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# Digital Image Processing: Image Enhancement in the Frequency Domain

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A course on Digital Image Processing based on Gonzalez book

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### Digital Image Processing: Image Enhancement in the Frequency Domain

1. 1. CSC447: Digital Image Processing Chapter 4: Prof. Dr. Mostafa Gadal-Haqq M. Mostafa Computer Science Department Faculty of Computer & Information Sciences AIN SHAMS UNIVERSITY
2. 2. Foundation  Fourier Theorem: Any function that periodically repeat itself can be represented by the some of sines and/or cosines of different frequencies, each multiplied by a different coefficient. )cossin()( 0 xbxaxf ii n i ii    2CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
3. 3. The Discrete Fourier Transform (DFT)  1-D Fourier Transform:  The Fourier transform, F(u), of a discrete 1- D function, f(x); x = 0, 1, 2, …, M-1, is:  Where u= 0, 1, 2, …, M-1  1-D Inverse Fourier Transform:      1 0 /2 )( 1 )( M x Muxj exf M uF      1 0 /2 )()( M u Muxj euFxf  3CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
4. 4. The Discrete Fourier Transform (DFT)  F(u) is called the frequency component of the Fourier Transform, and its domain (the values of u) is called the frequency domain, because u determines the frequency of the components of the transform:  Since F(u) is complex quantity It is convenient to express it in polar form  |F(u)| is called the magnitude, and (u) is the phase  The Power Spectrum P(u) = |F(u)|2 )](/)([tan(u)and,)]()([|)(|where, |)(|)( 1-2/122 )( uRuIuIuRuF euFuF uj     4CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
5. 5. The Discrete Fourier Transform (DFT)  1-D Fourier Transform: 5CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
6. 6. The Discrete Fourier Transform (DFT)  2-D Fourier Transform:  The Fourier transform, F(u,v), of a discrete 2-D function (MxN), f(x,y) is:  Where u= 0,1,2, …,M-1, and v = 0,1,2, …, N-1  2-D Inverse Fourier Transform:        1 0 1 0 )//(2 ),( 1 ),( M x N y NvyMuxj eyxf MN vuF          1 0 1 0 )//(2 ),(),( M u N v NvyMuxj evuFyxf  6CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
7. 7. The Discrete Fourier Transform (DFT)  the Fourier spectrum , phase angle, andpower spectrum , are defined as before: ),(),(|),(| and)],,(/),([tanv)(u, ,)],(),([|),(|where, |),(|),( 222 1- 2/122 ),( vuIvuRvuFP(u,v) vuRvuI vuIvuRvuF evuFvuF vuj       7CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
8. 8. The Discrete Fourier Transform (DFT)  2-D Fourier Transform: 8CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
9. 9. The Discrete Fourier Transform (DFT)  2-D Fourier Transform: 9CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
10. 10. The Discrete Fourier Transform (DFT)  2-D Fourier Transform: 10CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
11. 11. The Discrete Fourier Transform (DFT)  2-D Fourier Transform: 11CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
12. 12. The Discrete Fourier Transform (DFT)  Properties of the Fourier Transform: 12CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
13. 13. The Discrete Fourier Transform (DFT)  Properties of the Fourier Transform: 13CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
14. 14. The Discrete Fourier Transform (DFT)  Properties of the Fourier Transform: 14CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
15. 15. The Discrete Fourier Transform (DFT)  Properties of the Fourier Transform: 15CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
16. 16. The Discrete Fourier Transform (DFT)  Properties of the Fourier Transform: 16CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
17. 17. Filtering in the Frequency Domain  Basic Operations 17CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
18. 18. Filtering in the Frequency Domain  2-D Fourier Transform 18CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
19. 19. Filtering in the Frequency Domain  2-D Fourier Transform: 19CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
20. 20. Filtering in the Frequency Domain  Notch filter 20CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
21. 21. Filtering in the Frequency Domain  Ideal Low-pass Filter (ILPF) cutoff frequency 21CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
22. 22. Filtering in the Frequency Domain  How to find the cutoff frequency for a ILPF?  Find the circle that enclose a certain amount of the power spectrum of the image:  Where P(u,v) is the Power spectrum at frequencies (u,v) the. Then , a circle of radius r enclose a  percentage of the power, where 22CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
23. 23. Filtering in the Frequency Domain  Distribution of the power spectrum: 23CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
24. 24. Filtering in the Frequency Domain  Filtering with power cutoff 24CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
25. 25. Filtering in the Frequency Domain  Butterworth Low-pass Filter 25CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
26. 26. Filtering in the Frequency Domain  Butterworth Low-pass Filter 26CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
27. 27. Filtering in the Frequency Domain  Gaussian Low-pass Filter  Where D(u,v) id the distance from the origin 27CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
28. 28. Filtering in the Frequency Domain  Gaussian Low-pass Filter 28CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
29. 29. Filtering in the Frequency Domain  Gaussian Low-pass and High-pass filters: 29CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
30. 30. Filtering in the Frequency Domain  Gaussian Low-pass filters: 30CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
31. 31. Filtering in the Frequency Domain  Gaussian Low-pass filters: 31CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
32. 32. Filtering in the Frequency Domain  Ideal High-pass filters: 32CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
33. 33. Filtering in the Frequency Domain  Ideal High-pass filters: 33CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
34. 34. Filtering in the Frequency Domain  Ideal High-pass filters: 34CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
35. 35. Filtering in the Frequency Domain  Gaussian High-pass filters: 35CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
36. 36. Filtering in the Frequency Domain  High-pass filters: 36CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
37. 37. Filtering in the Frequency Domain  High-pass filters: 37CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
38. 38. Filtering in the Frequency Domain  Ideal Band-Pass Filter 38CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
39. 39. Filtering in the Frequency Domain  The Laplacian filters: 39CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
40. 40. Filtering in the Frequency Domain  Gaussian High-pass filters: 40CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
41. 41. Filtering in the Frequency Domain  Homomorphic filters: 41CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
42. 42. Filtering in the Frequency Domain  Homomorphic filters: 42CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
43. 43. Filtering in the Frequency Domain  Homomorphic filters: 43CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
44. 44. HW3  4.9 and 4.12 44CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.