彩色图像边缘检测Color Edge Detection in Presence of Gaussian Noise.docx

彩色图像边缘检测Color Edge Detection in Presence of Gaussian Noise.docx

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彩色图像边缘检测ColorEdgeDetectioninPresenceofGaussianNoise

ColorEdgeDetectioninPresenceofGaussianNoise

UsingNonlinearPre-filtering

Abstract:

AnewwayforedgedetectionincolorimagesdisturbedbyGaussiannoiseis

presented.Thispaperproposedamethodwhichusesamulti-passprocessingmethod

thatreducesthenoiseinthecolorimage.Themainideaforthispaperisthatadopttwo

differentpre-filteringthataimatavoidingfalseedgesproducedbynoiseandat

preservingtheimagedetailsduringnoiseremoval.Afterdoingthatweneeddesignan

algorithmforedgedetectionwhichaimsatdecreasingthesensitivitytonoiseofthe

overallmethodsothatwecansuccessfullyaccurateedgemapsevenencounterhighly

corrupteddata.Accordingtothesimulationanddata,wecanfindtheproposedmethod

fornoiseimagesarequitegood.

KeyWord:

edgedetection,fuzzysystems,Gaussiannoise,imageprocessing,nonlinear

filters.

INTRODUCTION

Edgedetectionisameasurewhichplaysaveryimportantroleintherealizationofa

completeimageunderstandingsystem（Likeimagesegmentationandobjectrecognition

arealldependsonthequalityoftheedgedetection）.Comparedwithgrayscalepicture,

colorimagehasrichermeasurementinformationthatcanimprovethequalityofthe

imageandextenditsrange.However,edgedetectionwillencountersometroublewhen

imagehassomenoiselikeGaussian.Therearesomemethodsfocuseonmonochrome

pictureslikeSobeltechniquetothethreeimagechannels.ActuallySobel’sedgepointin

thecolorimagecouldbeestimatedbyevaluatingthemaximumofthegradient

componentssumsothatitcanyieldanaccurateedgemap.SomemethodslikeDifference

vector（DV）methodusinggradient-basedalgorithmswhichisverysensitivetonoisehas

agoodperformancetotheedgedetection.Butthesemethodsarelimited,especiallywhen

theimagedataarehighlycorrupted,theresultwillbecomeveryannoyingsoweneed

newmethodtosolvethisproblem.Newmethodwhichusesmulti-passprocessingaimsat

reducingnoisebeforeextractingtheimageedges.Combiningpre-filteringschemewith

algorithmforedgedetectionwhichnotonlyimprovetheincreasetheaccuracyofthe

edgedetectionbutalsoconsiderthenoisesmoothingandspatialresolution.Sothe

followswillshowthemethodologyandtheresult.

Methodology

Pre-filteringforedgedetection

Firstweneedknowpre-filteringforedgedetectionisbasedonthedifferentkindsof

noisypixels:

TypeApixels:

pixelscorruptedbynoisewithamplitudenottoodifferentfromthatofthe

neighbors.

TypeBpixels:

pixelscorruptedbynoisewithamplitudemuchlargerthanthatofthe

neighbors.

Fig1Blockdiagramofthemulti-passprocessing

（p）

WedealwithdigitizedmultichannelRGBimages.xdenotetheMulti-channelimageat

thepass,andthexdenotestheinputnoisyimage,xkmeanswheretheR（red）,G

（0）

（green）,andB（blue）channelsarerespectively,denotedbyk=1,k=2,k=3.Putthem

togetherwhichconstructthisdiagram.Inthismethod,weuse24-bitcolorimage.Sowe

haveL=256.

Fig2colordivision

HereweneedmentionthathereweusetypeApre-filteringtwicewhichaimstoincrease

theeffectivenessofthesmoothingaction.

Theproposedmulti-passprocessinginvolvesthefollowingoperations:

TypeAPre-filtering:

Differencesbetweenthepixeltobedealwithanditsneighborsasbelow:

Smalldifferencesaresupposednoisetobereduced;largedifferencesaresupposededges

tobepreserved.

Theprocedureisdefinedbythefollowingrelationship:

1

8

1

8

1

1

∑∑

x

（1）

k

（i,j）=xk

（0）

（i,j）+

（i,j）+

ς

ς

（1）

（xk

k

（0）

（i+m,j+n）,xk

（0）

（i,j））

（i,j））

m=-1n=-1

1

1

∑∑

x

（2）

k

（i,j）=xk

（1）

（2）

（xk

k

（1）

（i+m,j+n）,xk

（1）

m=-1n=-1

Whereςkisaparameterizednonlinearfunction.Letk（p）（i,j）beaconstantpixel

（p）

x

value,0≤xk（i,j）≤L-1atlocation[i,j].

（p）

ThisformulameansweputthenoiseimageintothetypeApre-filteringandLetx（n）be

thepixelluminanceatlocationn=[n1,n2]andthepixelsprocessedcombinewithits

neighborconstruct3*3window.Withthismodelwhichshowsbelowwecanuse

parameterizednonlinearfunctiontoachievethesmoothingactionandexcludesthevalues

thatareverydifferentfromthecentralelement,inordertoavoiddetailblurduringnoise

cancellation.

Fig33*3window

Theς（p）

k

parameterizednonlinearfunctionprocedureshowsasfollow:

ς

k（p）（u,v）=u-v

|u-v|≤a

（p）

k

Whereurepresentstheneighborandvdenotedthecentralpixel.

Withthisvaluewhichmeanstheneighborarequiteclosetothecentralpixel.Sothefilter

performsthearithmeticmeanofthepixelluminanceintheneighborhood,thus

realizingthemaximumsmoothingaction.

ς

k（p）（u,v）=0

|u-v|≥3a

（p）

k

Inthisformwhichmeanstheneighborisquitedifferencewiththecentralpixelthefilter

behavesastheidentityfilter,thusperformingthemaximumdetailpreservation.

（u,v）=（3ak（p）

-u+v）sgm（u-v）

ς

（p）

k

ak（p）≤u-v|≤3ak（p）

|

2

Intermediatesituationsareprocessedasacompromisebetweentheseoppositeeffects.

（p）

Accordingtotheformula,wefindthatthebehaviordependsonthevalueofak.Inorder

togettheoptimalvalue,weusetheautomaticparametertuningtogetthebestvalue.

Fig4Automaticparametertuning

.

（p）

HereweusePinsteadofak,andthekisequaltotheinitialpvalue.Herepisfrom2to

L/4,eachimagehasitsownpvalue,usingthistuningprocedurewecangetthevalue.Why

wechooseMSE?

Becauseitisconceptuallysimpleandwidelyused,followthestepsevery

time,throughdifferentk1andk2wecangettheoptimalvalueanddothesecondtuning

procedure.Aftergettingthepvaluewecanuseittodothesmoothingaction.

ThisistheresultfortypeApre-filtering:

Fig.5GaussiannoiseimageandnoiseimagewithtypeApre-filtering

Accordingtotheimage,wecanfindthatthenoiseimagewithtypeApre-filtering

maketheimagemoresmoothingandleavetheoutlierfortypeBpre-filteringtodeal

with.

TypeBpre-filtering:

ActuallytypeBpre-filteringfocusontheoutlier,weknowoutlierpresentinthedataasan

effectofthetailofGaussiandistributiontheeffectcanbecomeveryannoyingTypeBpre-

filteringconsidersthedifferencesbetweenthepixeltobeprocessedanditsneighborsina

differentway:

ifallthesedifferencesareverylarge,thepixelisanoutliertobecancelled.It

isbrieflysummarizedasfollows.

µLA（u,v）denotesthemembershipfunctionthatdescribesthefuzzyrelation:

“uismuch

largerthanv”andtheformulashowsbelow:

Usingthesethreeformulascanachievethegoalwhichoutliercanberemovedbythisway.

Wecanseetheresultshowsbelow:

Fig6imageafterfilterAandimageafterfilterB

IfweaddpepperandsaltnoisetooriginalimageandseehowtypeBworks.

BecausepepperandsaltnoisebelongtotypeBnoise,typeBfiltershouldhaveaverygood

performance.

Fig7pepperandsaltfiltering

AccordingtotheFig6and7wecanfindthatusethetypeBpre-filteringwhichcanremove

theoutliersuccessfullybutactuallyitstillnotnoisefree.Howeverwecanstilldeclarethis

methodisquitegood.

Noise-protectededgedetector

Eventhoughthenonlinearpre-filteringsignificantlyreducestheGaussiannoise,according

totheimagewhichshowsbeforetheimagedataarenotnoise-free.Thus,anedgedetector

withverylowsensitivitytonoiseisnecessary.Weusethefollowrelationshiptodothe

detection:

y（i,j）=（L-1）[1-MIN{μSM（B1）,μSM（B2）}]

Suitablechoicesare:

S1={x

（3）

（i-1,j-1）,x

（i-1,j-1）,x

（i,j）,x（i,j+1）,x（i+1,j）,A1,A2}

（3）

（3）

（i-1,j）,x（i-1,j+1）}

（3）

S2={x

（3）

（3）

（3）

（i,j-1）,x（i+1,j-1）}

S1={x

（3）

（3）

μSM

isthemembershipfunctionoffuzzyset“small,”symbol||||denotestheEuclidean

distance,ShisthesetofNh

Whereb1andb2areintegerparameters.Smallervaluesperformastrongeractivationofthe

operatorinthepresenceofimageedgesatthepriceofanincreaseofthesensitivityto

noise,sothechoosingofthistwovaluesalsoneedtobeconsideredinourexperiment.

Typicallyb1=5,30

Theedgedetectionprocedureaimsatrepresentingobjectcontoursasbrightlinesinthe

resultingedgemap.Conversely,uniformregionsarerepresentedasdarkareas.

Itisknownthataccuratedetectionofedgesinnoisydatashouldsatisfiedtworequirements:

（1）Theedgedetectionprocessshouldavoidfalseedgesproducedbynoise

（2）Ensurethatactualedgesarecorrectlydetected.

Wewilldosummationbygrayandcolorimagetoshowhowouredgedetectorworks

Wecanseetheresultshowsbelow:

Fig8Summationbygrayimage

Fig9Summationbycolorimage

Fig10Summationbylenaimage

AccordingtotheFig8,9and10wecanfindthatbyusingmultichannel,colorimagehasa

betterpre-filteringandedgedetectionresult,becausethatcolorimagehas3channelsbut

grayimageonlyhas1channel,sothattypeAfilterwillhaveabetterperformancewhen

dealwithcolorimage.Wecandeclareourmethodisquitegood.

Wewillalsoshowtheedgedetectionresultwhenimageidcorruptedbypepperandsalt

noise.Theresultshowsbelow:

Fig11EDofpepperandsaltnoisyimage

Accordingtothefigure11,wecangetaverygoodedgedetectionresultbyusingtypeB

filter,italmostremoveallsuperimposingimpulsenoise（saltandpeppernoise）.Thethree

mainpartsofourmethod:

multichannel,typeAfilter,typeBf