Frequency Domain Filters for Smoothing - Ideal Lowpass Filters, Butterworth Lowpass Filters, Gaussian Lowpass Filters

1. IMAGE SMOOTHING
USING FREQUENCY
DOMAIN FILTERS
By,
H. Suhaila Afzana
C. Surega
T. Vaitheeswari
1

2. CONTENTS
 Frequency Domain Filters
 Lowpass Filters
 Ideal Lowpass Filters
 Butterworth Lowpass Filters
 Gaussian Lowpass Filters
 Lowpass Filters – Comparison
 Lowpass Filtering Examples
2

3. FREQUENCY DOMAIN FILTERS
 Smoothing(blurring) is achieved in the frequency domain by high-
frequency attenuation; that is, by lowpass filtering.
 Here, we consider 3 types of lowpass filters:
 Ideal lowpass filters
 Butterworth lowpass filters
 Gaussian lowpass filters
 These three categories cover the range from very sharp(ideal), to
very smooth(Gaussian) filtering.
3

4. FREQUENCY DOMAIN FILTERS
 The Butterworth filter has a parameter called the filter order.
 For high order values, the Butterworth filter approaches the ideal
filter. For low order values, Butterworth filter is more like a Gaussian
filter.
 Thus, the Butterworth filter may be viewed as providing a transition
between two “extremes”.
4

5. LOWPASS FILTERS
 The most basic of filtering operations is called “lowpass”.
 A lowpass filter is also called a “blurring” or smoothing filter.
 The simplest lowpass filter just calculates the average of a pixel and
all of its eight immediate neighbours.
 Lowpass is also called as blurring mask.
5

6. IDEAL LOWPASS FILTERS
 A 2-D lowpass filter that passes without attenuation all frequencies
within a circle of radius D0 from the origin and “cuts off” all
frequencies outside this circle is called an ideal lowpass filter(ILPF); it
is specified by the function:






0
0
),(if0
),(if1
),(
DvuD
DvuD
vuH
6

7. IDEAL LOWPASS FILTERS
 D0 is a positive constant and D(u,v) is the distance between a point
(u,v) in the frequency domain and the center of the frequency
rectangle; that is,
2/122
])2/()2/[(),( QvPuvuD 
7

8. IDEAL LOWPASS FILTERS
 The ideal lowpass filter is radially symmetric about the origin, which
means that the filter is completely defined by a radial cross section.
 Rotating the cross section by 360° yields the filter in 2-D.
 For an ILPF cross section, the point of transition between H(u,v)=1 and
H(u,v)=0 is called the cutoff frequency D0.
 Simply cut off all high frequency components that are at a specified
distance D0 from the origin of the transform, changing the distance
changes the behaviour of the filter.
8

9. IDEAL LOWPASS FILTERS
A)Perspective plot of an ideal lowpass filter transfer function
B)Filter displayed as an image
C)Filter radius cross section
9

10. IDEAL LOWPASS FILTERS
 When the lowpass filter is applied ringing occurs in the image.
 The narrower the filter in the frequency domain, the more severe
are the blurring and ringing.
 The more ringing in the image, the more blurring of the image.
10

11. IDEAL LOWPASS FILTERS
 Above we show an image, it’s Fourier spectrum and a series of ideal
low pass filters of radius 5, 15, 30, 80 and 230 superimposed on top
of it.
11

12. IDEAL LOWPASS FILTERS
Original
image
Result of filtering
with ideal low pass
filter of radius 5
Result of filtering
with ideal low pass
filter of radius 30
Result of filtering
with ideal low
pass filter of
radius 230
Result of filtering
with ideal low pass
filter of
radius 80
Result of filtering
with ideal low pass
filter of
radius 15
12

13. BUTTERWORTH LOWPASS
FILTERS
 The Butterworth lowpass filter is a type of signal processing filter
designed to have as flat a frequency response as possible in the
passband.
 It is also referred to as a maximally flat magnitude filter.
 It was first described in 1930 by the British Engineer and physicist
Stephen Butterworth.
13

14. BUTTERWORTH LOWPASS
FILTERS
 The transfer function of a Butterworth lowpass filter of order n with
cutoff frequency at distance D0 from the origin is defined as:
n
DvuD
vuH 2
0 ]/),([1
1
),(


14

15. BUTTERWORTH LOWPASS
FILTERS
A)Perspective plot of an Butterworth lowpass filter transfer function
B)Filter displayed as an image
C)Filter radius cross section of orders 1 through 4
15

16. BUTTERWORTH LOWPASS
FILTERS
Original
image
Result of filtering with
Butterworth filter of
order 2 and cutoff
radius 5
Result of filtering with
Butterworth filter of
order 2 and cutoff
radius 30
Result of filtering with
Butterworth filter of
order 2 and cutoff
radius 230
Result of filtering with
Butterworth filter of
order 2 and cutoff
radius 80
Result of filtering with
Butterworth filter of
order 2 and cutoff
radius 15
16

17. BUTTERWORTH LOWPASS
FILTERS
17

18. GAUSSIAN LOWPASS FILTERS
 The transfer function of a Gaussian lowpass filter is defined as:
 Here, is the standard deviation and is a measure of spread of the
Gaussian curve.
 If we put =D0 we get,
22
2/),(
),( vuD
evuH 

2
0
2
2/),(
),( DvuD
evuH 

18

19. GAUSSIAN LOWPASS FILTERS
A)Perspective plot of a GLPF transfer function
B)Filter displayed as an image
C)Filter radius cross section for various values of D0
19

20. GAUSSIAN LOWPASS FILTERS
 Main advantage of a Gaussian LPF over a Butterworth LPF is that
we are assured that there will be no ringing effects no matter what
filter order we choose to work with.
20

21. GAUSSIAN LOWPASS FILTERS
Original
image
Result of filtering
with Gaussian filter
with cutoff radius 5
Result of filtering
with Gaussian filter
with cutoff radius 30
Result of filtering
with Gaussian filter
with cutoff radius
230
Result of filtering
with Gaussian
filter with cutoff
radius 85
Result of filtering
with Gaussian filter
with cutoff radius
15
21

22. LOWPASS FILTERS-COMPARISON
Result of
filtering with
ideal low pass
filter of radius
15
Result of
filtering with
Butterworth
filter of order
2 and cutoff
radius 15
Result of
filtering with
Gaussian filter
with cutoff
radius 15
22

23. LOWPASS FILTERING EXAMPLES
 A low pass Gaussian filter is used to connect broken text
23

24. LOWPASS FILTERING EXAMPLES
 Different lowpass Gaussian filters used to remove blemishes in a
photograph 24

25. 25