Frequency Domain FIltering.pdf

Frequency Domain

1. ImageAnalysisandProcessing Image Enhancements
in the Frequency Domain
LaurentNajman
[email protected]
ESIEEParis
Universit´eParis-Est,Laboratoired’InformatiqueGaspard-Monge,´Equipe
A3SI
imageprocessing,transforms–p.1/46

2. Book
Chapter4(pages147–219)
DigitalImageProcessing,SecondEdition
authors:RafaelC.GonzalezandRichardE.Woods editor:PrenticeHall
–p.3/44

3. FrequencyDomainFilteringOperation
Frequencydomain:spacedenedbyvaluesoftheFouriertransformandits
frequencyvariables(u;v).
RelationbetweenFourierDomainandimage:
u=v=0correspondstothegray-levelaverage Lowfrequencies:image'scomponentwithsmoothgray-levelvariation(e.g.
areaswithlowvariance)
Highfrequencies:quickgray-levelchange(e.g.edges,noise)
Source:http://www.imageprocessingbook.com–p.4/44

4. FilteringintheFrequencyDomain
Let
H(u;v)
alter,alsocalledltertransferfunction.
Filter:suppresscertainfrequencieswhileleaving
othersunchanged
G(u;v)=H(u;v)F(u;v)
H(u;v)
inimageprocessing:
Ingeneral
H(u;v)
isreal:zero-phase-shiftlter
H
multiplyrealandimaginarypartsof
F
(u;v)=tan
1I(u;v)
R(u;v)
doesnotchangeif
H
isreal
–p.5/44

5. GeneralStepsforFiltering
1.Multiplyinputimageby
(1)
x+y
(centering)
2.Compute
F(u;v)
(DFT)
3.Multiply
F(u;v)
by
H(u;v)
(ltering)
4.ComputeinverseDFTof
H(u;v)F(u;v)
5.Obtaintherealpartoftheresult
6.Mutliplyby
(1)
x+y
(decentering)
–p.6/44

6. FromSpatialtoFrequencyDomain
f(x;y)h(x;y),F(u;v)H(u;v)
(x;y)h(x;y),F[(u;v)]H(u;v)
h(x;y),H(u;v)
Multiplicationinthefrequencydomainisaconvolution
inthespatialdomain.
Given
h(x;y)
,wecanobtain
H(u;v)
bytakingthe
inverseFouriertransform.
–p.7/44

7. Padding(1)
convolution:
f(x)h(x)=
1
M
M1X m=0
f(m)h(xm)
periodicity:
F(u;v)=F(u+M;v)=F(u;v+N)=F(u+M;v+N)
–p.8/44

8. Padding(2)
Source:http://www.imageprocessingbook.com–p.9/44

9. Padding(3)
Let
P
anidenticalperiodfor
f
and
g
:
f
e
(x)=
(
f(x)0xA1
0AxP
g
e
(x)=
(
g(x)0xB1
0BxP If
P<A+B1
,thetwosignalwilloverlap:
wraparounderror.
If
P>A+B1
,theperiodswillbeseparated.
If
P=A+B1
,theperiodswillbeadjacent.
Wecanavoidwraparounderrorusing
PA+B1
.
Ingeneral,weuse
P=A+B1
.
–p.10/44

10. Padding(4)
Source:http://www.imageprocessingbook.com–p.11/44

11. Padding(5)
Source:http://www.imageprocessingbook.com–p.12/44

12. Padding:Example
Source:http://www.imageprocessingbook.com–p.13/44

13. SpatialRepresentationofaFilter
1.Multiplylter
H(u;v)
by
(1)
u+v
(centering)
2.ComputetheinverseDFT
3.MultiplytherealpartoftheinverseDFTby
(1)
x+y
Source:http://www.imageprocessingbook.com–p.14/44

14. ASimpleFilter:NotchFilter(1)
Wewishtoforcetheaveragevalueofanimageto
zero:
F(0;0)
istheaveragevalueoftheimage
ifsizeoftheimageis
MN
thenthecentered
valueoftheFouriertransformistheaveragevalue
(
M
2
,
N2
) H(u;v)=
(
0if(u;v)=(
M
2
;
N2
)
1otherwise:
–p.15/44

15. ASimpleFilter:NotchFilter(2) notchlter(constantfunctionwithaholeattheorigin)
H(u;v)=
(
0if(u;v)=(
M
2
;
N2
)
1otherwise:
Source:http://www.imageprocessingbook.com–p.16/44

16. LowpassandHighpassFilter
Source:http://www.imageprocessingbook.com–p.17/44

17. SmoothingFrequency-DomainFilters
G(u;v)=H(u;v)F(u;v)
Smoothing:attenuatingspeciedrangeof
high-frequencycomponents
Threetypesoflowpasslter:
ideal(verysharp) Butterworth(tunable) Gaussian(verysmooth)
–p.18/44

18. IdealLowpassFilters(1)
H(u;v)=
(
1ifD(u;v)D
0
0ifD(u;v)>D
0
D
0
:nonnegativequantity
D(u;v)
:distancefromapoint
(u;v)
totheorigin
origin:
(
M
2
;
N2
)
(centered)
Then,
D(u;v)=
r
(u
M
2
)
2
+(v
N
2
)
2
–p.19/44

19. IdealLowpassFilters(2)
ideallter(ILPF):allfrequenciesinsideacircleof
radius
D
0
arepassedwithnoattenuation
Inourcase:
zero-phase-shiftlter radiallysymmetric
Transition:cutofffrequency Ringingbehaviorofthelter
Source:http://www.imageprocessingbook.com–p.20/44

20. IdealLowpassFilters:Example
Source:http://www.imageprocessingbook.com–p.21/44

21. CutoffFrequency
Computecirclesthatenclosespecicamountoftotal
imagepower
P
T P
T
=
M1X u=0
N1
X v=0
P(u;v)
P(u;v)=jF(u;v)j
2
=R
2
(u;v)+I
2
(u;v)
=
100
P
T
X
u
X
v
P(u;v)
Source:http://www.imageprocessingbook.com–p.22/44

22. RingingEffectofILPF
Source:http://www.imageprocessingbook.com–p.23/44

23. ButterworthLowpassFilters
BLPFoforder
n
,withacutofffrequencydistance
D
0
is
denedas
H(u;v)=
1
1+[
D(u;v)
D
0
]
2n
D(u;v)=
r
(u
M
2
)
2
+(v
N
2
)
2
noclearcutoffbetweenpassedandltered
frequencies
–p.24/44

24. ButterworthLowpassFilters(2)
Source:http://www.imageprocessingbook.com–p.25/44

25. ButterworthLowpassFilters:Example
n=2
,radii=
5;15;30;80;230
Source:http://www.imageprocessingbook.com–p.26/44

26. RingingEffectofBLPF
Source:http://www.imageprocessingbook.com–p.27/44

27. GaussianLowpassFilters(1)
TheformofagaussianlowpasslterGLPFin2Dis:
H(u;v)=e
D
2
(u;v)
2
2
D(u;v)=
r
(u
M
2
)
2
+(v
N
2
)
2
TheinverseFouriertransformofaGLPFisalsoa
Gaussian
AspatialGaussianlterwillhavenoringing
–p.28/44

28. GaussianLowpassFilters(2)

:mesureofthespreadoftheGaussiancurve
Let
=D
0
,then:
H(u;v)=e
D
2
(u;v)
2D
2
0
Source:http://www.imageprocessingbook.com–p.29/44

29. GaussianLowpassFilters:Example(1)
n=2
,radii=
5;15;30;80;230
Source:http://www.imageprocessingbook.com–p.30/44

30. GaussianLowpassFilters:Example(2)
Source:http://www.imageprocessingbook.com–p.31/44

31. HighpassFilters(1)
Highpasslter:imagesharpening(low-frequency
attenuation)
Inourcase:
zero-phase-shiftlter radiallysymmetric
H
hp
(u;v)=1H
lp
(u;v)
with:
H
lp
(u;v)
:transferfunctionofthecorresponding
lowpasslter
H
hp
(u;v)
:transferfunctionofthecorresponding
highpasslter
–p.32/44

32. HighpassFilters(2)
Source:http://www.imageprocessingbook.com–p.33/44

33. IdealHighpassFilters(1)
H(u;v)=
(
0ifD(u;v)D
0
1ifD(u;v)>D
0
D
0
:nonnegativequantity
D(u;v)
:distancefromapoint
(u;v)
totheorigin
origin:
(
M
2
;
N2
)
(centered)
Then,
D(u;v)=
r
(u
M
2
)
2
+(v
N
2
)
2
–p.34/44

34. IdealLowpassFilters:example
Source:http://www.imageprocessingbook.com–p.35/44

35. ButterworthHighpassFilters
BLPFoforder
n
,withacutofffrequencydistance
D
0
is
denedas
H(u;v)=
1
1+[
D
0
D(u;v)
]
2n
D(u;v)=
r
(u
M
2
)
2
+(v
N
2
)
2
noclearcutoffbetweenpassedandltered
frequencies
–p.36/44

36. ButterworthHighpassFilters:Example
Source:http://www.imageprocessingbook.com–p.37/44

37. GaussianHighpassFilters
TheformofagaussianlowpasslterGLPFin2Dis:
H(u;v)=1e
D
2
(u;v)
2D
2
0
D(u;v)=
r
(u
M
2
)
2
+(v
N
2
)
2
TheinverseFouriertransformofaGLPFisalsoa
Gaussian
AspatialGaussianlterwillhavenoringing
–p.38/44

38. GaussianHighpassFilters:Example
Source:http://www.imageprocessingbook.com–p.39/44

39. Laplacian:SpatialDomain(1) r
2
f=
@
2
f
@x
2
+
@
2
f
@y
2
@
2
f
@x
2
=f(x+1;y)+f(x1;y)2f(x;y)
@
2
f
@y
2
=f(x;y+1)+f(x;y1)2f(x;y)
r
2
f=f(x+1;y)+f(x1;y)+f(x;y+1)
+f(x;y1)4f(x;y)
Source:http://www.imageprocessingbook.com–p.40/44

40. Laplacian:SpatialDomain(2)
g(x;y)=
(
f(x;y)r
2
f(x;y)ifthecentercoecientisnegative
f(x;y)+r
2
f(x;y)ifthecentercoecientispositive
Source:http://www.imageprocessingbook.com–p.41/44

41. Laplacian:FrequencyDomain(1)
F

d
n
f(x)
dx
n

=(ju)
n
F(u)
Then,
F
h
@
2
f(x;y)
@x
2
+
@
2
f(x;y) @y
2
i
=(ju)
2
F(u;v)+(jv)
2
F(u;v)
=(u
2
+v
2
)F(u;v)
Finally,
F[r
2
f(x;y)]=(u
2
+v
2
)F(u;v)
TheLaplacianlterinthefrequencydomainis:
H(u;v)=(u
2
+v
2
)
–p.42/44

42. Laplacian:FrequencyDomain(2)
–p.43/44

43. Laplacian:Example
–p.44/44