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Unified mathematical framework for a compact and fully
parallel n-D skeletonization procedure
Antoine Manzanera, Thierry Bernard, Françoise Prêteux, Bernard Longuet
To cite this version:
Antoine Manzanera, Thierry Bernard, Françoise Prêteux, Bernard Longuet. Unified mathematical
framework for a compact and fully parallel n-D skeletonization procedure. Vision Geometry, Jul 1999,
Denver, United States. �10.1117/12.364113�. �hal-01245457�
parallel n-D skeletonization pro edure
Antoine Manzanera a d
, Thierry M. Bernard b
, Fran oise Pr^eteux
,and Bernard Longuet a
a
Aerospatiale E/SCS/V, 2 Rue Beranger, 92320 ChatillonCedex, Fran e b
ENSTA/LEI, 32 Bd Vi tor, 75015 Paris, Fran e
ARTEMIS Proje t Unit, INT, 9Rue Charles Fourier, 91011Evry Cedex, Fran e d
DCE/ETC4/CTA/GIP,16bis Avenue Prieur de la C^ote d'Or,94114 Ar ueil Cedex, Fran e
ABSTRACT
Wepresentinthispaperageneri algorithmto omputetheskeletonofann-dimensionalbinaryobje t. Considering the artesianhyper ubi grid,weprovideamathemati alframeworkinwhi haregiventheexpli itBoolean onditions underwhi htheiterativethinningpro edureremovesapoint. Thisalgorithmpreservesthetopologyinasensewhi h mat hesthepropertiesusuallyusedin2Dand3D.Furthermore,itisbasedonanoriginalkindofmedianhypersurfa e thatgivestotheskeletongoodbehaviorwithrespe ttobothshapepreservationandnoisesensitivity. Thealgorithm isfullyparallel,asnospatialsubiterationsareneeded. Thelatterproperty,togetherwiththesymmetryoftheboolean n-dimensional patternsleadsto aperfe tly isotropi skeleton. Thelogi alexpressionof thealgorithm is extremely on ise, and in 2D, a large omparative study shows that the overall number of elementary Boolean operations neededtogettheskeletonissmallerthanfortheotheriterativealgorithmsreported intheliterature.
Keywords: Skeleton,Fully ParallelAlgorithm,n-dimensional,Dis reteTopology,BooleanComplexity
1. INTRODUCTION
Representingashapewith asmallamountofinformation isamajor hallenge in omputervision. Skeletonization isoneoftheapproa hestothispurpose. Itarisesfromtheideathatashapeisfaithfully representedifitstopology ( onne ted omponents,holes...),aswellasitsgeometry(elongatedparts,rami ations...) andlo ationarepreserved. Theinterestinskeletafordigitalimagesismotivatedbytheusefulnessofthisrepresentationasaprepro essingstep in pattern re ognitionalgorithms. A largenumberof papershas been published in thesubje t,for 2Dimages for severalde ades,andfor3Dimagesalittlemorere ently. SkeletonizationinnDseemstobeamoreexoti issue,but itisofsomeinterestinroboti s,whereitprovidesthesafesttraje toriesinamulti-parameterspa e. Theskeletonis usuallyobtainedthroughaniterativepro edure alledthinning: theborder pointsareremovedaslongastheyare notjudgedsigni antforthefeaturesstatedabove,untilnomorepoint anbedeleted. Theremainingshapeisthen alledskeleton. Webelievethattheavailableknowledgeaboutdigitalskeletasu erstwoimportantproblems. Firstly, the hara terizationsthat aregivenin 2DarediÆ ultto adaptto 3D,whi h showstheneedforunifying on epts. Se ondly, these hara terizationsare ompli ated, and in parti ular in 3D, nevereasy to implement. We propose in the presentpaperanew thinningalgorithm, alled MB, whi h is de ned in the hyper ubi grid, independently of the dimension. Indeed, the onditions under whi h a point is removed are de ned by means of one removing ondition(Fun tionAlpha)andoneremaining ondition(Fun tionBeta),thatare hara terizedinanadimensional way. GettingtheBoolean hara terizationofthethinningpro essinanydimensionisthenstraightforward,andwe providethe orrespondingpatternmat hingform fordimensions1to3. Underthisform,ourpro edure lendsitself easilyto omparison,andits ompa tness anthenbeappre iated. Indeed,in2D,theoverallnumberofelementary Boolean operations needed to ompute the skeleton has proved, through a large study, to be lower than for all other thinning algorithm we know of. In3D, su h astudy is still to bedone, but the on iseness of theBoolean de nition,that makesitstraightforwardto implement,yetshowsup. Thispaperis organisedasfollows. Se tion2
E-mailaddresses: Antoine.Manzaneraet a.fr, Thierry.Bernardensta.fr, Fran oise.Preteuxint-evry.fr,
illustratingwith someresults. InSe tion 4wepresentthe properties ofthe MB algorithm,in thefollowing order: (1)Isotropyandfullparallelism(2)Topologypreservation(3)Medianhypersurfa epreservation(4)Noisesensitivity andre onstru tibility(5) Computationalspeed.
2. PRELIMINARIES 2.1. The hyper ubi grid.
LetZ betheset ofintegers,N thesetof naturalintegers,R thesetofreal numbers. LetP(A)bethe olle tionof allsubsetsofA,P
(A)=P(A)n;. Letn2N,Z n
isthen-dimensionaldis retespa e. The ubi grid isde nedby theimmersionofZ
n
intheaÆnespa eatta hedto R n
bymeansofthefollowingappli ation:
: Z n ! P(R n ) z 7 ! (z)= n Y i=1 [z i 1 2 ;z i + 1 2 ℄
Thusapoint z ofthedis retespa eisidenti ed toanhyper ube of thequanti ed aÆnespa e,i.e. to the artesian produ tofthe losedsegments enteredaroundz,z
i
beingthei-th oordinateofz inthe anoni basis.
2.2. Dis rete hyber ubi topologies. Let bethefun tion de ned onP
(R
n
)by (P)=V su hthat V isthelinearmanifoldgenerated byP. Thenn kindsofadja en yrelations(andthentopologiesaswell) anbede nedonthehyper ubi mesh,asfollows: De nitionLetz andz
0 betwopointsofZ n su hthat (z)\(z 0 )6=;. thenz andz 0
arek-adja ent (0kn)ifandonlyif:
dim( ((z)\(z 0
))=k (1)
De nitionLetzandz 0
betwopointsof Z n
. z andz 0
arek-neighbors ifthereexists j;kjn, su h thatz and z
0
arej-adja ent.
The adja en y of two points orresponds then to a non-empty interse tion of two hyper ubes, and the level of adja en y,tothedimensionofthemanifoldgeneratedbytheinterse tion. Pleasenotethedi eren ewiththenotion ofneighbor,whi hmat hesthede nitionsusuallygivenintheliterature. Anexamplein2DisprovidedonFigure1.
x
z
y
Figure1. Conne tivityrelationsindimension2:xandyare1-adja ent(andnot0-adja ent).xandzare0-adja ent. xandyare1-neighbors,therefore0-neighbors.
Letabinaryimage I beasubsetofZ n
.
De nitionInterior pointsofabinaryimage: I Z n
. z2I,z is ak-interior pointofI ifand onlyif: 8z 0 ;z and z 0 arek-neighbors)z 0 2I.
Countingthek-neighbors: ConsideringtheoriginO of Z n
, apoint is k-adja entto O ifand onlyif ithas n k oordinatesin thesetf 1;+1g,andtheothersequalto0. ThusapointhasA(n;k)k-adja entpoints,with:
A(n;k)=2 n k C n k n (2)
V(n;k)= n 1 X
i=k
A(n;i) (3)
Table1givestheV(n;k)numbersfornsmallerthan4. Fork=0,wemeetof oursetheoverallnumberofneighbors indimensionn: W n =V(n;0)= n 1 X i=0 2 n i C n i n =3 n 1 (4)
We presentthese numbersto point out the fa t that, in the literature, the terms that are usually employed are
Table1. Numbersofk-neighborsinnD,forn4.
knn 1 2 3 4 3 - - - 8 2 - - 6 32 1 - 4 18 64 0 2 8 26 80
V(n;k)- onne tivity,V(n;k)-neighbors,et . Nevertheless,in thispaper,withintentto meetbothhomogeneityand on ision,wewillalwaysusethesinglek-pre x.
2.3. Dis rete distan es and median hypersurfa es LetÆ
n k
denotethedistan eindu edbythek-thtopologyindimensionn. (orÆ k
whenthereisnoambiguityregarding thedimension). LetXZ
n . LetX denoteZ n nX,theba kgroundof X. De nitionThedistan emap asso iatedwithX andrelativetoÆ
k
isthefun tionthatasso iatesÆ k (x;X )toevery x inZ n .
De nitionLetr2N. Letx2Z n . LetB k (x;r)=fy2Z n ;Æ k
(x;y)rgbetheballof entre xand radiusr. Let X Z
n . B
k
(x;r)isamaximal ball ofX ifandonlyif:
8y2X;8q2N;B k (y;q)X)B k (x;r)6B k (y;q) (5) De nitionLetS k
(X) denotethe olle tionof allthe entresof maximal ballsof thedistan eÆ k
. S k
(X)is alled themedian hypersurfa e asso iatedwithdistan e Æ
k . Thisnotionhasbeentheonlysheerde nitionoftheskeleton.
1
Butitdoesnotmat hthemodernnotionofskeleton, sin e,inthegeneral ase,X andS
k
(X)donothavethesametopology. PropertyLetx2X. xbelongstoS
k
(X)ifandonlyifforeveryy k-neighborofx,Æ k (y;X )Æ k (x;X ).
Inotherwords,the olle tionofthe entresofmaximalballsisequaltothesetoflo almaximaofthe orresponding distan emap.
Theinterestofthesenotionsistoformalizetheneedtorepresentthegeometryandthelo ationoftheshape. Indeed, wewillguaranteethattheskeletonliesright\atthemiddle"oftheshapeforsomedistan eifit ontainsthemedian hypersurfa eforthisdistan e.
3. THINNING PROCEDURE
Ourthinningpro edureisde nedbymeansoftwobinaryfun tionsofthedis retespa e: fun tionAlpha andfun tion Beta. These twofun tions are respe tivelyde ned in Table2and in Table 3. The omplete algorithm is given in Table4.
To expressinformally thede nition given in Table2, apointz ofI su h that (z)=1must be k-adja ent toa (n 1)-interiorpoint. Ifthis onditionholds,thesetwopointshaveanon-emptyinterse tion,thatgeneratesalinear manifoldofdimensionk. Letusnow onsidertheimageofthismanifoldbythesymmetryof entrez. Itisaparallel
LetI Z n
. Fun tion :I !f0;1gisde nedasfollows: Letz2I. if thereexistsk;0kn 1,su hthat: (1)9z
0
(n 1)-interiorpoint,zandz 0 arek-adja ent (2)ft, t(n 1)-adja enttoz and \ t ((z)\(t))=S z ( ((z)\(z 0 )))gI . whereS z
denotesthesymmetryof entrez. then (z)=1.
else (z)=0.
Table3. Fun tionBeta.
LetI Z n
. Fun tion :I !f0;1gisde nedasfollows: Letz2I. if thereexistsk;0kn 2,su hthat:
There existtwo ouplesof k-adja entpoints(a;b)and( ;d)su hthat a;b; ;dareallk-neighborsofz,andsu h thatthetwofollowing onditionshold:
(1)(a)\(b)=( )\(d) (2)fa;bgI andf ;dgI
then (z)=1.
else (z)=0.
(n 1)-adja entstozandwhoseinterse tionwithzgeneratesoneofthosehyperplanesbelongtotheba kgroundof I.
To express informally the de nition given in Table 3, a point z of I su h that (z) = 1 must ontain in its k-neighborhoodtwo ouplesofk-adja entpointsthat havethesameinterse tion,andsu hthat one ouplebelong to theimage,andtheothertotheba kground.
Figure2displaysin atablethepatternsmat hed bythefun tionsAlpha andBeta,fordimensionsfrom1to3: in
Table4. TheMBalgorithm.
Repeatforallpointsz2Z n
within aparallelframework,until onvergen e: if (z)=1and (z)=0,removez.
dimensionn, apointis removedif: (1)it mat hesoneofthepattern i
(oroneof the=2rotatedversionaround oneoftheaxis). Herethegraypointsbelongtotheimage,thewhiteones totheba kground,thebla kpointisthe originofthepattern. (2)itdoesnot ontainanypattern
i
withinitsi-neighborhood.
Some resultsof the MB algorithm are displayed in 2D on Figure 3, and in 3D on Figure 4. They illustrate the propertiesandbehaviorofourthinningpro edure,that wedevelopinthefollowingse tion.
4. PROPERTIES AND BEHAVIOR 4.1. Isotropy and full parallelism
Inthehyper ubi grid,isotropy meansthatallthedire tionsrepresentedbythenaxismustbetreatedthesameway, aswellasthetwodire tionsoneveryaxis. Isotropyisoneofthefundamental onstraintsonwhi houralgorithmis built. Anaturalout omeofthis onstraintisthepresen eoftwo-pixelsthi ksurfa esintheskeleta,asit anbeseen onFigures3and4.Inthede nitionsofthetwoBooleanfun tions,nodire tiongetaspe ialtreatment. Thisproperty appears learlyonthepatternmat hingformofthepro edure,asallthepatternsare ompletelysymmetri al. With
α
2
α
1
α
0
α
α
1
0
0
α
β
1
β
0
β
0
Dimension
1
3
2
function Alpha
function Beta
Figure2. MB patternsindimension1to 3.
thesamea tion. Thisis animportantpropertyof theMB-algorithm: the pro edure isfully parallel, whi h means that all the iterationsa t exa tly thesame, removing pointsin the 2n ardinaldire tions at the same time. The onsequen eisthatthenumberofiterationsbefore onvergen eequalstheradius ofthebiggestball orresponding to the strongest adja en y relation. As we are going to see in thefollowing subse tion,removingpoints within a parallelframeworkinalldire tionswhilepreservingthetopologyisnottrivial,asnoBoolean hara terizationofthe pointstoremoveisavailable,unlikeinthe aseofasequentialframework.
4.2. Topology preservation
Toremove pointsfrom anobje twithout hangingits topologymeansthat weaim at preservingthe onne tivity relations that exist in the obje t and in its ba kground. We annot dis onne t a onne ted omponent, and we annot reateordeletea\hole". Nowinadis retetopology,spe ial aremustbetakentodealwiththese notions. For instan e, a onne ted omponent of the ba kground may run a ross a pie e of surfa e of the obje t only if there is aholein it ! Tokeepthis \natural"property meaningful,however,itis ne essaryto hoose twodi erent onne tivitylevelsfortheobje tanditsba kground. Inthispaperwealways onsider0- onne tivityfortheobje t (twohyper ubesoftheobje tareneighborsassoonastheyshareapoint)and(n 1)- onne tivityfortheba kground (twohyper ubesoftheba kgroundareneighborsonly iftheyshare afa e). Weusethe(0;n 1) pre x tore all the hoi eofthetopology.
Inthisse tion,wedonotintendtoprovethattheMB-algorithmpreservesthetopologyinn-D,as,toourknowledge, no hara terization in n-D hasbeen re ognized by the dis rete topology ommunity. Nevertheless, the formalism thatweuseinthefollowingresultsis ompletelygeneri asforthedimension,andtheresultsaretruefordimensions 2and 3.
Wemust rstgivesomemeaningto\preservingthetopology". Inthe2Dand3D ontinuousspa es,topology anbe representedbymeansofthefundamentalgroup,whi histhegroupoftheequivalent lassesofhomotopi ar s(two ar saresaidto behomotopi ifthere existsa ontinuousfun tion mappingonetotheother). Then twosubsetsof R
2
havethesametopologyifandonlyiftheirfundamentalgroupsareisomorphi ;inR 3
thispropertyisnolonger suÆ ient, as fundamental groupsof the ba kgrounds must be isomorphi too. In the nextdimensions, homotopy groupsofhigherordersarene essaryto hara terizethetopology.
Indis rete spa es,the entralnotionregardingtopologypreservationis simpli ity. A subsetA ofadis rete image X is saidto besimpleif X andX nA havethesametopology. Forapointx 2X,\x issimple" meansthat fxg
1
S
0
0
S0
S
1
1
Original
Guo & Hall: AFP1
Stewart
Guo & Hall: AFP2
Jang & Chin
Guo & Hall: AFP3
MB
Manzanera & Bernard
Figure 3. The results of the MB-2D algorithm. For omparison purposes are also displayed the results of the algorithmsofJang andChin,
2
Stewart, 3
thethree algorithmsof Guoand Hall, 4
andapreviousalgorithmofthe authors.
5
Original
MB-3D
Original
MB-3D
(1) (2)
Figure 4. TheresultsoftheMB-3Dalgorithm: onasimplehome-made3D obje t(1)andonathree-dimensional segmentedimageoflung(2).
importantresult isthat the de ision anbe madelo ally,examining onlya nite neighborhood. Wenowgivethe hara terization,forthe(0,n-1)- onne tivitymodel:
TheoremLetXZ n
beabinaryimage. Letx2X. LetX x 0
denotethesetofallthe0-neighborsofxinX,ex ept x itself,andX
x n 2
theset ofallthe(n-2)-neighbors ofxin X
. xis simple in X forthe(0,n-1)- onne tivitymodel ifandonlyifthetwofollowing onditionshold:
x is0-neighborwithonesingle0- onne ted omponentofX x 0 .
x is(n-1)-neighborwithonesingle(n-1)- onne ted omponentofX x n 2
.
In2D,thisresult orrespondtothe0- onne tivitynumber thathasbeende ned byYokoi, 6
butthathadbeenused earlier by Hildit h.
7
In 3D, it is aresult due to Bertrand and Malandain. 8
Thus, with the abovetheorem, we proposeauni edexpressionofthosetwofundamentalresults.
Unfortunately,aunionof simplepointsis notasimpleset, ingeneral,andthis isthemajordiÆ ultyfordesigning parallelthinningalgorithms. The rstproblemisto hara terizesimplesets. Ronse
9
didit rstfor2Dimages,itwas thengeneralisedbyKong
10
for higher-dimensionalimages. Inthese papers,itisshownthat asetis simpleforthe imageX ifandonlyifit anbeorderedinasequen eofpointsfx
1 ;:::;x
n
gsu hthatforeveryiinf1;:::;ng,x i
is individuallysimplewithrespe ttoXnfx
1 ;:::;x
i 1
g. Fromthisproperty,Ronseproposed 11
veryeÆ ientsuÆ ient onditionstoprovethesoundnessofparallelthinningalgorithmin2D.Thisresulthasbeenextendedtothe3D ase byMa.
12
Forthistopi also,wemaygiveauni edexpressionoftheseresultsasfollows: Letaunitlatti eelement of dimensionk,0kn,beasetof2
k
pointsofZ n
su hthateverypairofpointsisapairof(n k)-neighbors. Theorem(Ronse88,Ma 94)
LetX Z n
beabinary image. An algorithmthat removespointsin parallelfrom abinarynDshapeX preserves (0,n-1)- onne tivityifthetwofollowing onditionsaresatis ed:
EverysubsetofX thatis ontainedinaunit latti eelementofdimension(n 1) andthatisremovedbythe algorithmissimple.
No onne ted omponentofX ontainedin aunit latti eelementofdimensionn anbe ompletelyremoved.
This theorem allowsto provethe soundnessof aparallel thinningalgorithm by he kingonly alimitednumberof on gurations. Inthis ontext,thefollowingpropositionhasbeenprovedforn=2
13
andforn=3 14
: Proposition
Inthis se tion, weare on ernedwith the non-topologi alside of skeletonization,whi h is geometry onservation. Ideally, weensurethat the skeleton lies at themiddle of the shape, if the lo al maxima of theeu lidean distan e belongtotheskeleton. Thisdistan e is,however, omplexto ompute,andasourformerpurpose was on iseness, our hoi ewastodealonlywiththe anoni aldistan esofthehyper ubi grid.
Letpmn. Letusde nethe(m;p)-median surfa e asfollows: De nition S p m (X)=fx2X;8yp neighborofx;Æ m (y;X )Æ m (x;X )g (6)
Notethat the(m;m)-mediansurfa eofX orrespondstothe lassi alS m
(X)de ned inSe tion 2.3. Thebehavior ofMBwithrespe t tomediansurfa epreservationis hara terizedbythefollowingproposition:
PropositionLetn2N. LetX beawell-formedimage.
TheMB-nDskeletonofX ontainstheset S 0 n 1
(X).
This importantpropertyis illustratedin 2D onFigure 5and in 3D onFigure 6. On Figure 5, theupperpi tures show the distan e fun tions for the twodi erent onne tivity models, where the value is representedby the grey level. Onthe entre,theleftandtherightimagesshowthe orrespondinglo almaximaset,andatthemiddle,the original medianaxison whi h theMB skeleton(bottom, entre)is built. For omparison purposes arealso shown otherskeletaontheleftandontheright,basedonthelo almaximaforthe1-andthe0-distan erespe tively. On Figure 6, it an be seen aswell that di erent 3D skeleta anbe obtained, depending on themedian surfa e. On Figure(b),therestri tionoftheAlphafun tionoftheMBthinningalgorithmtothe
2
patternleadstoaskeleton basedon thelo al maxima of the2-distan e. OnFigure ( ), restri ting to patterns
2 and
1
leadsto a skeleton basedon the(2;1) median surfa e. Finally, the whole fun tion Alpha is applied onFigure (d), and theresulting skeletonisbasedonthe(2;0) mediansurfa e.
Wemustnowtellaboutthe\well-formedimages",forwhi htheskeleton ontainsthe(n 1;0) mediansurfa e. Awell-formedimageisanimageinwhi hthepointsaredeletedbythethinningoperatorintheorderofthe(n 1)-distan efun tion. Inthis ase,ifwe onsiderthepatterns
i
;0in 1ofFigure2,we anseethatthey anonly removeapointthat has in its0-neighborhood a(n 1)-interiorpoint,whose (n 1) distan eto theba kground is stri tly superior and thus thepointsof S
0 n 1
ne essarilybelong to the skeleton. It is worth observingthat, for usual images, points are examined a ordingto the order indu ed by the distan e fun tion. Nevertheless, there are ex eptions,likethose orresponding to ill- onstru tedimages. We giveanexampleof su h animage in 2Don Figure 7(a). Theseimages orrespondto a on gurationthat would \prote t"apie e ofsurfa e, forbidingathi k volumeto bethinned,asin theshapepresentintheupperright orneroftheimageonFigure3.
Althoughrigorousproofisstilltobeprovided,itseemsthatasuÆ ient onditionforan-Dimagetobewell-formed is notto ontainone-pixelholes mat hingone ofthe patterns
i
. An image X Z n
has aone-pixelholeif there existx2X
andV alinearmanifoldofdimensionk,2knsu hthatx2V and8y;y(n-1)-adja enttox,y2V, y2X.
Figure 7showsthe orrespondingpatterns: a2D imageiswell-formedifit doesnot ontainanyrotatedversionof pattern (b), a3D image ifitdoesnot ontainanyrotatedversionof patterns( ) or(d), whereat least one ofthe twosquarepointsdoesnotbelongtoX.
4.4. Noise sensitivity and re onstru tibility
TheMBskeletonisbasedonaspe ialkindofmedian surfa e,theaim ofwhi h istotreatmoresymmetri allythe di erent kindsofdistan es in the ubi grid. Inparti ular, thethinning operatordoesnotdistinguishtheballs of thedi erentdistan es,produ ingalwaysonesinglepointforanykindofball. Italsoprodu eonesinglepointfora familyof sets that arebounded (forthein lusion) bytwoballs of di erentdistan es, and ofthe sameradius. Let us allthese setsFuzzyballs. WeshowinFigure 8somefuzzyballsofradius7in2D,andtheirskeletonredu edto their entre(superimposed asawhite dot). In 2D,it anbe provedthat afuzzyball B (formally de ned asaset whose (1,0)-median axisis redu edto apoint, alled its entre ) is aset that is in luded betweena1-ball anda 0-ball,su hthat forallx inB, B ontainsthesmallestre tangle ontainingx and .
The behaviorof the MB skeleton with respe t to fuzzy balls is the ause of its good properties regarding noise immunityand=4 rotationinvarian e,as anbeappre iatedonFigure 9,where theresultsare omparedto other algorithms, alreadyreferen ed in Figure 3. Inreturn, it onlyallowspartial re onstru tion. This fa t isillustrated onFigure10: theshapeisre onstru tedfromtheweightedskeleton(here,thevalueofthedistan efun tion onthe 5
S
1
1
S
1
0
S
0
0
Local maxima
Distance map
MB
Manzanera & Bernard
Jang & Chin
1-connectivity
0-connectivity
Figure5. Di erentmedian axisleadingto di erentskeletain2D.
(a)
(b)
(c)
(d)
Figure6. Thedi erentmediansurfa esin 3D:(a)Original(b)S 2 ( )S 1 (d)S 0 .
Figure7. An ill- onstru tedimagein2D(a)and theresponsiblepatternsin 2D(b)and 3D( ,d)
Figure8. Somefuzzyballsofradius7in 2D,andtheirskeletonredu edtotheir entre.
thatisbasedonthe(1,1)-medianaxis,andthusallowsexa t re onstru tibility,theMB-algorithm,representedhere ontherightdoesnot. Weshowhereanapproximate re onstru tion,basedono tagonsofthe orrespondingradius.
4.5. Computational speed
Does on eptual on isenessimply omputationalspeed? Wehavearguedinearlierwork 5
thatthemostfundamental wayto ompare omputational eÆ ien yof dataparallel algorithmwasbytheso- alledShannonmeasure,i.e. the overallnumberof elementary operationsperpixelneededto performthealgorithm. Thenature ofthe elementary
Original
Guo & Hall: AFP3
Stewart
MB
Jang & Chin
fortheMB-skeletonbasedonthefuzzyballs(right).
operations depends on the omputer. However,weget asound generalmeasure by hoosing themost elementary operation to be omputed in a digital algorithm: the Boolean fun tion of two variables. Indeed, for any digital algorithm,the ostof everyelementaryBooleanoperationisfound eitherat asoftwarelevel,where ithasane e t onthe omputationtime,oratahardwarelevel,undertheformofanequivalenttwo-entrieslogi gate,whereithas ane e tontheexpenseinhardwareresour es. TheTable5displaystheShannonmeasureoftheMBalgorithm in 2Dandin3D.In2D,themeasureis omparedtosomeofthemostre entthinningalgorithmsfoundintheliterature.
Table 5. Comparingthe ostof parallel thinningalgorithms. r standsforthe maximalobje tthi kness(i.e. the radiusofthebiggest(n 1) ball ontainedintheimage). Thenumberofelementaryoperationsgivenfortheother algorithms is sometimes only an estimation, as the quoted papers did not alwaysprovide alogi minimization of theiralgorithm.
Algorithm Sizeofneighborhood Numberof elementary examined operationsrequired Jang&Chin
2
7 32r
Cardoner&Thomas 15 7 40r Stewart 3 19 64r Wu &Tsai 16 11 60r
Guo&Hall(AFP1) 4
11 73r
Guo&Hall(AFP2) 4
11 81r
Guo&Hall(AFP3) 4
11 91r
Manzanera&Bernard 5
13 18r
MB-2D 21 28r
MB-3D 81 148r
5. CONCLUSION
TheMB thinningalgorithmis, to ourknowledge,the rstthinning algorithmvalid bothin 2Dand in3D. Further resear hesindis retetopologyforhigherdimensionmesheswillallowto he kitsoverallvalidity,sin etheexpression in n-D is already available. More generally, we may hope that this work will suggest some resear h tra ks. In parti ular, we have proposed a uni ed formalism to express the theorems of homotopy in n-D propositions: the Yokoi/BertrandandMalandaintheoremforthesimplepoint hara terization,theRonse/Matheoremforthetopology preservationsuÆ ient onditions. Dothesepropositionsmakesensefordimensionshigherthan3? Oneofourgoals istoarousesomerea tionsthatwillbringustheanswer. Nevertheless,wethinkoneofthebesta hievementsofthe MBalgorithmisitsgreatsimpli ity,asthen-Dthinningpro edureisde nedthroughtwobinaryBooleanfun tions,
we anjusti ablyexpe tthesameadvantagein3D.Finallywehave hara terizedthemedianhypersurfa eonwhi h theskeletonis based,and shown that itprovided somenoiseimmunityto theskeleton. This median hypersurfa e impliesanewkindofshapedes riptors: thefuzzyballs,thatwehave hara terizedin2D.Furtherworkswillprovide amoregeneralandrigorousviewofthis lastissue.
ACKNOWLEDGMENTS
The rstauthorisextremelygratefultoBertrandCollin andDamienMer ier,fromCTA/GIP,forthevisualization tool and software support. The volumetri medi al image was a quired in the Servi e de Radiologie Central de l'H^opital delaSalpetriere(ProfessorPh. Grenier). ThelungwassegmentedbyCatalinFetita fromtheUPArtemis (INTEvry).
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