数字图像处理 外文翻译 外文文献 英文文献 数字图像处理.docx
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数字图像处理 外文翻译 外文文献 英文文献 数字图像处理.docx
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数字图像处理外文翻译外文文献英文文献数字图像处理
数字图像处理外文翻译外文文献英文文献数字图像处理
DigitalImageProcessing
1Introduction
Manyoperatorshavebeenproposedforpresentingaconnectedcomponentnadigitalimagebyareducedamountofdataorsimpliedshape.Ingeneralwehavetostatethatthedevelopment,choiceandmodi_cationofsuchalgorithmsinpracticalapplicationsaredomainandtaskdependent,andthereisno\bestmethod".However,itisinterestingtonotethatthereareseveralequivalencesbetweenpublishedmethodsandnotions,andcharacterizingsuchequivalencesordi_erencesshouldbeusefultocategorizethebroaddiversityofpublishedmethodsforskeletonization.Discussingequivalencesisamainintentionofthisreport.
1.1CategoriesofMethods
Oneclassofshapereductionoperatorsisbasedondistancetransforms.Adistanceskeletonisasubsetofpointsofagivencomponentsuchthateverypointofthissubsetrepresentsthecenterofamaximaldisc(labeledwiththeradiusofthisdisc)containedinthegivencomponent.Asanexampleinthis_rstclassofoperators,thisreportdiscussesonemethodforcalculatingadistanceskeletonusingthed4distancefunctionwhichisappropriatetodigitizedpictures.Asecondclassofoperatorsproducesmedianorcenterlinesofthedigitalobjectinanon-iterativeway.Normallysuchoperatorslocatecriticalpoints_rst,andcalculateaspeci_edpaththroughtheobjectbyconnectingthesepoints.
Thethirdclassofoperatorsischaracterizedbyiterativethinning.Historically,Listing[10]usedalreadyin1862thetermlinearskeletonfortheresultofacontinuousdeformationofthefrontierofaconnectedsubsetofaEuclideanspacewithoutchangingtheconnectivityoftheoriginalset,untilonlyasetoflinesandpointsremains.Manyalgorithmsinimageanalysisarebasedonthisgeneralconceptofthinning.Thegoalisacalculationofcharacteristicpropertiesofdigitalobjectswhicharenotrelatedtosizeorquantity.Methodsshouldbeindependentfromthepositionofasetintheplaneorspace,gridresolution(fordigitizingthisset)ortheshapecomplexityofthegivenset.Intheliteraturetheterm\thinning"isnotused
-1-
inauniqueinterpretationbesidesthatitalwaysdenotesaconnectivitypreservingreductionoperationappliedtodigitalimages,involvingiterationsoftransformationsofspeci_edcontourpointsintobackgroundpoints.AsubsetQ_Iofobjectpointsisreducedbyade_nedsetDinoneiteration,andtheresultQ0=QnDbecomesQforthenextiteration.Topology-preservingskeletonizationisaspecialcaseofthinningresultinginaconnectedsetofdigitalarcsorcurves.Adigitalcurveisapathp=p0;p1;p2;:
:
:
;pn=qsuchthatpiisaneighborofpi?
1,1_i_n,andp=q.Adigitalcurveiscalledsimpleifeachpointpihasexactlytwoneighborsinthiscurve.Adigitalarcisasubsetofadigitalcurvesuchthatp6=q.Apointofadigitalarcwhichhasexactlyoneneighboriscalledanendpointofthisarc.Withinthisthirdclassofoperators(thinningalgorithms)wemayclassifywithrespecttoalgorithmicstrategies:
individualpixelsareeitherremovedinasequentialorderorinparallel.Forexample,theoftencitedalgorithmbyHilditch[5]isaniterativeprocessoftestinganddeletingcontourpixelssequentiallyinstandardrasterscanorder.AnothersequentialalgorithmbyPavlidis[12]usesthede_nitionofmultiplepointsandproceedsbycontourfollowing.Examplesofparallelalgorithmsinthisthirdclassarereductionoperatorswhichtransformcontourpointsintobackgroundpoints.Di_erencesbetweentheseparallelalgorithmsaretypicallyde_nedbytestsimplementedtoensureconnectednessinalocalneighborhood.Thenotionofasimplepointisofbasicimportanceforthinninganditwillbeshowninthisreportthatdi_erentde_nitionsofsimplepointsareactuallyequivalent.SeveralpublicationscharacterizepropertiesofasetDofpoints(tobeturnedfromobjectpointstobackgroundpoints)toensurethatconnectivityofobjectandbackgroundremainunchanged.Thereportdiscussessomeofthesepropertiesinordertojustifyparallelthinningalgorithms.
1.2Basics
Theusednotationfollows[17].AdigitalimageIisafunctionde_nedonadiscretesetC,whichiscalledthecarrieroftheimage.TheelementsofCaregridpointsorgridcells,andtheelements(p;I(p))ofanimagearepixels(2Dcase)orvoxels(3Dcase).Therangeofa(scalar)imageisf0;:
:
:
GmaxgwithGmax_1.Therangeofabinaryimageisf0;1g.WeonlyusebinaryimagesIinthisreport.LethIibethesetofallpixellocationswithvalue1,i.e.hIi=I?
1
(1).Theimagecarrierisde_nedonanorthogonalgridin2Dor3D
-2-
space.Therearetwooptions:
usingthegridcellmodela2Dpixellocationpisaclosedsquare(2-cell)intheEuclideanplaneanda3Dpixellocationisaclosedcube(3-cell)intheEuclideanspace,whereedgesareoflength1andparalleltothecoordinateaxes,andcentershaveintegercoordinates.Asasecondoption,usingthegridpointmodela2Dor3Dpixellocationisagridpoint.
Twopixellocationspandqinthegridcellmodelarecalled0-adjacenti_p6=qandtheyshareatleastonevertex(whichisa0-cell).Notethatthisspeci_es8-adjacencyin2Dor26-adjacencyin3Difthegridpointmodelisused.Twopixellocationspandqinthegridcellmodelarecalled1-adjacenti_p6=qandtheyshareatleastoneedge(whichisa1-cell).Notethatthisspeci_es4-adjacencyin2Dor18-adjacencyin3Difthegridpointmodelisused.Finally,two3Dpixellocationspandqinthegridcellmodelarecalled2-adjacenti_p6=qandtheyshareatleastoneface(whichisa2-cell).Notethatthisspeci_es6-adjacencyifthegridpointmodelisused.AnyoftheseadjacencyrelationsA_,_2f0;1;2;4;6;18;26g,isirreexiveandsymmetriconanimagecarrierC.The_-neighborhoodN_(p)ofapixellocationpincludespandits_-adjacentpixellocations.Coordinatesof2Dgridpointsaredenotedby(i;j),with1_i_nand1_j_m;i;jareintegersandn;marethenumbersofrowsandcolumnsofC.In3Dweuseintegercoordinates(i;j;k).Basedonneighborhoodrelationswede_neconnectednessasusual:
twopointsp;q2Care_-connectedwithrespecttoM_CandneighborhoodrelationN_i_thereisasequenceofpointsp=p0;p1;p2;:
:
:
;pn=qsuchthatpiisan_-neighborofpi?
1,for1_i_n,andallpointsonthissequenceareeitherinMorallinthecomplementofM.AsubsetM_Cofanimagecarrieriscalled_-connectedi_MisnotemptyandallpointsinMarepairwise_-connectedwithrespecttosetM.An_-componentofasubsetSofCisamaximal_-connectedsubsetofS.Thestudyofconnectivityindigitalimageshasbeenintroducedin[15].ItfollowsthatanysethIiconsistsofanumberof_-components.Incaseofthegridcellmodel,acomponentistheunionofclosedsquares(2Dcase)orclosedcubes(3Dcase).Theboundaryofa2-cellistheunionofitsfouredgesandtheboundaryofa3-cellistheunionofitssixfaces.Forpracticalpurposesitiseasytouseneighborhoodoperations(calledlocaloperations)onadigitalimageIwhichde_neavalueatp2Cinthetransformedimagebasedonpixel
-3-
valuesinIatp2CanditsimmediateneighborsinN_(p).
2Non-iterativeAlgorithms
Non-iterativealgorithmsdeliversubsetsofcomponentsinspeciedscanorderswithouttestingconnectivitypreservationinanumberofiterations.Inthissectionweonlyusethegridpointmodel.
2.1\DistanceSkeleton"Algorithms
Blum[3]suggestedaskeletonrepresentationbyasetofsymmetricpoints.InaclosedsubsetoftheEuclideanplaneapointpiscalledsymmetrici_atleast2pointsexistontheboundarywithequaldistancestop.Foreverysymmetricpoint,theassociatedmaximaldiscisthelargestdiscinthisset.Thesetofsymmetricpoints,eachlabeledwiththeradiusoftheassociatedmaximaldisc,constitutestheskeletonoftheset.Thisideaofpresentingacomponentofadigitalimageasa\distanceskeleton"isbasedonthecalculationofaspeci_eddistancefromeachpointinaconnectedsubsetM_Ctothecomplementofthesubset.Thelocalmaximaofthesubsetrepresenta\distanceskeleton".In[15]thed4-distanceisspeciedasfollows.De_nition1Thedistanced4(p;q)frompointptopointq,p6=q,isthesmallestpositiveintegernsuchthatthereexistsasequenceofdistinctgridpointsp=p0,p1;p2;:
:
:
;pn=qwithpiisa4-neighborofpi?
1,1_i_n.
Ifp=qthedistancebetweenthemisde_nedtobezero.Thedistanced4(p;q)hasallpropertiesofametric.Givenabinarydigitalimage.Wetransformthisimageintoanewonewhichrepresentsateachpointp2hIithed4-distancetopixelshavingvaluezero.Thetransformationincludestwosteps.Weapplyfunctionsf1totheimageIinstandardscanorder,producingI_(i;j)=f1(i;j;I(i;j)),andf2inreversestandardscanorder,producingT(i;j)=f2(i;j;I_(i;j)),asfollows:
f1(i;j;I(i;j))=
8><>>:
0ifI(i;j)=0
minfI_(i?
1;j)+1;I_(i;j?
1)+1g
ifI(i;j)=1andi6=1orj6=1
-4-
m+notherwise
f2(i;j;I_(i;j))=minfI_(i;j);T(i+1;j)+1;T(i;j+1)+1g
TheresultingimageTisthedistancetransformimageofI.NotethatTisasetf[(i;j);T(i;j)]:
1_i_n^1_j_mg,andletT__Tsuchthat[(i;j);T(i;j)]2T_i_noneofthefourpointsinA4((i;j))hasavalueinTequaltoT(i;j)+1.Forallremainingpoints(i;j)letT_(i;j)=0.ThisimageT_iscalleddistanceskeleton.Nowweapplyfunctionsg1tothedistanceskeletonT_instandardscanorder,producingT__(i;j)=g1(i;j;T_(i;j)),andg2totheresultofg1inreversestandard
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