frequency Domain Filters

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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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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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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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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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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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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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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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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=

C54C54C54

C544C54

C54C54C54

s


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=

040

4D4

040

s


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=

4*4

*4**

4*4

s

frequency Domain Filters

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