filtering ○ DC coefficient:

33. 7- 33 ○ Separability:

34. 7- 34 ○ Conjugate Symmetry: F(u,v) = F*(-u,-v)

35. 7- 35 ○ Shifting

36. 7- 36 ○ Rotation Polor coordinates:

37. 7- 37 ○ Display: effect of log operation

38. 7- 38

39. 7- 39 ◎ Image Transform

40. 7- 40 ◎ Filtering in Frequency Domain ○ Low pass filtering I FT m IFT

41. 7- 41 D = 5 D = 30 ○ High pass filtering

42. 7- 42 Different Ds

43. 7- 43 ◎ Butterworth Filtering ○ Low pass filter ○ High pass filter

44. 7- 44 ○ Low pass filter ○ High pass filter

45. 7- 45 ◎ Homomorphic Filtering -- Deals with images with large variation of illumination, e.g., sunshine + shadows -- Both reduce intensity range and increases local contrast ○ Idea: I = LR, L: illumination, R: Reflectance logI = logL + logR low frequency high frequency

46. 7- 46

47. 7- 47

48. 7- 48 ○ Fast Fourier Transform (FFT) -- Successive doubling method Assume Let Let N = 2M.

49. 7- 49 = ] ∵ = ]

50. 7- 50 Let --------- (B) Consider

51. 7- 51

52. 7- 52 ---- (C)

53. 7- 53 F(u+M) = Recursively divide F(u) and F(u+M), ○ Analysis : The Fourier sequence F(u), u = 0, …, N-1 of f(x), x = 0, …, N-1 can be formed from sequences F(u) = Eventually, each contains one element F(w), i.e., w = 0, and F(w) = f(x). u = 0, ……, M-1

54. 7- 54

55. 7- 55 ○ Example: needs { f(0), f(2), f(4), f(6) } needs { f(1), f(3), f(5), f(7) } Computing Input { f(0), f(1), ……, f(7) }

56. 7- 56 Reorder the input sequence into {f(0), f(4), f(2), f(6), f(1), f(5), f(3), f(7)} Computing ＊ Bit-Reversal Rule