1. Digital Image Processing , 2008
Chapter 4:
Image Enhancement in the
Frequency Domain
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Update: Monday, April 11, 2016

2. Background: Fourier Series
Any periodic signals can be
viewed as weighted sum
of sinusoidal signals with
different frequencies
Frequency Domain:
view frequency as an
independent variable
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3. Fourier Tr. and Frequency Domain
Time, spatial
Domain
Signals
Frequency
Domain
Signals
Inv Fourier Tr.
1-D, Continuous case
Fourier Tr.:
Inv. Fourier Tr.:
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4. Fourier Tr. and Frequency Domain (cont.)
1-D, Discrete case
Fourier Tr.:
u = 0,…,M-1
Inv. Fourier Tr.:
x = 0,…,M-1
F(u) can be written as or
where
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5. Example of 1-D Fourier Transforms
Notice that the longer
the time domain signal,
The shorter its Fourier
transform
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6. Relation Between Dx and Du
For a signal f(x) with M points, let spatial resolution Dx be space
between samples in f(x) and let frequency resolution Du be space
between frequencies components in F(u), we have
Example: for a signal f(x) with sampling period 0.5 sec, 100 point,
we will get frequency resolution equal to
This means that in F(u) we can distinguish 2 frequencies that are
apart by 0.02 Hertz or more.
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7. 2-Dimensional Discrete Fourier Transform
For an image of size MxN pixels
2-D DFT
u = frequency in x direction, u = 0 ,…, M-1
v = frequency in y direction, v = 0 ,…, N-1
2-D IDFT
x = 0 ,…, M-1
y = 0 ,…, N-1
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8. 2-Dimensional Discrete Fourier Transform (cont.)
F(u,v) can be written as or
where
For the purpose of viewing, we usually display only the
Magnitude part of F(u,v)
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9. 2-D DFT Properties
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10. 2-D DFT Properties (cont.)
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11. 2-D DFT Properties (cont.)
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12. 2-D DFT Properties (cont.)
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13. Computational Advantage of FFT Compared to DFT
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14. Relation Between Spatial and Frequency Resolutions
where
Dx = spatial resolution in x direction
Dy = spatial resolution in y direction
Du = frequency resolution in x direction
Dv = frequency resolution in y direction
N,M = image width and height
( Dx and Dy are pixel width and height. )
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15. How to Perform 2-D DFT by Using 1-D DFT
f(x,y)
1-D
DFT
by row
F(u,y)
1-D DFT
by column
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F(u,v)

16. How to Perform 2-D DFT by Using 1-D DFT (cont.)
f(x,y)
Alternative method
1-D DFT
by column
1-D
DFT
by row
F(x,v)
F(u,v)
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17. Periodicity of 1-D DFT
From DFT: -N 2N N
We display only in this range
DFT repeats itself every N points (Period = N) but we usually
display it for n = 0 ,…, N-1
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18. Conventional Display for 1-D DFT
f(x)
DFT
N-1
N-1
Time Domain Signal
High frequency
area
Low frequency
area
The graph F(u) is not
easy to understand !
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19. Conventional Display for DFT : FFT Shift
FFT Shift: Shift center of the
graph F(u) to 0 to get better
Display which is easier to
understand.
N-1
High frequency area
Low frequency area
-N/2
N/2-1
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20. Periodicity of 2-D DFT
2-D DFT:
g(x,y) -M
For an image of size NxM pixels, its 2-D DFT repeats itself every N points in x-direction and every M points in y-direction. M
We display only
in this range 2M
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21. Conventional Display for 2-D DFT
F(u,v) has low frequency areas
at corners of the image while high
frequency areas are at the center
of the image which is inconvenient
to interpret.
High frequency area
Low frequency area
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22. 2-D FFT Shift : Better Display of 2-D DFT
2-D FFT Shift is a MATLAB function: Shift the zero frequency
of F(u,v) to the center of an image.
2D FFTSHIFT
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High frequency area
Low frequency area

23. 2-D FFT Shift (cont.) : How it works-M
Display of 2D DFT
After FFT Shift M
Original display
of 2D DFT 2M -N N 2N
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24. Example of 2-D DFT Notice that the longer the time domain signal,
The shorter its Fourier transform
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25. Example of 2-D DFT
Notice that direction of an object in spatial image and
Its Fourier transform are orthogonal to each other.
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26. Examples of 2DFT Image Fourier transform a a b b c c
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27. Example of 2-D DFT 2D DFT Original image 2D FFT Shift
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28. Example of 2-D DFT 2D DFT Original image 2D FFT Shift
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29. Basic Concept of Filtering in the Frequency Domain
From Fourier Transform Property:
We cam perform filtering process by using
Multiplication in the frequency domain
is easier than convolution in the spatial
Domain.
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30. Filtering in the Frequency Domain with FFT shift
F(u,v)
H(u,v)
(User defined)
g(x,y)
FFT shift X
2D IFFT
2D FFT
FFT shift
f(x,y)
G(u,v)
In this case, F(u,v) and H(u,v) must have the same size and
have the zero frequency at the center.
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31. Multiplication in Freq. Domain = Circular Convolution
f(x)
DFT
F(u)
G(u) = F(u)H(u)
IDFT
g(x)
h(x)
DFT
H(u)
Multiplication of
DFTs of 2 signals
is equivalent to
perform circular
convolution
in the spatial domain.
f(x)
h(x)
“Wrap around” effect
g(x)
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32. Multiplication in Freq. Domain = Circular Convolution
H(u,v)
Gaussian
Lowpass
Filter with
D0 = 5
Original
image
Filtered image
(obtained using
circular convolution)
Incorrect areas at image rims
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33. f(x) Zero padding DFT F(u) G(u) = F(u)H(u) h(x) Zero padding DFT H(u)
Linear Convolution by using Circular Convolution and Zero Padding
f(x)
Zero padding
DFT
F(u)
G(u) = F(u)H(u)
h(x)
Zero padding
DFT
H(u)
IDFT
Concatenation
g(x)
Padding zeros
Before DFT
Keep only this part
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34. Linear Convolution by using Circular Convolution and Zero Padding
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35. Linear Convolution by using Circular Convolution and Zero Padding
Filtered image
Zero padding area in the spatial
Domain of the mask image
(the ideal lowpass filter)
Only this area is kept.
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36. Filtering in the Frequency Domain : Example
In this example, we set F(0,0) to zero
which means that the zero frequency
component is removed.
Note: Zero frequency = average
intensity of an image
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37. Filtering in the Frequency Domain : Example
Lowpass Filter
Highpass Filter
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38. Filtering in the Frequency Domain : Example (cont.)
Result of Sharpening Filter
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39. Filter Masks and Their Fourier Transforms
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40. Ideal Lowpass Filter Ideal LPF Filter Transfer function
where D(u,v) = Distance from (u,v) to the center of the mask.
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41. Examples of Ideal Lowpass Filters
The smaller D0, the more high frequency components are removed.
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42. Results of Ideal Lowpass Filters
Ringing effect can be
obviously seen!
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43. How ringing effect happens
Surface Plot
Ideal Lowpass Filter
with D0 = 5
Abrupt change in the amplitude
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44. Spatial Response of Ideal
How ringing effect happens (cont.)
Surface Plot
Spatial Response of Ideal
Lowpass Filter with D0 = 5
Ripples that cause ringing effect
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45. How ringing effect happens (cont.)
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46. Butterworth Lowpass Filter
Transfer function
Where D0 = Cut off frequency, N = filter order.
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47. Results of Butterworth Lowpass Filters
There is less ringing
effect compared to
those of ideal lowpass
filters!
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48. Spatial Masks of the Butterworth Lowpass Filters
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Some ripples can be seen.

49. Gaussian Lowpass Filter Transfer function
Where D0 = spread factor.
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
Note: the Gaussian filter is the only filter that has no ripple and
hence no ringing effect.
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50. Gaussian Lowpass Filter (cont.)
filter with D0 = 5
Spatial respones of the
Gaussian lowpass filter
with D0 = 5
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Gaussian shape

51. Results of Gaussian Lowpass Filters
No ringing effect!
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52. Application of Gaussian Lowpass Filters
Original image
Better Looking
The GLPF can be used to remove jagged edges and “repair” broken characters.
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53. Application of Gaussian Lowpass Filters (cont.)
Remove wrinkles
Original image
Softer-Looking
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54. Application of Gaussian Lowpass Filters (cont.)
Original image : The gulf of Mexico and
Florida from NOAA satellite.
Filtered image
Remove artifact lines: this is a simple but crude way to do it!
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55. Hhp = 1 - Hlp Highpass Filters
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56. Ideal Highpass Filters
Ideal LPF Filter Transfer function
where D(u,v) = Distance from (u,v) to the center of the mask.
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57. Butterworth Highpass Filters Transfer function
Where D0 = Cut off frequency, N = filter order.
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58. Gaussian Highpass Filters Transfer function
Where D0 = spread factor.
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59. Gaussian Highpass Filters (cont.)
filter with D0 = 5
Spatial respones of the
Gaussian highpass filter
with D0 = 5
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60. Spatial Responses of Highpass Filters
Ripples
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61. Results of Ideal Highpass Filters
Ringing effect can be
obviously seen!
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62. Results of Butterworth Highpass Filters
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63. Results of Gaussian Highpass Filters
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64. Laplacian Filter in the Frequency Domain
From Fourier Tr. Property:
Then for Laplacian operator
We get
Image of
–(u2+v2)
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Surface plot

65. Laplacian Filter in the Frequency Domain (cont.)
Spatial response of –(u2+v2)
Cross section
Laplacian mask in Chapter 3
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66. Sharpening Filtering in the Frequency Domain
Spatial Domain
Frequency Domain Filter
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67. Sharpening Filtering in the Frequency Domain (cont.)
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68. Sharpening Filtering in the Frequency Domain (cont.)
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69. High Frequency Emphasis Filtering
Butterworth
highpass
filtered
image
Original
High freq. emphasis
filtered image
After
Hist
Eq.
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a = 0.5, b = 2

70. Homomorphic Filtering
An image can be expressed as
i(x,y) = illumination component
r(x,y) = reflectance component
We need to suppress effect of illumination that cause image
Intensity changed slowly.
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71. Homomorphic Filtering
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72. Homomorphic Filtering
More details in the room can be seen!
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73. Correlation Application: Object Detection
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