This document discusses image processing in the frequency domain using the Fourier transform. It explains that image enhancement can be performed by designing a transfer function in the frequency domain and multiplying it with the image's Fourier transform. Filtering an image corresponds to multiplying its Fourier transform by a filter transfer function. Common filters discussed include low-pass filters for smoothing and high-pass filters for sharpening. Ideal filters have abrupt cutoffs which cause ringing artifacts, while Butterworth and Gaussian filters provide smoother responses.

1.  
Prof. Duong Anh Duc

2.  The concept of filtering is easier to visualize in the frequency domain.
Therefore, enhancement of image f(m,n) can be done in the frequency
domain, based on its DFT F(u,v) .
 This is particularly useful, if the spatial extent of the point-spread
sequence h(m,n) is large. In this case, the convolution
g(m,n) = h(m,n)*f(m,n)
may be computationally unattractive.
2
Enhanced
Image
PSS
Given Image

3.  We can therefore directly design a transfer function H(u,v) and
implement the enhancement in the frequency domain as follows:
G(u,v) = H(u,v)*F(u,v)
3
Enhanced
Image
Transfer Function
Given Image

4.  Given a 1-d sequence s[k], k = {…,-1,0,1,2,…,}
 Fourier transform
 Fourier transform is periodic with 2
 Inverse Fourier transform
4

5.  How is the Fourier transform of a sequence s[k] related to the Fourier
transform of the continuous signal
 Continuous-time Fourier transform
5

6.  Given a 2-d matrix of image samples
s[m,n], m,n Z2
 Fourier transform
 Fourier transform is 2 -periodic both in x and y
 Inverse Fourier transform
6

7.  How is the Fourier transform of a sequence s[m,n] related to the
Fourier transform of the continuous signal
 Continuous-space 2D Fourier transform
7

8. 8
|F(u,v)| displayed as imagef(x,y)

9. 9
|F(u,v)| displayed in 3-D

10. 10
Image Magnitude Spectrum

11. 11
Image Magnitude Spectrum

12. 12
Image Magnitude Spectrum

13.  As the size of the box increases in spatial domain, the corresponding
“size” in the frequency domain decreases.
13

14. 14
|F(u,v)|f(x,y)

15. 15
F(u,v)

16. 16
|G(u,v)|g(x,y)

17. 17
G(u,v)

18.  Image formed from magnitude
spectrum of Rice and phase
spectrum of Camera man
18

19.  Image formed from magnitude
spectrum of Camera man and
phase spectrum of Rice
19

20.  For discrete images of finite extent, the analogous Fourier transform is
the DFT.
 We will first study this for the 1-D case, which is easier to visualize.
 Suppose { f(0), f(1), …, f(N – 1)} is a sequence/ vector/1-D image
of length N. Its N-point DFT is defined as
 Inverse DFT (note the normalization):
20

21.  Example: Let f(n) = {1, -1 ,2,3 } (Note that N=4)
21

22.  F(u) is complex even though f(n) is real. This is typical.
 Implementing the DFT directly requires O(N2) computations, where N
is the length of the sequence.
 There is a much more efficient implementation of the DFT using the
Fast Fourier Transform (FFT) algorithm. This is not a new transform (as
the name suggests) but just an efficient algorithm to compute the DFT.
22

23.  The FFT works best when N = 2m (or is the power of some integer
base/radix). The radix-2 algorithm is most commonly used.
 The computational complexity of the radix-2 FFT algorithm is Nlog(N)
adds and ½Nlog(N) multiplies. So it is an Nlog(N) algorithm.
 In MATLAB, the command fft implements this algorithm (for 1-D
case).
23

24.  The Fourier transform is suitable for continuous-domain images, which
maybe of infinite extent.
 For discrete images of finite extent, the analogous Fourier transform is
the 2-D DFT.
24

25.  Suppose f(m,n), m = 0,1,2,…M – 1, n = 0,1,2,…N – 1, is a discrete
N M image. Its 2-D DFT F(u,v) is defined as:
 Inverse DFT is defined as:
25

26.  For discrete images of finite extent, the analogous Fourier transform is
the 2-D DFT.
 Note about normalization: The normalization by MN is different than
that in text. We will use the one above since it is more widely used. The
Matlab function fft2 implements the DFT as defined above.
26

27.  Most often we have M=N (square image) and in that case, we define a
unitary DFT as follows:
 We will refer to the above as just DFT (drop unitary) for simplicity.
27

28. 28

29. 29
In matlab, if f and h are matrices representing two images,
conv2(f, h) gives the 2D-convolution of images f and h.

30.  Linearity (Distributivity and Scaling): This holds inboth discrete and
continuous-domains.
o DFT of the sum of two images is the sum of their individual DFTs.
o DFT of a scaled image is the DFT of the original image scaled by the same
factor.
30

31.  Spatial scaling (only for continuous-domain):
o If a, b > 1, image “shrinks” and the spectrum “expands.”
31

32.  Periodicity (only for discrete case): The DFT and its inverse are
periodic (in both the dimensions), with period N.
F(u,v) = F(u+N,v) = F(u,v+N) = F(u+N,v+N)
o Similarly,
is also N-periodic in m and n.
32

33.  Separability (both continuous and discrete): Decomposition of 2D DFT
into 1D DFTs
33

34. o Similarly,
34

35.  Convolution: In continuous-space, Fourier transform of the convolution
is the product of the Four transforms.
F[f(x,y)*h(x,y)] = F(u,v) H(u,v)
So if
g(x,y) = f(x,y)*h(x,y)
is the output of an LTI transformation with PSF h(x,y) to an input image
f(x,y), then
G(u,v) = F(u,v)*H(u,v)
35

36. o In other words, output spectrum G(u,v) is the product of the input
spectrum F(u,v) and the transfer function H(u,v).
o So the FT can be used as a computational tool to simplify the
convolution operation.
36

37.  Correlation: In continuous-space, correlation between two images
f(x,y) and h(x,y) is defined as:
 Therefore,
37

38.  rff(x,y) is usually called the auto-correlation of image f(x,y) (with
itself) and rff(x,y) is called the crosscorrelation between f(x,y) and
h(x,y).
 Roughly speaking, rfh(x,y) measures the degree of similarity between
images f(x,y) and h(x,y). Large values of rfh(x,y) would indicate that
the images are very similar.
38

39.  This is usually used in template matching, where h(x,y) is a template
shape whose presence we want to detect in the image f(x,y).
 Locations where rfh(x,y) is high (peaks of the crosscorrelation
function) are most likely to be the location of shape h(x,y) in image
f(x,y).
39

40.  Convolution property for discrete images: Suppose
f(m,n), m = 0,1,2,…M–1, n = 0,1,2,…N–1 is an N M image and
h(m,n), m = 0,1,2,…K–1, n = 0,1,2,…L–1 is an N M image.
then
g(m,n) = f(m,n)*h(m,n) is a (M+K–1) (N+L–1) image.
40

41. So if we want a convolution property for discrete images --- something
like
g(m,n) = f(m,n)*h(m,n)
we need to have G(u, v) to be of size (M+K–1) (N+L–1) (since
g(m, n) has that dimension).
Therefore, we should require that F(u, v) and H(u, v) also have the
same dimension, i.e. (M+K–1) (N+L–1)
41

42. So we zero-pad the images f(m, n), h(m, n), so that they are of size
(M+K–1 ) (N+L–1). Let fe(m,n) and he(m,n) be the zero-padded
(or extended images).
Take their 2D-DFTs to obtain F(u, v) and H(u, v), each of size
(M+K–1) (N+L– 1). Then
 Similar comments hold for correlation of discrete images as well.
42

43.  Translation: (discrete and continuous case):
Note that
so f(m, n) and f(m–m0, n–n0) have the same magnitude spectrum
but different phase spectrum.
Similarly,
43

44.  Conjugate Symmetry: If f(m, n) is real, then F(u, v) is conjugate
symmetric, i.e.
 Therefore, we usually display F(u–N/2,v–N/2), instead of F(u, v),
since it is easier to visualize the symmetry of the spectrum in this case.
 This is done in Matlab using the fftshift command.
44

45.  Multiplication: (In continuous-domain) This is the dual of the
convolution property. Multiplication of two images corresponds to
convolving their spectra.
F[f(x,y)h(x,y)] = F(u,v) H(u,v)
45

46. 46
f(m,n)

47. 47
|F(u–N/2,v–N/2)||F(u,v)|

48.  Average value: The average pixel value in an image:
 Notice that (substitute u = v = 0 in the definition):
48

49.  Differentiation: (Only in continuous-domain): Derivatives are normally
used for detecting edged in an image. An edge is the boundary of an
object and denotes an abrupt change in grayvalue. Hence it is a region
with high value of derivative.
49

50. 50

51. 51

52. 52

53.  Edges and sharp transitions in grayvalues in an image contribute
significantly to high-frequency content of its Fourier transform.
 Regions of relatively uniform grayvalues in an image contribute to low-
frequency content of its Fourier transform.
 Hence, an image can be smoothed in the Frequency domain by
attenuating the high-frequency content of its Fourier transform.
This would be a lowpass filter!
53

54. 54

55.  For simplicity, we will consider only those filters that are real and
radially symmetric.
 An ideal lowpass filter with cutoff frequency r0:
55

56.  Note that the origin (0, 0) is at the center and not the corner of the
image (recall the “fftshift” operation).
 The abrupt transition from 1 to 0 of the transfer function H(u,v) cannot
be realized in practice, using electronic components. However, it can
be simulated on a computer.
56
Ideal LPF with r0 = 57

57. 57
Ideal LPF with r0 = 57Original Image

58. 58
Ideal LPF with r0 = 26Ideal LPF with r0= 36

59.  Notice the severe ringing effect in the blurred images, which is a
characteristic of ideal filters. It is due to the discontinuity in the filter
transfer function.
59

60.  The cutoff frequency r0 of the ideal LPF determines the amount of
frequency components passed by the filter.
 Smaller the value of r0, more the number of image components
eliminated by the filter.
 In general, the value of r0 is chosen such that most components of
interest are passed through, while most components not of interest are
eliminated.
 Usually, this is a set of conflicting requirements. We will see some
details of this is image restoration
 A useful way to establish a set of standard cut-off frequencies is to
compute circles which enclose a specified fraction of the total image
power.
60

61.  Suppose
where is the total image power.
 Consider a circle of radius =r0(a) as a cutoff frequency with respect to
a threshold a such that
 We can then fix a threshold a and obtain an appropriate cutoff
frequency r0(a) .
61

62.  A two-dimensional Butterworth lowpass filter has transfer function:
n: filter order, r0: cutoff frequency
62

63. 63

64. 64

65.  Frequency response does not have a sharp transition as in the ideal
LPF.
 This is more appropriate for image smoothing than the ideal LPF, since
this not introduce ringing.
65

66. 66
LPF with r0= 18Original Image

67. 67
LPF with r0= 10LPF with r0= 13

68. 68
Image with false contouring
due to insufficient bits used
for quantization
Lowpass filtered version of
previous image

69. 69
Original Image Noisy Image

70. 70
LPF Image

71.  The form of a Gaussian lowpass filter in two-dimensions is given by
where
is the distance from the origin in the frequency plane.
 The parameter s measures the spread or dispersion of the Gaussian
curve. Larger the value of s, larger the cutoff frequency and milder the
filtering.
 When s = D(u, v), the filter is down to 0.607 of its maximum value of
1.
71
22
, vuvuD
22
2,
, vuD
evuH

72. 72

73.  Edges and sharp transitions in grayvalues in an image contribute
significantly to high-frequency content of its Fourier transform.
 Regions of relatively uniform grayvalues in an image contribute to low-
frequency content of its Fourier transform.
 Hence, image sharpening in the Frequency domain can be done by
attenuating the low-frequency content of its Fourier transform. This
would be a highpass filter!
73

74.  For simplicity, we will consider only those filters that are real and
radially symmetric.
 An ideal highpass filter with cutoff frequency r0:
74

75.  Note that the origin (0, 0) is at the center and not the corner of the
image (recall the “fftshift” operation).
 The abrupt transition from 1 to 0 of the transfer function H(u,v) cannot
be realized in practice, using electronic components. However, it can
be simulated on a computer.
75
Ideal HPF with r0= 36

76. 76
Ideal HPF with r0= 18Original Image

77. 77
Ideal HPF with r0= 26Ideal HPF with r0= 36

78.  Notice the severe ringing effect in the output images, which is a
characteristic of ideal filters. It is due to the discontinuity in the filter
transfer function.
78

79.  A two-dimensional Butterworth highpass filter has transfer function:
n: filter order, r0: cutoff frequency
79

80. 80

81.  Frequency response does not have a sharp transition as in the ideal
HPF.
 This is more appropriate for image sharpening than the ideal HPF,
since this not introduce ringing
81

82. 82
HPF with r0= 47Original Image

83. 83
HPF with r0= 81HPF with r0= 36

84.  The form of a Gaussian lowpass filter in two-dimensions is given by
where
is the distance from the origin in the frequency plane.
 The parameter s measures the spread or dispersion of the Gaussian
curve. Larger the value of s, larger the cutoff frequency and more
severe the filtering.
84
22
, vuvuD
22
2,
1, vuD
evuH

85. 85

86. 86