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ComputingMultidimensionalPersistenceGunnarCarlsson1GurjeetSingh1andAfr


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Document on Subject : "ComputingMultidimensionalPersistenceGunnarCarlsson1GurjeetSingh1andAfr"— Transcript:

1 ComputingMultidimensionalPersistence?Gun
ComputingMultidimensionalPersistence?GunnarCarlsson1,GurjeetSingh1,andAfraZomorodian21DepartmentofMathematics,StanfordUniversity,USA.2DepartmentofComputerScience,DartmouthCollege,USA.[ToappearinISAAC'09]Abstract.Thetheoryofmultidimensionalpersistencecapturesthetopol-ogyofamulti ltration{amultiparameterfamilyofincreasingspaces.Multi ltrationsarisenaturallyinthetopologicalanalysisofscienti cdata.Inthispaper,wegiveapolynomialtimealgorithmforcomputingmultidimensionalpersistence.1IntroductionInthispaper,wegiveapolynomialtimealgorithmforcomputingthepersistenthomologyofamulti ltration.Thecomputedsolutioniscompactandcomplete,butnotaninvariant.Theoretically,thisisthebestonemayhopeforsincecompletecompactinvariantsdonotexistformultidimensionalpersistence[1].1.1MotivationIntuitively,amulti ltrationmodelsagrowingspacethatisparameterizedalongmultipledimensions.Forexample,thecomplexwithcoordinate(3;2)inFigure1is lteredalongthehorizontalandverticaldimensions,givingrisetoabi ltra-tion.Multi ltrationsarisenaturallyintopologicalanalysisofscienti cdata.Often,scienti cdataisintheformofa nitesetofnoisysamplesfromsomeunderlyingtopologicalspace.Ourgoalistorobustlyrecoverthelostconnectiv-ityoftheunderlyingspace.Ifthesamplingisdenseenough,weapproximatethespaceasaunionofballsbyplacing-ballsaroundeachpoint.Asweincrease,weobtainagrowingfamilyofspaces,aone-parametermulti ltrationalsocalleda ltration.Thisapproximationisthecentralideabehindmanymethodsforcomputingthetopologyofapointset,suchasCech,Rips-Vietoris[2],orwit-ness[3]complexes.Often,theinputpointsetisalso lteredthroughmultiplefunctions.Wenowhavemultipledimensionsalongwhichourspaceis ltered.Thatis,wehaveamulti ltration. ?Theauthorswerepartiallysupportedbythefollowinggrants:G.C.byNSFDMS-0354543;A.Z.byDARPAHR0011-06-1-0038,ONRN00014-08-1-0908,andNSFCCF-0845716;allbyDARPAHR0011-05-1-0007. 2GunnarCarlsson,GurjeetSingh,andAfraZomorodian (0,2)(1,2)(3,2)(2,2) fbafab Fig.1.Abi ltration.Thecomplexwithlabeledverticesi

2 satcoordinate(3;2).Sim-plicesarehighligh
satcoordinate(3;2).Sim-plicesarehighlightedandnamedatthecriticalcoordinatesthattheyappear.1.2PriorWorkForone-dimensional ltrations,thetheoryofpersistenthomologyprovidesacompleteinvariantcalledabarcode,amultisetofintervals[4].Eachintervalinthebarcodecorrespondstothelifetimeofasingletopologicalattributewithinthe ltration.Sincefeatureshavelonglives,whilenoiseisshort-lived,aquickexaminationoftheintervalsgivesarobustestimationofthetopology.Theex-istenceofacompletecompactinvariant,aswellasecientalgorithmsandfastimplementationshaveledtosuccessfulapplicationofpersistencetoavarietyofproblems,suchasshapedescription[5],denoisingvolumetricdensitydata[6],detectingholesinsensornetworks[7],analyzingneuralactivityinthevisualcortex[8],andanalyzingthestructureofnaturalimages[9],tonameafew.Formulti ltrationsofdimensionhigherthanone,thesituationismuchmorecomplicated.Thetheoryofmultidimensionalpersistenceshowsthatnocompletediscreteinvariantexists,wherediscretemeansthatthestructureofthetargetfortheinvariantdoesnotdependonthestructureoftheunderlying eld[1].Instead,theauthorsproposeanincompleteinvariant,therankinvariant,whichcapturesimportantpersistentinformation.Unfortunately,thisinvariantisnotcompact,requiringlargestorage,soitsdirectcomputationusingtheone-dimensionalal-gorithmisnotfeasible.Avariantoftheproblemofmultidimensionalpersisthasappearedincomputervision[10].Apartialsolution,calledvineyards,hasbeeno ered[11].Afullsolution,however,hasnotbeenattemptedbyanypriorwork.1.3ContributionsInthispaper,weprovideacompletesolutiontotheproblemofcomputingmultidimensionalpersistence.Werecastpersistenceasaproblemwithincom-putationalalgebraicgeometry,allowingustoutilizepowerfulalgorithmsfrom ComputingMultidimensionalPersistence3thisarea.Wethenexploitthestructureprovidedbyamulti ltrationtogreatlysimplifythealgorithms.Finally,weshowthattheresultingalgorithmsarepoly-nomialtime,unliketheiroriginalcounterparts,whichareExpspace-complete,requiringexponentialspaceandtime.Webeg

3 inwithabriefreviewofnecessaryconceptsinS
inwithabriefreviewofnecessaryconceptsinSection2,andrecasttheproblemintoanalgebraicgeometricframe-work.Section3containsthemaincontributionofthispaper,whereweusethestructureofmulti ltrationstosimplifythetraditionalalgorithms.2Background&ApproachInthissection,wereviewconceptsfromalgebraictopologyandcomputationalalgebraicgeometry.Wethenpresentourapproachofcomputingmultidimen-sionalpersistenceusingalgorithmsfromthelatterarea.Duetolackofspace,weomitsigni cantportionsofourwork,referringtheinterestedreadertoourmanuscriptforacompletedescription[12].OurtreatmentofalgebraicgeometryanditsalgorithmsfollowChapter5ofCox,Little,andO'Shea[13].Ourgoalisthecomputationofthepersistenthomologyofamulti ltration.LetNbethesetofnon-negativeintegers.AtopologicalspaceXismulti- lteredifwearegivenafamilyofsubspacesfXugu,whereuNnandXuXsuchthatforu;w1;w2;vNn,thediagramsXuXw1Xw2Xv //  // (1)commutewheneveruw1;w2v.WecallthefamilyofsubspacesfXuguamulti ltration,suchastheexampleinFigure1.Inthispaper,weassumeourinputisamulti lteredsimplicialcomplexthathasthefollowingproperty:De nition1(one-critical).Amulti lteredcomplexwhereeachcellhasauniqueminimalcriticalgradeatwhichitentersthecomplexisone-critical.Thebi ltrationinFigure1isone-critical,asaremostmulti ltrationsthatariseinpractice[12].GivenasimplicialcomplexK,wemayde nechaingroupsCiasthefreeAbeliangroupsonorientedi-simplices.Theboundaryoperator@i:CiCi1connectsthechaingroupsintoachaincomplexC:!Ci+1@i+1!Ci@i!Ci1!:(2)Givenanychaincomplex,theithhomologygroupisHi=ker@i=im@i+1:(3)Givenamulti ltrationfXugu,foreachpairuvNn,XuXvbyde nition,soXu,Xv,inducingamapi(u;v)atthehomologylevelHi(Xu)Hi(Xv) 4GunnarCarlsson,GurjeetSingh,andAfraZomorodianthatmapsahomologyclassinXutotheonethatcontainsitinXv.Theithpersistenthomologyistheimageofiforallpairsuv.Ourworkrestsonthetheoryofpersistence[4,1].Thekeyinsightisthis:Persistenthomologyofamulti ltrationisstandardhomologyofasinglemulti-gradedmodulethatencodesthemulti ltrationusing

4 polynomialcoecients.LetAnk[x1;:::;xn]be
polynomialcoecients.LetAnk[x1;:::;xn]bethen-gradedpolynomialring,gradedbyAnvkxv;vNn.Wede neann-gradedmoduleoverthisringasfollows.De nition2(chainmodule).Givenamulti lteredsimplicialcomplexfKugu,theithchainmoduleisthen-gradedmoduleoverthegradedpolynomialringAnCiMuCi(Ku);(4)wherethek-modulestructureisthedirectsumstructureandxvu:Ci(Ku)Ci(Kv)istheinclusionKu,Kv.ThesegradedchainmodulesCiare nitelygenerated,andforone-critical ltra-tions,theyarealsofree,sowemaychoosebasesforthem.De nition3(standardbasis).ThestandardbasisfortheithchainmoduleCiisthesetofi-simplicesincriticalgrades.Givenstandardbases,wemaywritetheboundaryoperator@i:CiCi1explicitlyasamatrixwithpolynomialentries.Wenowhaveanewn-gradedchaincomplex(2)thatencodesthemulti ltration.Thehomologyofthischaincomplexisthepersistenthomologyofthemulti ltration[1].Byde nition(3),wemaycomputehomologyinthreesteps:1.Computeim@i+1:Thisisasubmoduleofthepolynomialmodule,anditscomputationisthesubmodulemembershipproblemincomputationalal-gebraicgeometry.WemaysolvethisproblembycomputingthereducedGrobnerbasisusingtheBuchbergerandreductionalgorithms,andthendividingusingtheDividealgorithm.2.Computeker@i:Theisthe( rst)syzygymodule,whichwemaycomputeusingSchreyer'salgorithm.3.ComputeHi:Thistaskissimple,oncetheabovetwotasksarecomplete.Weneedtotestwhetherthegeneratorsofthesyzygysubmoduleareintheboundarysubmodule,ataskwhichmaybecompletedusingthetoolsabove.Whiletheabovealgorithmssolvethemembershipproblem,theyhavenotbeenusedinpracticeduetotheircomplexity.ThesubmodulemembershipproblemisageneralizationofthePolynomialIdealMembershipProblem(PIMP)whichisExpspace-complete,requiringexponentialspaceandtime[14,15].Indeed,theBuchbergeralgorithm,initsoriginalformisdoubly-exponential.Therefore,whileourreformulationofmultidimensionalpersistencegivesusalgorithms,weneedtomakethemfastertomakethisapproachfeasible. ComputingMultidimensionalPersistence53MultigradedAlgorithmsInthissection,weshowthatmulti ltrationsprovideadditionalst

5 ructurethatmaybeexploitedtosimplifytheal
ructurethatmaybeexploitedtosimplifythealgorithmsforourthreetasks.Thesesimpli -cationsconverttheseintractablealgorithmsintopolynomialtimealgorithms.3.1ExploitingHomogeneityThekeypropertythatweexploitforsimpli cationishomogeneity.De nition4(homogeneous).LetMbeanmnmatrixwithmonomialentries.ThematrixMishomogeneousi 1.everycolumnofMisassociatedwithacoordinateinthemulti ltration(uf)andthusacorrespondingmonomialxuf,2.everynon-zeroelementMjkmaybeexpressedasthequotientofthemono-mialsassociatedwithcolumnkandrowj,respectively.Anyvectorendowedwithacoordinateufthatmaybewrittenasaboveishomogeneous,e.g.thecolumnsofM.Wewillshowthat(1)allboundarymatrices@imaybewrittenashomogeneousmatricesinitially,and(2)thealgorithmsforcomputingpersistenceonlyprohomogeneousmatricesandvectors.Thatis,wemaintainhomogeneityasaninvariantthroughoutthecomputation.Webeginwithour rsttask.Lemma1.Foraone-criticalmulti ltration,thematrixof@i:CiCi1writ-tenintermsofthestandardbasesishomogeneous.Proof.Recallthatwemaywritetheboundaryoperator@i:CiCi1explicitlyasami1mimatrixMintermsofthestandardbasesforCiandCi1,asshowninmatrix(Equation5)for@1.FromDe nition3,thestandardbasisforCiisthesetofi-simplicesincriticalgrades.Inaone-criticalmulti ltration,eachsimplexhasauniquecriticalcoordinateu(De nition(1)).Inturn,wemayrepresentthiscoordinatebythemonomialxu.Forinstance,simplexainFigure1hascriticalgrade(1;1)andmonomialx(1;1)x1x2.Weorderthesemonomialsusinglexandusethisorderingtorewritethematrixfor@i.ThematrixentryMjkrelatesk,thekthbasiselementforCito^j,thejthbasiselementforCi1.If^jisnotafaceofk,thenMjk=0.Otherwise,^jisafaceofk.Sinceafacemustprecedeaco-faceinamulti ltration,uklexu^jxuklexxu^j,andMjkxuk=xu^jxuku^j.Thatis,thematrixishomogeneous.Forexample,^1aisafaceof1ab,soM11x1x22=x1x2x2inthematrixfor@1forthebi ltrationinFigure1.Corollary1.Foraone-criticalmulti ltration,theboundarymatrix@iintermsofthestandardbaseshasmonomialentries. 6GunnarCarlsson,GurjeetSingh,andAfraZomorodian

6 Proof.Theresultisimmediatefromtheproofof
Proof.Theresultisimmediatefromtheproofofthepreviouslemma.Thematrixentryiseither0,amonomial,orxu(k)u(^j),amonomial.Below,weshowthehomogeneousmatrixfor@1forthebi ltrationinFigure1,whereweaugmentthematrixwiththeassociatedmonomials.Weassumewearecomputingover2.266666666664 abbccddeefafbfce x1x22x21x22x1x1x21x1x22x22x2 ax1x2 x20000x200dx1 00110000b x1x22x21x220000x22c x21x22x10000x2 000x1x2100x2 0000x21x1x22x22377777777775(5)Wenextfocusonoursecondtask,showingthatgivenahomogeneousmatrixasinput,thealgorithmsproducehomogeneousvectorsandmatrices.LetFbeanmnhomogeneousmatrix.Letfe1;:::;emgandf^e1;:::;^engbethestandardbasesforthegradedpolynomialringsRmandRn,respectively.Ahomogeneousmatrixassociatesacoordinateandmonomialtotherowandcolumnbasisele-ments.Forexample,sincex1isthemonomialforrow2ofmatrix(5),wehaveu2=(1;0)andxue2x1.EachcolumninFishomogeneousandmaybewrittenintermsofrows:mXi=1ixuf xueiei;(6)whereikandweallowi=0whenarowisnotused.Forexample,columngforedgeabinourbi ltrationmaybewrittenas:gx2e1x2x22e3x2x22 x1x2e1x2x22 1e3xug xue1e1xug xue3e3Xi2f1;3gxug xueiei:ConsidertheBuchbergeralgorithm[13].ThealgorithmrepeatedlycomputesS-polynomialsofhomogeneousvectors.Lemma2.TheS-polynomialS(;g)ofhomogeneousvectorsandgishomo-geneous.Proof.AzeroS-polynomialistriviallyhomogeneous.Anon-zeroS-polynomialS(;g)impliesthathlcm(lm();lm(g))isnon-zero.Bythede nitionoflcm,theleadingmonomialsofandgcontainthesamebasiselementej.Wehave,lm()=uf uejej,lm(g)=ug uejej,and:hlcm(lm();lm(g))=lcmxuf xuej;xug xuejejlcm(xuf;xug) xuejej: ComputingMultidimensionalPersistence7Letx`lcm(xuf;xug)=xlcm(uf;ug),givingush` uejej.Wenowhaveh lt()` uejej fuf uejejx` fxuf;wheref=0isthe eldconstantintheleadingtermof.Similarly,wegeth lt(g)x` gxug;cg=0:Puttingittogether,wehaveS(;g)=h lt()h lt(g)g x` fxufmXi=1ixuf xueiei! x` gxugmXi=10ixug xueiei!mXi=1dix` xueiei;wheredii=cf0i=cg.ComparingwithEquation(6),weseethatS(;g)ishomogeneouswithuS(f;g)`.HavingcomputedtheS-polynomial,Buchbergernextdividesitbyt

7 hecurrenthomogeneousbasisGusingacalltoth
hecurrenthomogeneousbasisGusingacalltotheDividealgorithm[13].Lemma3.Divide(;(1;:::;t))returnsahomogeneousremaindervectorrforhomogeneousvectors;iRm.Proof.Initially,wesetrandptobe0and,respectively,sotheyarebothtriviallyhomogeneous.Sincebothiandparehomogeneous,wehaveiPmj=1ijufi uejej,pPmj=1djup uejej.Sincelt(i)divideslt(p),thetermsmustsharebasiselementeandwehavelt(i)=ikufi uee,lt(p)=dkup uee,lt(p)=lt(i)=dk cikup ufi,wherexuplexxufisothatthedivisionmakessense.Then,pisassignedtop(lt(p)=lt(i))imXj=1djxup xuejejdk ikxup xufimXj=1ijxufi xuejejmXj=1djdkij ikxup xuejejmXj=1d0jxup xuejej;whered0jdjdkij=cikandd0k=0,sothesubtractioneliminatesthekthterm.The nalsummeansthatpisnowanewhomogeneouspolynomialwiththesamecoordinateupasbefore.Similarly,lt(p)isaddedtorandsubtractedfromp,andneitheractionchangesthehomogeneityofeithervector.Bothremainhomogeneouswithcoordinateup. 8GunnarCarlsson,GurjeetSingh,andAfraZomorodianTheorem1(homogeneousGrobner).TheBuchbergeralgorithmcom-putesahomogeneousGrobnerbasisforahomogeneousmatrix.Proof.Initially,thealgorithmsetsGtobethesetofcolumnsoftheinputma-trixF,sothevectorsinGarehomogeneousbyLemma1.ThealgorithmthencomputestheS-polynomialofhomogeneousvectors;gG.ByLemma2,theS-polynomialishomogeneous.ItthendividestheS-polynomialbyG.Sincetheinputishomogeneous,DivideproducesahomogeneousremainderrbyLemma3.SinceonlyhomogeneousvectorsareaddedtoG,itremainshomoge-neous.WemayextendthisresulteasilytothereducedGrobnerbasis.Usingsimilararguments,wemayshowthefollowingresult.Theorem2(homogenoussyzygy).Forahomogeneousmatrix,allmatricesencounteredinthecomputationofthesyzygymodulearehomogeneous.3.2OptimizationsWehaveshownthatthestructureinherentinamulti ltrationallowsustocomputeusinghomogeneousvectorsandmatriceswhoseentriesaremonomialsonly.Wenextexploretheconsequencesofthisrestrictiononboththedatastructuresandcomplexityofthealgorithms.ByDe nition(4),anmnhomogeneousmatrixnaturallyassociatesmono-mialstothestandardbasesforRmandRn.Moreove

8 r,everynon-zeroentryofthematrixisaquotie
r,everynon-zeroentryofthematrixisaquotientofthesemonomialsasthematrixishomogeneous.Therefore,wedonotneedtostorethematrixentries,butsimplythe eldel-ementsofthematrixalongwiththemonomialsforthebases.Wemaymodifytwostandarddatastructurestorepresentthematrix.{linkedlist:Eachcolumnstoresitsmonomialaswellasalinked-listofitsnon-zeroentriesinsortedorder.Thenon-zeroentriesarerepresentedbytherowindexandthe eldelement.Thematrixissimplyalistofthesecolumnsinsortedorder.{matrix:Eachcolumnstoresitsmonomialaswellasthecolumnof eldcoecients.Ifwearecomputingovera nite eld,wemaypackbitsforspaceeciency.Thelinked-listrepresentationisappropriateforsparsematricesasitisspaecientatthepriceoflinearaccesstime.Thisisessentiallytherepresentationusedforcomputingintheone-dimensionalsetting[4].Incontrast,thematrixrepresentationisappropriatefordensematricesasitprovidesconstantaccesstimeatthecostofstoringallzeroentries.Themultidimensionalsettingprovidesuswithdensermatrices,asweshallsee,sothematrixrepresentationbecomesaviablestructure.Inaddition,thematrixrepresentationisoptimallysuitedtocomputingoverthe eld2,the eldoftencommonlyemployedintopologicaldataanalysis.Thematrixentrieseachtakeonebitandthecolumnentriesmaybepacked ComputingMultidimensionalPersistence9intomachinewords.Moreover,theonlyoperationrequiredbythealgorithmsissymmetricdi erencewhichmaybeimplementedasabinaryXORoperationprovidedbythechip.Thisapproachgivesusbit-levelparallelismforfree:Ona64-bitmachine,weperformsymmetricdi erence64timesfasterthanonthelist.Thecombinationofthesetechniquesallowthematrixstructuretoperformbetterthanthelinked-listrepresentationinpractice.Wemayalsoexploithomogeneitytospeedupthecomputationofnewvec-torsandtheirinsertionintothebasis.Wedemonstratethisbrie\ryusingtheBuchbergeralgorithm.WeorderthecolumnsofinputmatrixGusingthePOTruleforvectors[13].Supposewehave;gGwithg.IfS(;g)=0,lt()andlt(g)containthesamebasis,whichtheS-polynomialeliminates.So,wehaveS(;g)g.ThisimpliesthatwhendividingS(;g)by

9 thevectorsinG,weneedonlyconsidervectorst
thevectorsinG,weneedonlyconsidervectorsthataresmallerthang.Sincethevectorsareinsortedorder,weconsidereachinturnuntilwecannolongerdivide.BythePOTrule,wemaynowinsertthenewremaindercolumnhereintothebasisG.Thisgivesusaconstanttimeinsertionoperationformaintainingtheordering,aswellasfastercomputationoftheGrobnerbasis.3.3ComplexityOuroptimizationsfromthelastsectionallowustogivesimplepolynomialboundsonourmultigradedalgorithms.Thesebounds,inturn,implythatwemaycomputemultidimensionalpersistenceinpolynomialtime.Lemma4.LetFbeanmnhomogeneousmatrixofmonomials.TheGrobnerbasisGcontainsO(n2m)vectorsintheworstcase.WemaycomputeGusingBuchbergerinO(n4m3)worst-casetime.Proof.Intheworstcase,Fcontainsnmuniquemonomials.EachcolumnFmayhaveanyofthenmmonomialsasitsmonomialwhenincludedintheGrobnerbasisGTherefore,thetotalnumberofcolumnsintheGisO(n2m).IncomputingtheGrobnerBasis,wecompareallcolumnspairwise,sothetotalnumberofcomparisonsisO(n4m2).DividingtheS-polynomialtakesO(m)time.Therefore,theworst-caserunningtimeisO(n4m3).Weomittheproofofthefollowingduetolackofspaceandreferthereadertothefullmanuscript[12].Lemma5.LetFbeanmnhomogeneousmatrixofmonomialsandGbetheGrobnerBasisofF.TheSyzygymoduleSforGmaybecomputedusingSchreyer'salgorithminO(n4m2)worst-casetime.Theorem3.Multidimensionalpersistencemaybecomputedinpolynomialtime.4ConclusionInthispaper,wedeveloppolynomialtimealgorithmsformultidimensionalper-sistencebyrecastingtheproblemintocomputationalalgebraicgeometry.Al-thoughtherecastproblemisExpspace-complete,weexploitthemultigraded 10GunnarCarlsson,GurjeetSingh,andAfraZomorodiansettingtodeveloppracticalalgorithms.Wehaveimplementedallouralgorithmsandprovidestatisticalexperimentstodemonstratetheirfeasibilityinthefullmanuscript[12].Foradditionalspeedup,weplantoparallelizethecomputa-tionbybatchingandthreadingtheXORoperations.Wealsoplantoapplyouralgorithmstowardstudyingscienti cdata.Forinstance,forzero-dimensionalhomology,multidimensionalpersistencecorrespondstoclusteringm

10 ultiparam-eterizeddata,Thisgivesusafresh
ultiparam-eterizeddata,Thisgivesusafreshperspective,aswellasanewarsenalofcomputationaltools,toattackanoldandsigni cantproblemindataanalysis.References1.Carlsson,G.,Zomorodian,A.:Thetheoryofmultidimensionalpersistence.Discrete&ComputationalGeometry42(1)(2009)71{932.Gromov,M.:Hyperbolicgroups.InGersten,S.,ed.:EssaysinGroupTheory.SpringerVerlag,NewYork,NY(1987)75{2633.deSilva,V.,Carlsson,G.:Topologicalestimationusingwitnesscomplexes.In:Proc.SymposiumonPoint-BasedGraphics.(2004)157{1664.Zomorodian,A.,Carlsson,G.:Computingpersistenthomology.Discrete&Com-putationalGeometry33(2)(2005)249{2745.Collins,A.,Zomorodian,A.,Carlsson,G.,Guibas,L.:Abarcodeshapedescriptorforcurvepointclouddata.ComputersandGraphics28(2004)881{8946.Gyulassy,A.,Natarajan,V.,Pascucci,V.,Bremer,P.T.,Hamann,B.:Topology-basedsimpli cationforfeatureextractionfrom3Dscalar elds.In:Proc.IEEEVisualization.(2005)275{2807.deSilva,V.,Ghrist,R.,Muhammad,A.:Blindswarmsforcoveragein2-D.In:ProceedingsofRobotics:ScienceandSystems.(2005).8.Singh,G.,Memoli,F.,Ishkhanov,T.,Sapiro,G.,Carlsson,G.,Ringach,D.L.:Topologicalanalysisofpopulationactivityinvisualcortex.JournalofVision8(8)(62008)1{189.Carlsson,G.,Ishkhanov,T.,deSilva,V.,Zomorodian,A.:Onthelocalbehaviorofspacesofnaturalimages.InternationalJournalofComputerVision76(1)(2008)10.Frosini,P.,Mulazzani,M.:Sizehomotopygroupsforcomputationofnaturalsizedistances.Bull.Belg.Math.Soc.SimonStevin6(3)(1999)455{46411.Cohen-Steiner,D.,Edelsbrunner,H.,Morozov,D.:Vinesandvineyardsbyup-datingpersistenceinlineartime.In:Proc.ACMSymposiumonComputationalGeometry.(2006)119{12612.Carlsson,G.,Singh,G.,Zomorodian,A.:Computingmultidimensionalpersistence.13.Cox,D.A.,Little,J.,O'Shea,D.:Usingalgebraicgeometry.Secondedn.Volume185ofGraduateTextsinMathematics.Springer,NewYork(2005)14.Mayr,E.W.:Somecomplexityresultsforpolynomialideals.JournalofComplexity13(3)(1997)303{32515.vonzurGathen,J.,Gerhard,J.:ModernComputerAlgebra.Secondedn.Cam-bridgeUniversityPress,Cambridge,UK(