Difference Between Search & Browse Methods in Odoo 17
Digital Image Processing: Image Enhancement in the Frequency Domain
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. 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. 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. 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. The Discrete Fourier Transform (DFT)
1-D Fourier Transform:
5CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
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. 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. The Discrete Fourier Transform (DFT)
2-D Fourier Transform:
8CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
9. The Discrete Fourier Transform (DFT)
2-D Fourier Transform:
9CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
10. The Discrete Fourier Transform (DFT)
2-D Fourier Transform:
10CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
11. The Discrete Fourier Transform (DFT)
2-D Fourier Transform:
11CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
12. The Discrete Fourier Transform (DFT)
Properties of the Fourier Transform:
12CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
13. The Discrete Fourier Transform (DFT)
Properties of the Fourier Transform:
13CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
14. The Discrete Fourier Transform (DFT)
Properties of the Fourier Transform:
14CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
15. The Discrete Fourier Transform (DFT)
Properties of the Fourier Transform:
15CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
16. The Discrete Fourier Transform (DFT)
Properties of the Fourier Transform:
16CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
17. Filtering in the Frequency Domain
Basic Operations
17CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
18. Filtering in the Frequency Domain
2-D Fourier Transform
18CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
19. Filtering in the Frequency Domain
2-D Fourier Transform:
19CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
20. Filtering in the Frequency Domain
Notch filter
20CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
21. Filtering in the Frequency Domain
Ideal Low-pass Filter (ILPF)
cutoff
frequency
21CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
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. Filtering in the Frequency Domain
Distribution of the power spectrum:
23CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
24. Filtering in the Frequency Domain
Filtering with power cutoff
24CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
25. Filtering in the Frequency Domain
Butterworth Low-pass Filter
25CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
26. Filtering in the Frequency Domain
Butterworth Low-pass Filter
26CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
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. Filtering in the Frequency Domain
Gaussian Low-pass Filter
28CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
29. Filtering in the Frequency Domain
Gaussian Low-pass and High-pass filters:
29CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
30. Filtering in the Frequency Domain
Gaussian Low-pass filters:
30CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
31. Filtering in the Frequency Domain
Gaussian Low-pass filters:
31CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
32. Filtering in the Frequency Domain
Ideal High-pass filters:
32CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
33. Filtering in the Frequency Domain
Ideal High-pass filters:
33CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
34. Filtering in the Frequency Domain
Ideal High-pass filters:
34CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
35. Filtering in the Frequency Domain
Gaussian High-pass filters:
35CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
36. Filtering in the Frequency Domain
High-pass filters:
36CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
37. Filtering in the Frequency Domain
High-pass filters:
37CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
38. Filtering in the Frequency Domain
Ideal Band-Pass Filter
38CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
39. Filtering in the Frequency Domain
The Laplacian filters:
39CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
40. Filtering in the Frequency Domain
Gaussian High-pass filters:
40CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
41. Filtering in the Frequency Domain
Homomorphic filters:
41CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
42. Filtering in the Frequency Domain
Homomorphic filters:
42CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
43. Filtering in the Frequency Domain
Homomorphic filters:
43CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.
44. HW3
4.9 and 4.12
44CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.