1. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
Joseph Fourier was born in 1768 (France)
“the analytic theory of heat”, 1822(his book)
Fourier’s contribution <<any function that periodically repeats itself can be expressed as the sum of sines/or cosines of different frequencies (Fourier series)>>
<<even functions that are not periodic (but whose area under the curve is finite) can be expressed as integral of sines and or cosines (Fourier transform)

2. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain

3. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
4.2. Introduction to the Fourier Transform and the Frequency domain
the One-Dimensional Fourier Transform and its inverse

4. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
روابط يک بعدي به سادگي قابل تعميم به دو بعد هستند

5. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
تبديل فوريه يک تابع گسسته يک بعدي از رابطه زير حاصل مي شود.
يکي از خصوصيات مهم تبديل فوريه گسسته اين است که همواره موجود است و بر خلاف حالت پيوسته وجود آن به شرط خاصي نياز ندارد.

6. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
تبديل فوريه يک تابع گسسته
(رابطه اويلر)
نکته مهم: براي محاسبه هر ترم از تبديل فوريه تمام مقادير f دخالت دارند.
تبديل فوريه به يک منشور رياضي شباهت دارد که يک سيگنال را به مولفه هايش تفکيک مي کند.

7. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
مولفه هاي تبديل فوريه مختلط هستند
F(u) in polar coordinate
Magnitude or spectrum of FT
Phase angel or phase spectrum of FT
Power spectrum or spectral density

8. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
Example 4.1:
M=1024
K=16
A=1

9. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
Two-Dimensional DFT and its inverse

10. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
Two-Dimensional DFT and its inverse
Magnitude or spectrum of FT
Phase angel of FT
Power spectrum

11. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
Two-Dimensional DFT and its inverse
Due to the properties of exponentials, it is not difficult to show
اين رابطه به اين معني است که مي توان تبديل فوريه يک تصوير را به وسط منتقل کرد
DC component

12. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
براي نمايش بهتر فوريه تصوير از تبديل لگاريتمي استفاده شده است.

13. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
Filtering in the Frequency domain
Some basic properties of the frequency domain

14. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
Basics of filtering in the frequency domain
Multiply the input image by (-1)x+y to center the transform
Compute F(u,v), the DFT of the image
Multiply F(u,v) by a filter function H(u,v)
Compute the inverse DFT
Obtain the real part of the result
Multiply the result by (-1)x+y

15. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain

16. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
Some basic filter and their properties
Notch filter (a constant function with a hole at the origin)
Low-pass filter
High-pass filter

17. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain

18. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain

19. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain

20. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
Correspondence between filtering in the spatial and frequency domains
The most fundamental relationship between the spatial and frequency domains is established by a well-known result called the “convolution theorem”

21. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
براي تکميل بحث به اطلاعات ديگري نيز نياز داريم که در ادامه به آنها مي پردازيم

22. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
براي تکميل بحث به اطلاعات ديگري نيز نياز داريم که در ادامه به آنها مي پردازيم

23. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
LOW PASS FILTER
Let H(u) denote a frequency domain, Gaussian filter function
Corresponding filter in the spatial domain is
1)هر دو تابع گوسي هستند با مقادير حقيقي
2) اگر انحراف معيار تابع گوسي در حوزه فرکانس زياد باشد، انحراف معيار حوزه مکان کم خواهد بود و بر عکس

24. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
HIGH PASS FILTER
Let H(u) denote a frequency domain, Gaussian filter function
Corresponding filter in the spatial domain is

25. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain

26. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
4.3 Smoothing Frequency-Domain Filters
Ideal Low-pass filter(ILPF)

27. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
D0 is Cut-off frequency
فرکانس قطع فيلتر ايده آل توسط ادوات الکترونيک قابل پياده سازي نيست، اما بصورت نرم افزاري مي توان آن را پياده کرد

28. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
D0 is Cut-off frequency
پيدا کردن نقطه قطع با استفاده از طيف قدرت امکان پذير است.
Power Spectrum
به ازاي شعاعهاي متفاوت، مقادير مختلفي براي آلفا بدست مي آيد.

29. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain

30. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain

31. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
استفاده از فيلتر ايده آل پايين گذر بخاطر وجود خواص بلورينگ و Ringing عملي و ميسر نيست. اين خصوصيت در شکل صفحه قبل مشهود است.
در شکل صفحه بعد دليل خصوصيت Ringing به تصوير کشيده شده است.

32. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain

33. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
Butterworth low-pass filters(BLPF)
If H(u,v)=0.5 then D(u,v)=D0
n is order of filter

34. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain

35. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain

36. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
فيلتر باترورث با مرتبه 20، خصوصياتي بسيار نزديک به فيلتر ايده آل دارد.

37. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
Gaussian Low-pass filter(GLPF)
Where D0 is the cutoff frequency. When D(u,v)=D0, the filter is down to of its maximum value.

38. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain

39. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain

40. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain

41. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
4.3.4.Additional Examples of Lowpass filter

42. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
نرم کردن تصوير ماهواره براي پردازش هاي بعدي

43. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
4.4. Sharpening Frequency Domain Filters

44. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain

45. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain

46. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
Ideal High-pass filter (IHPF)

47. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
Butterworth High-pass filter (BHPF)

48. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
Gaussian High-pass filter (GHPF)

49. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
The Laplacian in the Frequency Domain

50. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
The Laplacian in the Frequency Domain

51. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain

52. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
We form an enhanced image g(x,y) by subtracting the laplacian
from the original image:

53. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain

54. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
Unsharp masking, High-Boost filtering, and High Frequency
Emphasis Filtering
Unsharp masking: توليد تصوير شارپ از طريق کم کردن تصوير اصلي از تصوير بلور شده، لذا داريم:
لذا داريم:

55. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
Unsharp masking, High-Boost filtering, and High Frequency
Emphasis Filtering
در حوزه فوريه:
لذا داريم:

56. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
فيلتر high-boost در حوزه فوريه

57. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
High-frequency Emphasis filtering: گاهي اوقات لازم است که مقادير فرکانس بالا با اهميت بيشتري ارتقا يابد. در اين مواقع از اين تکنيک استفاده مي شود.

58. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
4.5. Homomorphic filtering
استفاده از مدل illumination-reflectance مي تواند در حذف نويز از تصاوير با رنج ديناميکي کم موثر باشد. اين فيلتر بر مبناي جداسازي مولفه هاي لومينانس و رفلکتانس و پردازش آنها عمل مي کند.

59. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
4.5. Homomorphic filtering

60. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
4.5. Homomorphic filtering

61. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
4.5. Homomorphic filtering

62. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain
مولفه illumination داراي تغييرات مکاني کم است در حاليکه مولفه reflectance داراي تغييات شديد خصوصا در نواحي نزديک به لبه ها است. با استفاده از اين روش مولفه illumination حذف و مولفه reflectance تقويت مي شود.

63. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain

64. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain

65. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain

66. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain

67. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain

68. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain

69. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain

70. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain

71. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain

72. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain

73. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain

74. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain

75. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain

76. Image Enhancement in the
Chapter 4
Image Enhancement in the
Frequency Domain