1. Computer Graphics & Image Processing Chapter # 4 Image Enhancement in Frequency Domain 2/26/20161

2. ALI JAVED Lecturer SOFTWARE ENGINEERING DEPARTMENT U.E.T TAXILA Email:: alijaved@uettaxila.edu.pkalijaved@uettaxila.edu.pk Office Room #:: 7 2/26/20162

3. Introduction 2/26/20163

4. Background (Fourier Series)  Any function that periodically repeats itself can be expressed as the sum of sines and cosines of different frequencies each multiplied by a different coefficient  This sum is known as Fourier Series  It does not matter how complicated the function is; as long as it is periodic and meet some mild conditions it can be represented by such as a sum  It was a revolutionary discovery 2/26/20164

5. 5

6. Background (Fourier Transform)  Even functions that are not periodic but Finite can be expressed as the integrals of sines and cosines multiplied by a weighing function  This is known as Fourier Transform  A function expressed in either a Fourier Series or transform can be reconstructed completely via an inverse process with no loss of information  This is one of the important characteristics of these representations because they allow us to work in the Fourier Domain and then return to the original domain of the function 2/26/20166

7. Fourier Transform ‘Fourier Transform’ transforms one function into another domain, which is called the frequency domain representation of the original function The original function is often a function in the Time domain In image Processing the original function is in the Spatial Domain The term Fourier transform can refer to either the Frequency domain representation of a function or to the process/formula that "transforms" one function into the other. 2/26/20167

8. Our Interest in Fourier Transform We will be dealing only with functions (images) of finite duration so we will be interested only in Fourier Transform 2/26/20168

9. Applications of Fourier Transforms  1-D Fourier transforms are used in Signal Processing  2-D Fourier transforms are used in Image Processing  3-D Fourier transforms are used in Computer Vision  Applications of Fourier transforms in Image processing: – –Image enhancement, –Image restoration, –Image encoding / decoding, –Image description 2/26/20169

10. One Dimensional Fourier Transform and its Inverse  The Fourier transform F (u) of a single variable, continuous function f (x) is  Given F(u) we can obtain f (x) by means of the Inverse Fourier Transform 2/26/201610

11. Discrete Fourier Transforms (DFT) 1-D DFT for M samples is given as The Inverse Fourier transform in 1-D is given as 2/26/201611

12. Discrete Fourier Transforms (DFT) 1-D DFT for M samples is given as The inverse Fourier transform in 1-D is given as 2/26/201612

13. Two Dimensional Fourier Transform and its Inverse  The Fourier transform F (u,v) of a two variable, continuous function f (x,y) is  Given F(u,v) we can obtain f (x,y) by means of the Inverse Fourier Transform 2/26/201613

14. 2-D DFT 2/26/201614

15. Fourier Transform 2/26/201615

16. 2-D DFT 2/26/201616

17. 2/26/201617 Shifting the Origin to the Center

18. 2/26/201618 Shifting the Origin to the Center

19. 2/26/201619 Properties of Fourier Transform  As we move away from the origin in F(u,v) the lower frequencies corresponding to slow gray level changes  Higher frequencies correspond to the fast changes in gray levels (smaller details such edges of objects and noise)  The direction of amplitude change in spatial domain and the amplitude change in the frequency domain are orthogonal (see the examples)

20. 2/26/201620 Properties of Fourier Transform The Fourier Transform pair has the following translation property

21. 2/26/201621 Properties of Fourier Transform

22. 2/26/201622 Properties of Fourier Transform

23. 2/26/201623 DFT Examples

24. 2/26/201624 DFT Examples

25. 2/26/201625 Properties of Fourier Transform

26. 2/26/201626 Properties of Fourier Transform

27. 2/26/201627 Properties of Fourier Transform  The lower frequencies corresponds to slow gray level changes  Higher frequencies correspond to the fast changes in gray levels (smaller details such edges of objects and noise)

28. 2/26/201628 Filtering using Fourier Transforms

29. 2/26/201629 Example of Gaussian LPF and HPF

30. 2/26/201630 Filters to be Discussed

31. 2/26/201631 Low Pass Filtering A low-pass filter attenuates high frequencies and retains low frequencies unchanged. The result in the spatial domain is equivalent to that of a smoothing filter; as the blocked high frequencies correspond to sharp intensity changes, i.e. to the fine-scale details and noise in the spatial domain image.smoothing filter

32. 2/26/201632 High Pass Filtering A high pass filter, on the other hand, yields edge enhancement or edge detection in the spatial domain, because edges contain many high frequencies. Areas of rather constant gray level consist of mainly low frequencies and are therefore suppressed.

33. 2/26/201633 Band Pass Filtering A bandpass attenuates very low and very high frequencies, but retains a middle range band of frequencies. Bandpass filtering can be used to enhance edges (suppressing low frequencies) while reducing the noise at the same time (attenuating high frequencies). Bandpass filters are a combination of both lowpass and highpass filters. They attenuate all frequencies smaller than a frequency Do and higher than a frequency D1, while the frequencies between the two cut-offs remain in the resulting output image.

34. 2/26/201634 Ideal Low Pass Filter

35. 2/26/201635 Ideal Low Pass Filter

36. 2/26/201636 Ideal Low Pass Filter (example)

37. 2/26/201637 Why Ringing Effect

38. 2/26/201638 Butterworth Low Pass Filter

39. 2/26/201639 Butterworth Low Pass Filter

40. 2/26/201640 Butterworth Low Pass Filter (example)

41. 2/26/201641 Gaussian Low Pass Filters

42. 2/26/201642 Gaussian Low Pass Filters

43. 2/26/201643 Gaussian Low Pass Filters (example)

44. 2/26/201644 Gaussian Low Pass Filters (example)

45. 2/26/201645 Sharpening Fourier Domain Filters

46. 2/26/201646 Sharpening Spatial Domain Representations

47. 2/26/201647 Sharpening Fourier Domain Filters (Examples)

48. 2/26/201648 Sharpening Fourier Domain Filters (Examples)

49. 2/26/201649 Sharpening Fourier Domain Filters (Examples)

50. 2/26/201650 Laplacian in Frequency Domain

51. 2/26/201651 Unsharp Masking, High Boost Filtering

52. 2/26/201652 Example of Modified High Pass Filtering

53. 2/26/201653 Homomorphic Filtering

54. 2/26/201654 Homomorphic Filtering

55. 2/26/201655 Homomorphic Filtering

56. 2/26/201656 Homomorphic Filtering

57. 2/26/201657 Homomorphic Filtering (Example)

58. 2/26/201658 Basic Filters And scaling rest of values.

59. 2/26/201659 Example (Notch Function)

60. Any question