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Filter Design
Lecture 20: 10-Sep-12
Dr. P P Das
Filtering in Frequency Domain
yxyxf
vuF
vuH
vuH
vuFvuHyxg
)1(by ),(y premultipl
- centered is ),( Needs
centerabout symmetric is ),(Usually
unctionTransfer F Filter :),(
),(),(),( 1
Example: Frequency Domain Filter
otherwise
QvPuvuH
,1
2/,2/,0),(
),(),(),( 1 vuFvuHyxg
Wraparound Effect & Zero Padding
Can we pad the filter?
Filtering Steps
f(x,y) is MxN. Pad to PxQ. Typically, P=2M, Q=2N Form fp(x,y) of size PxQ by adding necessary zeros to f(x,y) Multiply fp(x,y) by (–1)^(x+y) to center transform Compute F(u,v) by DFT of fp(x,y) Use real symmetric filter H(u,v), of size PxQ & center at (P/2,
Q/2). Form G(u,v)=H(u,v)F(u,v) Compute gp(x,y) by product of real part of IDFT of G(u,v) and
(–1)^(x+y). Extract g(x,y) taking MxN at left top corner of gp(x,y)
Filter Design
Lecture 21-22: 11-Sep-12
Dr. P P Das
Filtering Steps
f(x,y) is MxN. Pad to PxQ. Typically, P=2M, Q=2N Form fp(x,y) of size PxQ by adding necessary zeros to f(x,y) Multiply fp(x,y) by (–1)^(x+y) to center transform Compute F(u,v) by DFT of fp(x,y) Use real symmetric filter H(u,v), of size PxQ & center at (P/2,
Q/2). Form G(u,v)=H(u,v)F(u,v) Compute gp(x,y) by product of real part of IDFT of G(u,v) and
(–1)^(x+y). Extract g(x,y) taking MxN at left top corner of gp(x,y)
Filtering in Spatial vis-à-vis Frequency Domains
Filter(FIR) ResponseImpulse Finite:),(
: Hencefinite. are ),( in Quantities
),( of ResponseImpulse :y)(x,
)},({),(Output
: Hence.1),(),,(),(Let
),( : Filter DomainSpatial
),( : FilterDomainFrequency
1
yxh
yxh
vuHh
vuHyxh
vuFyxyxf
yxh
vuH
Example Gaussian Filter
222
22
2
2/
2)(
)(
: FilterLowpass
x
u
Aexh
AeuH
21
22
21
2/2/
and
22)(
)(
: FilterHighpass
222
2221
2
22
221
2
BA
BeAexh
BeAeuHxx
uu
versa- viceand
)( of profileNarrow
)( of profile Broad
High
xh
uH
222
22
2
2/
2)(
)(x
u
Aexh
AeuH
22
2222
12
22
221
2
22
21
2/2/
22)(
)(xx
uu
BeAexh
BeAeuH
Image Smoothing
Frequency Domain Filters
Image Smoothing – Low-pass Filter
Low-pass Filtering– Ideal: Very sharp– Butterworth– Gaussian: Very Smooth
Butterworth Filter is parameterized by Filter Order– High Order Ideal– Low Order Gaussian
Ideal Low-Pass Filter (ILPF)
Frequencyoff-Cut :
)2/()2/(),(
),(,0
),(,1),(
0
2/122
0
0
D
QvPuvuD
DvuD
DvuDvuH
Ideal Low-Pass Filter (ILPF)
Ideal Low-Pass Filter (ILPF)
),(),(),(),( where
),( : PowerImage Total
222
1
0
1
0
vuIvuRvuFvuP
vuPPP
u
Q
vT
}),(:,{ 0
/),(100
:filterby off-cutpower ofPercent
DvuDvuTPvuP
Ideal Low-Pass Filter (ILPF)
ILPF
Radius Power (α)
10 87.0
30 93.1
60 95.7
160 97.8
460 99.2
Blurring & Ringing decreases with increase of radius / power
Blurring & Ringing in ILPF
ILPF Box Filter Sinc in Spatial Domain
Butterworth Low-Pass Filter (BLPF)
BLPFofOrder :
Frequencyoff-Cut :
)2/()2/(),(
/),(1
1),(
0
2/122
20
n
D
QvPuvuD
DvuDvuH n
Butterworth Low-Pass Filter (BLPF)
Unlike ILPF, no sharp cut-off
BLPF
Radius
10
30
60
160
460
ILPF BLPF
- Smooth transition in blurring- No ringing
- Sharp transition in response causing heavy blurring & ringing
BLPF: Ringing increases with order
Gaussian Low Pass Filter (GLPF)
Frequencyoff-Cut :
),(
centerabout spread of Measure:
)2/()2/(),(
),(
0
2/),(
2/122
2/),(
20
2
22
D
evuH
QvPuvuD
evuH
DvuD
vuD
Gaussian Low Pass Filter (GLPF)
Radius
10
30
60
160
460
GLPF
- No ringing as IDFT of Gaussian is Gaussian
ILPF BLPF-Smooth transition in blurring- No ringing
GLPF- No ringing- Sharp transition in
response causing heavy blurring & ringing
Radius: 10, 30, 60, 160, 460
GLFP
BLFP
ILFP
Image Smoothing: Low-Pass Filters
LPF: Character Recognition
LFT: Printing & Publishing
LFT: Printing & Publishing
LPF: Satellite Imagery
Image Sharpening
Frequency Domain Filters
High-Pass Filter (HPF)
),(1),( vuHvuH LPHP
Image Sharpening – High-pass Filter
High-pass Filtering– Ideal: Very sharp– Butterworth– Gaussian: Very Smooth
Butterworth Filter is parameterized by Filter Order– High Order Ideal– Low Order Gaussian
Ideal High-Pass Filter (IHPF)
Frequencyoff-Cut :
)2/()2/(),(
),(,1
),(,0),(
0
2/122
0
0
D
QvPuvuD
DvuD
DvuDvuH
IHFP
BHFP
GHFP
Image Sharpening: High-Pass Filters
Ideal High-Pass Filter (IHPF)
Butterworth High-Pass Filter (BHPF)
BLPFofOrder :
Frequencyoff-Cut :
)2/()2/(),(
),(/1
1),(
0
2/122
20
n
D
QvPuvuD
vuDDvuH n
Butterworth High-Pass Filter (BHPF)
Gaussian High Pass Filter (GHPF)
Frequencyoff-Cutt :
)2/()2/(),(
1),(
0
2/122
2/),( 20
2
D
QvPuvuD
evuH DvuD
Gaussian High Pass Filter (GHPF)
IHPFBHPF GHPF
Radius: 30, 60, 160
Image Sharpening: High-Pass Filters
HPF: Finger Print
BHPF: n=4, D0=50
Thank you
Filter Design
Lecture 23: 17-Sep-12
Dr. P P Das
Laplacian in Spatial Domain
Laplacian– Isotropic – Rotation Invariant
),(4)1,()1,(
),1(),1(2
yxfyxfyxf
yxfyxff
),(),(),( 2 yxfcyxfyxg
2
2
2
22
z
f
t
ff
Laplacian in Frequency Domain
dudteeztfFztf ujtj 22),(),()},({
),()(4}{
),(4}{),(4}{
)},({4
),()2(
),()},({),(
2222
2
2
2
22
222
222
2
2
122
2222
2
221
Ff
z
f
t
ff
Fz
fF
t
f
F
ddeeFjt
f
ddeeFFztf
jtj
jtj
2
2
2
22
z
f
t
ff
Laplacian in Frequency Domain
)},(),{{),(
:image an of Laplacian
),(4
))2/()2/(4),(
:rectanglefrequency ofcenter respect to With
)(4),(
12
22
222
222
vuFvuHyxf
vuD
QvPuvuH
vuvuH
Laplacian in Frequency Domain
Enhancement Eq:
Scales of and as computed by DFT differ widely due to the DFT process
Normalize to [0,1] before DFT Normalize to [-1,1]
1
),(),(),( 2
c
yxfcyxfyxg
),( yxf ),(2 yxf
),( yxf
),(2 yxf
Laplacian in Frequency Domain
Comparative Laplacian in Spatial & Frequency Domains
Unsharp Mask, Highboost Filtering & High-Frequency-Emphasis Filtering
In spatial domain:
Masking Unsharpemphasized- De:1
Filtering Highboost:1
Masking Unsharp:1
),(*),(),(
),(),(),(
k
k
k
yxgkyxfyxg
yxfyxfyxg
mask
mask
Unsharp Mask, Highboost Filtering & High-Frequency-Emphasis Filtering
In frequency domain:
),(),(),(
),(),(),(1 vuFvuHyxf
yxfyxfyxg
LPLP
LPmask
)},(),(*1{
)},(),(1*1{
),(*),(),(
1
1
vuFvuHk
vuFvuHk
yxgkyxfyxg
FilterEmphasisFrequencyHigh
HP
LP
mask
Unsharp Mask, Highboost Filtering & High-Frequency-Emphasis Filtering
In frequency domain:
sfrequencie high of oncontributi theControls:0
origin fromoffset theControls:0
)},(),(*{),(
2
1
211
k
k
vuFvuHkkyxgFilterEmphasisFrequencyHigh
HP
Image: 416x596
D0=40 (5% of short side of padded image)
k1=0.5
k2=0.75
Homomorphic Filtering
Illumination-Reflectance Model in frequency domain Illumination Component
– Slow Spatial Variations– Attenuate contributions by illumination
Reflectance Component– Varies abruptly – junctions of dissimilar objects– Amplify contributions by reflectance
Simultaneous dynamic range compression & contrast enhancement
Homomorphic Filtering
),(),(),(
)},({ln)},({ln
)},({ln)},({
),(ln),(ln
),(ln),(
),(),(),(
vuFvuFvuZ
yxryxi
yxfyxz
yxryxi
yxfyxz
yxryxiyxf
ri
Homomorphic Filtering
),('),('
)},(),({)},(),({
)},({),(
),(),(),(),(
),(),(),(
),(),(),(
11
1
yxryxi
vuFvuHvuFvuH
vuSyxs
vuFvuHvuFvuH
vuZvuHvuS
vuFvuFvuZ
ri
ri
ri
Homomorphic Filtering
),(),(),( 00),('),('),( yxryxieeeyxg yxryxiyxs
Homomorphic Filtering
Illumination Component– Slow Spatial Variations– Low Frequencies log of illumination– attenuate contributions by illumination
Reflectance Component– Varies abruptly – junctions of dissimilar objects– High frequencies log of reflectance– amplify contributions by reflectance
Simultaneous dynamic range compression & contrast enhancement
1L
1H
LDvuDc
HL evuH 20
2 /),(1)(),(
Homomorphic Filtering
Image: 1162x746γL=0.25, γH=2, c=1, D0=80
Band-reject & Band-pass Filters
),(1),( vuHvuH BRBP
Band-reject & Band-pass Filters
D0=80, n=4
Notch Filters – Narrow Filtering
Notch Filters – Narrow Filtering
Q
kkkNR vuHvuHvuH
1
),(),(),(
2/122
2/122
)2/()2/(),(
)2/()2/(),(
kkk
kkk
vNvuMuvuD
vNvuMuvuD
Butterworth Notch Reject Filters
nkkk
nkk
NR
vuDDvuDD
vuH
20
3
12
0 ),(/1
1
),(/1
1
),(
),(1),( vuHvuH NRNP
D0=80, n=4
Thank you
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