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DisparityMapEstimationUsingATotalVariationBoundWidedMiled,Jean-ChristophePesquet,MichelParentTocitethisversion:WidedMiled,Jean-ChristophePesquet,MichelParent.
DisparityMapEstimationUsingATotalVariationBound.
ThirdCanadianConferenceonComputerandRobotVision,Jun2006,Quebec.
inria-00001255DisparityMapEstimationUsingATotalVariationBoundWidedMiled1,2,JeanChristophePesquet2andMichelParent11INRIA,IMARAProjectDomainedeVoluceau78150Lechesnay,CedexFranceemail:{wided.
miled,michel.
parent}@inria.
fr2InstitutGaspardMonge/UMR-CNRS8049UniversitéMarne-la-Vallée77454Champs-sur-Marne,Francee-mail:pesquet@univ-mlv.
frAbstractThispaperdescribesanewvariationalmethodforesti-matingdisparityfromstereoimages.
Thestereomatchingproblemisformulatedasaconvexprogrammingprobleminwhichanobjectivefunctionisminimizedundervariousconstraintsmodellingpriorknowledgeandobservedinfor-mation.
Thealgorithmproposedtosolvethisproblemhasablock-iterativestructurewhichallowsawiderangeofcon-straintstobeeasilyincorporated,possiblytakingadvan-tageofparallelcomputingarchitectures.
Inthiswork,weuseaTotalVariationboundasaregularizationconstraint,whichisshowntobewell-suitedtodisparitymaps.
Experi-mentalresultsforstandarddatasetsarepresentedtoillus-tratethecapabilitiesoftheproposeddisparityestimationtechnique.
1.
IntroductionStereovisionsystemsdeterminedepthinformationfromapairofleftandrightimageswhicharetakenatthesametime,butfromslightlydifferentviewpoints.
Themostim-portantprobleminstereoimageprocessingistondcorre-spondingpixelsfrombothimages,leadingtotheso-calleddisparityestimation.
Anumberofstudieshavebeenre-portedonthedisparityestimationproblemsincethe1970's[1]includingfeature-based,area-basedandenergy-basedapproaches.
Thefeature-basedapproachextractsfeaturesfromimagepairs(e.
g.
,edges,lines,corners)andthenestab-lishescorrespondencebetweenthesefeatures.
Theyyieldaccurateinformationbutthemaindrawbackofthemethodisthesparsenessoftherecovereddepthmap.
Thearea-basedapproachndscorrespondingpointsbymeasuringsimilaritybasedonwindowcorrelation.
Thisapproachat-temptstodeterminethecorrespondenceforeverypixel,whichresultsinadensedepthmap,buttendstofailatdepthdiscontinuitiesandlowtexturedareas.
Manyattemptshavebeenmadetoremedythisseriousproblembymak-ingthesizeandshapeofthematchingwindowadaptivetothelocalvariationofdisparitycharacteristics[2],[3],[4].
Energy-basedapproachesattempttoovercomethisproblembyminimizingvariationalformulations,whereadatatermandasmoothnesstermarepenalized.
Anexcellentsurveyondensestereomethodscanbefoundin[5].
Variationalapproacheswereintroducedinimagepro-cessingforrestorationanddenoisingproblemsandhavesubsequentlyattractedmuchinterestinthecomputervisioncommunitywheretheywererstdevisedforthepurposeofestimatingopticalowfromasequenceofimages.
Numer-icalstudiesonopticalowviavariationaltechniqueshavebeenperformedinthelastdecade.
Forareview,thereadercanreferto[6].
Thesestudiesshowedthatthevariationalopticalowmethodsareamongthemostpowerfultech-niques,whichnaturallymotivatestheirextensionfordis-paritymapestimation[7],[8].
VariationalmethodsarebasedontheminimizationofanenergyfunctionalE(u)whichconsistsofadatatermandaregularizationtermE(u)=Edata(u)+λEsmooth(u),(1)whereudenotestheeldtobeestimatedandλisapositivecoefcientthatweightsthesmoothnesstermrelativelytotherstdataterm.
Thisfunctionalisoftenminimizedviaaniterativeprocedurederivedfromtheassociatednon-linearEuler-Lagrangeequations.
Thisallowstoprove,inmanycases,theexistenceanduniquenessoftheoptimalsolutionbutrequirestheimplementationofsophisticatednumericalschemes[7].
Moreover,thediscretizationofthePDEisadelicateproblemandthechoiceoftheLagrangeparameterλisadifculttask.
Thelatterproblembecomesevenmoreinvolvedwhenasumofregularizationtermshastobecon-sideredtoaddressmultipleconstraintswhichmayariseintheproblem.
Theaimofthisstudyistoproposeanovelvariationalmethodfordisparitymapestimationbasedonthesettheo-reticapproach.
Thisapproachusesacriterionoffeasibilityratherthanapenalizedoptimalitycriterion.
Then,asolu-tionisacceptableifitisconsistentwithallavailableinfor-mationarisingfromthepriorknowledgeandtheobserveddata.
Eachpieceofinformationisrepresentedbyaconvexsetandthefeasibilityset,thatistheintersectionofthesesets,representsallacceptablesolutions.
Oneoftheadvan-tagesofthisapproachisthatitisofteneasiertodenetheconstraintsetsthantochoosetheoptimalvaluesoftheLa-grangeparametersinaregularizationmethod.
ThemostpopularmethodthatinvolvesthesettheoreticformulationistheProjectionOntoConvexSets(POCS).
Typically,thisiterativemethodsuccessivelyprojectspartialsolutionsontopropertysets.
POCSisoneofthemostpreva-lenttoolforsolvingrecoveryproblemsinimageprocessingandithasalsobeenusedfortherestorationoftheopticalow[9].
However,thismethodpresentsseveralshortcom-ings,includingitsslowconvergenceandthefactthatitisnotwellsuitedforanimplementationonparallelproces-sors.
Numerousapproacheshavebeendevelopedinordertoovercometheabovelimitations.
Thereaderisreferredto[10]forareviewofexistingprojectiontechniques.
Thebasicprincipleoftheapproachproposedinthisworkistoformulatethematchingproblemasaconvexpro-grammingproblem.
Moreprecisely,aquadraticobjectivefunctionisminimizedundercertainconstraintsandthere-sultingoptimizationproblemissolvedviaablock-iterativeparalleldecompositionmethod.
Thismethodallowsawiderangeofconstraintstobeeasilyincorporated,thusleadingtoimprovedresults.
WeproposeinthisworktousetheTo-talVariationasaregularizingconstraint.
Asshownintheseminalworkof[11],themainadvantageofthisregular-itymeasureisthatitpreservestheedgesintheimagewhilesmoothingthehomogeneousareas.
Itisthereforeappropri-atefordisparitymapsthatoftencontainlargehomogeneousregions.
Ourpaperisorganizedasfollows.
Section2describesthebinocularvisionsystemanddenesourstereomodel.
InSection3,weintroducethemathematicalbackgroundandnecessarynotationsforthemethodproposedinthispaper.
ThenwepresentthesettheoreticdisparitymapestimationanddescribetheconstraintsweincorporateintheprobleminSection4.
ExperimentresultsarepresentedinSection5,followedbyaconclusioninSection6.
2.
ProblemstatementInthissection,weformulatethematchingproblemanddescribeourstereomodel.
2.
1.
StereovisionsystemTheconsideredstereovisionsystemconsistsoftwocam-eraspositionedsidebysideinordertoobtainleftandrightimages.
ThecamerasareseparatedbyaxeddistanceT,calledbaseline,andshouldhavethesamefocallengthf.
Foreachpixelintheleftimage,acorrespondingpixelismatchedintherightimage.
Thecamerageometrycansig-nicantlyaffecttheamountofprocessingrequiredbythematchingstrategy.
Weconsidertheparallelcameracon-guration,sothatbothcamerasarelocatedonahorizontalplaneandhaveparallelopticalaxes.
Aconstraintthatisoftenconsideredinstereovisionsys-temsistheepipolargeometryconstraint.
Thisconstraintinvolvesthatanypointlyingonanepipolarlineinoneim-agenecessarilycorrespondstoapointlyingonthehomol-ogousepipolarlineintheotherimage.
Whenthecamerasareparallel,aswehaveassumedhere,theepipolarlinesarehorizontalasshowninFigure1.
Althoughsometimescon-vergentcamerasareused,suitablealgorithms,suchastheonedescribedin[12],canbeappliedtorectifytheimages.
Consequently,whetherthecamerasareparallelorrectica-tionisapplied,theverticalcomponentofthedisparityvec-torvanishes,sothatonlyascalarvaluehastobeestimated.
Figure1.
Epipolargeometrywithinaparallelcameraconguration.
LetIlandIrbetheleftandrightviewsofastereopair,respectively.
Twopointspl=(xl,yl)andpr=(xr,yr)intheimagesIlandIraresaidtobematchedifandonlyiftheyaretheprojectionofthesame3-Dpointonthetwoimageplanes.
Inthiscase,thedisparityupointingfrompltoprisgivenbyu=xlxr.
2.
2.
StereomodelAdisparityvalueforeachpixelofanimageisestimatedbysearchingacorrespondingpointintheotherimage.
Eachpossiblecorrespondingpoint,designatedasacandidate,isevaluatedusingacostfunction.
Thesumofsquaredinten-sitydifferences(SSD)iscommonlyusedasthesimilaritymeasureduetoitssimplicity:J(u)=(x,y)∈D[Il(x,y)Ir(xu,y)]2,(2)whereDN2istheimagesupport.
Inordertosimplifythenotations,wehavenotmadeexplicitthatuisafunctionof(x,y)intheaboveexpression.
GivenasetAofadmissiblecandidatedisparityvectors,anoptimaldisparityelduisgivenbyu=argminu∈AJ(u).
(3)Assumingthataninitialdisparityestimateuofuisavailable,forexamplefromapreviousestimation(possiblywithinaniterativeprocess)andthatthemagnitudediffer-enceoftheeldsuanduisrelativelysmall,wecanap-proximatethewarpedrightimagearoundubyaTaylorex-pansionasfollows:Ir(xu,y)Ir(xu,y)Ixr(xu,y)(uu),(4)whereIxr(xu,y)isthehorizontalgradientofthewarpedrightimage.
Introducingthenotations:L(x,y)=Ixr(xu,y),(5)andr(x,y)=Il(x,y)+Ir(xu,y)+uL(x,y),(6)weendupwiththefollowingquadraticcriterion,relatedtothelinearizedmodel(4):J(u)=(x,y)∈D[L(x,y)ur(x,y)]2.
(7)Now,ourpurposeistorecoverthetruedisparityimageufromtheobservedeldsLandr.
Thisproblemisill-posedanditmustthereforeberegularizedbyaddingsomeconstraintstothesolution.
Appropriateconvexconstraintswillbesubsequentlydenedsoastoformulatetheproblemwithintheframeworkwhichisdescribedinthenextsection.
3.
Convexsettheoreticframework3.
1.
FormulationofinverseproblemsManyproblemsinimageprocessinghavebeensuccess-fullyaddressedfromasettheoreticformulation.
Intheex-istingworks,themainconcernistondsolutionsthatareconsistentwithallinformationarisingfrompriorknowl-edgeandtheobserveddata.
Everyknownpropertyisrepre-sentedbyasetinthesolutionspaceandtheintersectionofthesesetsconstitutesthefamilyofacceptablesolutions.
Inmanyproblems,theaimistondanacceptablesolutionuminimizingacertaincostfunction[10].
Ageneralformula-tionofthisprobleminaHilbertimagespaceHisminu∈HJ(u)subjecttoconstraints(Ψi)1≤i≤m,(8)whereJ:Hrepresentstheobjectivefunc-tiontobeminimizedandtheconstraints(Ψi)1≤i≤marisefromtheavailableinformation.
AfamilyofpropertysetsofHcanbeconstructedasfollowsi∈{1,m},Si={u∈H|usatisesΨi}.
(9)ThusSiisthesetofallestimatesthatareconsistentwiththeconstraintΨi.
ThesolutionsetconsistingofsolutionsthatareconsistentwithallavailableinformationisthefeasibilitysetS=mi=1Si.
(10)Restrictingourstudytoconvexfeasibilityproblems,agen-eralformulationisthereforeFindu∈S=mi=1SisuchthatJ(u)=infJ(S),(11)wheretheobjectiveJisaconvexfunctionandthecon-straintssets(Si)1≤i≤mareclosedconvexsubsetsofH.
Itismoreconvenienttomodeltheconstraintsets(Si)1≤i≤maslevelsets:i∈{1,.
.
.
,m},Si={u∈H|fi(u)≤ψi},(12)where(fi)1≤i≤marecontinuousconvexfunctionsand(ψi)1≤i≤marerealparameters.
Weassumethattheprob-lemisconsistent,i.
e.
S=.
3.
2.
SubgradientprojectionsHere,webrieyrecallthebasicfactsonsubgradientpro-jectionswhicharenecessaryforourproblem.
Moredetailscanbefoundin[13].
TheimagespaceisarealHilbertspaceHwithscalarproduct.
|.
andnorm.
.
Avec-tort∈Hisasubgradientofacontinuousconvexfunctionf:H→Ratu∈Hifv∈H,vu|t+f(u)≤f(v).
(13)(Asfiscontinuous,italwayspossessesatleastonesubgra-dientateachpointu.
)Thesetofallsubgradientsoffatuisthesubdifferentialoffatuandisdenotedbyf(u).
Iffisdifferentiableatu,thenf(u)={f(u)}.
Letψ∈RandC={v∈H|f(v)≤ψ}beanonemptyclosedandconvexsubsetofH.
Fixu∈Handasubgradi-entt∈f(u),thesubgradientprojectionPCuofuontoCisgivenby:PCu=uf(u)ψt2t,iff(u)>ψ;u,iff(u)≤ψ.
(14)WenotethatcomputingPCuisoftenamucheasiertaskthancomputingtheexactprojectionontoC,asthelat-teramountstosolvingaconstrainedminimizationproblem[13].
However,whentheprojectioniseasytocompute,onecanuseitasasubgradientprojection.
3.
3.
AnecientalgorithmWenowproceedtothedescriptionofthequadraticprogrammingalgorithmdevelopedin[13]tosolvethequadraticconvexproblem(11)whichisequivalenttomini-mizingu→uu0,R(uu0)(15)overS,whenRisaself-adjointdenitepositiveoperatorandu0∈H.
3.
3.
1Algorithm1.
Fixε∈]0,1/m[andsetn=0.
2.
TakeanonemptyindexsetIn{1,.
.
.
,m}.
3.
Foreveryi∈In,setai,n=Pi,nunwherePi,nisasubgradientprojectionofunontoSiasin(14).
4.
Chooseweights{ξi,n}i∈In]ε,1]suchthati∈Inξi,n=1.
Setvn=i∈Inξi,nai,nandLn=i∈Inξi,nai,n2.
5.
IfLn=0,exititeration.
Otherwise,setbn=u0un,cn=Rbn,dn=R1vnandLn=Ln/dn,vn.
6.
Chooseλn∈[εLn,Ln]andsetdn=λndn.
7.
Setπn=cn,dn,n=bn,cn,νn=λndn,vnandρn=nνnπ2n.
8.
Setun+1=un+dn,ifρn=0,πn≥0;u0+(1+πnνn)dn,ifρn>0,πnνn≥ρn;un+νnρn(πnbn+ndn),ifρn>0,πnνnumax,ontheamountofalloweddisparity.
ThesetassociatedwiththisinformationisS2={u∈H|umin≤u≤umax}.
(20)Wenotethatinpracticeuminandumaxareoftenavailable.
Finally,weuseanotherconstraintinspiredfromtheworkin[7]thatusesaregularizationterm(u)D(Il)(u)basedontheNagel-Enkelmannoperator[17].
ThemainideaofthisconstraintisthatdiscontinuitiesinthedisparitymaparepreservedaccordinglytotheedgesoftheleftimageIl.
Thus,theconstrainthasanisotropicbehaviorwithinuni-formareas,butatobjectedgesitintroducesananisotropicsmoothing.
Detailsgivenin[7]aboutthisorientedsmooth-nessconstraintcanhelpforgivinganapproximationκofthisregularizationtermleadingtothefollowingconvexsetS3={u∈H|(u)D(Il)(u)≤κ}.
(21)WeremarkthattheexactprojectionontoS2isstraightfor-wardlyobtainedwhereasasubgradientprojectionontoS3canbeeasilycalculated.
Insummary,weformulatethecorrespondingproblemastheminimizationof(16)onS=∩3i=1Si,wherethesets(Si)1≤i≤3aregivenbyequations(19),(20)and(21).
5.
ExperimentalresultsInthissection,weevaluatethebenetswhichcanbedrawnfromourapproachusingfourstandarddatasetsfromtheMiddleburydatabase[5],namedTsukuba,Sawtooth,VenusandMap.
Forallexperiments,theparameterαinequation(16)wassetto50,whichensuresagoodtrade-offbetweenconvergencespeedandestimationaccuracy.
Wenotethatvaluesrangingfrom10to100alsoleadtoreliableresults.
Avalueof104forthetotalvariationboundτwasfoundtobeappropriatefortheconsidereddatasetoftestimages.
TheevaluationmeasureistheAverageAbsoluteDispar-ityError(AADE)betweenestimatedueandgroundtruthdisparitiesut:AADE=1N*M(x,y)|ut(x,y)ue(x,y)|,(22)whereN*Misthetotalnumberofpixels.
Theerrorcriterionismeasuredtroughdifferentareasintheimage,classiedasuntextured(Buntex),discontinuous(Bdisc)andtheentireimage(Ball).
Followingtheevaluationprocedurein[5],onlynon-occludedpixelsareconsideredinallthreecases.
ThequantitativeresultsinTable1andtheextracteddis-paritymapsinFigure3demonstratethehighqualityper-formanceofourapproach.
Theobservedimprovementsdemonstratetheabilityofthealgorithmtocomputesmoothdisparitymapswithaccuratedepthdiscontinuitiesinawidevarietyofsituations.
ThisisaconsequenceofusingtheTotalVariationconstraint.
Furthermore,Figure2showsthecomparisonofrecoveredocclusionmapswithgroundtruthones.
Ourresultsareveryclosetothegroundtruthoc-clusionmapswhichprovesthatourmethodcanefcientlyhandletheuniquenessandorderingconstraintsforaboutallvisiblepixels.
Forcomparisonpurposes,theresultsobtainedfromotherdensestereoalgorithms,availableathttp://www.
middelbury.
edu/stereoandhavingthetoprankintheonlineevaluationonthesamewebsite,arealsoincludedinTable1.
Thiscompari-sonshowsthatourapproachisquitecompetitivewithstate-of-the-artmethods.
Inaddition,Table2reportstheerrorvalues,onallnon-occludedimagepixels,obtainedwhenoneoftheconstraints(eitherS1orS3)ismissing.
Thisconrmsthatitisusefultoincorporatemultipleconstraints.
6.
ConclusionInthispaper,wehaveintroducedaconvexsettheoreticformulationfortheestimationofthedisparitymapfromastereopair.
Theresultingoptimizationproblemissolvedviaablockiterativeparalleldecompositionmethod.
ThisefcientalgorithmallowedustocombineaTotalVaria-tionconstraintwithadditionalconvexconstraintssoastosmoothhomogeneousregionswhilepreservingdiscontinu-ities.
Experimentsonvariousdatasetsdemonstratethattheperformanceoftheproposedapproachiscomparablewithstate-of-the-artmethods.
Inourfuturework,wewillcon-tinuetoinvestigatethesettheoreticframeworkthatprovidesanewperspectiveformatchingproblemswhileleadingtoefcientnumericalsolutions.
Figure2.
OcclusionmapsfortheMiddleburystereodata.
Leftcolumn:ourresults.
Rightcolumn:groundtruth.
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27Table1.
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Figure3.
ResultsonMiddleburydataset.
Fromtoptobottom:Tsukuba,Sawtooth,Venus,Map.
Fromlefttoright:Referenceimages,extracteddisparitymapsandgroundtruthdisparitymaps.
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