Digital Image Processing Chapter # 4 Image Enhancement in Frequency Domain

1. Digital Image Processing Chapter # 4 Image Enhancement in Frequency Domain

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

3. Introduction

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

6. Background (Fourier Transform) • Even functions that are not periodic 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

7. Fourier Transform • ‘Fourier Transform’ transforms one function into another domain , which is called the frequency domainrepresentation 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 termFourier transformcan refer to either the Frequency domain representation of a function or to the process/formula that "transforms" one function into the other.

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

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

10. One Dimensional Fourier Transform and its Inverse • The Fourier transform F (u) of a single variable, continuousfunction f (x) is • Given F(u) we can obtain f (x) by means of the Inverse Fourier Transform

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

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

13. Two Dimensional Fourier Transform and its Inverse • The Fourier transform F (u,v) of a two variable, continuousfunction f (x,y) is • Given F(u,v) we can obtain f (x,y) by means of the Inverse Fourier Transform

14. 2-D DFT

15. Fourier Transform

16. 2-D DFT

17. Shifting the Origin to the Center

18. Shifting the Origin to the Center

19. 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)

20. DFT Examples

21. DFT Examples

22. Filtering using Fourier Transforms

23. Example of Gaussian LPF and HPF

24. Filters to be Discussed

25. 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. 1/1/2020 25

26. High Pass Filtering A highpass 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. 1/1/2020 26

27. 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. 1/1/2020 27

28. Ideal Low Pass Filter

29. Ideal Low Pass Filter

30. Ideal Low Pass Filter (example)

31. Butterworth Low Pass Filter

32. Butterworth Low Pass Filter

33. Butterworth Low Pass Filter (example)

34. Gaussian Low Pass Filters 1/1/2020 34

35. Gaussian Low Pass and High Pass Filters 1/1/2020 35

36. Gaussian Low Pass Filters 1/1/2020 36

37. Gaussian Low Pass Filters (example) 1/1/2020 37

38. Gaussian Low Pass Filters (example) 1/1/2020 38

39. Sharpening Fourier Domain Filters 1/1/2020 39

40. Sharpening Spatial Domain Representations 1/1/2020 40

41. Sharpening Fourier Domain Filters (Examples) 1/1/2020 41

42. Sharpening Fourier Domain Filters (Examples) 1/1/2020 42

43. Sharpening Fourier Domain Filters (Examples) 1/1/2020 43