1. 1 © 2010 Cengage Learning Engineering. All Rights Reserved. 1 Introduction to Digital Image Processing with MATLAB ® Asia Edition McAndrew ‧ Wang ‧ Tseng Chapter 7: The Fourier Transform

2. 2 © 2010 Cengage Learning Engineering. All Rights Reserved. 7.1 Introduction The Fourier transform allows us to perform tasks that would be impossible to perform any other way It is more efficient to use the Fourier transform than a spatial filter for a large filter The Fourier transform also allows us to isolate and process particular image frequencies Ch7-p.145

3. 3 7.2 Background © 2010 Cengage Learning Engineering. All Rights Reserved. Ch7-p.145

4. 4 FIGURE 7.2 © 2010 Cengage Learning Engineering. All Rights Reserved. A periodic function may be written as the sum of sines and cosines of varying amplitudes and frequencies Ch7-p.146

5. 5 7.2 Background © 2010 Cengage Learning Engineering. All Rights Reserved. These are the equations for the Fourier series expansion of f (x), and they can be expressed in complex form Ch7-p.147

6. 6 7.2 Background © 2010 Cengage Learning Engineering. All Rights Reserved. If the function is nonperiodic, we can obtain similar results by letting T → ∞, in which case Ch7-p.147

7. 7 7.2 Background © 2010 Cengage Learning Engineering. All Rights Reserved. Fourier transform pair Further details can be found, for example, in James [18]. Ch7-p.148

8. 8 7.3 The One-Dimensional Discrete Fourier Transform © 2010 Cengage Learning Engineering. All Rights Reserved. Ch7-p.148

9. 9 7.3 The One-Dimensional Discrete Fourier Transform © 2010 Cengage Learning Engineering. All Rights Reserved. Definition of the One-Dimensional DFT Ch7-p.148

10. 10 7.3 The One-Dimensional Discrete Fourier Transform © 2010 Cengage Learning Engineering. All Rights Reserved. This definition can be expressed as a matrix multiplication where F is an N × N matrix defined by Given N, we shall define Ch7-p.149

11. 11 © 2010 Cengage Learning Engineering. All Rights Reserved. e.g. suppose f = [1, 2, 3, 4] so that N = 4. Then 7.3 The One-Dimensional Discrete Fourier Transform Ch7-p.150

12. 12 © 2010 Cengage Learning Engineering. All Rights Reserved. 7.3 The One-Dimensional Discrete Fourier Transform Ch7-p.150

13. 13 © 2010 Cengage Learning Engineering. All Rights Reserved. THE INVERSE DFT If you compare Equation (7.3) with Equation 7.2 you will see that there are really only two differences: 1.There is no scaling factor 1/ N 2.The sign inside the exponential function has been changed to positive. 3.The index of the sum is u, instead of x 7.3 The One-Dimensional Discrete Fourier Transform Ch7-p.150

14. 14 © 2010 Cengage Learning Engineering. All Rights Reserved. 7.3 The One-Dimensional Discrete Fourier Transform Ch7-p.151

15. 15 © 2010 Cengage Learning Engineering. All Rights Reserved. 7.3 The One-Dimensional Discrete Fourier Transform Ch7-p.151

16. 16 7.4 Properties of the One-Dimensional DFT © 2010 Cengage Learning Engineering. All Rights Reserved. LINEARITY This is a direct consequence of the definition of the DFT as a matrix product Suppose f and g are two vectors of equal length, and p and q are scalars, with h = pf + qg If F, G, and H are the DFT’s of f, g, and h, respectively, we have Ch7-p.152

17. 17 7.4 Properties of the One-Dimensional DFT © 2010 Cengage Learning Engineering. All Rights Reserved. SHIFTING Suppose we multiply each element x n of a vector x by (−1) n. In other words, we change the sign of every second element Let the resulting vector be denoted x’. The DFT X’ of x’ is equal to the DFT X of x with the swapping of the left and right halves Ch7-p.152

18. 18 7.4 Properties of the One-Dimensional DFT © 2010 Cengage Learning Engineering. All Rights Reserved. e.g. Ch7-p.152

19. 19 7.4 Properties of the One-Dimensional DFT © 2010 Cengage Learning Engineering. All Rights Reserved. Notice that the first four elements of X are the last four elements of X1 and vice versa Ch7-p.153

20. 20 SCALING where k is a scalar and F= f If you make the function wider in the x -direction, it's spectrum will become smaller in the x -direction, and vice versa Amplitude will also be changed 7.4 Properties of the One-Dimensional DFT © 2010 Cengage Learning Engineering. All Rights Reserved. F Ch7-p.153

21. 21 7.4 Properties of the One-Dimensional DFT © 2010 Cengage Learning Engineering. All Rights Reserved. CONJUGATE SYMMETRY CONVOLUTION Ch7-p.153

22. 22 7.4 Properties of the One-Dimensional DFT © 2010 Cengage Learning Engineering. All Rights Reserved. THE FAST FOURIER TRANSFORM Appendix B Ch7-p.157 2n2n

23. 23 7.5 The Two-Dimensional DFT © 2010 Cengage Learning Engineering. All Rights Reserved. The 2-D Fourier transform rewrites the original matrix in terms of sums of corrugations Ch7-p.157

24. 24 7.5.1 Some Properties of the Two- Dimensional Fourier Transform © 2010 Cengage Learning Engineering. All Rights Reserved. SIMILARITY THE DFT AS A SPATIAL FILTER SEPARABILITY Ch7-p.159

25. 25 7.5.1 Some Properties of the Two- Dimensional Fourier Transform © 2010 Cengage Learning Engineering. All Rights Reserved. LINEARITY THE CONVOLUTION THEOREM Suppose we wish to convolve an image M with a spatial filter S 1.Pad S with zeroes so that it is the same size as M ; denote this padded result by S’ 2.Form the DFTs of both M and S’ to obtain (M) and (S’) Ch7-p.160

26. 26 7.5.1 Some Properties of the Two- Dimensional Fourier Transform © 2010 Cengage Learning Engineering. All Rights Reserved. 3.Form the element-by-element product of these two transforms: 4.Take the inverse transform of the result: Put simply, the convolution theorem states or equivalently that Ch7-p.161

27. 27 7.5.1 Some Properties of the Two- Dimensional Fourier Transform © 2010 Cengage Learning Engineering. All Rights Reserved. THE DC COEFFICIENT SHIFTING Ch7-p.162 DC coefficient

28. 28 7.5.1 Some Properties of the Two- Dimensional Fourier Transform © 2010 Cengage Learning Engineering. All Rights Reserved. CONJUGATE SYMMETRY DISPLAYING YRANSFORMS Ch7-p.163

29. 29 7.6 Fourier Transforms in M ATLAB © 2010 Cengage Learning Engineering. All Rights Reserved. fft, which takes the DFT of a vector, ifft, which takes the inverse DFT of a vector, fft2, which takes the DFT of a matrix, ifft2, which takes the inverse DFT of a matrix, and fftshift, which shifts a transform Ch7-p.164

30. 30 7.6 Fourier Transforms in M ATLAB © 2010 Cengage Learning Engineering. All Rights Reserved. e.g. Note that the DC coefficient is indeed the sum of all the matrix values Ch7-p.164

31. 31 7.6 Fourier Transforms in M ATLAB © 2010 Cengage Learning Engineering. All Rights Reserved. e.g. Ch7-p.165

32. 32 7.6 Fourier Transforms in M ATLAB © 2010 Cengage Learning Engineering. All Rights Reserved. e.g. Ch7-p.165

33. 33 7.7 Fourier Transforms of Images © 2010 Cengage Learning Engineering. All Rights Reserved. Ch7-p.167

34. 34 FIGURE 7.10 © 2010 Cengage Learning Engineering. All Rights Reserved. Ch7-p.168

35. 35 FIGURE 7.11 © 2010 Cengage Learning Engineering. All Rights Reserved. Ch7-p.169

36. 36 FIGURE 7.12 © 2010 Cengage Learning Engineering. All Rights Reserved. Ch7-p.169

37. 37 FIGURE 7.13 © 2010 Cengage Learning Engineering. All Rights Reserved. EXAMPLE 7.7.2 Ch7-p.170

38. 38 FIGURE 7.14 © 2010 Cengage Learning Engineering. All Rights Reserved. EXAMPLE 7.7.3 Ch7-p.170

39. 39 FIGURE 7.15 © 2010 Cengage Learning Engineering. All Rights Reserved. EXAMPLE 7.7.4 Ch7-p.171

40. 40 7.7 Fourier Transforms of Images © 2010 Cengage Learning Engineering. All Rights Reserved. Ch7-p.172

41. 41 7.8 Filtering in the Frequency Domain © 2010 Cengage Learning Engineering. All Rights Reserved. Ideal Filtering LOW-PASS FILTERING Ch7-p.172

42. 42 FIGURE 7.16 © 2010 Cengage Learning Engineering. All Rights Reserved. Ch7-p.173

43. 43 FIGURE 7.17 © 2010 Cengage Learning Engineering. All Rights Reserved. D = 15 Ch7-p.174

44. 44 7.8 Filtering in the Frequency Domain © 2010 Cengage Learning Engineering. All Rights Reserved. >> cfl = cf.*b >> cfli = ifft2(cfl); >> figure, fftshow(cfli, ’abs’) Ch7-p.174

45. 45 FIGURE 7.18 © 2010 Cengage Learning Engineering. All Rights Reserved. D = 5D = 30 Ch7-p.175

46. 46 7.8 Filtering in the Frequency Domain © 2010 Cengage Learning Engineering. All Rights Reserved. HIGH-PASS FILTERING Ch7-p.174

47. 47 FIGURE 7.19 © 2010 Cengage Learning Engineering. All Rights Reserved. Ch7-p.176

48. 48 FIGURE 7.20 © 2010 Cengage Learning Engineering. All Rights Reserved. Ch7-p.176

49. 49 7.8.2 Butterworth Filtering © 2010 Cengage Learning Engineering. All Rights Reserved. Ideal filtering simply cuts off the Fourier transform at some distance from the center It has the disadvantage of introducing unwanted artifacts (ringing) into the result One way of avoiding these artifacts is to use as a filter matrix, a circle with a cutoff that is less sharp Ch7-p.177

50. 50 FIGURE 7.21 © 2010 Cengage Learning Engineering. All Rights Reserved. Ch7-p.177

51. 51 FIGURE 7.22 & 7.23 © 2010 Cengage Learning Engineering. All Rights Reserved. Ch7-p.178

52. 52 FIGURE 7.24 © 2010 Cengage Learning Engineering. All Rights Reserved. Ch7-p.179

53. 53 FIGURE 7.25 © 2010 Cengage Learning Engineering. All Rights Reserved. Ch7-p.179

54. 54 FIGURE 7.26 © 2010 Cengage Learning Engineering. All Rights Reserved. >> cfbli = ifft2(cfbl); >> figure, fftshow(cfbli, ’abs’) >> bl = lbutter(c,15,1); >> cfbl = cf.*bl; >> figure, fftshow(cfbl, ’log’); Ch7-p.180

55. 55 FIGURE 7.27 © 2010 Cengage Learning Engineering. All Rights Reserved. Ch7-p.181

56. 56 7.8.3 Gaussian Filtering © 2010 Cengage Learning Engineering. All Rights Reserved. A wider function, with a large standard deviation, will have a low maximum Ch7-p.181

57. 57 FIGURE 7.28 © 2010 Cengage Learning Engineering. All Rights Reserved. Ch7-p.182

58. 58 FIGURE 7.29 © 2010 Cengage Learning Engineering. All Rights Reserved. Ch7-p.183

59. 59 7.9 Homomorphic Filtering © 2010 Cengage Learning Engineering. All Rights Reserved. where f(x, y) is intensity, i(x, y) is the illumination and r(x, y) is the reflectance Ch7-p.184

60. 60 FIGURE 7.31 © 2010 Cengage Learning Engineering. All Rights Reserved. Ch7-p.185

61. 61 FIGURE 7.32 © 2010 Cengage Learning Engineering. All Rights Reserved. Ch7-p.185

62. 62 FIGURE 7.33 © 2010 Cengage Learning Engineering. All Rights Reserved. Ch7-p.186

63. 63 FIGURE 7.34 © 2010 Cengage Learning Engineering. All Rights Reserved. Ch7-p.186

64. 64 FIGURE 7.35 © 2010 Cengage Learning Engineering. All Rights Reserved. Ch7-p.187