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Course Website: http://www.comp.dit.ie/bmacnamee
Digital Image Processing
Image Enhancement:
Filtering in the Frequency Domain
http://www.comp.dit.ie/bmacnameehttp://www.comp.dit.ie/bmacnamee
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"#Contents
In this lecture we will loo$ at imageenhancement in the !requency domain
% &ean 'aptiste &oseph Fourier
% (he Fourier series ) the Fourier trans!orm
% Image Processing in the !requency domain
* Image smoothing
* Image sharpening
% Fast Fourier (rans!orm
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+
o!
"#&ean 'aptiste &oseph Fourier
Fourier was born in ,u-erreFrance in 012
% 3ost !amous !or his wor$ 4La Théorie Analitique de la Chaleur” published in
2 % (ranslated into English in 202: 4The
Analytic Theory of Heat”
5obody paid much attention when the wor$ was
!irst published6ne o! the most important mathematical theoriesin modern engineering
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#
o!
"#(he 'ig Idea
7
,ny !unction that periodically repeats itsel! can be
e-pressed as a sum o! sines and cosines o!
di!!erent !requencies each multiplied by a di!!erent
coe!!icient % a Fourier series I m a g e s t a $ e n ! r o m 8
o n 9 a l e 9 ) W o o d s D i g i t a l
I m a g e P r o c e s s i n g : ; ; <
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"
o!
"#(he 'ig Idea cont=<
5otice how we get closer and closer to the
original !unction as we add more and more!requencies
( a $ e n ! r o m w w w . t ! h > b e r l i n . d e / ? s c h w e n $ / h o b b y / ! o u r i e r / W e l c o m e . h t m l
http://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.htmlhttp://www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.html
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"#(he 'ig Idea cont=<
Frequencydomain signal
processing
e-ample in E-cel
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"#(he Discrete Fourier (rans!orm DF(<
(he Discrete Fourier Transform o! f(x, y) !or x 7; =M> and y 7 ;= N> denoted by
F(u, v), is gi@en by the equation:
!or u 7 ; =M> and v 7 ; = N>.
∑∑−
=
−
=
+−=1
0
1
0
)//(2),(),( M
x
N
y
N vy M ux je y x f vu F π
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"#DF( ) Images
(he DF( o! a two dimensional image can be@isualised by showing the spectrum o! the
images component !requencies
I m a g e s t a $ e n ! r o m 8
o n 9 a l e 9 ) W o o d s D i g i t a l I m a g e P r o c e s s i n g : ; ; <
DFT
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A
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"#DF( ) Images
I m a g e s t a $ e n ! r o m 8
o n 9 a l e 9 ) W o o d s D i g i t a l I m a g e P r o c e s s i n g : ; ; <
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;
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"#DF( ) Images
I m a g e s t a $ e n ! r o m 8
o n 9 a l e 9 ) W o o d s D i g i t a l I m a g e P r o c e s s i n g : ; ; <
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"#DF( ) Images cont=<
I m a g e s t a $ e n ! r o m 8
o n 9 a l e 9 ) W o o d s D i g i t a l I m a g e P r o c e s s i n g : ; ; <
DFT
Bcanning electron microscope
image o! an integrated circuitmagni!ied ?";; times
Fourier spectrum o! the image
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"#DF( ) Images cont=<
I m a g e s t a $ e n ! r o m 8
o n 9 a l e 9 ) W o o d s D i g i t a l I m a g e P r o c e s s i n g : ; ; <
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"#DF( ) Images cont=<
I m a g e s t a $ e n ! r o m 8
o n 9 a l e 9 ) W o o d s D i g i t a l I m a g e P r o c e s s i n g : ; ; <
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#
o!
"#(he In@erse DF(
It is really important to note that the Fouriertrans!orm is completely reversible
(he in@erse DF( is gi@en by:
!or x 7 ; =M> and y 7 ; = N>
∑∑−
=
−
=
+=1
0
1
0
)//(2),(1),( M
u
N
v
N vy M ux jevu F MN
y x f π
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"
o!
"#(he DF( and Image Processing
(o !ilter an image in the !requency domain:. Compute F(u,v) the DF( o! the image
. 3ultiply F(u,v) by a !ilter !unction H(u,v)
+. Compute the in@erse DF( o! the result
I m a g e s t a $ e n ! r o m 8
o n 9 a l e 9 ) W o o d s D i g i t a l I m a g e P r o c e s s i n g : ; ; <
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"#Bome 'asic Frequency Domain Filters
I m a g e s t a $ e n ! r o m 8
o n 9 a l e 9 ) W o o d s D i g i t a l I m a g e P r o c e s s i n g : ; ; <
ow Pass Filter
igh Pass Filter
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0
o!
"#Bome 'asic Frequency Domain Filters
I m a g e s t a $ e n ! r o m 8
o n 9 a l e 9 ) W o o d s D i g i t a
l I m a g e P r o c e s s i n g : ; ; <
2
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2
o!
"#Bome 'asic Frequency Domain Filters
I m a g e s t a $ e n ! r o m 8 o n 9 a l e 9 ) W o o d s D i g i t a
l I m a g e P r o c e s s i n g : ; ;
<
A
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A
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"#Bmoothing Frequency Domain Filters
Bmoothing is achie@ed in the !requency domainby dropping out the high !requency components
(he basic model !or !iltering is:
G(u,v) = H(u,v)F(u,v)
where F(u,v) is the Fourier trans!orm o! the
image being !iltered and H(u,v) is the !ilter
trans!orm !unction
Low pass filters % only pass the low !requencies
drop the high ones
;
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"#Ideal ow Pass Filter
Bimply cut o!! all high !requency components thatare a speci!ied distance D; !rom the origin o! the
trans!orm
changing the distance changes the beha@iour o!
the !ilter I m a g e s t a $ e n ! r o m 8 o n 9 a l e 9 ) W o o d s D i g i t a
l I m a g e P r o c e s s i n g : ; ;
<
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"#Ideal ow Pass Filter cont=<
(he trans!er !unction !or the ideal low pass !iltercan be gi@en as:
where D(u,v) is gi@en as:
>
≤=
0
0
),(if 0
),(if 1),(
Dvu D
Dvu Dvu H
2/122
])2/()2/[(),( N v M uvu D −+−=
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"#Ideal ow Pass Filter cont=<
,bo@e we show an image its Fourier spectrum
and a series o! ideal low pass !ilters o! radius "
" +; 2; and +; superimposed on top o! it
I m a g e s t a $ e n ! r o m 8 o n 9 a l e 9 ) W o o d s D i g i t a
l I m a g e P r o c e s s i n g : ; ;
<
+
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+
o!
"#Ideal ow Pass Filter cont=<
I m a g e s t a $ e n ! r o m 8 o n 9 a l e 9 ) W o o d s D i g i t a
l I m a g e P r o c e s s i n g : ; ;
<
#
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#
o!
"#Ideal ow Pass Filter cont=<
I m a g e s t a $ e n ! r o m 8 o n 9 a l e 9 ) W o o d s D i g i t a
l I m a g e P r o c e s s i n g : ; ;
<
"
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"
o!
"#Ideal ow Pass Filter cont=<
I m a g e s t a $ e n ! r o m 8 o n 9 a l e 9 ) W o o d s D i g i t a
l I m a g e P r o c e s s i n g : ; ;
<
6riginal
image
esult o! !ilteringwith ideal low pass
!ilter o! radius "
esult o! !iltering
with ideal low pass
!ilter o! radius +;
esult o! !iltering
with ideal low pass
!ilter o! radius +;
esult o! !iltering
with ideal low pass
!ilter o! radius 2;
esult o! !iltering
with ideal low pass
!ilter o! radius "
1
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1
o!
"#Ideal ow Pass Filter cont=<
I m a g e s t a $ e n ! r o m 8 o n 9 a l e 9 ) W o o d s D i g i t a
l I m a g e P r o c e s s i n g : ; ;
<
esult o! !ilteringwith ideal low pass
!ilter o! radius "
0
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"#Ideal ow Pass Filter cont=<
I m a g e s t a $ e n ! r o m
8 o n 9 a l e 9 ) W o o d s D i g i t a l I m a g e P r o c e s s i n g : ; ;
<
esult o! !ilteringwith ideal low pass
!ilter o! radius "
2
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o!
"#'utterworth owpass Filters
(he trans!er !unction o! a 'utterworth lowpass!ilter o! order n with cuto!! !requency at distance
D0 !rom the origin is de!ined as:
I m a g e s t a $ e n ! r o m
8 o n 9 a l e 9 ) W o o d s D i g i t a l I m a g e P r o c e s s i n g : ; ;
<
n Dvu Dvu H 20 ]/),([1
1
),( +=
A
' tt th Filt t
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o!
"#'utterworth owpass Filter cont=<
I m a g e s t a $ e n ! r o m
8 o n 9 a l e 9 ) W o o d s D i g i t a l I m a g e P r o c e s s i n g : ; ; <
6riginal
image
esult o! !ilteringwith 'utterworth !ilter
o! order and cuto!!
radius "
esult o! !iltering
with 'utterworth
!ilter o! order and
cuto!! radius +;
esult o! !iltering
with 'utterworth !ilter
o! order and cuto!!
radius +;
esult o! !iltering with
'utterworth !ilter o!
order and cuto!!
radius 2;
esult o! !iltering with
'utterworth !ilter o!
order and cuto!!
radius "
+;
' tt th Filt t
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o!
"#'utterworth owpass Filter cont=<
I m a g e s t a $ e n ! r o m
8 o n 9 a l e 9 ) W o o d s D i g i t a l I m a g e P r o c e s s i n g : ; ; <
6riginal
image
esult o! !iltering
with 'utterworth !ilter
o! order and cuto!!
radius "
+
' tt th Filt t
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"#'utterworth owpass Filter cont=<
I m a g e s t a $ e n ! r o m
8 o n 9 a l e 9 ) W o o d s D i g i t a l I m a g e P r o c e s s i n g : ; ; <
esult o! !iltering with
'utterworth !ilter o!
order and cuto!!
radius "
+
8 i Filt
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"#8aussian owpass Filters
(he trans!er !unction o! a 8aussian lowpass!ilter is de!ined as:
I m a g e s t a $ e n ! r o m
8 o n 9 a l e 9 ) W o o d s D i g i t a l I m a g e P r o c e s s i n g : ; ; <
20
2 2/),(),(
Dvu Devu H
−=
++
8 i Filt t
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"#8aussian owpass Filters cont=<
I m a g e s t a $ e n ! r o m
8 o n 9 a l e 9 ) W o o d s D i g i t a l I m a g e P r o c e s s i n g : ; ; <
6riginal
image
esult o! !ilteringwith 8aussian !ilter
with cuto!! radius "
esult o! !iltering
with 8aussian !ilter
with cuto!! radius +;
esult o! !iltering
with 8aussian !ilter
with cuto!! radius
+;
esult o! !iltering
with 8aussian
!ilter with cuto!!
radius 2"
esult o! !ilteringwith 8aussian
!ilter with cuto!!
radius "
+#
! owpass Filters Compared
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"#owpass Filters Compared
esult o! !iltering
with ideal low pass
!ilter o! radius "
esult o! !iltering
with 'utterworth
!ilter o! order
and cuto!! radius
"
esult o! !iltering
with 8aussian
!ilter with cuto!!
radius "
I m a g e s t a $ e n ! r o m
8 o n 9 a l e 9 ) W o o d s D i g i t a l I m a g e P r o c e s s i n g : ; ; <
+"
! owpass Filtering E-amples
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"#owpass Filtering E-amples
, low pass 8aussian !ilter is used to connect
bro$en te-t
I m a g e s t a $ e n ! r o m
8 o n 9 a l e 9 ) W o o d s D i g i t a l I m a g e P r o c e s s i n g : ; ; <
+1
o! owpass Filtering E-amples
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"#owpass Filtering E-amples
I m a g e s t a $ e n ! r o m
8 o n 9 a l e 9 ) W o o d s D i g i t
a l I m a g e P r o c e s s i n g : ; ; <
+0
o! owpass Filtering E-amples cont
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o!
"#owpass Filtering E-amples cont=<
Di!!erent lowpass 8aussian !ilters used to
remo@e blemishes in a photograph
I m a g e s t a $ e n ! r o m
8 o n 9 a l e 9 ) W o o d s D i g i t a l I m a g e P r o c e s s i n g : ; ; <
+2
o! owpass Filtering E-amples cont
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"#owpass Filtering E-amples cont=<
I m a g e s t a $ e n ! r o m
8 o n 9 a l e 9 ) W o o d s D i g i t a l I m a g e P r o c e s s i n g : ; ;
<
+A
o! owpass Filtering E-amples cont
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o!
"#owpass Filtering E-amples cont=<
6riginal
image
8aussian lowpass
!ilter
Processed
image
Bpectrum o!
original image
I m a g e s t a $ e n ! r o m
8 o n 9 a l e 9 ) W o o d s D i g i t a l I m a g e P r o c e s s i n g : ; ;
<
#;
o! Bharpening in the Frequency Domain
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"#Bharpening in the Frequency Domain
Edges and !ine detail in images are associated
with high !requency components
High pass filters % only pass the high
!requencies drop the low ones
igh pass !requencies are precisely the re@erse
o! low pass !ilters so:
H hp(u, v) = 1 – H lp(u, v)
#
o! Ideal igh Pass Filters
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"#Ideal igh Pass Filters
(he ideal high pass !ilter is gi@en as:
where D; is the cut o!! distance as be!ore
>
≤=
0
0
),(if 1
),(if 0),(
Dvu D
Dvu Dvu H
I m a g e s t a $ e n ! r o m
8 o n 9 a l e 9 ) W o o d s D i g i t a l I m a g e P r o c e s s i n g : ; ;
<
#
o! Ideal igh Pass Filters cont
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o!
"#Ideal igh Pass Filters cont=<
esults o! ideal
high pass !iltering
with D0 7 "
esults o! ideal
high pass !iltering
with D0 7 +;
esults o! ideal
high pass !iltering
with D0 7 2; I m a g e s t a $ e n ! r o m
8 o n 9 a l e 9 ) W o o d s D i g i t a l I m a g e P r o c e s s i n g : ; ;
<
#+
o! 'utterworth igh Pass Filters
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"#'utterworth igh Pass Filters
(he 'utterworth high pass !ilter is gi@en as:
where n is the order and D0 is the cut o!!distance as be!ore
nvu D Dvu H
2
0 )],(/[1
1),(
+=
I m a g e s t a $ e n ! r o m
8 o n 9 a l e 9 ) W o o d s D i g i t a l I m a g e P r o c e s s i n g : ; ;
<
##
o! 'utterworth igh Pass Filters cont
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"#'utterworth igh Pass Filters cont=<
esults o!
'utterworth
high pass
!iltering o!
order with D0 7 "
esults o!
'utterworth
high pass
!iltering o!
order with D0 7 2;
esults o! 'utterworth high pass
!iltering o! order with D0 7 +; I m a g e s t a $ e n ! r o m
8 o n 9 a l e 9 ) W o o d s D i g i t a l I m a g e P r o c e s s i n g : ; ;
<
#"
o! 8aussian igh Pass Filters
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"#8aussian igh Pass Filters
(he 8aussian high pass !ilter is gi@en as:
where D0 is the cut o!! distance as be!ore
20
2 2/),(1),(
Dvu Devu H
−−=
I m a g e s t a $ e n ! r o m
8 o n 9 a l e 9 ) W o o d s D i g i t a l I m a g e P r o c e s s i n g : ; ;
<
#1
o! 8aussian igh Pass Filters cont <
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"#8aussian igh Pass Filters cont=<
esults o!
8aussian
high pass
!iltering with
D0 7 "
esults o!
8aussian
high pass
!iltering with
D0 7 2;
esults o! 8aussian high pass
!iltering with D0 7 +; I m a g e s t a $ e n ! r o m
8 o n 9 a l e 9 ) W o o d s D i g i t a l I m a g e P r o c e s s i n g : ; ;
<
#0
o! ighpass Filter Comparison
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"#ighpass Filter Comparison
esults o! ideal
high pass !iltering
with D0 7 "
I m a g e s t a $ e n ! r o m
8 o n 9 a l e 9 ) W o o d s D i g i t a l I m a g e P r o c e s s i n g : ; ;
<
#2
o! ighpass Filter Comparison
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"#ighpass Filter Comparison
esults o! 'utterworth
high pass !iltering o! order with D0 7 "
I m a g e s t a $ e n ! r o m
8 o n 9 a l e 9 ) W o o d s D i g i t a l I m a g e P r o c e s s i n g : ; ; <
#A
o! ighpass Filter Comparison
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"#ighpass Filter Comparison
esults o! 8aussianhigh pass !iltering with
D0 7 "
I m a g e s t a $ e n ! r o m
8 o n 9 a l e 9 ) W o o d s D i g i t a l I m a g e P r o c e s s i n g : ; ; <
";
o!
"#ighpass Filter Comparison
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"#g p p
esults o! ideal
high pass !iltering
with D0 7 "
esults o! 8aussian
high pass !iltering with
D0 7 "
esults o! 'utterworth
high pass !iltering o! order
with D0 7 " I m a g e s t a $ e n ! r o m
8 o n 9 a l e 9 ) W o o d s D i g i t a l I m a g e P r o c e s s i n g : ; ; <
"
o!
"#ighpass Filter Comparison
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"#g p p
esults o! idealhigh pass !iltering
with D0 7 "
I m a g e s t a $ e n ! r o m
8 o n 9 a l e 9 ) W o o d s D i g i t a l I m a g e P r o c e s s i n g : ; ; <
"
o!
"#ighpass Filter Comparison
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"#g p p
esults o! 'utterworthhigh pass !iltering o! order
with D0 7 "
I m a g e s t a $ e n ! r o m
8 o n 9 a l e 9 ) W o o d s D i g i t a l I m a g e P r o c e s s i n g : ; ; <
"+
o!
"#ighpass Filter Comparison
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"#g p p
esults o! 8aussianhigh pass !iltering with
D0 7 "
I m a g e s t a $ e n ! r o m
8 o n 9 a l e 9 ) W o o d s D i g
i t a l I m a g e P r o c e s s i n g : ; ; <
"#
o!
"#ighpass Filtering E-ample
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"#g p g p
6 r i g i n a l i m a g e
ig
hpas
s!ilteringresu lt
i g h ! r e q u e n c y
e m p h a s i s r e s u l t ,
!terh
is togram
equa
lisation
I m a g e s t a $ e n ! r o m
8 o n 9 a l e 9 ) W o o d s D i g
i t a l I m a g e P r o c e s s i n g : ; ; <
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o!
"#ighpass Filtering E-ample
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"1
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"#
"0
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"#
"2
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"#
"A
o!
"#aplacian In (he Frequency Domain
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C a p l a c i a n i n t h e
! r e q u e n c y d o m a
i n
>D
ima
geo!Caplacia
n
inthe!r e
quency
domain
I n @ e r s e
D F ( o !
C a p l a c i a n i n t h e
! r e q u e n c
y d o m a i n
Goomed section o
the image on the
le!t compared to
spatial !ilter I m a g e s t a $ e n ! r o m 8 o n 9 a l e 9 ) W o o d s D i g
i t a l I m a g e P r o c e s s i n g : ;
; <
1;
o!
"#Frequency Domain aplacian E-ample
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6riginal
image
aplacian
!iltered
image
aplacian
image scaled
Enhanced
image
1
o!
"#Fast Fourier (rans!orm
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(he reason that Fourier based techniques ha@e
become so popular is the de@elopment o! theFast Fourier Transform (FFT algorithm
,llows the Fourier trans!orm to be carried out in
a reasonable amount o! timeeduces the amount o! time required to per!orm
a Fourier trans!orm by a !actor o! ;; % 1;;
timesH
1
o!
"#
Frequency Domain Filtering ) BpatiaDomain Filtering
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Domain Filtering
Bimilar obs can be done in the spatial and
!requency domains
Filtering in the spatial domain can be easier to
understand
Filtering in the !requency domain can be much!aster % especially !or large images
1+
o!
"#Bummary
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In this lecture we e-amined image
enhancement in the !requency domain
% (he Fourier series ) the Fourier trans!orm
% Image Processing in the !requency domain
* Image smoothing* Image sharpening
% Fast Fourier (rans!orm
5e-t time we will begin to e-amine image
restoration using the spatial and !requency
based techniques we ha@e been loo$ing at
1#o!
"#
Imteresting ,pplication 6! FrequencyDomain Filtering
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o a e g
1"o!
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Imteresting ,pplication 6! FrequencyDomain Filtering
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g
11o!
"#JuestionsK
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