Image Processing

Ch4:
Filtering in frequency domain
part2

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 Ch4, lesson 7: summary of properties

Usefulness of FT Concepts:

 FT concepts are essential for understanding

the spectral (FD) properties of signals and
systems and have very wide applications such
as filtering, image compression, image
feature representation etc.

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Frequency Domain Filtering & Spatial
Domain Filtering
Similar jobs can be done in the spatial and
frequency domains

Filtering in the spatial domain can be easier

to understand

Filtering in the frequency domain can be

much faster – especially for large images
Frequency Domain Filtering

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 Ch4, lesson 8: The basics of filtering in the Frequency Domain (FD)

introduction
 It consists of modifying the FT of an input image and
then finding the IFT to get the output image.
 Mathematically its given by:
G(u,v)= H(u,v)F(u,v)

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 Ch4, lesson 8: The basics of filtering in the Frequency Domain (FD)

Steps of frequency domain filter

Multiply the input image by (-1)x+y

2. Compute F(u,v)
3. Multiply F(u,v) by a filter function
H(u,v)
4. Compute the inverse DFT
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6. Multiply by (-1)x+y
 Ch4, lesson 9: correspondence between FD & SD filters

Correspondence between FD and SD filtering:

 SD: filtering is given by discrete convolution

 FD: Discrete convolution in FD is equivalent to

multiplication

 Filtering can be done in either domain.

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 Ch4, lesson 10: low pass & high pass filters in FD

Low pass & High pass filters in FD

 Low-pass filter: A filter that attenuates high frequencies while
passing low frequencies.- used for blurring (smoothing)

 High-pass filter: A filter that attenuates low frequencies while

passing high frequencies. used for sharpening

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 Ch4, lesson 10: low pass & high pass filters in FD

Low pass & High pass filters in FD

1) Low pass - Smoothing filters
1.1) Ideal lowpass filters (very sharp) More smooth
in the edge of
1.2) Butterworth lowpass filters cut-off
1.3) Gaussian lowpass filters (very smooth) frequency

 Butterworth filter parameter: filter order

 High values: filter has the form of the ideal filter.
 Low values: filter has the form of the Gaussian filter.

2) High pass- Sharpening filters

2.1) Ideal highpass filters
2.2) Butterworth highpass filters
2.3) Gaussian highpass filters 9
 Ch4, lesson 10: low pass & high pass filters in FD

1) Low pass – image smoothing

Low-pass filter: A filter that attenuates high frequencies while passing
low frequencies.- used for blurring (smoothing)

Image smoothing using FD filters

 Noise is usually high frequency.
 Image details and edges have high frequency characteristics.

 Hence noise and details removal is usually termed smoothing or

blurring.

 Smoothing is achieved by lowpass filters (LPFs).

 3 types of LPFs will be studied i.e. Ideal LPF, Butterworth LPF and
Gaussian LPF.

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 Ch4, lesson 10: low pass filters

1) Low pass - image Smoothing

Smoothing filters:
1.1) Ideal low pass filters (ILPF)
1 if D(u, v)  D0
H (u, v)  
0 if D(u, v)  D0

1.2) Butterworth low pass filters (BLPF)

1
H (u, v) 
1  [ D(u, v) / D0 ]2 n

1.3) Gaussian low pass filters (GLPF)

 D 2 ( u ,v ) / 2 D0 2
H (u, v)  e
 Ch4, lesson 11: ILPF

1.1) 2-D Ideal low pass filter ILPF

The simplest lowpass filter is ILPF, it (cuts off) all high
frequency components that are at distance greater
than a specified distance: D0 from the center

1 if D(u, v)  D0
H (u, v)  
0 if D(u, v)  D0
Where
 D(u,v) is the distance from (u,v) to the center of the frequency rectangle
 Image size: MxN
 Center of the frequency rectangle: (u,v) = (M/2, N/2)
 Distance to the center: D(u,v) = [(u – M/2)2 + (v – N/2)2]1/2
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 H(u,v)
Example on 2-D Ideal
low pass filter ILPF

D(u,v)
D0
1 if D(u, v)  D0
H (u, v)  
(u,v)
0 if D(u, v)  D0

1 2 3 4 5 6 7 8 9

1
Center of the frequency rectangle:
2
(M/2, N/2) = (5,5)
3
Suppose D0 = 3
4 The value of H1 = [(5-5)2 + (5-7)2]1/2
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H1
= 2<3 then H1 = 1
6

7 The value of H2 = [(5-9)2 + (5-2)2]1/2

8 = 5>3 then H2 = 0
9
H2

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 ‫‪Ch4, lesson 11: ILPF‬‬

‫?‪How does ILPF function works‬‬

‫•نضرب قيم هذا الماسك )‪ H(u,v‬بالقيم المكافئة لها بالصورة )‪ .F(u,v‬فنحصل فقط على ال ‪ frequencies‬داخل‬
‫الدائرة‬

‫•دائما منتصف الصورة تحمل اهم المعلومات عنها ( اي معظم معلومات الصورة)‪ ,‬بينما تتوزع التفاصيل‬
‫•كل ما‪14‬كبر نصف قطر الدائرة (اي ‪ ,)D0‬كل ما حصلنا على صورة اقرب لالصل‪ (.‬انظر الشريحة التالية)‬
 ‫‪Ch4, lesson 11: ILPF‬‬

‫كل ما كبر نصف قطر الدائرة (اي ‪,)D0‬‬

‫كل ما حصلنا على صورة اقرب لالصل‬

‫‪Radius‬‬ ‫‪total image power %‬‬ ‫دائما منتصف الصورة تحمل اهم‬
‫‪5‬‬ ‫‪92.0‬‬ ‫المعلومات عنها ( اي معظم معلومات‬
‫‪15‬‬ ‫‪94.6‬‬ ‫الصورة)‪ ,‬بينما تتوزع التفاصيل‬
‫‪30‬‬ ‫‪96.4‬‬
‫‪80‬‬ ‫‪98‬‬
‫‪230 pixels‬‬ ‫‪99.5‬‬
‫‪15‬‬
ILPF: example

original Freq.

94.6
92

96.4

98

99.5
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 Ch4, lesson 11: ILPF

Ideal Low Pass Filter results

Result of filtering
Original with ideal low
image pass filter of
Ringing is a characteristics of radius 5
ideal Filter

Result of filtering
Result of filtering with ideal low
with ideal low pass filter of
pass filter of radius 30
radius 15

Result of filtering Result of filtering

with ideal low with ideal low
pass filter of pass filter of
radius 80 radius 230
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Ideal lowpass filtering is not very practical but they can be implemented on a computer to study
their behavior.
Ringing
effect

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 Ch4, lesson 11: ILPF

Blurring and ringing feature of ILPFs:

The ILPF has sharp cutoffs or discontinuities which cause ringing.
 The center of lobe is the cause for blurring but the outer smaller lobes
cause ringing.
 We want to achieve blurring with little ringing.

ILPF: Freq.
blurring
ILPF: spatial

F-1

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ringing
 Ch4, lesson 12: BLPF

1.2) Butterworth low pass filter - BLPF:

ILPF transfer function has sharp discontinuities.
BLPF transfer function does not have sharp discontinuities in order
to reduce ringing.

The transfer function of a Butterworth lowpass filter of order n with

cutoff frequency at distance D0 from the origin is defined as:

1
H (u, v) 
1  [ D(u, v) / D0 ]2 n

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 Ch4, lesson 12: BLPF

Butterworth low pass filter - BLPF:

1
H (u, v) 
1  [ D(u, v) / D0 ]2 n

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 Ch4, lesson 12: BLPF

Butterworth Lowpass Filter (cont…)

Result of
Original filtering with
image Butterworth
filter of order 2
Ringing is not visible in and cutoff radius
any of these images.
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Result of filtering Result of
with Butterworth filtering with
filter of order 2 Butterworth
and cutoff radius filter of order 2
15 and cutoff
radius 30
Result of
Result of filtering
filtering with
with Butterworth
Butterworth
filter of order 2
filter of order 2
and cutoff radius
and cutoff radius
80
 Ch4, lesson 12: BLPF

Ringing increases with filter order (n) as

seen in SD
 Ch4, lesson 13: GLPF

1.3) Gaussian Lowpass Filters -GLPF

The transfer function of a Gaussian
lowpass filter is defined as
 D ( u ,v ) / 2 D0 2
2
H (u, v)  e
 Ch4, lesson 12: BLPF

Gaussian low pass filter - GLPF:

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 Ch4, lesson 13: GLPF

Gaussian Lowpass Filters (cont…)

Result of
Original filtering with
image Gaussian filter
No ringing with cutoff
radius 5

Result of Result of
filtering with filtering with
Gaussian filter Gaussian filter
with cutoff with cutoff
radius 15 radius 30

Result of Result of
filtering with filtering with
Gaussian Gaussian filter
filter with with cutoff
cutoff radius radius 230
 Lowpass Filters Compared
Result of
Result of
filtering with
filtering with
Butterworth
ideal low pass
filter of order
filter of radius
2 and cutoff
15
radius 15

Result of
filtering with
Gaussian filter
with cutoff
radius 15
Practical applications
A lowpass Gaussian filter is used to connect
broken text

Machine recognition systems have

difficulty in reading broken characters. GLPF with D0 = 80
Practical applications:
Different Lowpass Gaussian filters used
to remove blemishes in a photograph

GLPF, D0=80

1028x732

GLPF, D0=100
Practical applications:
Scan line

588x600 GLPF, D0=30 GLPF, D0=10

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2)High pass - image Sharpening
Low-pass filter
High-pass filter

High
High
freq.
freq.

Low Low
freq. freq.

High High
freq. freq.

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 Ch4, lesson 14: high pass

2)High pass - image Sharpening

 Blurring (smoothing) is achieved by attenuating
‫ التخلص‬the HF(high frequency) components of DFT of an
image. (low pass filters)

 Sharpening is achieved by attenuating the LF (low

frequency) components of DFT of an image. Where
Edges and fine detail in images are associated with
high frequency components (high pass filters)

High pass filters – only pass the high frequencies,

drop the low ones. High pass frequencies are precisely
the reverse of low pass filters, so:
Hhp(u, v) = 1 – Hlp(u, v)
 Ch4, lesson 14: high pass

2)High pass - image Sharpening

Sharpening filters:
2.1) Ideal high pass filters (IHPF)
0 if D(u, v)  D0
H (u, v)  
1 if D(u, v)  D0

2.2) Butterworth high pass filters (BHPF)

1
H (u, v) 
1  [ D0 / D(u, v)]2 n

2.3) Gaussian high pass filters (GHPF)

 D 2 ( u ,v ) / 2 D0 2
H (u, v)  1  e
 Ch4, lesson 15: IHPF

2.1) Ideal High Pass Filters

The ideal high pass filter is given as:
0 if D(u, v)  D0
H (u, v)  
1 if D(u, v)  D0
where D0 is the cut off distance as before
 Ch4, lesson 15: IHPF

Ideal High Pass Filters (cont…)

Results of ideal Results of ideal Results of ideal

high pass filtering high pass filtering high pass filtering
with D0 = 15 with D0 = 30 with D0 = 80
 Ch4, lesson 16: BHPF

Butterworth High Pass Filters

The Butterworth high pass filter is given as:
1
H (u, v) 
1  [ D0 / D(u, v)]2 n
where n is the order and D0 is the cut off
distance as before
 Ch4, lesson 16: BHPF

Butterworth High Pass Filters (cont…)

Results of Results of
Butterworth Butterworth
high pass high pass
filtering of filtering of
order 2 with order 2 with
D0 = 15 D0 = 80

Results of Butterworth high pass

filtering of order 2 with D0 = 30
 Ch4, lesson 17: GHPF

Gaussian High Pass Filters

The Gaussian high pass filter is given as:
 D 2 ( u ,v ) / 2 D0 2
H (u, v)  1  e
where D0 is the cut off distance as before
 Ch4, lesson 17: GHPF

Gaussian High Pass Filters (cont…)

Results of Results of
Gaussian Gaussian
high pass high pass
filtering with filtering with
D0 = 15 D0 = 80

Results of Gaussian high

pass filtering with D0 = 30
 Highpass Filter Comparison

BF represents a transition
between the sharpness
of the IF and the
smoothness of the GF

GF ‫ و‬IF ‫ مرحلة وسطية بين‬BF ‫يعني يعتبر‬

GF ‫ حاد للغاية بينما يمتاز‬IF ‫حيث يعتبر‬
‫بنعومته‬
Highpass Filter Comparison

Results of ideal Results of Butterworth Results of Gaussian

high pass filtering high pass filtering of high pass filtering with
with D0 = 15 order 2 with D0 = 15 D0 = 15
The 3 HPF SD representation
 Periodic Noise

Noise components

Periodic noise can be reduced in

via frequency domain

Are generated due to electrical or electromechanical interference during image

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acquisition
noisy spectrum
filtered

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bandreject
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Horizontal
Scan lines

Notch
DFT pass

Notch Notch
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pass reject