Chapter 7: Fourier Analysis

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

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

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. ○ In complex form: Compare with

6. Continuous case

7. Discrete case: ◎ Fourier Transform

8. Matrix form Let

9. 。Example: f = {1,2,3,4}. Then, N = 4,

11. Let ○ Inverse DFT

12. 。Example:

13. ◎ Properties ○ Linearity: 。 Example: Noise removal f’ = f + n, n: noise, ○ Scaling

14. ○ Periodicity:

15. ○ Shifting:

16. 。 Example:

17. ◎ Convolution: Convolution theorem:

18. ◎ Correlation Correlation theorem

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

20. ＊ 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

21. 。Example: Let The coefficients of form the convolution

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

24. 7-24

25. 。 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

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

27. ◎ 2D Fourier Transform ○ FT: IFT:

28. ◎ 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:

29. ○ Separability:

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

31. ○ Shifting

32. ○ Rotation Polor coordinates:

33. ○ Display: effect of log operation

35. ◎ Image Transform

36. ◎ Filtering in Frequency Domain ○ Low pass filtering I FT m IFT

37. D = 5 D = 30 ○ High pass filtering

38. Different Ds

39. ◎ Butterworth Filtering ○ Low pass filter ○ High pass filter

40. ○ Low pass filter ○ High pass filter

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

42. low frequency high frequency

44. Euler’s formula: 7-44

45. 7-45

46. 7-46

47. ○ Fast Fourier Transform (FFT) -- Successive doubling method Let Assume Let N = 2M. 7-47

48. ] = ] ∵ = 7-48

49. Let --------- (B) Consider 7-49

50. 7-50