A desirable set of properties for such algorithms includes incremental updating representation of directional uncertainty the ability to 64257ll gaps in the reconstruction and robustness in the presence of outliers Prior algo rithms possess subsets ID: 23043
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AVolumetricMethodforBuildingComplexModelsfromRangeImagesBrianCurlessandMarcLevoyStanfordUniversityAbstractAnumberoftechniqueshavebeendevelopedforreconstructingsur-facesbyintegratinggroupsofalignedrangeimages.Adesirablesetofpropertiesforsuchalgorithmsincludes:incrementalupdating,representationofdirectionaluncertainty,theabilitytoÞllgapsinthereconstruction,androbustnessinthepresenceofoutliers.Prioralgo-rithmspossesssubsetsoftheseproperties.Inthispaper,wepresentavolumetricmethodforintegratingrangeimagesthatpossessesalloftheseproperties.Ourvolumetricrepresentationconsistsofacumulativeweighted AuthorsÕAddress:ComputerScienceDepartment,StanfordUniversity,Stanford,CA94305E-mail:curless,levoy@cs.stanford.eduWorldWideWeb:http://www-graphics.stanford.eduimagesintoasingledescriptionofthesurface.Asetofdesirablepropertiesforsuchasurfacereconstructionalgorithmincludes: SurfaceCCDLaser(a) Direction of travelObjectCCDCCD image planeLaserCylindrical lensLaser sheetzx(b) (c)(d) Figure1.Fromopticaltriangulationtoarangesurface.(a)In2D,anarrowlaserbeamilluminatesasurface,andalinearsensorimagesthereectionfromanobject.Thecenteroftheimagepulsemapstothecenterofthelaser,yieldingarangevalue.Theuncertainty,,indeterminingthecenterofthepulseresultsinrangeuncertainty,alongthelaserslineofsight.Whenusingthespacetimeanalysisforopticaltriangulation[6],theuncertaintiesrunalongthelinesofsightoftheCCD.(b)In3D,alaserstripetriangulationscannerrstspreadsthelaserbeamintoasheetoflightwithacylindricallens.TheCCDobservesthereectedstripefromwhichadepthproleiscomputed.Theobjectsweepsthroughtheeldofview,yieldingarangeimage.Otherscannerconrotatetheobjecttoobtainacylindricalscanorsweepalaserbeamorstripeoverastationaryobject.(c)Arangeimageobtainedfromthescannerin(b)isacollectionofpointswithregularspacing.(d)Byconnectingnearestneighborswithtriangles,wecreateapiecewiselinearrangesurface.twobasicdirections:reconstructionfromunorganizedpoints,andreconstructionthatexploitstheunderlyingstructureoftheacquireddata.Thesetwostrategiescanbefurthersubdividedaccordingtowhethertheyoperatebyreconstructingparametricsurfacesorbyre-constructinganimplicitfunction.Amajoradvantageoftheunorganizedpointsalgorithmsisthefactthattheydonotmakeanypriorassumptionsaboutconnectivityofpoints.Intheabsenceofrangeimagesorcontourstoprovideconnec-tivitycues,thesealgorithmsaretheonlyrecourse.Amongthepara-metricsurfaceapproaches,Boissanat[2]describesamethodforDe-launaytriangulationofasetofpointsin3-space.Edelsbrunneranducke[9]generalizethenotionofaconvexhulltocreatesurfacescalledalpha-shapes.Examplesofimplicitsurfacereconstructionin-cludethemethodofHoppe,etal[16]forgeneratingasigneddistancefunctionfollowedbyanisosurfaceextraction.Morerecently,Bajaj,etal[1]usedalpha-shapestoconstructasigneddistancefunctiontowhichtheytimplicitpolynomials.Althoughunorganizedpointsal-gorithmsarewidelyapplicable,theydiscardusefulinformationsuchassurfacenormalandreliabilityestimates.Asaresult,thesealgo-rithmsarewell-behavedinsmoothregionsofsurfaces,buttheyarenotalwaysrobustinregionsofhighcurvatureandinthepresenceofsystematicrangedistortionsandoutliers.Amongthestructureddataalgorithms,severalparametricap-proacheshavebeenproposed,mostofthemoperatingonrangeimagesinapolygonaldomain.SoucyandLaurendeau[25]de-scribeamethodusingVenndiagramstoidentifyoverlappingdatare-gions,followedbyre-parameterizationandmergingofregions.TurkandLevoy[30]devisedanincrementalalgorithmthatupdatesare-constructionbyerodingredundantgeometry,followedbyzipperingalongtheremainingboundaries,andnallyaconsensusstepthatreintroducestheoriginalgeometrytoestablishnalvertexpositions.Rutishauser,etal[24]useerrorsalongthesensorslinesofsighttoestablishconsensussurfacepositionsfollowedbyare-tessellationthatincorporatesredundantdata.Thesealgorithmstypicallyperformbetterthanunorganizedpointalgorithms,buttheycanstillfailcatas-trophicallyinareasofhighcurvature,asexempliedinFigure9.Severalalgorithmshavebeenproposedforintegratingstructureddatatogenerateimplicitfunctions.Thesealgorithmscanbeclassiastowhethervoxelsareassignedoneoftwo(orthree)statesoraresamplesofacontinuousfunction.Amongthediscrete-statevolumet-ricalgorithms,Connolly[4]castsraysfromarangeimageaccessedasaquad-treeintoavoxelgridstoredasanoctree,andgeneratesresultsforsyntheticdata.Chien,etal[3]efcientlygenerateoctreemodelsunderthesevereassumptionthatallviewsaretakenfromthedirectionscorrespondingtothe6facesofacube.LiandCrebbin[19]andTarboxandGottschlich[28]alsodescribemethodsforgenerat-ingbinaryvoxelgridsfromrangeimages.Noneofthesemethodshasbeenusedtogeneratesurfaces.Further,withoutanunderlyingcontinuousfunction,therearenomechanismforrepresentingrangeuncertaintyorforcombiningoverlapping,noisyrangesurfaces.Thelastcategoryofourtaxonomyconsistsofimplicitfunctionmethodsthatusesamplesofacontinuousfunctiontocombinestruc-tureddata.Ourmethodfallsintothiscategory.PreviouseffortsinthisareaincludetheworkofGrosso,etal[12],whogeneratedepthmapsfromstereoandaveragethemintoavolumewithoccupancyrampsofvaryingslopescorrespondingtouncertaintymeasures;theydonot,however,performanalsurfaceextraction.Succi,etal[26]createdepthmapsfromstereoandopticalowandintegratethemvolumet-ricallyusingastraightaverage.Thedetailsofhismethodareunclear,buttheyappeartoextractanisosurfaceatanarbitrarythreshold.InboththeGrossoandSuccipapers,therangemapsaresparse,thedi-rectionsofrangeuncertaintyarenotcharacterized,theyusenotimeorspaceoptimizations,andthenalmodelsareoflowresolution.Recently,Hilton,etal[14]havedevelopedamethodsimilartooursinthatitusesweightedsigneddistancefunctionsformergingrangeimages,butitdoesnotaddressdirectionsofsensoruncertainty,incre-mentalupdating,spaceefciency,andcharacterizationofthewholespaceforpotentialholelling,allofwhichwebelievearecrucialforthesuccessofthisapproach.Otherrelevantworkincludesthemethodofprobabilisticoccu-pancygridsdevelopedbyElfesandMatthies[10].TheirvolumetricspaceisascalarprobabilityeldwhichtheyupdateusingaBayesianformulation.Theresultshavebeenusedforrobotnavigation,butnotforsurfaceextraction.Adifcultywiththistechniqueisthefactthatthebestdescriptionofthesurfaceliesatthepeakorridgeoftheprobabilityfunction,andtheproblemofridge-ndingisnotonewithrobustsolutions[8].Thisisoneofourprimarymotivationsfortakinganisosurfaceapproachinthenextsection:itleveragesoffofwell-behavedsurfaceextractionalgorithms.Thediscrete-stateimplicitfunctionalgorithmsdescribedabovealsohavemuchincommonwiththemethodsofextractingvolumesfromsilhouettes[15][21][23][27].Theideaofusingbackdropstohelpcarveouttheemptinessofspaceisonewedemonstrateinsection4.3VolumetricintegrationOuralgorithmemploysacontinuousimplicitfunction,,rep-resentedbysamples.Thefunctionwerepresentistheweighted SensorNearFarVolumeRangesurfaceZero-crossing(isosurface)Newzero-crossingDistancesurface(a)(b)Figure2.Unweightedsigneddistancefunctionsin3D.(a)Arangesensorlookingdownthex-axisobservesarangeimage,shownhereasarecon-structedrangesurface.Followingonelineofsightdownthex-axis,wecangenerateasigneddistancefunctionasshown.Thezerocrossingofthisfunctionisapointontherangesurface.(b)Therangesensorre-peatsthemeasurement,butnoiseintherangesensingprocessresultsinaslightlydifferentrangesurface.Ingeneral,thesecondsurfacewouldinterpenetratetherst,butwehaveshownitasanoffsetfromthesurfaceforpurposesofillustration.Followingthesamelineofsightasbefore,weobtainanothersigneddistancefunction.Bysummingthesefunctions,wearriveatacumulativefunctionwithanewzerocrossingpositionedmidwaybetweentheoriginalrangemeasurements.signeddistanceofeachpointtothenearestrangesurfacealongthelineofsighttothesensor.Weconstructthisfunctionbycom-biningsigneddistancefunctions,...andweightfunctions,...obtainedfromrangeimages.Ourcombiningrulesgiveusforeachvoxelacumulativesigneddistancefunction,,andacumulativeweight.Werepre-sentthesefunctionsonadiscretevoxelgridandextractanisosurfacecorrespondingto)=0.Underacertainsetofassumptions,thisisosurfaceisoptimalintheleastsquaressense.Afullproofofthisoptimalityisbeyondthescopeofthispaper,butasketchappearsinappendixA.Figure2illustratestheprincipleofcombiningunweightedsigneddistancesforthesimplecaseoftworangesurfacessampledfromthesamedirection.Notethattheresultingisosurfacewouldbethesur-facecreatedbyaveragingthetworangesurfacesalongthesensorlinesofsight.Ingeneral,however,weightsarenecessarytorepre-sentvariationsincertaintyacrosstherangesurfaces.Thechoiceofweightsshouldbespecictotherangescanningtechnology.Forop-ticaltriangulationscanners,forexample,Soucy[25]andTurk[30]maketheweightdependonthedotproductbetweeneachvertexnor-malandtheviewingdirection,reectinggreateruncertaintywhentheilluminationisatgrazinganglestothesurface.Turkalsoarguesthattherangedataattheboundariesofthemeshtypicallyhavegreateruncertainty,requiringmoredown-weighting.Weadoptthesesameweightingschemesforouropticaltriangulationrangedata.Figure3illustratestheconstructionandusageofthesigneddis-tanceandweightfunctionsin1D.InFigure3a,thesensorisposi-tionedattheoriginlookingdownthe+xaxisandhastakentwomea-.Thesigneddistanceproles,mayextendindenitelyineitherdirection,buttheweightfunctions,,taperoffbehindtherangepointsforreasonsdis-cussedbelow.Figure3bistheweightedcombinationofthetwoproles.Thecombinationrulesarestraightforward: (1))=(2) 1d1(x)w1(x) r2w2(x)d2(x)W(x)D(x)R (a)(b) Figure3.Signeddistanceandweightfunctionsinonedimension.(a)Thesensorlooksdownthex-axisandtakestwomeasurements,arethesigneddistanceproles,andaretheweightfunctions.In1D,wemightexpecttwosensormeasure-mentstohavethesameweightmagnitudes,butwehaveshownthemtobeofdifferentmagnitudeheretoillustratehowtheprolescombineinthegeneralcase.(b)isaweightedcombinationofisthesumoftheweightfunctions.Giventhisformulation,thezero-crossing,,becomestheweightedcombinationofrepresentsourbestguessofthelocationofthesurface.Inpractice,wetruncatethedistancerampsandweightstothevicinityoftherangepoints.arethesigneddistanceandweightfunctionsfromthethrangeimage.Expressedasanincrementalcalculation,therulesare: (3)(4)wherearethecumulativesigneddistanceandweightfunctionsafterintegratingthethrangeimage.Inthespecialcaseofonedimension,thezero-crossingofthecu-mulativefunctionisatarange,givenby: (5)i.e.,aweightedcombinationoftheacquiredrangevalues,whichiswhatonewouldexpectforaleastsquaresminimization.Inprinciple,thedistanceandweightingfunctionsshouldextendindenitelyineitherdirection.However,topreventsurfacesonop-positesidesoftheobjectfrominterferingwitheachother,weforcetheweightingfunctiontotaperoffbehindthesurface.Thereisatrade-offinvolvedinchoosingwheretheweightfunctiontapersoff.Itshouldpersistfarenoughbehindthesurfacetoensurethatalldistancerampswillcontributeinthevicinityofthenalzerocrossing,but,itshouldalsobeasnarrowaspossibletoavoidinuencingsurfacesontheotherside.Tomeettheserequirements,weforcetheweightstofalloffatadistanceequaltohalfthemaximumuncertaintyintervaloftherangemeasurements.Similarly,thesigneddistanceandweightfunctionsneednotextendfarinfrontofthesurface.Restrictingthefunctionstothevicinityofthesurfaceyieldsamorecompactrep-resentationandreducesthecomputationalexpenseofupdatingthevolume.Intwoandthreedimensions,therangemeasurementscorrespondtocurvesorsurfaceswithweightfunctions,andthesigneddistancerampshavedirectionsthatareconsistentwiththeprimarydirectionsofsensoruncertainty.Theuncertaintiesthatapplytorangeimageintegrationincludeerrorsinalignmentbetweenmeshesaswellaser-rorsinherentinthescanningtechnology.Anumberofalgorithmsforaligningsetsofrangeimageshavebeenexploredandshowntoyieldexcellentresults[11][30].Theremainingerrorliesinthescannerit-self.Foropticaltriangulationscanners,forexample,thiserrorhasbeenshowntobeellipsoidalabouttherangepoints,withthemajoraxisoftheellipsealignedwiththelinesofsightofthelaser[13][24].Figure4illustratesthetwo-dimensionalcaseforarangecurvederivedfromasinglescancontainingarowofrangesamples.In (e)(f) 2n1max Dmin(a)(d)Sensor Figure4.Combinationofsigneddistanceandweightfunctionsintwodimensions.(a)and(d)arethesigneddistanceandweightfunctions,re-spectively,generatedforarangeimageviewedfromthesensorlineofsightshownin(d).Thesigneddistancefunctionsarechosentovarybe-tween,asshownin(a).Theweightingfallsoffwithincreasingobliquitytothesensorandattheedgesofthemeshesasin-dicatedbythedarkerregionsin(e).Thenormals,shownin(e),areorientedatagrazingangleandfacingthesensor,respectively.Notehowtheweightingislower(darker)forthegrazingnormal.(b)and(e)arethesigneddistanceandweightfunctionsforarangeimageofthesameobjecttakenata60degreerotation.(c)isthesigneddistancefunc-correspondingtothepervoxelweightedcombinationof(a)and(b)constructedusingequations3and4.(f)isthesumoftheweightsateachvoxel,.Thedottedgreencurvein(c)istheisosurfacethatrepresentsourcurrentestimateoftheshapeoftheobject.practice,weuseaxedpointrepresentationforthesigneddistancefunction,whichboundsthevaluestoliebetweenasshowninthegure.Thevaluesofmustbeneg-ativeandpositive,respectively,astheyareonoppositesidesofasigneddistancezero-crossing.Forthreedimensions,wecansummarizethewholealgorithmasfollows.First,wesetallvoxelweightstozero,sothatnewdatawilloverwritetheinitialgridvalues.Next,wetessellateeachrangeim-agebyconstructingtrianglesfromnearestneighborsonthesampledlattice.Weavoidtessellatingoverstepdiscontinuities(cliffsintherangemap)bydiscardingtriangleswithedgelengthsthatexceedathreshold.Wemustalsocomputeaweightateachvertexasdescribedabove.Oncearangeimagehasbeenconvertedtoatrianglemeshwithaweightateachvertex,wecanupdatethevoxelgrid.Thesigneddistancecontributioniscomputedbycastingarayfromthesensorthrougheachvoxelneartherangesurfaceandthenintersectingitwiththetrianglemesh,asshowningure5.Theweightiscomputedbylinearlyinterpolatingtheweightsstoredattheintersectiontrianglevertices.Havingdeterminedthesigneddistanceandweightwecanapplytheupdateformulaedescribedinequations3and4.Atanypointduringthemergingoftherangeimages,wecanex-tractthezero-crossingisosurfacefromthevolumetricgrid.Were-strictthisextractionproceduretoskipsampleswithzeroweight,gen-eratingtrianglesonlyintheregionsofobserveddata.Wewillrelaxthisrestrictioninthenextsection.4HoleÞllingThealgorithmdescribedintheprevioussectionisdesignedtore-constructtheobservedportionsofthesurface.Unseenportionsofthesurfacewillappearasholesinthereconstruction.Whilethisre-sultisanaccuraterepresentationoftheknownsurface,theholesareestheticallyunsatisfyingandcanpresentastumblingblocktofollow-onalgorithmsthatexpectcontinuousmeshes.In[17],forexample, SensorRange surface wawbwc w d Voxel Viewingray Figure5.Samplingtherangesurfacetoupdatethevolume.Wecom-putetheweight,,andsigneddistance,,neededtoupdatethevoxelbycastingarayfromthesensor,throughthevoxelontotherangesurface.Weobtaintheweight,,bylinearlyinterpolatingtheweights()storedatneighboringrangevertices.Notethatforatranslatingsensor(likeourCyberwarescanner),thesensorpointisdifferentforeachcolumnofrangepoints.theauthorsdescribeamethodforparameterizingpatchesthatentailsgeneratingevenlyspacedgridlinesbywalkingacrosstheedgesofamesh.Gapsinthemeshpreventthealgorithmfromcreatingafairparameterization.Asanotherexample,rapidprototypingtechnolo-giessuchasstereolithographytypicallyrequireawatertightinordertoconstructasolidreplica[7].Oneoptionforllingholesistooperateonthereconstructedmesh.Iftheregionsofthemeshneareachholeareverynearlyplanar,thenthisapproachworkswell.However,holesinthemeshescanbe(andfrequentlyare)highlynon-planarandmayevenrequireconnectionsbetweenunconnectedcomponents.Instead,weofferaholellingapproachthatoperatesonourvolume,whichcontainsmoreinforma-tionthanthereconstructedmesh.Thekeytoouralgorithmliesinclassifyingallpointsinthevol-umeasbeinginoneofthreestates:unseen,empty,ornearthesur-face.Holesinthesurfaceareindicatedbyfrontiersbetweenunseenregionsandemptyregions(seeFigure6).Surfacesplacedatthesefrontiersofferaplausiblewaytoplugtheseholes(dottedinFigure6).Obtainingthisclassicationandgeneratingtheseholellersleadstoastraightforwardextensionofthealgorithmdescribedintheprevioussection:1.Initializethevoxelspacetothestate.2.Updatethevoxelsnearthesurfaceasdescribedintheprevi-oussection.Asbefore,thesevoxelstakeoncontinuoussigneddistanceandweightvalues.3.Followthelinesofsightbackfromtheobservedsurfaceandmarkthecorrespondingvoxelsasempty.Werefertothisstepasspacecarving4.Performanisosurfaceextractionatthezero-crossingofthesigneddistancefunction.Additionally,extractasurfacebe-tweenregionsseentobeemptyandregionsthatremainunseen.Inpractice,werepresenttheunseenandemptystatesusingthefunctionandweighteldsstoredonthevoxellattice.Werepresenttheunseenstatewiththefunctionvaluesandtheemptystatewiththefunctionvalues)=0,asshowninFigure6b.Thekeyadvantageofthisrepre-sentationisthatwecanusethesameisosurfaceextractionalgorithmweusedintheprevioussectionwithouttherestrictiononinterpo-latingvoxelsofzeroweight.Thisextractionndsboththesigneddistanceandholellisosurfacesandconnectsthemnaturallywheretheymeet,i.e.,atthecornersinFigure6awherethedottedredlinemeetsthedashedgreenline.Notethatthetrianglesthatarisefrom UnseenEmptyNear surface x) = 0W(x�) 0W(x) = 0 Sensor UnseenEmptyObservedisosurfaceHole fillisosurface surface(a)(b)D(x) = DminD(x) = Dmaxminmax D (x) Figure6.Volumetricgridwithspacecarvingandholelling.(a)Theregionsinfrontofthesurfaceareseenasempty,regionsinthevicinityofthesurfacerampthroughthezero-crossing,whileregionsbehindremainunseen.Thegreen(dashed)segmentsaretheisosurfacesgeneratedneartheobservedsurface,whilethered(dotted)segmentsareholellers,gen-eratedbytessellatingoverthetransitionfromemptytounseen.In(b),weidentifythethreeextremalvoxelstateswiththeircorrespondingfunctionvalues.interpolationsacrossvoxelsofzeroweightaredistinctfromtheoth-ers:theyareholellers.Wetakeadvantageofthisdistinctionwhensmoothingsurfacesasdescribedbelow.Figure6illustratesthemethodforasinglerangeimage,andpro-videsadiagramforthethree-stateclassicationscheme.Theholellerisosurfacesarefalseinthattheyarenotrepresentativeoftheobservedsurface,buttheydoderivefromobserveddata.Inpartic-ular,theycorrespondtoaboundarythatconneswherethesurfacecouldplausiblyexist.Inpractice,wendthatmanyoftheseholellersurfacesaregeneratedincrevicesthatarehardforthesensortoBecausethetransitionbetweenunseenandemptyisdiscontinuousandholelltrianglesaregeneratedasanisosurfacebetweenthesebi-narystates,withnosmoothtransition,wegenerallyobservealiasingartifactsintheseareas.Theseartifactscanbeeliminatedbyprelter-ingthetransitionregionbeforesamplingonthevoxellatticeusingstraightforwardmethodssuchasanalyticlteringorsuper-samplingandaveragingdown.Inpractice,wehaveobtainedsatisfactoryre-sultsbyapplyinganothertechnique:post-lteringthemeshafterre-constructionusingweightedaveragesofnearestvertexneighborsasdescribedin[29].Theeffectofthislteringstepistoblurtheholellsurface.Sinceweknowwhichtrianglescorrespondtoholellers,weneedonlyconcentratethesurfacelteringonthetheseportionsofthemesh.Thislocalizedlteringpreservesthedetailintheob-servedsurfacereconstruction.Toachieveasmoothblendbetweenlteredholellverticesandtheneighboringsurface,weallowthelterweightstoextendbeyondandtaperoffintothevicinityoftheholellboundaries.Wehavejustseenhowspacecarvingisausefuloperation:ittellsusmuchaboutthestructureoffreespace,allowingustoholesinanintelligentway.However,ouralgorithmonlycarvesbackfromobservedsurfaces.Therearenumeroussituationswheremorecarvingwouldbeuseful.Forexample,theinteriorwallsofahollowcylindermayeludedigitization,butbyseeingthroughthehollowportionofthecylindertoasurfaceplacedbehindit,wecanbetterapproximateitsgeometry.Wecanextendthecarvingparadigmtocoverthesesituationsbyplacingsuchabackdropbehindthesurfacesbeingscanned.Byplacingthebackdropoutsideofthevoxelgrid,weutilizeitpurelyforcarvingspacewithoutintroducingitsgeometryintothemodel.5Implementation5.1HardwareTheexamplesinthispaperwereacquiredusingaCyberware3030MSlaserstripeopticaltriangulationscanner.Figure1billustratesthescanninggeometry:anobjecttranslatesthroughaplaneoflaserlightwhilethereectionsaretriangulatedintodepthprolesthroughaCCDcamerapositionedoffaxis.Toimprovethequalityofthedata,weapplythemethodofspacetimeanalysisasdescribedin[6].Thetsofthisanalysisincludereducedrangenoise,greaterimmu-nitytoreectancechanges,andlessartifactsnearrangediscontinu-ities.WhenusingtraditionaltriangulationanalysisimplementedinhardwareinourCyberwarescanner,theuncertaintyintriangulationforoursystemfollowsthelinesofsightoftheexpandinglaserbeam.Whenusingthespacetimeanalysis,however,theuncertaintyfollowsthelinesofsightofthecamera.Theresultsdescribedinsection6ofthispaperwereobtainedwithoneortheothertriangulationmethod.Ineachcase,weadheretotheappropriatelinesofsightwhenlayingdownsigneddistanceandweightfunctions.5.2SoftwareThecreationofdetailed,complexmodelsrequiresalargeamountofinputdatatobemergedintohighresolutionvoxelgrids.Theex-amplesinthenextsectionincludemodelsgeneratedfromasmanyas70scanscontainingupto12millioninputverticeswithvolumet-ricgridsranginginsizeupto160millionvoxels.Clearly,timeandspaceoptimizationsarecriticalformergingthisdataandmanagingthesegrids.5.2.1Run-lengthencodingThecoredatastructureisarun-lengthencoded(RLE)volumewiththreeruntypes:empty,unseen,andvarying.Thevaryingeldsarestoredasastreamofvaryingdata,ratherthanrunsofconstantvalue.Typicalmemorysavingsvaryfrom10:1to20:1.Infact,thespacerequiredtorepresentoneofthesevoxelgridsisusuallylessthanthememoryrequiredtorepresentthenalmeshasalistofverticesandtriangleindices.5.2.2FastvolumetraversalUpdatingthevolumefromarangeimagemaybelikenedtoinversevolumerendering:insteadofreadingfromavolumeandwritingtoanimage,wereadfromarangeimageandwritetoavolume.Asare-sult,weleverageoffofasuccessfulideafromthevolumerenderingcommunity:forbestmemorysystemperformance,streamthroughthevolumeandtheimagesimultaneouslyinscanlineorder[18].Ingeneral,however,thescanlinesofarangeimagearenotalignedwiththescanlinesofthevoxelgrid,asshowninFigure7a.Bysuitablyresamplingtherangeimage,weobtainthedesiredalignment(Fig-ure7b).Theresamplingprocessconsistsofadepthrenderingoftherangesurfaceusingtheviewingtransformationspecictothelinesofsightoftherangesensorandusinganimageplaneorientedtoalignwiththevoxelgrid.Weassigntheweightsasvertexcolorstobelinearlyinterpolatedduringtherenderingstep,anapproachequiva-lenttoGouraudshadingoftrianglecolors.Tomergetherangedataintothevoxelgrid,westreamthroughthevoxelscanlinesinorderwhilesteppingthroughthecorrespondingscanlinesintheresampledrangeimage.WemapeachvoxelscanlinetothecorrectportionoftherangescanlineasdepictedinFigure7d,andweresampletherangedatatoyieldadistancefromtherangesurface.Usingthecombinationrulesgivenbyequations3and4,weupdatetherun-lengthencodedstructure.TopreservethelinearmemorystructureoftheRLEvolume(andthusavoidusinglinkedlistsofrunsscatteredthroughthememoryspace),wereadthevoxelscanlinesfromthecurrentvolumeandwritetheupdatedscanlinestoasecondRLEvolume;i.e.,wedouble-bufferthevoxelgrid.Notethatdependingonthescannergeometry,themappingfromvoxels (c) VoxelslicesRangeimageSensor(a)(d) VoxelslicesRangeimageSensor VolumeRange image Resampledrange image(b) Figure7.Rangeimageresamplingandscanlineordervoxelupdates.(a)Rangeimagescanlinesarenotingeneralorientedtoallowforcoherentlystreamingthroughvoxelandrangescanlines.(b)Byresamplingtherangeimage,wecanobtainthedesiredrangescanlineorientation.(c)Castingraysfromthepixelsontherangeimagemeanscuttingacrossscanlinesofthevoxelgrid,resultinginpoormemoryperformance.(d)Instead,werunalongscanlinesofvoxelmappingthemtothecorrectpositionsontheresampledrangeimage.torangeimagepixelsmaynotbelinear,inwhichcasecaremustbetakentoresampleappropriately[5].Forthecaseofmergingrangedataonlyinthevicinityofthesurface,wetrytoavoidprocessingvoxelsdistantfromthesurface.Tothatend,weconstructabinarytreeofminimumandmaximumdepthsforeveryadjacentpairofresampledrangeimagescanlines.Beforeprocessingeachvoxelscanline,wequerythebinarytreetodecidewhichvoxels,ifany,areneartherangesurface.Inthisway,onlyrelevantpiecesofthescanlineareprocessed.Inasimilarfash-ion,thespacecarvingstepscanbedesignedtoavoidprocessingvox-elsthatarenotseentobeemptyforagivenrangeimage.Theresult-ingspeed-upsfromthebinarytreearetypicallyafactorof15withoutcarving,andafactorof5withcarving.Wedidnotimplementabrute-forcevolumeupdatemethod,howeverwewouldexpecttheoverallalgorithmdescribedherewouldbemuchfasterbycomparison.5.2.3FastsurfaceextractionTogenerateournalsurfaces,weemployaMarchingCubesalgo-rithm[20]withalookuptablethatresolvesambiguouscases[22].Toreducecomputationalcosts,weonlyprocessvoxelsthathavevaryingdataorareattheboundarybetweenemptyandunseen.6ResultsWeshowresultsforanumberofobjectsdesignedtoexplorethero-bustnessofouralgorithm,itsabilitytollgapsinthereconstruction,anditsattainablelevelofdetail.Toexplorerobustness,wescannedathindrillbitusingthetraditionalmethodofopticaltriangulation.Duetothefalseedgeextensionsinherentindatafromtriangulationscanners[6],thisparticularobjectposesaformidablechallenge,yetthevolumetricmethodbehavesrobustlywherethezipperingmethod[30]failscatastrophically.ThedragonsequenceinFigure11demon-stratestheeffectivenessofcarvingspaceforholelling.Theuseofabackdrophereisparticularlyeffectiveinllingthegapsinthemodel.Notethatwedonotusethebackdropatalltimes,inpartbecausetherangeimagesaremuchdenserandmoreexpensivetoprocess,andalsobecausethebackdroptendstoobstructthepathoftheobjectwhenautomaticallyrepositioningitwithourmotioncon-trolplatform.Finally,theHappyBuddhasequenceinFigure12showsthatourmethodcanbeusedtogenerateverydetailed,hole-freemodelssuitableforrenderingandrapidmanufacturing.StatisticsforthereconstructionofthedragonandBuddhamodelsappearinFigure8.Withtheoptimizationsdescribedintheprevioussection,wewereabletoreconstructtheobservedportionsofthesur-facesinunderanhourona250MHzMIPSR4400processor.Thespacecarvingandholellingalgorithmisnotcompletelyoptimized,buttheexecutiontimesarestillintherangeof3-5hours,lessthanthetimespentacquiringandregisteringtherangeimages.Forbothmodels,theRMSdistancebetweenpointsintheoriginalrangeim-agesandpointsonthereconstructedsurfacesisapproximately0.1mm.Thisgureisroughlythesameastheaccuracyofthescanningtechnology,indicatinganearlyoptimalsurfacereconstruction.7DiscussionandfutureworkWehavedescribedanewalgorithmforvolumetricintegrationofrangeimages,leadingtoasurfacereconstructionwithoutholes.Thealgorithmhasanumberofdesirableproperties,includingtherepre-sentationofdirectionalsensoruncertainty,incrementalandorderin-dependentupdating,robustnessinthepresenceofsensorerrors,andtheabilitytollgapsinthereconstructionbycarvingspace.Ouruseofarun-lengthencodedrepresentationofthevoxelgridandsynchro-nizedprocessingofvoxelandresampledrangeimagescanlinesmakethealgorithmefcient.Thisinturnallowsustoacquireandintegratealargenumberofrangeimages.Inparticular,wedemonstratetheabilitytointegrateupto70scansintoahighresolutionvoxelgridtogeneratemillionpolygonmodelsinafewhours.Thesemodelsarefreeofholes,makingthemsuitableforsurfacetting,rapidproto-typing,andrendering.Thereareanumberoflimitationsthatpreventusfromgeneratingmodelsfromanarbitraryobject.Someoftheselimitationsarisefromthealgorithmwhileothersarisefromthelimitationsofthescanningtechnology.Amongthealgorithmiclimitations,ourmethodhasdif-cultybridgingsharpcornersifnoscanspansbothsurfacesmeetingatthecorner.Thisislessofaproblemwhenapplyingourhole-llingalgorithm,butwearealsoexploringmethodsthatwillworkwith-outholelling.Thinsurfacesarealsoproblematic.Asdescribedinsection3,theinuencesofobservedsurfacesextendbehindtheirestimatedpositionsforeachrangeimageandcaninterferewithdis-tancefunctionsoriginatingfromscansoftheoppositesideofathinsurface.Inthisrespect,theapexesofsharpcornersalsobehavelikethinsurfaces.Whilewehavelimitedthisinuenceasmuchaspos-sible,itstillplacesalowerlimitonthethicknessofsurfacethatwecanreliablyreconstructwithoutcausingartifactssuchasthickeningofsurfacesorroundingofsharpcorners.Wearecurrentlyworkingtoliftthisrestrictionbyconsideringtheestimatednormalsofsurfaces.Otherlimitationsarisefromthescanningtechnologiesthemselves.Opticalmethodssuchastheoneweuseinthispapercanonlyprovidedataforexternalsurfaces;internalcavitiesarenotseen.Further,verycomplicatedobjectsmayrequireanenormousamountofscanningtocoverthesurface.Opticaltriangulationscanninghastheadditionalproblemthatboththelaserandthesensormustobserveeachpointonthesurface,furtherrestrictingtheclassofobjectsthatcanbescannedcompletely.Thereectancepropertiesofobjectsarealsoafactor.Opticalmethodsgenerallyoperatebycastinglightontoanobject,butshinysurfacescandeectthisillumination,darkobjectscanabsorbit,andbrightsurfacescanleadtointerreections.Tominimizetheseeffects,weoftenpaintourobjectswithaat,graypaint.Straightforwardextensionstoouralgorithmincludeimprovingtheexecutiontimeofthespacecarvingportionofthealgorithmanddemonstratingparallelizationofthewholealgorithm.Inaddition, 472.4 MModelScans Figure8.StatisticsforthereconstructionofthedragonandBuddhamod-els,withandwithoutspacecarving.moreaggressivespacecarvingmaybepossiblebymakinginferencesaboutsensorlinesofsightthatreturnnorangedata.Inthefuture,wehopetoapplyourmethodstootherscanningtechnologiesandtolargescaleobjectssuchasterrainandarchitecturalscenes.AcknowledgmentsWewouldliketothankPhilLacrouteforhismanyhelpfulsugges-tionsindesigningthevolumetricalgorithms.AfraZomorodianwrotethescriptinginterfaceforscanningautomation.HomanIgehywrotethefastscanconversioncode,whichweusedforrangeimageresam-pling.ThankstoBillLorensenforhismarchingcubestablesandmeshdecimationsoftware,andforgettingthe3Dhardcopymade.MattPharrdidtheaccessibilityshadingusedtorenderthecolorBud-dha,andPatHanrahanandJulieDorseymadehelpfulsuggestionsforRenderMantricksandlightingmodels.ThanksalsotoDavidAddle-manandGeorgeDabrowskiofCyberwarefortheirhelpandfortheuseoftheirscanner.ThisworkwassupportedbytheNationalSci-enceFoundationundercontractCCR-9157767andIntervalResearchCorporation.References[1]C.L.Bajaj,F.Bernardini,andG.Xu.Automaticreconstructionofsurfacesandscalareldsfrom3Dscans.InProceedingsofSIGGRAPHÕ95(LosAngeles,CA,Aug.6-11,1995),pages109118.ACMPress,August1995.[2]J.-D.Boissonnat.Geometricstructuresforthree-dimensionalshaperepresentation.ACMTransactionsonGraphics,3(4):266286,October1984.[3]C.H.Chien,Y.B.Sim,andJ.K.Aggarwal.Generationofvolume/surfaceoctreefromrangedata.InTheComputerSocietyConferenceonComputerVisionandPatternRecognition,pages25460,June1988.[4]C.I.Connolly.Cumulativegenerationofoctreemodelsfromrangedata.InPro-ceedings,Intl.Conf.Robotics,pages2532,March1984.[5]B.Curless.Betteropticaltriangulationandvolumetricreconstructionofcomplexmodelsfromrangeimages.PhDthesis,StanfordUniversity,1996.[6]B.CurlessandM.Levoy.Betteropticaltriangulationthroughspacetimeanalysis.ProceedingsofIEEEInternationalConferenceonComputerVision,pages987994,June1995.[7]A.Dolenc.Softwaretoolsforrapidprototypingtechnologiesinmanufactur-ActaPolytechnicaScandinavica:MathematicsandComputerScienceSeriesMa62:1111,1993.[8]D.Eberly,R.Gardner,B.Morse,S.Pizer,andC.Scharlach.Ridgesforimageanalysis.JournalofMathematicalImagingandVision,4(4):353373,Dec1994.[9]H.EdelsbrunnerandE.P.Mucke.Three-dimensionalalphashapes.InWorkshoponVolumeVisualization,pages75105,October1992.[10]A.ElfesandL.Matthies.Sensorintegrationforrobotnavigation:combiningsonarandrangedatainagrid-basedrepresentation.InProceedingsofthe26thIEEEConferenceonDecisionandControl,pages18021807,December1987.[11]H.Gagnon,M.Soucy,R.Bergevin,andD.Laurendeau.Registrationofmulti-plerangeviewsforautomatic3-Dmodelbuilding.InProceedings1994IEEEComputerSocietyConferenceonComputerVisionandPatternRecognition,pages586,June1994.[12]E.Grosso,G.Sandini,andC.Frigato.Extractionof3Dinformationandvolumet-ricuncertaintyfrommultiplestereoimages.InProceedingsofthe8thEuropeanConferenceonArtiÞcialIntelligence,pages683688,August1988.[13]P.Hebert,D.Laurendeau,andD.Poussart.Scenereconstructionanddescription:geometricprimitiveextractionfrommultipleviewedscattereddata.InProceedings (a)(b)(e)(f)(c)(d) Figure9.Mergingrangeimagesofadrillbit.Wescanneda1.6mmdrillbitfrom12orientationsata30degreespacingusingtraditionalopticaltriangulationmethods.Illustrations(a)-(d)eachshowaplan(top)viewofaslicetakenthroughtherangedataandtworeconstructions.(a)Therangedatashownasunorganizedpoints:algorithmsthatoperateonthisformofdatawouldlikelyhavedifcultyderivingthecorrectsurface.(b)Therangedatashownasasetofwireframetessellationsoftherangedata:thefalseedgeextensionsposeachallengetobothpolygonandvolumetricmethods.(c)Aslicethroughthereconstructedsurfacegeneratedbyapolygonmethod:thezipperingalgorithmofTurk[31].(d)Aslicethroughthereconstructedsurfacegeneratedbythevolumetricmethoddescribedinthispaper.(e)Arenderingofthezipperedsurface.(f)Arenderingofthevolumetricallygeneratedsurface.Notethecatastrophicfailureofthezipperingalgorithm.Thevolumetricmethod,however,producesawatertightmodel.(g)Aphotographoftheoriginaldrillbit.Thedrillbitwaspaintedwhiteforscanning. ofIEEEConferenceonComputerVisionandPatternRecognition,pages286June1993.[14]A.Hilton,A.J.Toddart,J.Illingworth,andT.Windeatt.Reliablesurfacerecon-structionfrommultiplerangeimages.InFourthEuropeanConferenceonCom-puterVision,volumeI,pages117126,April1996.[15]Tsai-HongHongandM.O.Shneier.Describingarobotsworkspaceusingase-quenceofviewsfromamovingcamera.IEEETransactionsonPatternAnalysisandMachineIntelligence,7(6):721726,November1985.[16]H.Hoppe,T.DeRose,T.Duchamp,J.McDonald,andW.Stuetzle.Surfacere-constructionfromunorganizedpoints.InComputerGraphics(SIGGRAPHÕ92Proceedings),volume26,pages7178,July1992.[17]V.KrishnamurthyandM.Levoy.Fittingsmoothsurfacestodensepolygonmeshes.Intheseproceedings.[18]P.LacrouteandM.Levoy.Fastvolumerenderingusingashear-warpfactorizationoftheviewingtransformation.InProceedingsofSIGGRAPHÕ94(Orlando,FL,July24-29,1994),pages451458.ACMPress,July1994.[19]A.LiandG.Crebbin.Octreeencodingofobjectsfromrangeimages.PatternRecognition,27(5):727739,May1994.[20]W.E.LorensenandH.E.Cline.Marchingcubes:Ahighresolution3Dsurfaceconstructionalgorithm.InComputerGraphics(SIGGRAPHÕ87Proceedings)volume21,pages163169,July1987.[21]W.N.MartinandJ.K.Aggarwal.Volumetricdescriptionsofobjectsfrommul-tipleviews.IEEETransactionsonPatternAnalysisandMachineIntelligence5(2):150158,March1983.[22]C.Montani,R.Scateni,andR.Scopigno.Amodiedlook-uptableforimplicitdisambiguationofmarchingcubes.VisualComputer,10(6):353355,1994.[23]M.Potmesil.Generatingoctreemodelsof3Dobjectsfromtheirsilhouettesinasequenceofimages.ComputerVision,Graphics,andImageProcessing,40(1):129,October1987.[24]M.Rutishauser,M.Stricker,andM.Trobina.Mergingrangeimagesofarbitrar-ilyshapedobjects.InProceedings1994IEEEComputerSocietyConferenceonComputerVisionandPatternRecognition,pages573580,June1994.[25]M.SoucyandD.Laurendeau.Ageneralsurfaceapproachtotheintegrationofasetofrangeviews.IEEETransactionsonPatternAnalysisandMachineIntelligence17(4):344358,April1995.[26]G.Succi,G.Sandini,EGrosso,andM.Tistarelli.3Dfeatureextractionfromsequencesofrangedata.InRoboticsResearch.FifthInternationalSymposiumpages117127,August1990.[27]R.Szeliski.Rapidoctreeconstructionfromimagesequences.CVGIP:ImageUnderstanding,58(1):2332,July1993.[28]G.HTarboxandS.N.Gottschlich.IVIS:Anintegratedvolumetricinspectionsys-tem.InProceedingsofthe1994SecondCAD-BasedVisionWorkshop,pages220227,February1994.[29]G.Taubin.Asignalprocessingapproachtofairsurfacedesign.InProceedingsofSIGGRAPHÕ95(LosAngeles,CA,Aug.6-11,1995),pages351358.ACMPress,August1995.[30]G.TurkandM.Levoy.Zipperedpolygonmeshesfromrangeimages.InProceed-ingsofSIGGRAPHÕ94(Orlando,FL,July24-29,1994),pages311318.ACMPress,July1994.[31]RobertWeinstock.TheCalculusofVariations,withApplicationstoPhysicsandEngineering.DoverPublications,1974.AIsosurfaceasleastsquaresminimizerItispossibletoshowthattheisosurfaceoftheweightedsigneddistancefunctionisequivalenttoaleastsquaresminimizationofsquareddistancesbetweenpointsontherangesurfacesandpointsonthedesiredreconstruction.Thekeyassumptionsarethattherangesensorisorthographicandthattherangeerrorsareindependentlydis-tributedalongsensorlinesofsight.Afullproofisbeyondthescopeofthispaper,butweprovideasketchhere.See[5]fordetails.Consideraregion,,onthedesiredsurface,,whichisobservedrangeimages.Wedenetheerrorbetweenanobservedrangesurfaceandapossiblereconstructedsurfaceastheintegraloftheweightedsquareddistancesbetweenpointsontherangesurfaceandthereconstructedsurface.Thesedistancesaretakenalongthelinesofsightofthesensor,commensuratewiththepredominantdirectionsofuncertainty(seeFigure10).Thetotalerroristhesumoftheintegralsfortherangeimages: x;yx;y;zFigure10.Tworangesurfaces,,aretessellatedrangeimagesacquiredfromdirections.Thepossiblerangesurface,x;y,isevaluatedintermsoftheweightedsquareddistancestopointsontherangesurfacestakenalongthelinesofsighttothesensor.Apoint,x;y;z,isshownherebeingevaluatedtonditscorrespondingsigneddistances,,andweights,s;t;fs;t;f(6)whereeachs;tcorrespondstoaparticularsensorlineofsightforeachrangeimage,isthedomainofintegrationforthethrangeimage,ands;t;fs;t;faretheweightsandsigneddis-tancestakenalongthethrangeimageslinesofsight.Now,consideracanonicaldomain,,onaparameterplane,x;y,overwhichisafunctionx;y.Thetotalerrorcanbere-writtenasanintegrationoverthecanonicaldomain:x;y;zx;y;z @x (7)whereisthesensingdirectionofthethrangeimage,andtheweightsanddistancesareevaluatedateachpoint,x;y;z,bymappingthemtothelinesofsightofthecorrespondingrangeimage.Thedotproductrepresentsacorrectiontermthatrelatesdifferentialareasintodifferentialareasin.Applyingthecalculusofvari-ations[31],wecanconstructapartialdifferentialequationforthethatminimizesthisintegral.Solvingthisequationwearriveatthefollowingrelation::wi(x;y;zx;y;z]=0(8)whereisthedirectionalderivativealong.Sincetheweightassociatedwithalineofsightdoesnotvaryalongthatlineofsight,andthesigneddistancehasaderivativeofunityalongthelineofsight,wecansimplifythisequationto:x;y;zx;y;z(9)Thisweightedsumofsigneddistancesisthesameaswhatwecomputeinequations1and2,withoutthedivisionbythesumoftheweights.Sincethethisdivisorisalwayspositive,theisosurfaceweextractinsection3isexactlytheleastsquaresminimizingsurfacedescribedhere. (a)(b)(f)(g)(h)(i)(j)(k) Figure11.Reconstructionofadragon.Illustrations(a)-(d)arefullviewsofthedragon.Illustrations(e)-(h)aremagniedviewsofthesectionhighlightedbythegreenboxin(a).Regionsshowninredcorrespondtoholelltriangles.Illustrations(i)-(k)areslicesthroughthecorrespondingvolumetricgridsatthelevelindicatedbythegreenlinein(e).(a)(e)(i)Reconstructionfrom61rangeimageswithoutspacecarvingandholelling.Themagniedrenderinghighlightstheholesinthebelly.Theslicethroughthevolumetricgridshowshowthesigneddistancerampsaremaintainedclosetothesurface.Thegapintherampsleadstoaholeinthereconstruction.(b)(f)(j)Reconstructionwithspacecarvingandholellingusingthesamedataasin(a).Whilesomeholesarelledinareasonablemanner,somelargeregionsofspaceareleftuntouchedandcreateextraneoustessellations.Theslicethroughthevolumetricgridrevealsthattheisosurfacebetweentheunseen(brown)andempty(black)regionswillbeconnectedtotheisosurfaceextractedfromthedistanceramps,makingitpartoftheconnectedcomponentofthedragonbodyandleavinguswithasubstantialnumberoffalsesurfaces.(c)(g)(k)Reconstructionwith10additionalimagesusingsurfacestoeffectmorecarving.Noticehowtheextraneousholelltrianglesnearlyvanish.Thevolumetricsliceshowshowwehavemanagedtoemptyoutthespacenearthebelly.Thebumpinessalongtheholellregionsofthebellyin(g)correspondstoaliasingartifactsfromtessellatingoverthediscontinuoustransitionbetweenunseenandemptyregions.(d)(h)Reconstructionasin(c)(g)withlteringoftheholellportionsofthemesh.Thelteringoperationblursoutthealiasingartifactsintheholellregionswhilepreservingthedetailintherestofthemodel.Carefulexaminationof(h)revealsafaintridgeinthevicinityofthesmoothedholell.Thisridgeisactualgeometrypresentinalloftherenderings,(e)-(h).Thenalmodelcontains1.8millionpolygonsandiswatertight. (a)(b) Figure12.Reconstructionand3DhardcopyoftheHappyBuddha.Theoriginalisaplasticandrosewoodstatuettethatstands20cmtall.Notethatthecameraparametersforeachoftheseimagesisdifferent,creatingaslightlydifferentperspectiveineachcase.(a)Photographoftheoriginalafterspraypaintingitmattegraytosimplifyscanning.(b)Gouraud-shadedrenderingofonerangeimageofthestatuette.ScanswereacquiredusingaCyberwarescanner,modiedtopermitspacetimetriangulation[6].Thisgureillustratesthelimitedandfragmentarynatureoftheinformationavailablefromasinglerangeimage.(c)Gouraud-shadedrenderingofthe2.4millionpolygonmeshaftermerging48scans,butbeforehole-lling.Noticethatthereconstructedmeshhasatleastasmuchdetailasthesinglerangeimage,butislessnoisy;thisismostapparentaroundthebelly.Theholeinthebaseofthemodelcorrespondstoregionsthatwerenotobserveddirectlybytherangesensor.(d)RenderManrenderingofan800,000polygondecimatedversionofthehole-lledandlteredmeshbuiltfrom58scans.Byplacingabackdropbehindthemodelandtaking10additionalscans,wewereabletoseethroughthespacebetweenthebaseandtheBuddhasgarments,allowingustocarvespaceandlltheholesinthebase.(e)Photographofahardcopyofthe3Dmodel,manufacturedby3DSystems,Inc.,usingstereolithography.Thecomputermodelwasslicedinto500layers,150micronsapart,andthehardcopywasbuiltuplayerbylayerbyselectivelyhardeningaliquidresin.Theprocesstookabout10hours.Afterwards,themodelwassandedandbead-blastedtoremovethestair-stepartifactsthatariseduringlayeredmanufacturing.