Medial faces from a concise 3D thinning algorithm

HAL Id: hal-01245449

https://hal.archives-ouvertes.fr/hal-01245449

Submitted on 17 Dec 2015

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Medial faces from a concise 3D thinning algorithm

Antoine Manzanera, Thierry Bernard, Françoise Prêteux, Bernard Longuet

To cite this version:

Antoine Manzanera, Thierry Bernard, Françoise Prêteux, Bernard Longuet. Medial faces from a

concise 3D thinning algorithm. International Conference on Computer Vision, Sep 1999, Corfou,

Greece. �10.1109/ICCV.1999.791239�. �hal-01245449�

Antoine Manzanera ThierryM.Bernard Fran oise Pr^eteux Bernard Longuet

DCE/ETC4/CTA/GIP ENSTA/LEI INT/UPArtemis Aerospatiale

16bAv. P.delaC^oted'Or 32BdVi tor 9RueCharlesFourier 2RueBeranger

94114Ar ueilCedex 75015Paris 91011EvryCedex 92320ChatillonCedex

Abstra t

We propose in this paper a new 3D fully parallel thinning algorithmthat webelievetobethe most on- ise duetoitssimple hara terization. Thealgorithm is indeed ompletely denedby aset of ve patterns, threeremoving onditionsandtwonon-removing on-ditions. Thesepatternsaredesignedfromthetwo fun-damental and ompatible onstraints usually expe ted in skeleta: (1) Topology preservation and (2)Medial surfa e. Fromthesetwo onstraints,theremoving pat-terns ( 1 ,  2 and  3

) dete t the non-lo al maxima, whereas the non-removing patterns (

1

and 

2 ) pre-ventanytopology hangethattheremoving onditions ould imply. We show that the three mentioned on-straints are respe ted. The logi al on iseness of our pro edure, alled MB-3D, makes it to our knowledge theeasiest3Dthinningalgorithmtoimplement. Some results are displayed, that illustrate the relevan e of our approa h.

Keywords

3Dfullyparallel thinningalgorithm-Dis rete topo-logy -Con ise Booleanexpression.

1 Introdu tion

Skeletonizationisavery ommonwaytorepresent binary shapeswith alimitedamount of information. A skeleton that faithfully represents a shape is ex-pe tedto(1)betopologi allyequivalenttothatshape and (2)renderitsgeometryand lo ation. Skeletaare usually obtainedthrough aniterativeredu tion ope-rator alled thinning: ertain types of border points are iteratively removed until no more points an be deleted: the remaining image is alled the skeleton. Thinning algorithmshavebeenanimportantsubje t of resear hforyearsin 2D, andmorere entlyin 3D. Lotsofeortshavebeendonetoprovidethesimplest hara terizationofthenon-skeletalpointsremovedby an elementary thinning iteration. In 3D,the hara -terizations remain ompli ated,withgreatnumberof

withspe ialrulestoavoiddis onne tiondueto paral-lelremoval[5℄.

We present in this paper what we believe to be the most omputationallyeÆ ienttodateBoolean expres-sionofa fullyparallel 3Dthinning pro ess: the non-skeletalpointsareentirely hara terizedthroughaset ofthreeBooleanremoving onditionsandtwoBoolean remaining onditions,every onditionbeingdenedby asimplepattern,whi hmakesouralgorithm straight-forwardtoimplement. Ouralgorithmmeetstwo fun-damental (yet ompatible) onstraints: (1) Topology preservation (2) presen e of thelo al maxima. Con-straint (2) ensures that the skeleton is lo ated right atthe\middle"ofshapes,andrenderstheirmost sig-ni ant geometri al features. The algorithm, alled MB-3D,is ompletelydenedbytwosmallfamiliesof patterns:  Patterns  1 ,  2 and  3

are designed to remove nonlo almaximapointsforthedistan eindu ed bythe6-topology,withinthe26-neighborhood.

 Patterns 1

and 2

aredesignedto avoid dis on-ne tionof18-and26- onne tedpoints respe tive-ly.

Forself- ontainedness purposes,the followingse tion re alls some preliminaries. In Se tion 3, we present ouralgorithm,givingtheBooleanexpressionandthe visualrepresentation of the patterns. Then we show thatwith thetwosimplepattern familiesthat dene it,theMB-3Dalgorithmrespe tsthetwo onstraints statedabove. Atthesametime, weillustratethe pa-perwithsomeresultsanddis ussthebehaviorofthe algorithm.

2 Theoreti al ba kground

In this se tion we set out the mathemati al tools ne essarytohandlethenotionswearedealingwith. In therstsubse tion,wepresentthedis rete geometry

B6

B18

B26

Figure 1: Unity sized balls for the three dierent topologiesinthe ubi grid.

of topology preservation, and present the way it has been addressed for the ubi grid in the litterature. Finally, weintrodu e themorphologi aloperators to beusedforthedenitionofourthinningpro edure.

2.1 Dis rete topologies in the ubi grid

LetZ 3

bethedis retespa e. LetX Z 3

a(binary) (three-dimensional) image. Let X

=Z

3

nX denote

the ba kground of X. We are working in the ubi

grid, this meansthat thereal spa e R 3

is dis retized into Z

3

by meansof the ubi quantization: A point

z 2Z

3

representsanelementaryvolumewhi h isthe unit ube entered aroundz. Inthismesh, three dif-ferent onne tivityrelations anbedened. Figure1 showsthedierenttopologiesinthe ubi grid,as de-nedbytheunitysizedballs. Thetopology(andthe indu eddistan e)isusuallydenotedusingthenumber ofneighborsinthe orrespondingtypeof onne tivity. Namely,apoint,i.e. a ubeinourrepresentation,has 6 (respe tively 18, 26) neighbors in the onne tivity

dened by B 6 (respe tively B 18 , B 26

) whi h are the pointsitsharesafa e(respe tivelyanedge,avertex) with. Letx;y betwopointsof Z

3

. Wesay that x is N-adja ent toy(N =6,18or26)ifxisaN-neighbor of y. Let A;B betwosubsets of Z

3

. We saythat A is N-adja ent to B if there exists a 2 A and b 2 B su hthat aisN-adja entto b. LetX Z

3

. X isan N- onne ted omponent (N- )ofZ

3

iftheredoesnot exist anypartitionofX intotwosubsetsthatarenot N-adja ent. LetX Z

3

beanimage. x2X is said tobeN-interior toX ifallitsN-neighborsbelongto X.

Denition 1 Let d

N

be the distan e indu ed by the

N-topology. Let X  Z

3

. A ball B is maximal in

X if B  X and there does not exist a ball B 0 su h that BB 0 X. Let S N

(X)be the olle tion of the entresofmaximumballsasso iatedwithd

N .

Denition 2 Let X  Z

3

. The distan e fun tion asso iatedwith d N on X is N (x)=d N (x;X ). Property 1 S N (X) = fx 2 X;8y N-adja ent to x; N (x) N

(y)g: In other words, the olle tion of the entresof maximal balls orresponds tothe setof

This formalism aims at giving a sound basis to the notionof medial surfa e. Indeed, we ensurethat the skeletonlies \atthemiddle" oftheshapeif weknow thatit ontainsthelo almaximaofthe orresponding distan efun tion.

2.2 Topology preservation

The topologi al equivalen e isa well known prop-erty. A doughnut is equivalent to a oee up be- ause they have both exa tlyone \hole of the same type"(the handle). In2Dand 3D, thetopology an be hara terizedbytheso- alled fundamental group, i.eapartitionofthe urvesbythehomotopi relation (two urvesarehomotopi ifthereexistsa ontinuous morphingfromonetotheother).

Toget asound denition of su h topologi al proper-ty in our ubi grid, spe ial are must be taken in the hoi e of the onne tivity. In parti ular, an ob-je tmaybe rossedbya onne ted omponentofthe ba kgroundonlyifthereisaholethroughit! Inthis respe t,itisusually hosenthestrongest onne tivity forthe ba kground (i.e. fa e sharing), and aweaker onefor theobje titself (i.e. edge orvertexsharing). The onne tivity model that is used in this paper is (26,6)- onne tivity, whi h means 26- onne tivity for theimageand 6- onne tivityfortheba kground. Our thinning pro ess works by iterative deletion of setsofpoints. The entralnotionaroundthe hara -terizationof thedeleted pointsis simpli ity. A point issimple ifitsdeletion doesnot hange thetopology. Asin 2D, the omputation of simpli ity anbedone within anite neighborhood of thepoint. Themost on ise hara terization is provided byBertrand and Malandainin[1℄:

Theorem1 (BertrandandMalandain 94)

Let X Z

3

be a binary image. Let x2 X. Let X x 26 denote the set of all the 26-neighbors of x, ex ept x itself,that belong toX,andX

x 18

theset ofall the 18-neighborsofx thatdonotbelong toX. x issimplein X for the (26,6)- onne tivity model if andonly ifthe twofollowing onditionshold:

 xis26-adja ent toonly one26- ofX x 26

.

 xis6-adja ent toonlyone6- ofX x 18

.

This hara terization uses onne ted omponents

ounting only, asin the 2D ase. It is important to noti e,however,thatsimpli ityisapropertywhi his stri tly individual with respe t to a point of the u-bi grid. In general, simultaneouslyremovingsimple pointsfromashapeleadstotopology hanges. From thisproblemarosethenotionofsimplesets,whi hare

on ept in [8℄ for 2D images, on ept that was then generalisedbyKongin [3℄forhigher-dimensional im-ages. Inthesepapers,itisshownthat asetissimple for theimage X if andonly ifit an be orderedin a sequen e of points fx

1 ;:::;x

n

gsu h that foreveryi in f1;:::;ng,x

i

isindividually simple(in theformer sense) with respe t to X nfx

1 ;:::;x

i 1

g. From this property, Ronse proposed in [9℄ suÆ ient onditions that wereveryeÆ ienttoprovethesoundnessof par-allel thinning algorithm in 2D. This result has been extended to the 3D aseby Ma in [4℄. We now give Ma'sresultforthe(26,6)- onne tivity. Letaunit lat-ti e square betheset offour ornersofaunit square of the ubi grid,andaunitlatti e ube betheset of eight ornersofaunit ubeofthe ubi grid.

Theorem2 (Ma94)

Let X  Z

3

be a binary image. An algorithm that removes points in parallel from a binary3D shape X preserves(26,6)- onne tivity ifthe twofollowing on-ditions aresatised:

 Every subsetofX that is ontainedinaunit lat-ti e square and that is removedby the algorithm issimple.

 No onne ted omponentofX ontainedinaunit latti e ubeis ompletely removed.

This theoremallowsto provethesoundnessofa par-allelthinningalgorithmby he kingalimitednumber of ongurations.

2.3 Morphologi al operators

We dene hereunder the morphologi al operators neededtoprovidetheBooleanexpressionofour thin-ning pro edure.

The morphologi al erosion of an image X by a set

B  Z

3

, denoted X B is the set of all pointsx of Z

3

su h that the translated set of B by ve tor x is ompletely in ludedin X.

The morphologi al dilation of an image X by a set

B  Z

3

, denoted XB is the set of all pointsx of Z

3

su h that the interse tion of thetranslated set of B byve torx withX isnon-empty.

A pattern ofZ 3

is atuple (H;M)of nitesubsets of Z

3

su hthatH\M =;.

TheHit-Or-MissTransform(HMT)ofanimageX by

apattern =(H;M)istheimage: X~ =(X H)\(X

M).

We will say that x mat hes every time that x 2

X~ .

Ifwedenote B thesetofalltheN-neighborsofthe

neanother transformationthat we allHit-Or-Miss Neighborhood Transform (HMNT) relativeto theN -neighborhood,thatwedenoteX}

N

,anddeneby:

X} N =(X~ )(B N (H[M)) NotethatX} N isasupersetofX~ .

Thesenotionsaregoingtobeusedinthedenitionof MB-3D.HMT orrespondstoa ongurationthatthe neighborhoodof apointmust exa tly mat h,whereas

HMNT orresponds to a onguration that must be

ontained inthementionedneighborhood.

3 The thinning pro edure

MB-3Dis aniterativeparallelthinning algorithm, where ea h iteration deletes from an image X a set of points denoted mb(X), orresponding to ertain neighborhood onditions. These onditionsarebased onpatternsthatareshownin Table1. Everypattern a tually omeswithallits=2rotatedversionsaround thethree axesOx,Oy, andOz.

z2mb(X)ifand onlyif: (1)9i2f1;2;3g;z2X~ i (2)z62X} 18  1 and(3)z62X} 26  2 LetX 0 =X,X n+1 =X n nmb(X n ). TheMB-3DskeletonofX isX 1 .

α

1

α

2

α

3

β

₁

β

₂

Table 1: Denition of the MB-3D algorithm,based on5 lassesofpatterns.

Therst olle tion(the i

family)isusedinHMTs. Everypattern representstwosubsetsofZ

3

, The Hit-set orrespondstothegrey ubes,whi harethepoints whose value is 1. The Miss-set orresponds to the transparent ubes, whi h are the points whose value is0. Thedark ube orrespondstotheorigin. No ori-entationisgiven,aseverypatternmustbe onsidered in allits possibleorientations,indeed, thepro edure is ompletelyisotropi . Note right away that 

1 , 

2

and 

3

are based on the unity sized ball B 6

of Fi-gure1. Thusthesepatternsnaturallylendthemselves to omputationally eÆ ient des ription and manipu-lation. These ond olle tion(the

i

family)isusedin

HMNTs,

1

istobedete tedinthe18-neighborhood, 

2

in the 26-neighborhood. Note that no origin is ne essaryhere, sin e both patterns are symmetri al. To simplify in the following se tions we shall say \x

mat hes

1

"(resp.  2

)everytimethat x2X} 18

 1 (resp. x2X}  ).

(1)

(2)

Figure 2: Twoexamplesto illustrate thene essity of patterns 

i .

The thinninga tion learlyresultsfrom theshapeof the 

i

patterns. Wea tually believethat the deni-tion of these patternsis averypure hara terization of apeeling pro ess: anypointthat mat hesan

i is adja entto a6-interiorpoint, su h that allthe fa es opposite to this interior point are on the frontier of theimage. Still the

i

areabit greedy: some topolo-gy hanges would o urwithout the safety provided bythe

i

patterns. Figure2showswhythe i

are ne- essary throughtwoexamples. The bla k points be-longto theimage,thewhiteonesto theba kground. (1) The square point mat h pattern 

1

, but its re-movalwouldleadto26-dis onne tion:MB-3Dwillnot removeitsin e

1

is ontainedinits18-neighborhood. (2)Thetwosquarepointsmat hpattern

1

,buttheir simultaneousremovalwould 6- onne t thetwowhite points, whi h is forbidden: MB-3D will not remove them sin e

2

is ontainedin the26-neighborhoodof thebla ksquarepoints.

4 Results and behavior

Someresultsofourthinningalgorithm anbeseen on Figure 3. Theresultsof MB-3D are displayedon the left olumn (Images(1.a)to (4.a)). Asexpe ted, there are two pixel-thi k surfa es. This is anatural out omeoftheisotropy onstraint.

Inthisse tion,weestablishthesoundnessofthe pro- edure,rstly,byprovingthatthealgorithmpreserves the(26,6)-topology,andse ondlybyshowingthat, un-der a ertain onditionwhi h isexpli ited, the skele-ton ontainsthemaximaofthed

6

distan eswithinthe 26-neighborhood. Wenextdis ussthebehaviorofthe algorithm asitisappliedto somesigni antshapes.

4.1 Topologi al properties

Weproveinthisse tionthattheMB-3Dalgorithm preservesthe(26-6)-topologyofthebinaryshapes. If x 2 X, weuse the two sets X

x 26 and X x 18 dened in Theorem1. Theproofisbasedonvelemmae. Lem-mae 1to 3dealwiththe 26-topologypreservation of

obje ts, whereas Lemmae 4 and 5 deal with the

6-topology preservation of the ba kground. Lemma 1

and4provethatoneiterationoftheMB-3D

algorith-(1.a)

(1)

(2)

(2.a)

(3)

(3.a)

(4)

(4.a)

Figure3: Someresultsofthethinningalgorithm.The left olumn ontains the original images. The right olumndisplaystheresultsofMB-3D.

areusedtoproveLemma3. Lemma4isusedtoprove

Lemma 5. Lemmae 3 and 5 prove that any pair of

6-adja entpointsremovedbyMB-3D isasimpleset. Finally,theproofis ompleted inProposition1.

Lemma1 Letx2X,betweentwo6-neighborsaand b, with a 62 X and b 2 X ( f Figure 6). If x is 26-adja ent to more than one 26- of X

x 26 , then either x is ontainedin pattern  1 ,or x is ontained inthe patternrepresentedonFigure5.

proof

Ifxis26-adja enttomorethanone 26- of X

x 26

, then there must exist a point y in X

x 26

whi h is not26-adja ent to b. y annot be a 6-neighbor of x, but it may be an 18-neighbor, as illustrated by on

Figure5: Pattern.

Figure6(1). Inthat ase,sin e andbarenotinthe same26- ,x mat hes

1

. If thereis nosu h ,then y is only a 26-neighborof x, as illustrated by d on

Figure 4: Resultof MB-3Don asegmentedimage of lung.

a

x

b

b

x

a

d

(2)

c

(1)

Figure6: ProvingLemma 1.

Corollary 1 Any point removed by one iteration of the algorithm fulls ondition1ofTheorem1.

Indeed, any point that mat hes pattern  1

or  2

is ne essarily between two 6-neighbors, one in X, the other in theba kground. Thesameholds forapoint that mat hes

3

, andnot 1

. Then Lemma1applies

and,sin epatternisaparti ular aseofpattern 2

, thepointis26-adja enttoonlyone26- ofX

x 26

.

Lemma2 Let x 2 X. Let Y be a subset of X su h

that Y  mb(X)and Y [fxg is ontained in a unit latti e square. Then x 2 (X nY)}

26  implies x 2 X} 26  2 . proof Let us onsider x 2 (XnY)} 26 . If x 2 X } 26 , then x 2 X } 26  2

. If not, the situation is that of

Figure 7(1), where Y  fy 1 ;y 2 ;y 3

g. Note that the three points represented by squares belong either to

Y or to X . If y 1 2 X or y 3 2X , then obviously x 2 X } 26  2 . If not, fy 1 ;y 3 ;zg  X. It follows that y 2 maymat h an i

onlywith aninterior point

within the ube drawn on Figure 7(1). But for

ea h of the seven possibilities, one an easily he k that this is not possible. Then y

2 62 Y, soy 2 2 X , andthefourpointsfx;t;y

2

;zgmakeupa 2

pattern2

Lemma3 Let x andy be two6-neighbors su hthat

fx;yg mb(X). Then x is 26-adja ent to only one 26- of (Xnfyg) x .

y2

y

1

y3

(1)

z

x

_t

e

(2)

f

c

a

x

b

y

Figure7: ProvingLemmae2and3.

proof

Under the premises of Lemma 3, it an easily be

he ked that whatever the 

i

it mat hes, x is al-ways betweentwo6-neighbors su h that onebelongs toXnfygandtheotherto X

. Nowsupposethatx is26-adja enttomorethanone 26- of (Xnfyg)

x 26

. FromLemma1,xmustmat honeofthetwo pattern-s 

1

or  within (X nfyg). But Lemma 2 shows it

annotbesin exwouldhavemat hed 2

beforethe removalofy,in ontradi tionwithxbeingremovedby

MB-3D.Sox mat hes

1

within(Xnfyg);more pre- isely, the situation of x is that of Figure 6(1), with and b in distin t 26- s. Sin e x does not mat h 

1

within X, y asaremoved point, is partof  1

, as

shown on Figure 7(2). Besides, e and f must both

belongtoX

. Butthen,y ouldnothavemat hedan 

i

pattern, whi h isin ontradi tionwithitsremoval

byMB-3D2

Lemma4 Letx2X,betweentwo6-neighborsaand b, witha62X andb2X. If x is 6-adja ent to more thanone 6- of X

x 18

, then x is ontained in pattern 

1 .

proof

SeeFigure8(1). Ifthereexists 62X su hthataand belongtotwodistin t6- sofX

x 18

,thenpointdsu h thatd6=x,d6-adja entto bothaand mustbelong toX. Sox mat hespattern

1 2

Corollary2 Any point removed by one iteration of the algorithmfulls ondition2ofTheorem1.

Lemma5 Let x and y be two6-neighbors su h that

fx;yg  mb(X). Then x is 6-adja ent to only one

6- of(Xnfyg) x 18

.

proof

Thepremisesof Lemma5(identi al tothoseof Lem-ma3),impliesthatxisbetweentwo6-neighborssu h thatonebelongstoXnfygandtheothertoX

. Now

supposethatx is6-adja enttomorethanone6- of (Xnfyg)

x

. FromLemma4,xmustmat h 1

(1)

(2)

d

a

x

c

b

e

a

d

b

x

c

y

Figure8: ProvingLemmae4and5.

(X nfyg). SeeFigure 8(2),where a and y belong to distin t6- sof(Xnfyg)

x 18

. Ifband bothbelongto X,theny ouldnothavemat hedan

i

pattern,sob or belong to X

. Let ussupposeit is b. Sin e x is removed,itdoesnotmat hpattern

2

,andsod2X

. Sin e x doesnotmat h pattern

1 ,e2X

also, and nallyaandy belongtothesame6- . Thatleadsto a ontradi tion2

Wemaynowgivethemain proposition.

Proposition1 The MB-3D algorithm preserves the

(26,6) topology.

proof

As mentionedearlier, Lemma 1and Lemma 4prove

that one iteration of the MB-3D algorithm removes only simple points. Nowlet fx

1 ;x

2

gbe apair of 6-adja ent points, simultaneously removed by MB-3D.

Lemma 3andLemma 5provethat fx

1 ;x

2

gisa sim-ple set. Moregenerally,let Y be aset of pointssu h

that Y  mb(X) and Y is ontained in a unit

lat-ti e square. Let x 2 Y su h that x is not simplein (Xn(Y nfxg)). ThenLemmae 1and4,showthatx mat hespattern

1

or,butthelatterisforbidenby

Lemma 2. Thenx mat hes

1

within(Xn(Y nfxg)). Now let us onsider fx

1 ;x

2

g  mb(X) apair of 18-adja ent,not6-adja entpoints. Itis easytoseethat if x 1 62 X } 18  1 , then x 1 62 (Xnfx 2 g)} 18  1 . So x 1 is simple in (X nfx 2 g), and then fx 1 ;x 2 g is a

simple set. Let fx

1 ;x 2 ;x 3 g  mb(X) be a triplet

of points ontainedin aunit latti esquaresu hthat x

1

and x

2

are 6-adja ent. Then fx 1

;x 2

g is simple, and it is easy to see that if x

3 62 X } 18  1 , then x 3 62(Xnfx 1 ;x 2 g)} 18  1 ,sofx 1 ;x 2 ;x 3 gis asimple set. Letfx 1 ;x 2 ;x 3 ;x 4

gmb(X)bethe four orners ofaunitlatti esquare. fx

1 ;x

2 ;x

3

gisasimpleset,and ifx 4 62X} 18  1 ,thenx 4 62(Xnfx 1 ;x 2 ;x 3 g)} 18  1 ,so fx 1 ;x 2 ;x 3 ;x 4

gis asimpleset. Thus wehaveproved thatanyset ontainedwithinaunitlatti esquareisa simpleset. Atlast,itisobviousthataniterationofthe MB-3Dalgorithm annotentirelyremovea onne ted omponent ontainedin aunit latti e ube, sin eno 

i

tsinto this elementary ube. Sowehaveproved

ls onditions(1)and(2)ofTheorem2. ThenMB-3D preserves(26,6)-topology2

4.2 Non-topologi al properties

As we have seen in Se tion 2.1, geometry preser-vation is related to the notion of medial surfa e. In the ubi grid, there exist three anoni al distan es, namelyd 6 ,d 18 andd 26

,leadingtothreedierentlo al maximasets. A fully parallelthinning algorithmhas to favor the 6-distan e, sin e a removed point must be a 6- ontour point (i.e. have a 6-neighbor in the

ba kground). Let k = 6, 18 or 26. We dene the

(6;k) medialsurfa easthefollowingset: S k 6 (X)=fx2X;8yk-adja enttox; 6 (x) 6 (y)g

Note that the ase k = 6 orresponds to the set

S 6

(X)denedinSe tion2.1. Inordertogetafaithful shape representation featuringsome noiseimmunity, theMB-3D algorithmis basedonthe(6;26) medial surfa e,i.e. S

26 6

(X).

Weillustrate the sele tivea tion of the i

by apply-ingtheMB-3Dtoaparallelepiped,rstlyrestri tedto pattern 

1

, se ondly to thetwopatterns  1

and 

2 , and nally the omplete algorithm. Results an be seenonFigure9. Weseethatdierentskeletaare ob-taineda ordingto themedial surfa etheyarebuilt on. The skeleton (b) (resp. ( ), (d)) isbased onthe medialsurfa e S 6 (X)(resp. S 18 6 (X), S 26 6 (X)). Thus theMBalgorithm anleadtodierentskeletabythe restri tion to ertain 

i

patterns. This an be very usefulfor the versatile representation of omplex3D obje ts.

As every removed point is adja en-t to a 6-interior point, it an be for-mallyshownthattheskeleton ontains the set S

26 6

(X) dened above, as long asthepointsareexaminedintheorder indu edbythedistan e fun tion. This

is what appends with usual images.

Nevertheless,thereareex eptions,

or-Figure10: Ill- onstru ted 2Dimage.

responding to ill- onstru ted images. These images are the3D equivalent of the better known

patholog-(a)

(b)

(c)

(d)

Figure9: Dierent hoi esofthemedialsurfa e lead-ingtodierentskeleta.

(1)

(2)

Figure11: Ill- onstru tedpatterns.

i al images in 2D, of whi h we give an example on Figure 10. These images orrespond to a ongura-tionthat would \prote t"apie eof surfa e,

prevent-ing a thi k volume from being thinned. In 3D, an

image is ill- onstru ted if it ontains one of the two patterns shown onFigure 11(at leastoneofthetwo square points does not belong to X). Note that it orrespondstoone-pixelholesmat hing

1 or

2 .

The last, but not least, property of MB-3D to be emphasizedonisits omputationaleÆ ien y. Firstly, the on isenessoftheBoolean denitionsof the pro- edure leadstoa ompa t omputationaldes ription, whi hmeanseÆ ien yinthe omputationofone itera-tion. Se ondly, the full parallelism of the algorithm implies that the overall numberof iterations needed to a hievethe omputation oftheskeletonequalsthe radius of the largest 6-ball ontained as many itera-tions.

5 Con lusion

A new thinning algorithm for 3D digital pi tures hasbeenproposed. WehavegiveninTable1its om-plete expression. Compared to the other algorithms weknowof,MB-3Dseemstobethemost on iseand thenthesimplesttoimplement. Indeed,thepoints

re-movedbythe

i

patternsarethosethat areadja ent to a6-interiorpoint, and forwhi h every fa e oppo-siteto thisinteriorpointis afrontierfa e. With this very short hara terization, the 

i

patterns allow to obtainthemedialsurfa ethroughafullyparalleland isotropi pro edurewhilepreserving onne tivity, ex- ept in afew ases,taken areof bytheevensimpler 

i

patterns. Although the denition of the thinning algorithm is mu h shorter than all other algorithms weareawareof,theresultsprovetobesatisfying.

A knowledgments

The authors are extremely grateful to Bertrand Collin and Damien Mer ier,from CTA/GIP,for the visualizationtoolandsoftwaresupport. Theywishto thank also the anonymous reviewers for their

help-ful omments. The volumetri medi al image was

a quired in the Servi e de Radiologie Central de

UPArtemis(INTEvry).

Referen es

[1℄ GillesBertrandandGregoireMalandain.Anew har-a terizationofthree-dimensionalsimplepoints. Pat-ternRe ognitionLetters,15:169{175,1994.

[2℄ WeiXin Gong and Gilles Bertrand. A simple par-allel 3d thinning algorithm. In International Con-feren e on Pattern Re ognition, Atlanti City - NJ, pages188{190,61990.

[3℄ T.Y.Kong.Ontheproblemofdeterminingwhethera parallel redu tion operator for n-dimensionalbinary imagesalwayspreservetopology.InSPIEConferen e onVision Geometry-Boston,MA, 1993.

[4℄ CherngMinMa.Ontopologypreservationin3d thin-ning. CVGIP: Image Understanding, 59-3:328{339, 1994.

[5℄ C.MinMa. A3dfullyparallelthinningalgorithmfor generatingmedialfa es. PatternRe ognitionLetters, 16:83{87,1995.

[6℄ C.MinMaandMilanSonka.Afullyparallel3d thin-ningalgorithmanditsappli ations.ComputerVision andImage Understanding,64-3:420{433,1996.

[7℄ KalmanPalagyiandAttilaKuba.A3d6-subiteration thinningalgorithm for extra ting mediallines. Pat-ternRe ognitionLetters,19:613{627,1998.

[8℄ Christian Ronse. A topologi al hara terization of thinning. Theoreti al Computer S ien e, 43:31{41, 1986.

[9℄ Christian Ronse. Minimal test patterns for onne -tivitypreservationinparallelthinningalgorithmsfor binarydigitalimages. Dis reteAppliedMathemati s, 21:67{79,1988.

[10℄ P.K. Saha, B.B. Chaudhuri, and D. Dutta Ma jum-ber. A new shape preserving parallel thinning al-gorithmfor 3ddigital images. Pattern Re ognition, 30-12:1939{1955,1997.