1. Image Enhancement in the Frequency Domain Spring 2006, Jen-Chang Liu

2. Outline Introduction to the Fourier Transform and Frequency Domain Magnitude of frequencies Phase of frequencies Fourier transform and DFT Filtering in the frequency domain Smoothing Frequency Domain Filters Sharpening Frequency Domain Filters Homomorphic Filtering Implementation of Fourier transform

3. Background 1807, French math. Fourier Any function that periodically repeats itself can be expressed as the sum of of sines and/or cosines of different frequencies, each multiplied by a different coefficient (Fourier series)

4. Periodic function f(t) = f(t+T), T: period (sec.) 1/T: frequency (cycles/sec.)

5. Periodic function f Frequency Weight f 1 w 1 f 2 w 2 f 3 w 3 f 4 w 4

7. How to measure weights? Assume f 1, f 2, f 3, f 4 are known How to measure w 1, w 2, w 3, w 4 ? min Minimize squared error

8. Minimize MSE calculation min

9. Orthogonal condition f 1 and f 2 are orthogonal if f 1, f 2, f 3, f 4 are orthogonal to each other 正交

10. Minimization calculation To satisfy min We have => Recall in linear algebra: projection

11. Weight = Projection magnitude Represent input f(x) with another basis functions v Vector space (1,0) projection Functional space f f1f1

12. Summary 1 A function f can be written as sum of f 1, f 2, f 3, … If f1, f2, f3, … are orthogonal to each other Weight (magnitude)

13. Summary 1: sine, cosine bases Let f 1, f 2, f 3, … carry frequency information Let them be sines and cosines n, k:integers => They all satisfy orthogonal conditions

14. Summary 1: orthogonal

15. Fourier series For (Assume periodic outside) DC 頻率 =1 頻率 =2 頻率 =3

16. Outline Introduction to the Fourier Transform and Frequency Domain Magnitude of frequencies Phase of frequencies Fourier transform and DFT Filtering in the frequency domain Smoothing Frequency Domain Filters Sharpening Frequency Domain Filters Homomorphic Filtering Implementation of Fourier transform

17. Correlation with different phase Weight calculation 相關係數 f1f1 f 相位

18. Correlation with different phase (cont.) Weight calculation 相關係數 ? f1f1 f

19. Deal with phase: method 1 For example, expand f(t) over the cos(wt) basis function Consider different phases  0 22  Corr(  Problem: weight(w,  )

20. Deal with phase: method 2 Complex exponential as basis j 1 real With frequency w: Advantage: Derive magnitude and phase  simultaneously

21. Deal with phase 2: example Input magnitude phase

22. Fourier series with phase For (Assume periodic outside) DC 頻率 k=1 k=2 k=3 Complex weight

23. Outline Introduction to the Fourier Transform and Frequency Domain Magnitude of frequencies Phase of frequencies Fourier transform and DFT Filtering in the frequency domain Smoothing Frequency Domain Filters Sharpening Frequency Domain Filters Homomorphic Filtering Implementation of Fourier transform

24. Fourier transform Functions that are not periodic can be expressed as the integral of sines and/or cosines multiplied by a weighting functions Frequency up to infinity Perfect reconstruction Functions  --  Fourier transform Operation in frequency domain without loss of information

25. 1-D Fourier Transform Fourier transform F(u) of a continuous function f(x) is: Inverse transform: Forward Fourier transform:

26. 2-D Fourier Transform Fourier transform F(u,v) of a continuous function f(x,y) is: Inverse transform: x y u v F

27. Future development 1950, fast Fourier transform (FFT) Revolution in the signal processing Discrete Fourier transform (DFT) For digital computation

28. 1-D Discrete Fourier Transform f(x), x=0,1, …,M-1. discrete function F(u), u=0,1, …,M-1. DFT of f(x) Inverse transform: Forward discrete Fourier transform:

29. 0123 … M-1 f(x) x Assume periodic outside 0123M-1 x 頻率 =0 0123 M-1 x 頻率 =1/M ( 基頻 ) … … 0123 M-1 x 頻率 =2/M … … 0123 M-1 x 頻率 =(M-1)/M …

30. Frequency Domain 頻率域 Where is the frequency domain? j 1 Euler ’ s formula: frequency u F(u)

31. Fourier transform

33. Physical analogy Mathematical frequency splitting Fourier transform Physical device Galss prism 三稜鏡 Split light into frequency components

34. F(u) Complex quantity? Polar coordinate real imaginary m magnitude phase Power spectrum

35. Some notes about sampling in time and frequency axis Time index Frequency index Also follow reciprocal property

36. Extend to 2-D DFT from 1-D 2-D: x-axis then y-axis

37. Complex Quantities to Real Quantities Useful representation magnitude phase Power spectrum

38. Some notes about 2-D Fourier transform Frequency axis x y u v u v Fshift 0

39. DFT: example log(F)

40. Properties in the frequency domain Fourier transform works globally No direct relationship between a specific components in an image and frequencies Intuition about frequency Frequency content Rate of change of gray levels in an image

41. +45,-45 degree artifacts