A Tensor VotingApproach for the Hierarchical Segmentation o3-D f Acoustic Images LinmiTao 1Vittorio , Murino
1
, and Gérard Medioni
2
1
2
DipartimentodInformatica, i University oVerona, f 37134 Verona, Italy Institute forRoboticsand IntelligentSystems, University oSouther f C n alifornia LosAngeles, CA 90089-0273, USA {tao,murino}@sci.univr.it,
[email protected] wavelength,whichim s uchlongerthanthatofthev isible light[2].Besidestheseintrinsiccharacteristics ofthe camera,thequalityoacoustic f imagesiaslsoaffe ctedby theenvironmentconditions,andthe pose ofsurfaces relativetothedirectionofsonarpulsetransmissi on,that may leadto imageswith a variabledensity opoints. f Although preliminary a lowlevel signalprocessing is directly performedintheacousticcamera,ithasbeen provedusefultofiltertheacousticimages.Thesi mplest wayonoise f filteringitsodeterminethreshold a levelby assumingthatechoeshavestrongerresponsethanot her clutteringinterferences, although differentapplications andsensingconfigurationsfacedifferentproblems and severalsimilar techniquesareutilizedingeneral.Inthe caseom f easuringtheseabedtopography,Okino[3] set thethresholdforseparatingtheseabedechoesand reverberation.Henriksen[4]addressedthesamepro blem inreal-timeunderwaterobjectdetectionforautono mous underwater vehiclebsyettingtwothresholds tored ucethe amountofdataandtoremovenoiseefficiently.In [15],a methodforthesegmentationofsimulated3-Dsonar imagesusingsurfacefittingprocedureswaspropose dto recover 3-Dsurfacestructuresfrom sparsedata. Traditionalopticalimageprocessingtechniquesare alsofrequentlyusedin3-Dacousticimageprocessi ng. Filtersormasks,suchassquareGaussianweighted averagefilter,squareselectiveweightedaveragef ilter, etc.,areusedtoconvolvetherawacousticdatafo r smoothing theimages or reducing thenoise[5].Sta tistical approaches,typicallyMarkovRandomFields[6],are appliedtomodelthephysicaldataacquisition proc ess,the propertiesoftheunderwaterenvironment,andthen to restoreoto rsegmenttheacousticimage[7,8,9].I ntensity informationofthepointscomesintouseas reliability information duringtheestimationprocess of these statisticalapproaches [10,11]. Wideinformationius tilizedintheseapproaches,b ut the intrinsicstructuralinformationin3-Ddataisnot exploitedyet.Inotherwords, pointscapturedbythe
Abstract Wepresentahierarchicalandrobustalgorithm addressingtheproblemoffilteringandsegmentatio nof three-dimensional acousticimages.Thisalgorithmis basedonthetensorvotingapproach–aunified computationalframeworkfortheinferenceofmultip le salientstructures.Unlikemostpreviousapproaches n, o models or prior information of the underwater environment,northeintensity informationofacoustic imagesicsonsideredinthisalgorithm.Salientstr uctures andoutliernoisy pointsare directly clusteredintwosteps according to both the density and the structural informationofinputdata.Ourexperimentaltrials show promising results, very robust despite the low computationalcomplexity.
1.Introduction This paper addresses the filtering and the segmentationof noisythree-dimensional(3-D)points’ setsacquired byahigh-resolutionacousticcamera[1]. Suchcameraifsormedbya two-dimensional(2-D)array of acoustic transducerssensibletosignalsbackscattered fromthescenepreviouslyinsonifiedbyahigh-freq uency acousticpulse.Thewholesetof theacquired rawsignals isprocessedinordertoestimatesignalscomingfr om 2a Dsetof fixedsteeringdirections (called beamsignals ) andtoattenuatethosecomingfromotherdirections . In thisway,the3-Dpointscanbe computed by detectingthe timeinstantaw t hichthemaximumpeakoccursinea ch beamsignal.Inaddition,the amplitude ofthemaximum peak ofeachsignalcanbe associatedwiththe3-Dpoint to getan intensity imageregisteredwith the3-Dimage. Unfortunately,rawimagesobtainedbyanacoustic cameraaretypically of quitepoorquality,duetostrongly degradingspecklenoise,thenon-ideal characteristicsof thesensortransferfunction(i.e., sidelobes),andthe
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(λ1 − λ 2 )S + (λ 2 − λ 3(2)) P + λ 3 B
acousticcamera,althoughalow t resolution(3cma best) t andverynoisy,succeedstocapture theinherentstructural inthescene.Inthispaper,a informationofthesurfaces new approachisproposedtoextractthestructural informationinacousticimagesbyconvolvingtheim ages withawell-definedkernel.Thisapproach,called tensor voting,isdevelopedbyGuyandMedioniforsegmentation[12,13]purposes.Theinputofthealgorithm isthe positionotfhepointsandnopriorinformationabo utthe physicalcharacteristicof the acousticcameraor the underwaterenvironment are employedintheprocess.The original points’ sets are segmented hierarchically accordingtotheirinherentstructuralinformation inthree groups:highdensitystructuredpoints,lowdensity structuredpoints,andnoisy points. Therestofthepaperisorganizedafsollows.Sect ion 2presents short a overviewofthetensorvotingap proach. InSection3,theproposedmethodisproposedaimed at thesegmentationof3-Dacousticdata.Experimental resultswithrealdata,anditsrobustnesstohigh percentageofnoisearepresentedinSection4,and , finally,conclusions aredrawniS n ection 5.
where S defines satick tensor, and Bdefines baalltensor (Fig.1):
Pdefines palatetensor
S = eˆ1eˆ1T P = eˆ1eˆ1T + eˆ2 eˆ2T B = eˆ1eˆ1T + eˆ2 eˆ 2T + eˆ3 eˆ3T
(3)
Thesetensorsdefinethethreebasistensorsforan y generalsecond-ordersymmetric3-Dtensor.ByEquat ion (1),alinearcombinationofthesebasistensorsde fines anysecond-ordersymmetrictensorand,onthecontr ary, anysecond-ordersymmetrictensorcanbe decomposed intothreeindependentelements:sticktensor,plat teensor, andballtensor.FromEquation(2), some features canbe estimated,specifically:(a)thenormaldirectiono fa surface, estimated bytheeigenvector eˆ1andtherelative ,encodedby the sticktensor;(b) eigenvalue (λ1 − λ 2 )is thetangentdirectionofacurve,indicatedbythe eigenvalue eigenvector eˆand (λ2 − λ3 ) , isencoded 3 the bytheplatetensor;(c)therelativeconfidenceof junctionofcurves,estimated bytheeigenvalue
2. A briefreviewofthe tensor voting
the λ3 , is
encodedbtyheballtensor (Fig.1). Tensorvotingisaunifiedcomputationalframework developedbyGuyandMedioni[12,13]forinferring multiplesalientstructuresfromsparsenoisydata in2-D or3-Dspace.Themethodologyisgroundedontwo elements: tensorcalculusforrepresentation,andlinear votingforcommunication.Localstructuresareunif ormly representedbyasecondordersymmetrictensor,whi ch effectivelyencodespreferreddirection,whileavoi ding earlydecisionon ormalorientationsandmaintenan ceof globalorientationconsistency.Datacommunication is accomplished by a linear voting process, which simultaneouslyignoresoutliernoise,correctserro neous orientation(ifpresent),anddetectssurfaceorien tation discontinuities.Themethodologyisnon-iterativea nd robusttoconsiderableamountofoutliernoise.The only freeparameter is thescaleothe f sizeothe f voti ng kernel.
2.2 Thevoting kernel Differentfromtheclassicalconvolutionkernels,t he votingkerneldefinesthemostlikelynormalbysel ecting amostlikelycontinuationcurvebetweentwopoints ( O and PinFig.2).Thelengthofthenormalvectorat P, representingthestrengthofthevote,isdefinedb ythe following equation isnphericalcoordinates: r 2 + cϕ 2 (4) D F (r , ϕ, σ ) = exp − 2 σ where risthearclength OP, cisaconstantchosena priori, ϕisthecurvature,and σisthescaleoaf nalysis, whichistheonlyfreeparameterinthebasicforma lism. Inthevotecollectionstageotfensorvoting,the second order momentcontributionsfromeachvectorvoteare aggregated,andtheresultisasecond-ordersymmet ric tensor.
2.1Second-ordersymmetric tensor In3-D,thesecondordersymmetrictensorisan ellipsoid,whichisfullydescribedbyitsassociat eigensystem,withthreeeigenvectors eˆ1 , eˆ 2and
2.3 Thetensor votingprocessing
ed eˆ(Fig. 3
In2-Dor3-Dspace,eachinputsitecanbpoint ae a, pointwithanassociatednormaldirection,orany combinationothe f above. This inputisfirstencodedinto secondordersymmetrictensors,namelythe tokens.Ifthe inputsiteshaveonlypositionaldata,thisinforma tionis transformedintoanisotropictensorwithunitradi us, i.e., baall.
1), and the three corresponding eigenvalues , is, λ1 ≥ λ2 ≥ λ3 ≥ 0that λ1 0 0 eˆ1 ˆe3 ] 0 λ 2 0 eˆ2T 0 0 λ 3 eˆ3T Rearranging theeigensystem,theellipsoidigsiven T
(1) [eˆ1 eˆ2
by
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Thisencodingprocedureisfollowedbythefirst voting stage, in which tokens communicate their informationwitheachotherinaneighborhoodaccor ding apredefinedvotingkernel(see Eq.4andFig.2),and refinetheinformationtheycarry.Infact,thisst ageitshe same of theclassicalconvolution.Thedifferenceotfhe votingfromtheconvolutionisthatthevotingkern el definestherelationshipbetweenthetensorsbased not onlyonthedistances,butalsoontheirrelativen ormal directions.Afterthisprocess,eachtokeninsowa generic second-order symmetric tensor, encoding both the strength andtheorientation othe f inputpoints.
Therefinedtokenarevotedagaininthesecondsta ge. Thesegenerictensortokenspropagatetheirinforma tion in allofneighbors,despite thepresenceof tokenson thesepositionsornot,basedonthepredefinedvot ing kernel.Thus,thisprocedureleadstoadensetenso m r ap, which recovers every point in the domain. This processingisbuiltontheassumptionthatsomedat a pointsarelostinthesampling.Inpractice,thed omain spaceivs oxelizedintoauniformarrayovf oxelsa ndthe secondvoting procedureivery s timeconsuming. Theresultingdensetensormapisthendecomposed tosticktensors,platetensors,andballtensors. Points, whichhavelocalmaximumstrengthalongadirection of thesticktensors,areselectedaspoints belongingtoa surface.Points,whichhavelocalmaximumstrength along direction a of the platetensor,are labelled aspoints locatedoncurves. Junctionsare identifiedas thepoints havingthe localmaximum strengthoftheballtensors with nopreferentialdirection. Therest ofthe pointsare labelled asnoisypoints.Thefinaloutputisthe set of points locatedosnurfaces,curves,andtheir junct ions. Itisverydifficulttoclearlydescribetheproced ures ofthetensorvotinginveryfewpages.Hopefully, the basicideacan breeachedviathefollowing example . Anacousticimage isrepresentedby saetofpointsin thethree-dimensionalspace Ω.Thefirststepothe f tensor voting procedure istovoxelizethe3-Dspace Ωtosmall cubicvolumes. Somecubes containoriginal3-D points, but others maybe empty.Supposethat A, Band Care threeof these cubes,and that onepoint iscontained in A, onepoint iscontained in B,whereasnopointsarepresent in C. Subsequently,thepointsareencodedasballtensor s sincethereisonlypositioninformationassociated with thepoints.Inthefirstvotingstage, Aand Bexchange theirinformationorvote withotherpointsand,asthe resultofthetensorvotingprocessing,theprimary ball tensorsarerefinedtogenericsecond-ordersymmetr ic tensors. Inthesecondvotingstage, Aand Bexchangetheir informationwithotherpointsagain.Theprocessing here isthatcube Calsocollectsinformationfrom itsneighbors evenithere f ins oanypointinit.Theresultof thisstage ofvotingitshatallofthecubesin the space Ωhasbeen encodedbygeneraltensors,whichitsheso-called dense tensormap. Finally,allofthetensorsaredecomposedand various structures, i.e., surfaces,curves,andjunctions,are extractedfrom stick,plate,andballtensors.
Figure1.Asecondsymmetrictensor,its eigensystemandthedecomposition.
3. Hierarchicalfiltering and Figure2.Therelationshipbetweentwotokens, and P. Nisthenormalin O.
segmentation
Theproposed hierarchicalfilteringandsegmentation procedure basedontensorvotinghasbeendeveloped tryingtolimitthecomputationalcomplexityofthe
O
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informationplay importantrolesin derivingthepolarity andthestrength othe f resultvector. In general a convolution,thenumber of the neighbors isrestrictedaseightandthelengthofavectori snot alwaysone.Inthiscase,thedistancebetween yand xi shouldbetakenintheconvolutionalgorithm,since the neighborsindifferentdistancescontributediffere nt values to the convolution process. This general convolving algorithm is describedas:
originaltechnique,i.e.,avoiding thetime-consuming secondvotingstage. Thistechnique hasbeennamed tensorconvolution to indicatetherelationship with the classicalconvolutionoperation,whiledifferentiat ingit from thestandard tensor voting.
3.1Theconcept oftensor convolution Intheterminologyofconvolution,pointscollect informationfromtheirneighborsaccordingapredef ined convolutionkernel.Figure3 shows asimpleexampleof thisnewconvolutionprocess acting onboth points’ strength (intensity) anddirections. Theconvolutionprocessconsidersalocalarea aroundeachimagepoint,carryingoutanaverageof the pointsinthislocalareaweightedbythevaluesco ntained in theconvolution kernel(e.g.,seeFig.3a). Ageneralconvolutionmask can bexpressedas:
n v v P(• ) = ∑ β (d ( xi , y ))xi + αy ,
(9)
i =1
where d(xi, y)is thedistancebetween thepoints
and xi
y.
n
P(• ) = ∑ βxi + αy , i =1
where xisoi neothe f n eighborsopafoint y. x=1oi 0r meansifthereisapointornotonalocationoft he neighbors, while y=1meansthereisalwaysapoint, whichicsonvolved. α= -8, β=1and n=8inthiscaseof theconvolution kernelof Fig.3a. WithreferencetoFig.3band3c, itis assumedthat thesitesmarkedbyzerohavenopointsandtheoth ers, markedby1or P)( havepointsonit.Suppose that the pointsonthesites P)( areconvolvedbytheconvolution kernelshowninFig.3a,thedifferencebetweenthe convolvingresultso3(b) f and3(c)isthat P(b)> P(a)since , thepoints in Fig. 3(c)aredenser than thosein Fig. 3(b). Further,supposethatthepointshavetheirassocia ted directions and this information is utilized in the convolution process,Equation(5)is changedto:
(a)
(b)
(c)
n v v P(•) = ∑ βxi + αy , i =1
v where xand i
v yarevectors.
Simulation examples of this new convolution algorithmarepresentedin3(d)and3(e),wherethe points havethesamedensitybutassociateddifferentnorm al directions.Foreasilycomparingtheresultsofthe two differentconvolution algorithms,supposethat:
0 v xi = 1 v y =1.
(d)
No point (7) , else
(e)
Figure3.Thedensityandstructuralinformation inaconvolutionprocess( P(b)> P(a)).Seetextfor explanation.
(8)
Onemoreproblemisthatthisdefinitionofthe convolutioncanonlyworkwiththepoints,whichal ready hasbeenassociatednormaldirections.Forthepoin ts, whichhaveonlypositioninformation,amethodshou ld bedefinedforinducinga preferreddirectiononthe
where||·||isthelengthofavector.Letthecent erpoints onFig.3dand3be ceonvolvedbythealgorithm(Eq 6), . andtheresultsareshownbythedotarrowsonthe same figures.Theseresultssuggestthatboththestruct ure information(thenormaldirections)andthedensity
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convolvingprocessing.Thismethodips roposedint ensor votingbysettingupthemostlikelycontinuationc urves between twopoints (Fig.2). Finally, the convolving kernel of the tensor convolutionisdefinedbyEquation(9)andFigure( 2), while β(d(xi, y))ids efinedbyEquation(4)indetails.By thisconvolutiontechnique,someinherentstructure information hideditnheclouds of points can bex plored. Inthecaseoaf cousticimages,theinputisonlyt he locationinformationof pointsin3-Dspaceandis encodedbyabundleofunitvectorswithallofthe possibledirections.Thisbundleofunitvectorsis representedbyaunitballtensor.However,theenc oding processingdoesnotproduceanypriordirectional informationonthepoints,whichcanbeutilizedin the convolution process. Accordingtothedefinitionothe f convolutionkern el, themostlikelycontinuationcurves areusedtoconnect the anytwopointsinvolvingintheconvolution(e.g., O and PinFig.4).Inthisexample(Fig.4),theinformat ion generatedby Oat Pisexpressedby vectors,their directionsaredefinedbythenormalofthemostli kely continuationcurvesandtheirmodulescanbecalcul ated by theconvolvingalgorithmin Eq.(9).Thus,the convolution process derived the most preferred orientationin Pbasedontherelativepositionbetween O and P. Fortheacousticimages,theinduceddirection representsthenormaldirectionsothe f points, embedding theinherentsurfacestructure,from whichtheacoustical pulses arereflected.In other words,theinherent structural information iexploited s itnhis process.
elementsoftheeigensystemofthesecondsymmetric tensors.Every convolution anddecomposition ifsol lowed bythesegmentation using differentthresholdsbasedon theeigenvalueofthedecomposedsticktensors.The result ofthetwosegmentationprocessesitshatthehighdensitystructuredpointsare separatedfromthelowdensitystructuredpoints,andnoisypointsarefil tered out fromthelow-densitystructuredpoints.Thistechni queis robustto several levels ofnoise. Firstofall,thepointsinanacousticimageare transformedintothe isotropic tensorsouf nitradius(ball tensors),sincethereisonlythe three-dimensional pose informationofthepoints,which isutilizedinthis approach.
Figure4.Theconvolutionbetween Oand P in the3-Dspace.Thedotlinesarethemostlikely continuationcurvesbetween Oand Pandthe vectorsaretheirnormalat P.
3.2Hierarchicalsegmentation and filtering
Figure5.Theframeworkofthehierarchical segmentationbasedontensorvoting.
Theframeworkothe f proposedhierarchicalacoustic imagesegmentationapproachisshowninFig.5.In this framework,theacousticimagesareencodedtoball tensors,convolvedanddecomposedtwiceinto thethree
Inthefirstconvolutionstage,everypointis processedusing thepredefinedconvolutionkernel(Eq.9, Fig.2).Thisprocedureisthesameasthegeneral
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convolution operationexcept for theconvolvingkernels. Theresult ofthisprocess is thateachpoint is now encodedby agenericsecond-ordersymmetrictensor (ellipsoidtensor)insteadothe f primary balltens or. Asmentionedabove,thestructuralinformationin acousticimagesindicatesthesurfaces,whichbacks catter sonarpulses. Theassumedreasonable assumption isthat thestructuralinformationinacousticimagesisen coded bythesimilarityothe f normaldirectionsothe fs tructured points. Actually, normaldirectionsotfhepointslocatedon thesamesurfacechangesmoothly.ThisisthewellknownsmoothnessconstraintproposedbyMarr[17]. Thisinformation,thesimilarityofthenormaldire ctions ofthestructuredpoints,canbeextractedbythet ensor convolutionand,assimulatedinFig.3(d)andFig. 3(e), thesimilarityinorientationstendstogeneratemu ch larger values than therandomly distributeddirecti ons. Afterthefirsttensorconvolution,theresulting generic tensorsaredecomposedintothethreeelements: stick,plate,andballtensors(Fig.1).Inthispa per,weare onlyinterestedinextractingthe set ofstructuredpoints, or,inotherwords,pointslyingonsurfacesfromt he randomlydistributedpoints,andthisinformationi s encodedinthesticktensors.Thus,basedonlyont he eigenvaluesotfhesticktensors, (λ1 − λ2 )t,heacoustic imageissegmented intotwoimages:high-density structuredpoints(HDSPoints, Fig.5)andmixedpoints (MIXPoints, Fig.5),whichincludethelowdensity structuredpoints andnoisy points. InordertofilterMIXPoints,thepointsintheim age areencodedtotheballtensorsagain.Inotherwor ds,only thepositionalinformationofthepointsare employedto encodethepoints,meanwhileallinformationfromt he firsttensor voting processing is notused. Similartothefirsttensorconvolutionstage,the ball tensors are convolved,andtheresultisdecomposedagain. Thesegmentationis still basedontheeigenvaluesotfhe stick tensors,andfinally,noisy points arefilter edout.
3.3Thesegmentations’
rd
(a)Referenceimage:Fig.7a,3 imagefromleft (T1=18.5, σ=10,TotalPoints=2459)
rd
(b)Referenceimage:Fig.7c,3 imagefromleft (T2=0.41, σ=10,TotalPoints=1773)
thresholds
Thethresholdusedinthefirstsegmentation is dependentonthedifferenceofthedistributionden sities ofthepoints associatedto thehigh-densityandthelowdensitypoints’sets.Thethreshold usedinthesecond segmentationisbasicallybasedonthedensityoft he noisy points’ set. Inourexperiments,thethresholdsareheuristicall y setupas10to20percentofthe largest eigenvalue f (λ1 − λ2 ) ofthesticktensors,sincethehistogramsothe eigenvaluesareverysharpbetweenthestructuredp oints andnoisy points in our experiments (Fig.6).
Figure6.Thedistributionof
VGxλ(VALUE)after
thefirst(a)andsecond(b)convolutionstage. AccordingtoEquation(4),theresultinggeneric tensorsaresensitivetothedistancesbetweenthe points andasindicatedin[2],3-Dacousticimageshave typicallyaveryvariablepoints’density duetothe differentrelationshipbetweensurfacenormalsand the acoustic pulse directions. The high-density points’ distribution occurswhentheacousticpulsedirectionis nearly paralleltothesurfacenormal,whereaslowdensity points’ distributioncomesacrosswhentheanglebetween
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thesedirectionsislarge.Inshort,athreshold,T 1,is necessaryforseparatehigh-densitystructured(HDS ) pointsfromlow-densitystructured(LDS)points,an d anotherthreshold, T2,shouldalsobseettosegmentLDS points from non-structurednoisy points. Unfortunately,thesetwothresholds(T1>T2) cannot be directly appliedtotheconvolutionresults. Actually, pointsin the acousticimagescanbegrouped intofour groups:G1)pointsin high-densitystructuredsets,G2) noisypointsnear high-densitystructuredsets,G3)points in low-densitystructuredsets,G4) noisypoints.Afterthe firstconvolution,the variation trendsotfheeigenvalues ofthedecomposedsticktensors, , (λ 1 − λ 2 ) = VGiλ along thegroups Gi(i=1,..,4) λ
λ
ThepointsinMIXimageareencodedtoballtensors again.Thesecondconvolutionisperformedwiththe sameconvolutionkernelandtheresultsare again decomposedintotheir threeelements accordingly. Thesecondsegmentationisstillbasedonthe strength values of the stick tensors from the decomposition.Heuristically,thesecondthreshold (T2)is settoseparateLDSpoints(G3)fromnoisepoints( G2, G4).Finally,LDSimage(Fig.7(d)),which includesLDS points,is segregatedfrom MIXimage.
3.4The differencebetween hierarchical segmentation and tensor voting
is thefollowing: λ
VG1 ≥ VG 2
and VG1 ≥ VGλ3
(9)
VGλ2 ≥ VGλ4
and
VGλ3 ≥ VGλ4
(10)
AtypicalexampleishowninFig.6a,inwhich
Itshouldbe emphasizedthatthesecondtensor convolutioninthehierarchicalsegmentationisdif ferent fromthesecondstageotfhetensorvoting,inwhic hthe generictensorspropagatetheirinformationinto allof theirneighbors,inthree respects.Firstofall,thesecond tensorconvolutionistillbasedonthesparseinp utpoints, whereasthesecondstageoftensorvotingisdense. Second,thethree-dimensionalspaceisnotvoxelize dto unitcubesinthetensorconvolution,whichiistv oxelized inpracticeforgeneratingthedensetensormapin the secondstageoftensorvoting.Third,theinformati on generatedbythefirstconvolutionins otbroughti ntothe secondconvolutionprocess, exceptforseparatingHDS andMIXpoints’sets, whereasthesecondvotingutilizes therefinedtensors.
VGλ2
and VGλare bothcloseto zero(lessthan20).Obviously, 3 thereisnogoodwaytosetupanappropriatethres hold λ (T2)forsegmentingG2fromG3.Indeed VG 2ismuch larger than VGλ3(Fig.6aand6b)inthisexample.The values (VGλ2 , VGλ3 )aredependentonboththedensityof G3andthedistancebetweenG2andG1. Thisproblemcanbesolvedinthehierarchical convolutionandsegmentationapproach.Athreshold (T1) issetheuristicallytoseparateG1fromtheother groups. InthisexperimentT1isimplysetto10%oftheb iggest value,sincethehistogramisverysharpintheare anear zero. Notethat therearesomepointsthattheirvaluesare biggerthen2inFig.6a,buttherearenoanypoin tsthat their values arebigger then 2in Fig.6b. Theonlydifferencebetweentwoconvolutionsitsha t theHDSpoints(G1)areremovedfromtheotherpoin ts. Thatmeans,after thefirstsegmentation, G2ischanged backto"normal"noisepoints sincetherearenoHDS pointsnearG2,andtheassociatedvaluesotfhese points ) backto "normal"( VGλ2 ∈ (0, 0.41) , (e.g., VGλ2 ∈ (2, T 1)go whicharelessthan
4. Experimentalresults Wedemonstratethegeneralusefulnessofthe hierarchicalsegmentation andfiltering approachwi th four real 3-Dacousticimages(Fig.7).Theoriginalimage(F ig. 7a,nonoise) isacquiredby anacousticcamera (Echoscope1600,OmnitechA/S).Theobjectsinthe image constituteanoffshorerig madebypipelines,anda pieceofseabed.Theotherimages(Fig.7a50%Nois e, 100%Noise,and200%Noise)aresynthesizedbasedo n theoriginalimagesbyaddingdifferentpercentages of clutter points. The noisy points are uniformally distributedinthe3-Dspace,andareaddedwithre ference totheoriginalnumberofpoints. Suchpercentagesare 50%,100%,and200%. Intheseimages,theHDSpointsrepresentthe pipelinestructure nearlyperpendiculartotheimageplane oftheacousticcamera,whiletheLDScloudsaredi vided intotwoparts:twotubesabovetheHDScloud,and a pieceoseabed f under theHDScloud. In theseexperiments,thescaleparameterofEquation (4)isalwayssetto10( σ=10),sincethesegmentation resultsarenotsensitivetoareasonablerangeof this parameter.Thethreshold,asdiscussedaboveindet ails, forthefirstsegmentationis10%ofthemaximumof
. resultisinferredevidently VGλ3This
bycomparingT1inFig.6awith
VGxλdistributioninFig.
6b. Asaresult,ahierarchicalconvolutiontechniquei s proposedforsegmentingHDSpoints,LDSpointsand noisypointsinacousticimages.Thefirstsegmenta tion separatestheoriginalimageintotwoimages:HDSi mage (Fig.7(b)),whichincludesHDSpoints(G1)andMIX image(Fig.7(c)),which containsthe othergroups(G2, G3,G4).
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, thethresholdforthesecondsegmentation (λ1 − λ2 )and (afterHDSpointsremoval) issetto20%ofthemaximum oftheeigenvaluesotfhesticktensors.Thesegmen tation resultsarenotsensitivetorange a othese f thres holdstoo, becausetheconvolutionprocessingleadstoabig differencebetweentheeigenvaluesof structuredpoints andthatofnoisy points. Theacousticimagesaresegmented intotwopartsin thefirstsegmentation(Fig.7a,X%Noise →Fig.7bX% Noise+Fig.7cX%Noise).Evidently,theseresults are hardly affectedbythedifferentpercentagesofnoise points.Indeed,thesegmentationisstillstableup to 1000%noiseinourexperiments.Inthiscase,HDS points setarealmostindistinguishableitnheimage. ThepointsintheimagesoF f ig.7careencodedto balltensorsandconvolvedagain.The resultingtensors aredecomposedandLDScloudsies xtractedfromnoi sy pointsbasedon (λ1 − λ2 )values.Theoutcomeis illustratedinFig.7d.Allofthelow-densitystru cturesare wellseparatedfromnoise.Thisresultdemonstrates that thealgorithmis robust atover200%noisefortheLDS points.
NoNoise
100%Noise
50%Noise
200%Noise (a)
5.Conclusions A tensor convolution approach based on the computationalframeworkotfensorvotingips ropose dto filterandtosegmentnoisy 3-D acousticimages.This approachextractsbothpointdistributioninformati onand theinherentstructuralinformationdirectlyfromt heinput databy using a hierarchicalconvolution andsegmentation scheme. Unlike traditionalconvolutionmethods,thenew convolutionkernel, basedontensorvoting,isintroduced tothe3-Ddatafilteringandsegmentation,inwhic hthe inherentstructuralinformationis exploited. Experiments demonstratethatthisnewapproachcanextractlow densitystructuredpointsfromnoisyimages,while thisis usuallyadifficultytaskforsomethreedimensiona data l filteringalgorithms.Theexperimentsalsoshowtha the tensorconvolutionisrobust withrespecttodifferent noiselevels.
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50%Noise
200%Noise (b)
ACKNOWLEDGMENTS
NoNoise
ThisworkissupportedbytheEuropeanCommission undertheprojectno.GRD1-2000-25409namedARROV – AugmentedRealityforRemotelyOperatedVehicles based on 3D acoustical and optical sensors for underwater inspectionand survey . AcousticimageshavebeenacquiredbyEchoscope160 acousticcamera,andarecourtesyofDr.R.K.Hanse OmnitechA/S(Norway).
50%Noise
0, n,
6. References [1]R.K.HansenandP.A.Andersen, “A3-DUnderwater AcousticCamera –PropertiesandApplications”,InP. TortoliandL.Masotti(eds.), AcousticalImaging P, lenum Press,London, 1996,pp. 607 –611.
100%Noise
[2] V.Murino,AT . rucco,“Three-dimensional Image Generationand ProcessinginUnderwater AcousticVision”, Proceedings ofthe IEEE 2000, , 88(12), pp. 19031946. –
200%Noise (c)
[3]M.Okino,Y.Higashi, “MeasurementofSeabedTopographybyMultibeamSonarUsingCFFT”, IEEEJ.Oceanic Engineering, 1986, 11(4), pp. 474 –479. [4]L.Henriksen, “Real-timeUnderwaterObjectDetection BasedonanElectricallyScannedHigh-resolutionSo nar”, Proc.IEEESymposiumonAutonomousUnderwaterVehic le Technology, Cambridge,1994.
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100%Noise
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FigureFiltering 7. andsegmentation experiments.(a)Noisyacousticimages.(b),(c) Resultsofthefirstsegmentationstage:HDS points’set(b),andMIXpoints’set(c)are extracted.(d)Resultsofthesecond segmentationstage:noisypointsarefiltered, andLDSpoints’setisextracted.
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