1.
Unit 4
Image Restoration
Frequency Domain Filters
(Part III)
Kalyan Acharjya
Jaipur National University, Jaipur
1
Lecture by Kalyan Acharjya

2.
Lecture by Kalyan Acharjya
2
Disclaimer
All images/contents used in this presentation
are copyright of original owner. This PPT is
use for academic purpose only.

3.
Filter
 Filter: A device or material for suppression or
minimizing waves or oscillations of certain
frequencies
 Frequency: The number of times that a periodic
function repeats the same sequence of values
during a unit variation of the independent
variable.
 Filters are classified as (Frequency Domain):
(1) Low-pass (2) High-pass
(3) Band-pass (4) Band-stop ….many more
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Lecture by Kalyan Acharjya

4.
Filters Types
Original signal
Low-pass filtered
High-pass filtered
Band-pass filtered
Band-stop filtered
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Lecture by Kalyan Acharjya

5.
Image Restoration?
 Objective: To restore a degraded/distorted image to its original content
and quality.
 Spatial Domain: g(x,y)=h(x,y)*f(x,y)+ ŋ(x,y)
 Frequency Domain: G(u,v)=H(u,v)F(u,v)+ ŋ(u,v)
 Matrix: G=HF+ŋ
Degradation
Function h
Restoration
Filters
g(x,y)
f(x,y)
ŋ(x,y)
f(x,y)
^
Degradation Restoration
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Lecture by Kalyan Acharjya

6.
Low-Pass Filters (Smoothing filters)
 Preserve Low Frequencies-Useful For Noise Suppression
Frequency Domain Time Domain
Example:
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Lecture by Kalyan Acharjya

7.
High-Pass Filters (Sharpening Filters)
 Preserves High Frequencies - Useful for Edge Detection
Frequency Domain Time Domain
Example:
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Lecture by Kalyan Acharjya

8.
Band-Pass and Band Stop Filters
 Preserves Frequencies Within a Certain Band
Frequency Domain Time Domain
Example:
Band Stop/ Reject
8
Lecture by Kalyan Acharjya

9.
Lecture by Kalyan Acharjya
9
Correction:
Before Integration
1/2pi

10.
Image Processing and Fourier Transform
Lecture by Kalyan Acharjya
10
Input
Image
Fourier
Transform
Do
Operations
Inverse
Fourier
Transform
Fourier Transform:
Inverse Fourier Transform

11.
Fourier Transform
Lecture by Kalyan Acharjya
11

12.
Fourier Spectrum
Lecture by Kalyan Acharjya
12
Percentage of image power enclosed in circles (Small to
Large): 90, 95, 98, 99, 99.5, 99.9

13.
Fourier Transform
Lecture by Kalyan Acharjya
13
f(x,y) F(u,v)
H(u,v)g(x,y)
G(u,v)=F(u,v) • H(u,v)g(x,y) =f(x,y) * h(x,y)

14.
Ideal Low Pass Filters
Lecture by Kalyan Acharjya
14
u
v
H(u,v)
0 D0
1
D(u,v)
H(u,v)
H(u,v) =
1 D(u,v)  D0
0 D(u,v) > D0
D(u,v) =  u2 + v2
D0 = cut off frequency

15.
Blurring-Ideal Low Pass Filter
98.65%
99.37%
99.7%
15Lecture by Kalyan Acharjya

16.
Image Sharpening - High Pass Filter
H(u,v) - Ideal Filter
H(u,v) =
0 D(u,v)  D0
1 D(u,v) > D0
D(u,v) =  u2 + v2
D0 = cut off frequency
0 D0
1
D(u,v)
H(u,v)
u
v
H(u,v)
16Lecture by Kalyan Acharjya

17.
H(u,v)
D(u,v)0 D0
1
D(u,v) =  u2 + v2
High Pass Gaussian Filter
u
v
H(u,v)
H(u,v) = 1 - e
-D2(u,v)/(2D2
0)
e/11 
17Lecture by Kalyan Acharjya

18.
High Pass Filtering - Example
Original High pass Emphasis
High Frequency Emphasis + Histogram Equalization
18Lecture by Kalyan Acharjya

19.
Band Pass Filtering
H(u,v) = 1 D0-  D(u,v)  D0 +
0 D(u,v) > D0 +
D(u,v) =  u2 + v2
D0 = cut off frequency
u
v
H(u,v)
0
1
D(u,v)
H(u,v)
D0- w
2 D0+ w
2D0
0 D(u,v)  D0 -w
2
w
2
w
2
w
2
w = band width
19Lecture by Kalyan Acharjya

20.
Band Reject Filters
 Removing periodic noise form an image involves removing a particular range of
frequencies from that image.
 Band reject filters can be used for this purpose.
 An ideal band reject filter is given as follows:












2
),(1
2
),(
2
0
2
),(1
),(
0
00
0
W
DvuDif
W
DvuD
W
Dif
W
DvuDif
vuH
20
Lecture by Kalyan Acharjya

21.
Band Reject Filters contd..
 The ideal band reject filter is shown below, along with Butterworth
and Gaussian versions of the filter.
Ideal Band
Reject Filter
Butterworth
Band Reject
Filter (of order 1)
Gaussian
Band Reject
Filter
21
Lecture by Kalyan Acharjya

22.
Result of Band Reject Filter
Fig: Corrupted by Sinusoidal Noise Fig: Fourier spectrum of Corrupted Image
Fig: Butterworth Band Reject Filter Fig :Filtered image
22
Lecture by Kalyan Acharjya

23.
Adaptive Filters
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Lecture by Kalyan Acharjya

24.
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Lecture by Kalyan Acharjya

25.
Adaptive Median Filter
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Lecture by Kalyan Acharjya

26.
Algorithm
Objectives:
Remove salt and
pepper (Impulse)
noise
Provide smoothing
Reduce distortion,
such as excessive
thinning or
thickening of object
boundaries
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Lecture by Kalyan Acharjya

27.
Results
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Lecture by Kalyan Acharjya

28.
Band Pass Filters
),(1),( vuHvuH brbp 
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Lecture by Kalyan Acharjya

29.
Notch Filter
Lecture by Kalyan Acharjya
29
Are used to remove repetitive "Spectral" noise
from an image.
Are like a narrow High pass filter, but they
"notch" out frequencies other than the dc
component.
Attenuate a selected frequency (and some of
its neighbors) and leave other frequencies of
the Fourier transform relatively unchanged.

30.
Notch Filters
 Notch filters
 Ideal Notch Reject Filter


 

otherwise1
Dv)(u,DorDv)(u,Dif0
),( 0201
vuH
  2/12
0
2
01 )2/()2/(),( vNvuMuvuD 
  2/12
0
2
02 )2/()2/(),( vNvuMuvuD 
30
Lecture by Kalyan Acharjya

31.
Notch Filters
 Butterworth Notch Reject Filter of order n
 Gaussian notch reject filter
n
vuDvuD
D
vuH








),(),(
1
1
),(
21
2
0










2
0
21 ),(),(
2
1
1),(
D
vuDvuD
evuH
31
Lecture by Kalyan Acharjya

32.
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Lecture by Kalyan Acharjya

33.
Notch Filter Result
33
Lecture by Kalyan Acharjya

34.
Thank You!
Any Question Please?
kalyan5.blogspot.in
34Lecture by Kalyan Acharjya