image_deblurring.ppt

7/23/2019 image_deblurring.ppt

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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




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




)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




6estoration o ,ST Images




)nother 78ample




The 6eal !ptical+ Solution

'eore the repair )ter the repair




)pplication II+" Law

7norcement

$otion#blurred license plate image




6estoration 78ample




)pplication into 'iometrics

out#o#ocus iris image




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




'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




'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




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)




*"m!n: 1D *oriontal motion %lurring 1 1 1 1 1 1 17

./40d.

Image 78ample

./10d.8m>n+




'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+




Image Deblurring Introduction

Inverse fltering Suer rom noise amplifcation

Wiener fltering Tradeo between image recovery and


Landweber algorithm




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




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




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




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




Image E,am-le


deblurred image ater9seudo#inverse fltering

δ<02.+




Image Deblurring

Introduction




Landweber algorithm




(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




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




Image E,am-le


deblurred image aterwiener fltering

J<020.+




Image 78ample -on;t+

J<02. J<0200.J<020.




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




Image 78ample

γ <02. γ <0200.γ <020.




Image Deblurring

Introduction




Landweber algorithm




$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




(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"




(umerical 78ample -on;t+

(ote that iteration Auic@ly converges to 8<.




$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




>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



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

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image_deblurring.ppt