1. Methods for Digital Image Processing

2. Basic ideas of Image Transforms

3. Spatial Frequency or Fourier Transform
Jean Baptiste Joseph Fourier

4. Why are Spatial Frequencies important?
Efficient data representation
Provides a means for modeling and removing noise
Physical processes are often best described in “frequency domain”
Provides a powerful means of image analysis

5. What is spatial frequency?
Instead of describing a function (i.e., a shape) by a series of positions
It is described by a series of cosines

6. What is spatial frequency?
g(x) = A cos(x)
g(x) 2 A x

7. What is spatial frequency?
A cos(x  2/L)
g(x) = A cos(x  2/)
A cos(x  2f)
g(x)
Period (L)
Wavelength ()
Frequency f=(1/ )
Amplitude (A)
Magnitude (A) x

8. What is spatial frequency?
g(x) = A cos(x  2f)
g(x) A x
(1/f)
period

9. But what if cosine is shifted in phase?
g(x) = A cos(x  2f + )
g(x) x 

10. What is spatial frequency?
Let us take arbitrary g(x)
x g(x)
cos(0.25) =
cos(0.50) = 0.0
cos(0.75) =
cos(1.00) = -1.0
cos(1.25) = …
cos(1.50) = 0
cos(1.75) =
cos(2.00) = 1.0
cos(2.25) =
g(x) = A cos(x  2f + )
A=2 m
f = 0.5 m-1
= 0.25 = 45
g(x) = 2 cos(x  2(0.5) )
2 cos(x   )
We calculate discrete values of g(x) for various values of x
We substitute values of A, f and 

11. What is spatial frequency?
g(x) = A cos(x  2f + )
g(x)
We calculate discrete values of g(x) for various values of x x

12. Now we take discrete values of Ai , fi and i
gi(x) = Ai cos(x  2fi + i), i = 0,1,2,3,... x

13. Now we substitute fi = i/N
gi(x) = Ai cos(x  2fi + i), i = 0,1,2,3,...
gi(x) = Ai cos(x  2i/N + i), i = 0,1,2,3,…,N-1
f=i/N N
N = time interval

14. Values for various values of i
gi(x) = Ai cos(x  2i/N + i), i = 0,1,2,3,…,N-1
f=i/N
We calculate values of function for various values of i N

15. gi(x) = Ai cos(x  2i/N + i), i = 0,1,2,3,…,N-1
Substituting various values of i to the formula we get various cosinusoides
gi(x) = Ai cos(x  2i/N + i), i = 0,1,2,3,…,N-1 A2 A1 A0
i=1
i=2
i=0

16. gi(x) = Ai cos(x  2i/N + i), i = 0,1,2,3,…,N/2 - 1
Changing N to N/2
gi(x) = Ai cos(x  2i/N + i), i = 0,1,2,3,…,N/2 - 1
If N equals the number of pixel in a line, then...
i=0
i=N/2 - 1
Lowest frequency Highest frequency

17. What is spatial frequency?
gi(x) = Ai cos(x  2i/N + i), i = 0,1,2,3,…,N/2-1
If N equals the number of pixels in a line, then...
i=0
i=N/2-1
Lowest frequency Highest frequency

18. What will happen if we take N/2?
gi(x) = Ai cos(x  2i/N + i), i = 0,1,2,3,…,N/2-1
If N equals the number of pixel in a line, then...
i=0
i=N/2
Lowest frequency Too high
Redundant frequency

19. What is spatial frequency?
g(x) = A cos(x  2f + )
gi(x) = Ai cos(x  2i/N + i), i = 0,1,2,3,…,N/2-1

22. We try to approximate a periodic function with standard trivial (orthogonal, base) functions
Low frequency
Medium frequency + = +
High frequency

23. We add values from component functions point by point+ = +

24. g(x)
i=1
i=2
i=3
i=4
i=5
i=63 x
Example of periodic function created by summing standard trivial functions
127

25. g(x)
i=1
i=2
i=3
i=4
i=5
i=10 x
127
Example of periodic function created by summing standard trivial functions

26. 64 terms
g(x)
10 terms
g(x)
Example of periodic function created by summing standard trivial functions

27. Fourier Decomposition of a step function (64 terms)
g(x)
i=1
i=2
i=3
i=4
i=5
Example of periodic function created by summing standard trivial functions x
i=63
127

28. Fourier Decomposition of a step function (11 terms)
g(x)
i=1
i=2
i=3
Example of periodic function created by summing standard trivial functions
i=4
i=5
i=10 x 63

29. Main concept – summation of base functions
Any function of x (any shape) that can be represented by g(x) can also be represented by the summation of cosine functions
Observe two numbers for every i

30. Information is not lost when we change the domain
Spatial Domain
gi(x) = 1.3, 2.1, 1.4, 5.7, …., i=0,1,2…N-1
N pieces of information
Frequency Domain
N pieces of information
N/2 amplitudes (Ai, i=0,1,…,N/2-1) and
N/2 phases (i, i=0,1,…,N/2-1) and

31. What is spatial frequency?
Information is not lost when we change the domain
What is spatial frequency?
gi(x)
and
Are equivalent
They contain the same amount of information
The sequence of amplitudes squared is the SPECTRUM

32. EXAMPLE

33. Substitute values
A cos(x2i/N)
frequency (f) = i/N
wavelength (p) = N/I
N=512
i f p
infinite / /
256 1/
Assuming N we get this table which relates frequency and wavelength of component functions

34. More examples to give you some intuition….

35. Fourier Transform Notation
g(x) denotes an spatial domain function of real numbers
(1.2, 0.0), (2.1, 0.0), (3.1,0.0), …
G() denotes the Fourier transform
G() is a symmetric complex function
(-3.1,0.0), (4.1, -2.1), (-3.1, 2.1), …(1.2,0.0) …, (-3.1,-2.1), (4.1, 2.1), (-3.1,0.0)
G[g(x)] = G(f) is the Fourier transform of g(x)
G-1() denotes the inverse Fourier transform
G-1(G(f)) = g(x)

36. Power Spectrum and Phase Spectrum
complex
Complex conjugate
|G(f)|2 = G(f)G(f)* is the power spectrum of G(f)
(-3.1,0.0), (4.1, -2.1), (-3.1, 2.1), … (1.2,0.0),…, (-3.1,-2.1), (4.1, 2.1)
9.61, , 14.02, …, 1.44,…, 14.02, 21.22
tan-1[Im(G(f))/Re(G(f))] is the phase spectrum of G(f)
0.0, , , …, 0.0, , 27.12

37. 1-D DFT and IDFT Discrete Domains Discrete Fourier Transform
Equal time intervals
Discrete Domains
Discrete Time: k = 0, 1, 2, 3, …………, N-1
Discrete Frequency: n = 0, 1, 2, 3, …………, N-1
Discrete Fourier Transform
Inverse DFT
Equal frequency intervals
n = 0, 1, 2,….., N-1
k = 0, 1, 2,….., N-1

38. Fourier 2D Image Transform

39. Another formula for Two-Dimensional Fourier
Image is function of x and y
A cos(x2i/N) B cos(y2j/M)
fx = u = i/N, fy = v =j/M
Lines in the figure correspond to real value 1
Now we need two cosinusoids for each point, one for x and one for y
Now we have waves in two directions and they have frequencies and amplitudes

40. Fourier Transform of a spot
Original image
Fourier Transform

41. Transform Results
image
transform
spectrum

42. Two Dimensional Fast Fourier in Matlab

43. Filtering in Frequency Domain
… will be covered in a separate lecture on spectral approaches…..

44. H(u,v) for various values of u and v
These are standard trivial functions to compose the image from

46. <
<
image
..and its spectrum

47. Image and its spectrum

48. Image and its spectrum

49. Image and its spectrum

50. This is a very important result
Convolution Theorem
Let g(u,v) be the kernel
Let h(u,v) be the image
G(k,l) = DFT[g(u,v)]
H(k,l) = DFT[h(u,v)]
Then
This is a very important result
where means multiplication
and means convolution.
This means that an image can be filtered in the Spatial Domain or the Frequency Domain.

51. Convolution Theorem Let g(u,v) be the kernel Let h(u,v) be the image
G(k,l) = DFT[g(u,v)]
H(k,l) = DFT[h(u,v)]
Then
Instead of doing convolution
in spatial domain we can do multiplication
In frequency domain
Multiplication in spectral domain
Convolution in spatial domain
where means multiplication
and means convolution.

52. v
Image u
Spectrum
Noise and its spectrum
Noise filtering

53. Image
Spectrum v u

54. Image x(u,v) v u
Spectrum log(X(k,l)) l k

55. Image of cow with noise
Spectrum log(X(k,l)) k l v u
Image x(u,v)

56. white noise
white noise spectrum
kernel spectrum (low pass filter)
red noise
red noise spectrum

57. Filtering is done in spectral domain. Can be very complicated

58. Image Transforms Fast Fourier Fast Cosine Radon Transform Slant
2-D Discrete Fourier Transform
Fast Cosine
2-D Discrete Cosine Transform
Radon Transform
Slant
Walsh, Hadamard, Paley, Karczmarz
Haar
Chrestenson
Reed-Muller

59. Discrete Cosine Transform (DCT)
Used in JPEG and MPEG
Another Frequency Transform, with Different Set of Basis Functions

60. Discrete Cosine Transform in Matlab

61. “Statistical” Filters
Median Filter also eliminates noise
preserves edges better than blurring
Sorts values in a region and finds the median
region size and shape
how define the median for color values?

62. “Statistical” Filters Continued
Minimum Filter (Thinning)
Maximum Filter (Growing)
“Pixellate” Functions
Now we can do this quickly in spectral domain

63. Thinning
Growing

64. Pixellate Examples

65. DCT Features
Can be used for face recognition, tell my story from Japan.
Fringe Pattern
DCT Coefficients
DCT
Zonal Mask 1 2 3 4 5
(1,1)
(1,2)
(2,1)
(2,2) .
Artificial
Neural
Network
Feature
Vector

66. Noise Removal Transforms for Noise Removal
Image with Noise Transform Image reconstructed

67. Image Segmentation Recall: Edge Detection
f(x,y)
Gradient
Mask
fe(x,y) -1 -2 1 2
Now we do this in spectral domain!!

68. Image Moments
2-D continuous function f(x,y), the moment of order (p+q) is:
Central moment of order (p+q) is:

69. Image Moments (contd.) Now we do this in spectral domain!!
Normalized central moment of order (p+q) is:
A set of seven invariant moments can be derived from gpq
Now we do this in spectral domain!!

70. Image Textures Now we do this in spectral domain!!
Grass Sand Brick wall
Now we do this in spectral domain!!
The USC-SIPI Image Database

71. Problems
There is a lot of Fourier and Cosine Transform software on the web, find one and apply it to remove some kind of noise from robot images from FAB building.
Read about Walsh transform and think what kind of advantages it may have over Fourier
Read about Haar and Reed-Muller transform and implement them. Experiment

72. Sources Howard Schultz, Umass Herculano De Biasi Shreekanth Mandayam
ECE Department, Rowan University

73. Please visit the website
Image Compression
Please visit the website