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7/26/2019 frequency Domain Filters
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FREQUENCY DOMAIN
FILTERS
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Introduction Make use of convoution !ro!ert" of fourier
transfor#$
I#a%e FFT I#a%e s!ectru#
Fiter #ask in
fourier do#ain
&i'e (" !i'e
#uti!ication
Fourier
s!ectru#
Inverse FFTFitered I#a%e
Co#!utationa" faster to !erfor# t)o *D Fourier
transfor#s and a fiter #uti!" t+an to !erfor# a
convoution in t+e i#a%e ,s!atia- do#ain$
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S#oot+in% Fre.uenc" Do#ain
Fiters Idea Lo) &ass Fiters ,IL&F-/
Si#!est fiter
Cuts off a +i%+ fre.uenc"co#!onents t+at are at a distance%reater t+an t+e s!ecified distance D0fro# t+e ori%in of t+e
centered s!ectru#$
T+e *1D Idea Lo) &ass Fiter +as a transfer function
2+ere
=
0
0
-3,0
-3,4-3,
DvuDif
DvuDifvuH
** -*5,-*5,-3, NvMuvuD +=
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S#oot+in% Fre.uenc" Do#ain
Fiters Cont$$ D0is t+e cut off fre.uenc"
One )a" to t+e s!ecif" cutoff fre.uenc" is ("
cacuatin% t+e a#ount of !o)er encosed (" t+es!ectru#$
T+e !ercenta%e of !o)er 6 is cacuated (" takin% t+esu##ation over t+e vaues of u and v t+at ie inside a
circe of radius D0
2+ere !Tis
=
u v
Tpvup 5-3,400
=
=
+==4
0
4
0
** -3,-3,-3,-73,M
u
N
v
T vuIvuRvupvupp
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IL&F
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With
D0 =
8,16,32 =
90.55,
91.27,
91.59
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E'a#!e
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Ringing in ILPF
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S#oot+in% Fre.uenc" Do#ain
Fiters Cont$$ 8utter)ort+ Lo) &ass fiters ,8&LF-/
T+e transfer function of 8utter)ort+ o) !ass fiter is %iven ("
S#oot+ transition (et)een o) and +i%+ fre.uencies3 so no
rin%in% effect$
[ ] nDvuDvuH
*
05-3,4
4-3,
+=
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8L&F
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With
D0 =
8,16,32
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S#oot+in% Fre.uenc" Do#ain
Fiters Cont$$ 9aussian Lo) &ass fiters ,9&LF-/
T+e transfer function of 9aussian o) !ass fiter is
%iven ("
S#oot+ transition (et)een o) and +i%+ fre.uencies3
so no rin%in% effect$
( )*0* *5-3,e'!-3, DvuDvuH =
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9L&F
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With
D0 =8,16,32
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Co#!arison (5) IL&F3 8L&F and 9L&F
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A!!ications of o) !ass fiterin%
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A!!ications of o) !ass fiterin%
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A!!ications of o) !ass fiterin%
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S+ar!enin% Fre.uenc" Do#ain
Fiters
Idea :i%+ &ass Fiters,I:&F-/
T+e transfer function for a*1D idea +i%+ !ass fiter isdefined as
8utter)ort+ :i%+ &assFiters ,8:&F-/ T+e transfer function is
%iven ("
=
0
0
-3,4
-3,0-3,
DvuDif
DvuDifvuH
[ ] nvuDDvuH
*
0 -3,544-3,
+=
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S+ar!enin% Fre.uenc" Do#ain
Fiters Cont$$
9aussian :i%+ &ass Fiters ,9:&F-/
T+e transfer function of a 9aussian :&F is %iven
("( )*0* *5-3,e'!4-3, DvuDvuH =
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I:&F
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IHPF
D0=
8,16,32
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8:&F
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BHPF
D0=
8,16,32
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9:&F
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GHPF
D0=
8,16,32
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Co#!arison (5) I:&F3 8:&F and 9:&F
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T+e La!acian in Fre.uenc"
do#ain
Fro# t+e differentiation !ro!ert" of Fourier transfor#
T+e La!acian o!erator in Fourier do#ain can t+en (e
found as
-,-,
-,
uFjudx
xfd
F
n
n
n
=
[ ]
-3,-,
-3,-,-3,-,
**
**
*
*
*
**
vuFvu
vuFjvvuFjuy
f
x
fFfF
+=
+=
+
=
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T+e La!acian in Fre.uenc"
do#ain Cont$$
So )e can i#!e#ent t+e La!acian usin% t+e fiter3
:,u3v- ; 1,u*
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:o#o#or!+ic Fiterin%
T+e +o#o#or!+ic fiter is an a!!roac+ (ased on aniu#ination1refectance i#a%e #ode3 )+ere t+e i#a%ef,'3"- is taken as t+e !roduct of iu#ination andrefectance co#!onents$ f,'3"- ; i,'3"- r,'3"-
To co#!ute t+e Fourier re!resentation )e take =,'3"- ; n f,'3"- ;n i,'3"- < n r,'3"- F >=,'3"-? ; F >n f,'3"-? ; F >n i,'3"-? n r,'3"-? @,u3v- ; Fi,u3v- < Fr,u3v-
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:o#o#or!+ic Fiterin% Cont$$
No) if )e !rocess @,u3v- (" #eans of a fiter function
:,u3v-3 t+en
S,u3v- ; :,u3v- @,u3v- ; :,u3v- Fi,u3v- < :,u3v- Fr,u3v-
Takin% t+e inverse transfor#3 to %et t+e !rocessed
i#a%e3 s,'3"- ; F14>S,u3v-?
; F14>:,u3v- Fi,u3v- ?:,u3v- Fr,u3v-?
; i,'3"- < r,'3"-
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:o#o#or!+ic Fiterin% Cont$$
T+e en+anced i#a%e %,'3"- is o(tained (" takin%
t+e e'!onentia3
%,'3"- ; e s,'3"- ; e iA,'3"- < rA,'3"- ; e iA,'3"- e rA,'3"-
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:o#o#or!+ic Fiterin% Cont$$
A fiter function used for si#utaneous d"na#ic ran%e
co#!ression and contrast en+ance#ent is %iven
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:o#o#or!+ic Fiterin% Cont$$
T+e fiter function is %iven ("
( ) L
DvuDc
LH evuH += ?4>-3, -5-3,,
*0
*
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E'a#!e
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S!atia #asks and corres!ondin%
fre.uenc" do#ain fiters
=
444
444
444
B54s
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S!atia #asks and corres!ondin%
fre.uenc" do#ain fiters
=
C54C54C54
C544C54
C54C54C54
s
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S!atia #asks and corres!ondin%
fre.uenc" do#ain fiters
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S!atia #asks and corres!ondin%
fre.uenc" do#ain fiters
=
040
4D4
040
s
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S!atia #asks and corres!ondin%
fre.uenc" do#ain fiters
=
4*4
*4**
4*4
s