Image processing with nonseparable multiwavelets

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multiwavelets

Ana M.C. Ruedin

Departamentode Computaion

Faultad de CieniasExatas y Naturales

Universidad de Buenos Aires

E-mail: anitad.uba.ar

Deember 22,2000

Abstrat

Inageneralontextweintrodueimageproessingwith1-dimensional

waveletsandshowthediereneswithnonseparable2-dimensionalwavelets

havingquinunxdeimation,intherstplae,andwithbalaned

nonsep-arable2-dimensionalmultiwavelets,intheseondplae. Allofthemare

orthogonal. Formulaefor analysisand synthesisaregivenfor thelatter.

Therststepsareillustratedwithimages. Thedeompositionofthe

orig-inalimageinto2inputimagesisexplained. Weillustratewithexamples

2appliations: zoom-in(interpolation)andompressionofimages.

Keywords: multiwavelets,nonseparable,quinunx.

1 INTRODUCTION

Wavelettransforms havegoodtime-frequeny loalization, whih makesthem

aneÆienttoolforsignalproessing: theyareusedforompression,denoising,

zoominganimage. Generalizationsofwaveletsinludei)nonseparablewavelets

andii) multiwavelets. We showhowaombination ofbothworksin pratie,

andgiveexamplesoftwoappliations.

Insetion2webrieyintrodueimageproessingwith1-dimensionalwavelets.

In setion 3 we show how these are used to proess images {the separable

wavelets{ and how a more general approah inludes 2-d lters and

deima-tionwithadilationmatrix. Weillustratethisdening quinunxdeimation.

Insetion4multiwaveletsareintrodued,andinsetion5nonseparable

mul-tiwaveletsareintrodued. Formulaeforanalysisandsynthesisaregivenforthe

latterin setion6. Therststepsareillustratedwith images. The

deomposi-tionoftheoriginalimageinto2inputimagesisexplained. Finallyweillustrate

with examples two appliations: zoom-in (interpolation) and ompression of

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2 Wavelet proessing a 1-d signal

Givenaone-dimensionalsignal (0)

k

,ndingonestepofthewavelettransformof

thesignalonsistsini)lteringthesignalwithalow-passlterH 0 =[::h 3 h 2 h 1 h 0

::℄andarryingoutadeimationby2,givingasmoothedversionofthesignal

( 1)

k

(approximationoeÆients),andii)lteringthesignalwithahigh-pass

lterG 0

andarryingoutadeimationby2,givingthedetailoeÆientsd ( 1)

k ,

whih apturethe nedetails of thesignal. Ingure1wehavethetransform

andinversetransformsheme. Twofuntions denethewavelettransform: the

salingfuntion(x)andthewavelet (x),whihverify:

(x)= X

k 2Z h

k

(2x k ) (x)=

X

k 2Z g

k

(2x k )

CoeÆients (0)

k

orrepond to a ertain funtion f(x) written in the basis of

integertranslationsofthesalingfuntion: f(x k)g,andunderertain

on-ditionsthef (0)

k

gareveryloseto thesamplesoff(x)(see[Daubehies1992℄).

f(x)= X k (0) k (x k) CoeÆients ( 1) k

orrepondto thebasis f 1 p 2 ( x 2

k)g andd ( 1)

k

orrepond

tothebasisf 1 p 2 ( x 2 k)g

Inaseondstepthewholeproessisrepeatedon ( 1)

k

: wenowobtainthe

oarserapproximationoeÆients ( 2)

k

,andthedetailoeÆientsd ( 2)

k

,whih

orrespondto less ne detail. Weend uphavingaveryoarse approximation

oftheoriginalsignal,andaseriesofdetailsatdierentsales.

f(x)= X k ( L) k L;k (x)+ 1 X j= L+1 X k d (j) k j;k (x) (1) where j;k (x)=2

j=2

(2 j

x k) ;

j;k (x)=2

j=2

(2 j

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Figure2: Originalimage

Figure 3: OnestepoftheseparableDaubehies4transform

Salingfuntions andwaveletsanbeonstrutedtohaveertainuseful

prop-ertiessuh asshort support,orthogonality,symmetry, andpolynomial

approx-imation. However,exeptfortheHaar waveletthere anbenosymmetryand

orthogonalitysimultaneouslyforreallters.

3 Wavelets for image proessing

The way to proess an image with 1d- wavelets is to apply the transform to

therowsand then to theolumns of the resultingimage. In gure2we have

the original image of a phone, and and in gure 3 we have one step of the

separable Daubehies 4 transform. At the upper left orner is the phone at

aoarser resolution; the 3other submatries orrespond to details,whih are

mostlyorientedinthevertialandhorizontaldiretions. However,thisdoesnot

agreewithourvisualpereption.

Toavoidthis,imagesaretreatedwithnonseparablelters,andthe

deima-tionisdonewithageneraldilationmatrix. Examplesofnonseparable

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3.1 Nonseparable wavelets :Quinunx (diagonal)

deima-tion and upsampling

We onsider two possible dilation matries: D

1

, a reetion followed by an

expansionof p

2, andD

2

,arotationfollowedbyanexpansionof p

2. Forboth

matriesjDj=jdet (D)j=2.

D 1 = 1 1 1 1 D 2 = 1 1 1 1 BothD 1 andD 2

indue adeompositionofthesetsofallpairsofintegersZ 2

into 2osets:

1 and

2

; forming thequinunxsublatties -blak and white

squaresofahess-table:

Z 2 = 1 [ 2 ; 1 =fDZ 2 g ; 2 = DZ 2 + 1 0

LetD bethedilationmatrix.

WedenedeimationofanimagewithD as

Y =X #D () Y

k =X

Dk :

WedeneupsamplinganimagewithDas:

Y =X "D,Y

k =

X

j

; if k=Dj for somej2Z 2

0 if D 1

k2=Z 2

For example, if we have image X =

1 2

3 4

; then X upsampled with

D

1 isY =

2

4

0 1 0

2 0 3

0 4 0 3

5

:Thedilationmatrixis areetion,anditindues a

reetionintheupsampledimageY. IfY isnowdeimatedwithD

1

,wereover

X. Quinunxdeimation isequivalenttoeliminatingallthewhitesquaresofa

hess-table.

4 Multiwavelets

Inamoregeneralontext,weonsider theapproximationimages asgenerated

bytheintegertranslationsof2ormoresalingfuntions:theygiveriseto

multi-salingfuntionsandmultiwavelets. Multiwaveletsin1-dhavebeenonstruted

to have suitableproperties, suh as orthogonality, polynomialapproximation,

short support and symmetry. They have given good results for signal

om-pression: see [Strela1996℄ [Strelaetal.toappear℄ and [PlonkaandStrela1998℄.

Balanedmultiwavelets[LebrunandVetterli1997℄ wereintrodued in order to

avoid prelteringthe inputdata, so that the lowpassbranhof thetransform

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Inpreviousworks[Ruedin1999℄ [Ruedin2000℄ theauthoronstrutedexamples

of ontinuous balaned nonseparable orthogonal multisaling funtions, in an

attempttoombineadvanesmadein bothdiretions: multiwavelets,and

non-separable bidimensional wavelets. They are ompatly supported, have

quin-unxdeimation, andhavepolynomialapproximationorders (i.e. auray) 2

and3. Theirorrespondingmultiwaveletswerealsofound. (Toobtainthelter

oeÆientspleasee-mailtheauthor.) Wegiveashortoutlineofthetheory.

Let

1 and

2

be2ontinuous salingfuntions denedoverR 2

and

asso-iatedtoadilationmatrixD. Invetorformthedilationequationbeomes

(x)= X k 2Z 2 H k

(D x k )

1 (x) 2 (x) = X k [H k ℄ 1

( D x k)

2

( D x k)

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whereH

k

are2x2matrieswithindies

M 0= 2 4 0 H 1;1 H 2;1 0 H 0;0 H 1;0 H 2;0 H 3;0 0 H 1; 1 H 2; 1 0 3 5

andD isthedilationmatrix. Notationisratherlooseand

1

(Dx k)means

thatweapply

1

tothe2entriesofD x k= d 11 d 12 d 21 d 22 x 1 x 2 k 1 k 2

Thenumberofwavelets isjDj 1=1in bothases. Theequationfor the

multiwavelet,in vetorform,is:

(x)= X

k G

k

(D x k ) (3)

6 Analysis-synthesis formulae:

Assume that wedeomposethe originalimage into twoinputimages (0) 1;k and (0) 2;k (k2Z

2

),andletf(x)bethefuntionthat veries:

f(x)= X k 2Z 2 (0)T ;k

(x k), where (0) ;k = " (0) 1;k (0) 2;k # :

Theanalysissheme(seegure6) outputs2approximationimages: ( 1)

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( 1)

2;k

, and2 detailimages: d ( 1)

1;k

and d ( 1)

2;k

;whose expression we wantto nd.

Weset ( 1) ;k = " ( 1) 1;k ( 1) 2;k # d ( 1) ;k = " d ( 1) 1;k d ( 1) 2;k # . ( 1) ;k = 1 p jDj X j2Z 2 H j Dk (0) ;j (4) d ( 1) ;k = 1 p jDj X j2Z 2 G j Dk (0) ;j : (5)

Similarlyitan beshownthat thesynthesis formula is:

(0) ;k = 1 p jDj 2 4 X j2Z 2 H T k Dj ( 1) ;j + X j2Z 2 G T k Dj d ( 1) ;j 3 5 (6)

Ingure5wehave4images: theyaretheoeÆientsofonestepofthe

mul-tiwavelettransform: d ( 1)

1;k

(top left), ( 1)

1;k

(topright)andd ( 1) 2;k (bottomleft), ( 1) 2;k

(bottomright). Thedilationmatrix wasD

1

Theeetof downsampling

withD

1

intheanalysisformulaistoreetandontrattheimage.

Ingure6aretheoeÆientsof2stepsofthesametransform: d ( 1) 1;k ;d ( 2) 1;k , ( 2) 1;k

(top) and d ( 1) 2;k ; d ( 2) 2;k , ( 2) 2;k

(bottom). After 2 steps the image has

reovereditsoriginalorientation. Ittakes4stepstodosoifthedilationmatrix

isD

2 :

At eah step, before the images are proessed they haveto be periodized,

otherwise there are artifats at the borders. Periodization is dierent if the

frames of the images are normally oriented squares (after the even steps) or

diamond-orientedsquares(aftertheoddsteps).

Atthe beginning of theproessthe originalimage may be opiedso asto

get 2input images. Otherwise, the original image anbe deomposed into 2

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Figure8: Zoom-inwithD1-a3-bal

NotiethatinthiswayalltheoeÆientsofthemultiwavelettransformneed

asmuhstorageastheoriginalimage,andanbemadetotintoit.

7 Appliations:

7.1 Zoom in

Tozoom-inanimagewithone-dimensionalwavelets,weupsampletheimageand

thenonvolveitwithalow-passlter{i.eweapplythesynthesisalgorithmof

sheme1to theimage,after settingtozerothedetailoeÆients.

Tozoom-inanimagewithnonseparablemultiwavelets,weapplythe

synthe-sisalgorithmofsheme6: theimageis opiedonto ( 1)

1;

and ( 1)

2;

,thedetails

d ( 1)

1;

andd ( 1)

2;

aresetto zeroandthesynthesisformula6is applied.

The original image of a phone was deimated eliminating 3 pixels out of

4. It was then zoomed-in with separable Daubehies 4wavelet, on one hand

(seedetailingure7.1),andwithanonseparablebalanedmultiwavelethaving

auray3withdilationmatrixD

1

{D1 a3 bal{,ontheother(seedetailin

gure7.1). Comparingbothimages,weseethatthelatterhasmoreresolution;

thisis beausethedilationmatrix takes2steps tozoom-in animage to twie

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Reconstruction 15% Daubechies4 coefficients

Figure9: ReonstrutionwithDaubehies4-detail

7.2 Compression

Wavelet transforms are eÆient for image ompression: they deorrelate the

dataand giveasparse representation ofthe image. A threshold is applied to

thetransformedoeÆients;thethresholdedoeÆientshavemanyzeroesand

thisis usefulforompression. Thequality ofthereonstruted imageis good.

Sinethesystemisorthonormal,anerrorinthetransformedoeÆientsisequal

totheerrorin thereonstrutedimage(in2-norm).

The originalimage of aameraman washosenfor ompression. The

sep-arable wavelet transform Daubehies 4and the multiwavelet transform D1

a3 bal were applied to it and ompared. The number of steps takenwas

suhthat thenal oarseapproximationmatrixwasofsize 88. Theoriginal

images were ompressed retaining in all ases 15%of the largest transformed

oeÆients in absolute value {adeuate quantization and entropy oding will

improvetheompressionrate. Ingure9wehaveadetailofthereonstruted

imageafter transformingwith Daubehies 4andthresholding. Ingure10we

haveadetailofthereonstrutedimageaftertransformingwithD1 a3 bal

multiwavelet andthresholding. Thelatterlookssmoother, but theformer has

moreenergy ompation: thePSNRwas35.51 forDaubehies4and 32.20for

D1 a3 bal.

8 Conlusion

We have shown how image proessing is ahieved with nonseparable

multi-waveletshavingquinunxdeimation. Theexamplesrevealsatisfatoryresults

at interpolation ( zoom- in) for these wavelets. Forfuture work remains the

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Reconstruction 15% D1−acc3−bal1 coefficients

Figure10: ReonstrutionwithD1 a3 bal{detail

Referenes

[CohenandDaubehies1993℄ A.CohenandI.Daubehies. Non-separable

bidi-mensionalwaveletbases.RevistaMatematiaIberoameriana,9:51{137,1993.

[Daubehies1992℄ I.Daubehies.Tenleturesonwavelets.SoietyforIndustrial

and ApplMathematis,1992.

[KovaeviandVetterli℄ J.KovaeviandM.Vetterli. Newresultson

multidi-mensionallterbanksandwavelets. preprint.

[KovaeviandVetterli1992℄ J.KovaeviandM.Vetterli. Nonseparable

mul-tidimensional perfet reonstrution lter banks and wavelet bases for r n

.

IEEE,Transationson InformationTheory, 38(2):533{555,1992.

[Lebrunand Vetterli1997℄ J.Lebrunand M.Vetterli. Balanedmultiwavelets:

Theoryanddesign. IEEETransationson SignalProessing,1997.

[PlonkaandStrela1998℄ G.PlonkaandV.Strela. Construtionofmultisaling

funtionswithapproximationandsymmetry.SIAMJournalofMathematial

Analysis,29:481{510,1998.

[Ruedin1999℄ A. Ruedin. Nonseparableorthogonalmultiwaveletswith2and 3

vanishing moments on the quinunxgrid. Proeedings SPIE Wavelet Appl.

Signal ImagePro. VII,3813:455{466,1999.

[Ruedin2000℄ A.Ruedin.Balanednonseparableorthogonalmultiwaveletswith

2and3vanishingmomentsonthequinunxgrid. Proeedings SPIEWavelet

Appl.Signal ImagePro. VIII,4119:toappear,2000.

[Strelaetal.to appear℄ V.Strela,P.Heller,G.Strang,P.Topiwala,andC.Heil.

The appliationof multiwavelet lterbanksto signalandimage proessing.

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Figure

Figure9:Re
onstru
tionwithDaube
hies4-detail
Figure9:Re onstru tionwithDaube hies4-detail p.10
Figure10:Re
onstru
tionwithD1�a

3�bal{detail
Figure10:Re onstru tionwithD1�a 3�bal{detail p.11

Referencias

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