数字图像处理 外文翻译 外文文献 英文文献 数字图像处理文档格式.docx
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数字图像处理 外文翻译 外文文献 英文文献 数字图像处理文档格式.docx
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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;
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;
pn=qwithpiisa4-neighborofpi?
1,1_i_n.
Ifp=qthedistancebetweenthemisde_nedtobezero.Thedistanced4(p;
q)hasallpropertiesofametric.Givenabinarydigitalimage.Wetransformthisimageintoanewonewhichrepresentsateachpointp2hIithed4-distancetopixelshavingvaluezero.Thetransformationincludestwosteps.Weapplyfunctionsf1totheimageIinstandardscanorder,producingI_(i;
j)=f1(i;
I(i;
j)),andf2inreversestandardscanorder,producingT(i;
j)=f2(i;
I_(i;
j)),asfollows:
f1(i;
j))=
8>
<
>
0ifI(i;
j)=0
minfI_(i?
j)+1;
j?
1)+1g
ifI(i;
j)=1andi6=1orj6=1
-4-
m+notherwise
f2(i;
j))=minfI_(i;
j);
T(i+1;
T(i;
j+1)+1g
TheresultingimageTisthedistancetransformimageofI.NotethatTisasetf[(i;
j)]:
1_i_n^1_j_mg,andletT__Tsuchthat[(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;
T_(i;
j)),andg2totheresultofg1inreversestandard
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