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systemdict /pop get put userdict /pdfmark systemdict /cleartomark get put}ifelseps:SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if endps:SDict begin [ /Producer (dvips + Distiller) /Title () /Subject () /Creator (LaTeX with hyperref package) /Author () /Keywords () /DOCINFO pdfmark end򽍠e̍Ncolor push Blackcolor push gray 0ps:SDict begin H.S endcolor push gray 0 color popps:SDict begin H.R endFps:SDict begin [ /View [/XYZ H.V] /Dest (page.1) cvn /DEST pdfmark end color popG4 color pop,4Necps:SDict begin [ /Count 4 /Dest (chapter.1) cvn /Title (Multi-level decomposition) /OUT pdfmark endYps:SDict begin [ /Count -0 /Dest (section.1.1) cvn /Title (Introduction) /OUT pdfmark endSps:SDict begin [ /Count -4 /Dest (section.1.2) cvn /Title (Method) /OUT pdfmark endxps:SDict begin [ /Count -0 /Dest (subsection.1.2.1) cvn /Title (Overview of the proposed decomposition) /OUT pdfmark endlps:SDict begin [ /Count -0 /Dest (subsection.1.2.2) cvn /Title (Usage of the decomposition) /OUT pdfmark end|ps:SDict begin [ /Count -0 /Dest (subsection.1.2.3) cvn /Title (Decomposing a SE using the proposed method) /OUT pdfmark endbps:SDict begin [ /Count -0 /Dest (subsection.1.2.4) cvn /Title (Method for 3D SE) /OUT pdfmark endaps:SDict begin [ /Count -0 /Dest (section.1.3) cvn /Title (Results and analysis) /OUT pdfmark endfps:SDict begin [ /Count -0 /Dest (section.1.4) cvn /Title (Discussion and conclusion) /OUT pdfmark endTps:SDict begin [ /Page 1 /View [ /FitV ] /PageMode /UseOutlines /DOCVIEW pdfmark end/ps:SDict begin [ {Catalog}<<>> /PUT pdfmark endps:SDict begin H.S endps:SDict begin 12 H.A endIps:SDict begin [ /View [/XYZ H.V] /Dest (Doc-Start) cvn /DEST pdfmark endpapersize=614.295pt,794.96999ptps:SDict begin H.S endps:SDict begin 12 H.A endIps:SDict begin [ /View [/XYZ H.V] /Dest (chapter.1) cvn /DEST pdfmark end color push Black color pop`GGecrb1728Multi-levfeldecompsositionofEuclideanspheresZV eccc1000MichaelSarjuVhaz`^T2ecrm07001d 1 ecrm1000,U A tillaPeterKiralyl*M^2sand RZussellManningMersereauo̟^3#-:(kTecsl06001Iecsl0800Barco(MedicalImagingSystems,BeaÒverton,(OR,USA )&ecst0800michael.vaz@barco.com-:2Siemens(Corp cmmi10Union7 :DGil2`ationc:MorGphol2`og[icalClosing N\:InterGsection0 :Erosion0:Morphol2`og[icalOpeningNps:SDict begin H.S endps:SDict begin 12 H.A endLps:SDict begin [ /View [/XYZ H.V] /Dest (definition.1) cvn /DEST pdfmark endcolor push Black.]f ecbx1000Denition1.1(ConvexySE).Y color popIfthexdiscreteSERisequaltothesetofallvoxelsops:SDict begin H.S endps:SDict begin 12 H.A endKps:SDict begin [ /View [/XYZ H.V] /Dest (section.1.2) cvn /DEST pdfmark end Iڍcolor push Black2. ? color popMetho_d)Qps:SDict begin H.S endps:SDict begin 12 H.A endPps:SDict begin [ /View [/XYZ H.V] /Dest (subsection.1.2.1) cvn /DEST pdfmark end Lcolor push Black2.1 ( color 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begin H.R endps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0]BorderArrayPatch /Subtype /Link /Dest (figure.1.1) cvn H.B /ANN pdfmark end color popillustratesEquationcolor push rgb 1 0 0ps:SDict begin H.S endcolor push rgb 1 0 01a color pop `ps:SDict begin H.R end `ps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0]BorderArrayPatch /Subtype /Link /Dest (equation.1.2.1a) cvn H.B /ANN pdfmark end color popandshowsthethreepartitionsobtainedbyK>usingtheK?propGosedmethodtodecomposetheexampleK?2DK;SEKright?handside.N5ThecubicandsparsefactorsofeachofthesepartitionsareU showninFigurecolor push rgb 1 0 0ps:SDict begin H.S endcolor push rgb 1 0 02 color popps:SDict begin H.R endps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0]BorderArrayPatch /Subtype /Link /Dest (figure.1.2) cvn H.B /ANN pdfmark end color pop,whichillustratesEquationcolor push rgb 1 0 0ps:SDict begin H.S endcolor push rgb 1 0 01b color pop ps:SDict begin H.R end ps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0]BorderArrayPatch /Subtype /Link /Dest (equation.1.2.1a) cvn H.B /ANN pdfmark end color pop.Ncolor push BlackG4 color popL⟷ps:SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end򽍠e̍Ncolor push Blackcolor push gray 0ps:SDict begin H.S endcolor push gray 0 color popps:SDict begin H.R endFps:SDict begin [ /View [/XYZ H.V] /Dest (page.4) cvn /DEST pdfmark end color popG4 color pop,4Ru,Ncolor push Blacko`a$KPSfile="./images/FIG1.png" llx=0 lly=0 urx=254.20726 ury=65.19667 rwi=2542 )1/, ecti0900Figur$e5.1.7scolor push gray 0ps:SDict begin H.S endcolor push gray 0 color popps:SDict begin H.R endJps:SDict begin [ /View [/XYZ H.V] /Dest (figure.1.1) cvn /DEST pdfmark end color popThepropAosedmethoddecomposestheexampleSE(righpAossibleduetotheidempotency>propertps:SDict begin H.S endps:SDict begin 12 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TNps:SDict begin H.S endps:SDict begin 12 H.A endPps:SDict begin [ /View [/XYZ H.V] /Dest (subsubsection*.5) cvn /DEST pdfmark end T estingforsparsenesstW*eenforcethatoursparsefactorisnotface-connectedfor3Doredgeconnectedfor2D.Thereareafewexceptionstothisrule,suchasifthereisconnectivityLHwiththecentralvoxel(loGcalLIorigin)whileintherstiterationofU decompGosition,i.e.,duringtheprocessofdeterminingS1|s.Ncolor push BlackG4 color pop⟷ps:SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end򽍠e̍Ncolor push Blackcolor push gray 0ps:SDict begin H.S endcolor push gray 0 color popps:SDict begin H.R endFps:SDict begin [ /View [/XYZ H.V] /Dest (page.7) cvn /DEST pdfmark end color popG4 color pop,4㍒Ncolor push Blackh9󍍑lLPSfile="./images/FIG5.png" llx=0 lly=0 urx=269.34795 ury=189.92078 rwi=2693 U1Figur$efL5.color push gray 0ps:SDict begin H.S endcolor push gray 0 color popps:SDict begin H.R endJps:SDict begin [ /View [/XYZ H.V] /Dest (figure.1.5) cvn /DEST pdfmark end color popDeterminingS2Cfortheconsidered2Dexample.#Obtaintestcubicfactors CTj,bSE2,+RTj,*andcandidatesparsefactorsSTj.}Determine5the5subsetofSTj thatisindeed5sparseandfromthissubsetpicTjtoRSEi,r. color popN|ps:SDict begin H.S endps:SDict begin 12 H.A endPps:SDict begin [ /View [/XYZ H.V] /Dest (subsection.1.2.4) cvn /DEST pdfmark end color push Black2.4 ( color popMetho_dfor3DSEtcolor push Black color popOnce8the8decompGositionisavqailable,>usageisidenticaltothe2D8case discussedU abGove.Mcolor push Black color popDecompGosition<@issphere=isshoLimagesegmentation,BSE>Fmay>LbGedenedinthe>KcontextofanatomyN(ex:3hmmsphere)hinsteadofdiscretely*,andthediscreteSEhcouldthenNbGeCobtainedatBrun-timebasedonthescaleofthemedicalimageunderNconsideration:'thechestCTimageusedtoobtaintheresultsshowninNFigureU color push rgb 1 0 0ps:SDict begin H.S endcolor push rgb 1 0 05 color popps:SDict begin H.R endps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0]BorderArrayPatch /Subtype /Link /Dest (figure.1.5) cvn H.B /ANN pdfmark end color popwasquantizedatammscaleof0:6780:670:80. NAQJcomparisonQofQpropGosedmethodwithothersisunderway*.AnQempiricalNassessmentJofIthegainsofthemethoGdincontextofrealmedicalapplicationsNwillJbGeperformed.Sincethemethodlendsitselftovectorization,anevenNfasterU GraphicsProGcessingUnit(GPU)implementationispossible."ONReferencesN ps:SDict begin H.S endps:SDict begin 9.5 H.A endFps:SDict begin [ /View [/XYZ H.V] /Dest (Item.9) cvn /DEST pdfmark end color push Black6"ecrm0800[ps:SDict begin H.S end1ps:SDict begin 9.5 H.A endVps:SDict begin [ /View [/XYZ H.V] /Dest (cite.Vaz:Serra:1982:SE) cvn /DEST pdfmark end] color pop 8J.wSerra,Y7 {}ecti0800ImageuAnalysistandMathematicalMorphology,XAÒcademic Press,YLondon, 81982.ps:SDict begin H.S endps:SDict begin 9.5 H.A endGps:SDict begin [ /View [/XYZ H.V] /Dest (Item.10) cvn /DEST pdfmark end }color push Black[ps:SDict begin H.S end2ps:SDict begin 9.5 H.A endYps:SDict begin [ /View [/XYZ H.V] /Dest (cite.Vaz:Gonzalez:1992:SE) cvn /DEST pdfmark end] color pop 8R.{GonzaleziandjR.WJoMedicalImaging(Lund,@SwÒeden,Septemb710,A1997),ComputersinCardi- 8ology1997(A.KMurraÒyandS.JSwiryn,eds.),VJol.24,IEEE,PiscataÒwayJ,1997,1997, 8pp.(525528.ps:SDict begin H.S endps:SDict begin 9.5 H.A endGps:SDict begin [ /View [/XYZ H.V] /Dest (Item.16) cvn /DEST pdfmark end color push Black[ps:SDict begin H.S end8ps:SDict begin 9.5 H.A endWps:SDict begin [ /View [/XYZ H.V] /Dest (cite.Vaz:Soille:1996:SE) cvn /DEST pdfmark end] color pop 8PJ.Soille,vE.J.Breen,andpR.Jones,Recursive)implementationof)erosionsanddila- 8tionsUalongTdiscretelinesatarbitraryangles,IEEETJransactionsonPÒatternAnalysis 8and(MacÒhineIntelligence18(1996),no.5,562567.ps:SDict begin H.S endps:SDict begin 9.5 H.A endGps:SDict begin [ /View [/XYZ H.V] /Dest (Item.17) cvn /DEST pdfmark end color push Black[ps:SDict begin H.S end9ps:SDict begin 9.5 H.A endWps:SDict begin [ /View [/XYZ H.V] /Dest (cite.Vaz:Soille:2001:SE) cvn /DEST pdfmark end] color pop 8PJ.$SoilleĶandĵH.Talb> /Subtype /Link H.B /ANN pdfmark end color pop>(AÒccessin:2006-01-05.ps:SDict begin H.S endps:SDict begin 9.5 H.A endGps:SDict begin [ /View [/XYZ H.V] /Dest (Item.19) cvn /DEST pdfmark end color push Black[ps:SDict begin H.S end11ps:SDict begin 9.5 H.A endVps:SDict begin [ /View [/XYZ H.V] /Dest (cite.Vaz:Jones:1994:SE) cvn /DEST pdfmark end] color popwR.PJonesandI.Sv$alb> /Subtype /Link H.B /ANN pdfmark end color pop>(AÒccessin:2006-03-10.Ncolor push BlackG4 color pop;[{ 9&ectt08008Yecbx08007 {}ecti08006"ecrm08002 cmsy915" cmmi90o cmr9/, ecti0900.]f ecbx1000-!N ecbx1200+u ecbx0900*. ecrm0900)&ecst0800(kTecsl0600;cmmi6Aacmr6Iecsl0800T2ecrm0700V eccc1000`GGecrb1728 1 ecrm1000 !", cmsy10 b> cmmi10 0ercmmi7K`y cmr10ٓRcmr70