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EE465: Introduction t 1
Image Deblurring Introduction
Inverse fltering
Suer rom noise amplifcation Wiener fltering
Tradeo between image recovery and
noise suppression Iterative deblurring*
Landweber algorithm
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EE465: Introduction t 2
Introduction Where does blur come rom
!ptical blur" camera is out#o#ocus $otion blur" camera or ob%ect is moving
Why do we need deblurring &isually annoying
Wrong target or compression 'ad or analysis (umerous applications
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EE465: Introduction t 3
)pplication I+"
)stronomical Imaging The Story o ,ubble
Space Telescope
,ST+ ,ST -ost at Launch
.//0+" 1.23 billion
$ain mirror
imperections due tohuman errors
4ot repaired in .//5
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EE465: Introduction t 4
6estoration o ,ST Images
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EE465: Introduction t 5
)nother 78ample
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EE465: Introduction t 6
The 6eal !ptical+ Solution
'eore the repair )ter the repair
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EE465: Introduction t 7
)pplication II+" Law
7norcement
$otion#blurred license plate image
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EE465: Introduction t 8
6estoration 78ample
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EE465: Introduction t 9
)pplication into 'iometrics
out#o#ocus iris image
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EE465: Introduction t 10
h(m,n) + x(m,n) y(m,n)
!" nmw
# $inear degradation model
!" nmh %lurring &ilter
!0"'!" 2w N nmw σ additi(e )*ite aussian noise
$odeling 'lurring 9rocess
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EE465: Introduction t 11
'lurring :ilter 78ample
2
e,-"!"2
2
2
2
121
σ
wwww H
+−=
2e,-"!"
2
22
σ
nmnmh
+−=
:T
4aussian flter can be used to appro8imate out#o#ocus blur
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EE465: Introduction t 12
'lurring :ilter 78ample
-on;t+
!" 21 ww H
:T
$otion blurring can be appro8imated by .D low#pass flter along the moving
$)TL)' code" h<:S97-I)L=motion=>/>50+?
!" nmh
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EE465: Introduction t 13
2
2
10log10w
z BSNR
σ
σ =
.lurring /
The -urse o (oiseh(m,n) + x(m,n) y(m,n)
!0"'!" 2w
N nmw σ
z(m,n)
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EE465: Introduction t 14
*"m!n: 1D *oriontal motion %lurring 1 1 1 1 1 1 17
./40d.
Image 78ample
./10d.8m>n+
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EE465: Introduction t 15
'lind vs2 (onblind
Deblurring 'lind deblurring deconvolution+"
blurring @ernel hm>n+ is
un@nown (onblind deconvolution"
blurring @ernel hm>n+ is @nown
In this course> we only cover thenonblind case the easier case+
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EE465: Introduction t 16
Image Deblurring Introduction
Inverse fltering Suer rom noise amplifcation
Wiener fltering Tradeo between image recovery and
noise suppression Iterative deblurring*
Landweber algorithm
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EE465: Introduction t 17
In(erse ilter
h(m,n)
%lurring &ilter
h I (m,n) x(m,n)
y(m,n)
in(erse &ilter
∑ ∑
∞
−∞=
∞
−∞= ∀=−−
=⊗=
k l
I
I
combi
nmnml k hl nk mh
nmhnmhnmh
!"!!"!"!"
!"!"!"
δ
!"
1!"
21
21ww H
ww H I =
To compensate the blurring> we reAuire
hcombi (m,n)
x(m,n)B
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EE465: Introduction t 18
Inverse :iltering -on;t+
h(m,n) + x(m,n) y(m,n)
!" nmw
h I (m,n)
in(erse &ilter
x(m,n)B
Spatial"
!"!"!"!""!"!"!" nmhnmwnmhnm xnmhnm ynm x I I ⊗+⊗=⊗=
:reAuency"
!"
!"!"
!"!"!"!"!"!"!"8
21
2121
21
212121212121
ww H
wwW ww X
ww H wwW ww H ww X ww H wwY ww X I
+=
+==
amplifed noise
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EE465: Introduction t 19
Image E,am-le
: *; does t*e am-li&ied noise loo< so %ad=
>: eros in H(w1 ,w2 ) corres-ond to -oles in H I (w1 ,w2 )
motion blurred imageat 'S(6 o C0d'
deblurred image aterinverse fltering
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EE465: Introduction t 20
Pseudo?in(erse ilter
'asic idea"
To handle eros in ,w.>wE+> we treat them separately
when perorming the inverse fltering
≤
>=−
δ
δ
@!"@0
@!"@!"
1
!"
21
21
2121
ww H
ww H ww H ww H
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EE465: Introduction t 21
Image E,am-le
motion blurred imageat 'S(6 o C0d'
deblurred image ater9seudo#inverse fltering
δ<02.+
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EE465: Introduction t 22
Image Deblurring
Introduction
Inverse fltering Suer rom noise amplifcation
Wiener fltering Tradeo between image recovery and
noise suppression Iterative deblurring*
Landweber algorithm
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EE465: Introduction t 23
(orbert Wiener .F/C#./GC+
The renowned $IT proessor (orbert Wienerwas amed or his absent#mindedness2 Whil
crossing the $IT campus one day> he wasstopped by a student with a mathematicalproblem2 The perple8ing Auestion answered>(orbert ollowed with one o his own" HIn whdirection was I wal@ing when you stopped mhe as@ed> prompting an answer rom the
curious student2 H)h>H Wiener declared>Hthen I=ve had my lunch
)necdote o (orbert Wiener
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EE465: Introduction t 24
iener iltering
K ww H
ww H
ww H mmse += 221
21
21 @!"@
!"A
!"
)lso called $inimum $ean SAuare 7rror $$S7+ or Least#SAuare LS+ f
constant
78ample choice o J"2
2
z
w K
σ
σ = noise energy
signal energy
J<0 → inverse fltering
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EE465: Introduction t 25
Image E,am-le
motion blurred imageat 'S(6 o C0d'
deblurred image aterwiener fltering
J<020.+
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EE465: Introduction t 26
Image 78ample -on;t+
J<02. J<0200.J<020.
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EE465: Introduction t 27
Bonstrained $east /Cuare iltering
2
21
2
21
2121
@!"@@!"@!"A!"
ww ww H ww H ww H mmse
γ +=
Similar to Wiener but a dierent way o balancing the tradeo betwee
78ample choice o -"
−−−
−
=010141
010
!" nm
Laplacian operatorγ <0 → inverse fltering
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EE465: Introduction t 28
Image 78ample
γ <02. γ <0200.γ <020.
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EE465: Introduction t 29
Image Deblurring
Introduction
Inverse fltering Suer rom noise amplifcation
Wiener fltering Tradeo between image recovery and
noise suppression Iterative deblurring*
Landweber algorithm
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EE465: Introduction t 30
$ethod o SuccessiveSubstitution
) powerul techniAue or fnding theroots o any unction 8+
'asic idea 6ewrite 8+<0 into an eAuivalent eAuation
8<g8+ 8 is called f8ed point o g8++
Successive substitution" 8iK.<g8i+
nder certain condition> the iteration willconverge to the desired solution
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EE465: Introduction t 31
(umerical 78ample
23" 2 +−= x x x !
2!1 21 == x x Two roots"
"3
2023"
22 x "
x x x x x ! =
+=⇒=+−=
3
22
1
+=+
ii
x xsuccessive substitution"
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EE465: Introduction t 32
(umerical 78ample -on;t+
(ote that iteration Auic@ly converges to 8<.
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EE465: Introduction t 33
$and)e%er Iteration
/uccessi(e su%stitution:
00 = X
"1 nnn HX Y X X −+=+ β
HX Y X ! −="
!"!"!" 212121 ww X ww H wwY =Linear blurring
We want to fnd the root o
"""0" X " HX Y X X ! X X X ! =−+=+=⇒= β β
β rela8ation parameter M controls convergence property
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EE465: Introduction t 34
>s;m-totic >nal;sis
H I R RX Y X nn β β −=+=+ !1
Y R I R I Y R X nn
i
i
n "" 11
0
+−
=
−−== ∑ β β
>ssume con(ergence condition: 0lim 1 =+
∞→
Y Rk
k
X Y H Y R I X X nn
==−== −−
∞→∞11"lim β
in(erse &iltering
"1 nnn HX Y X X −+=+ β
)e *a(e
1"!1
11
0
≠−−
=∑−
=
# #
# #
N N
n
n
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EE465: Introd ction t 35
)dvantages o LandweberIteration
(o inverse operation e2g2> division+is involved
We can stop the iteration in themiddle way to avoid noiseamplifcation
It acilitates the incorporation o apriori @nowledge about the signalN+ into solution algorithm$ore detailed analysis is included in 773G3" )dvanced Image 9roces