Isthmus-based Parallel and Asymmetric 3D Thinning Algorithms

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Isthmus-based Parallel and Asymmetric 3D Thinning

Algorithms

Michel Couprie, Gilles Bertrand

To cite this version:

Michel Couprie, Gilles Bertrand. Isthmus-based Parallel and Asymmetric 3D Thinning Algorithms.

Discrete Geometry for Computer Imagery, Sep 2014, Siena, Italy. pp.51-62. �hal-01112000�

Thinning S heme and Algorithms

Mi helCouprieandGillesBertrand

UniversitéParis-Est,LIGM,ÉquipeA3SI,ESIEEParis,Fran e

⋆⋆

e-mail:mi hel. ouprieesiee.fr,gilles.bertrandesiee.fr

Abstra t. Criti alkernels onstituteageneralframeworksettledinthe ontext ofabstra t omplexesfor the studyofparallel thinninginany dimension.We take advantage of the properties of this framework, to propose a generi thinnings heme for obtainingthin skeletons from obje ts made of voxels. From this s heme, we derive algorithms that produ e urvilinear or surfa e skeletons, based onthe notion of 1D or 2Disthmus.

1 Introdu tion

Whendealingwithskeletons,onehastofa etwomainproblems:topology preser-vation, andpreservationofmeaningfulgeometri alfeatures.Here,weare inter-estedin theskeletonizationofobje tsthat aremadeof voxels(unit ubes) ina regular3Dgrid,i.e.,inabinary3Dimage.Inthis ontext,topologypreservation is usually obtainedthroughtheiteration ofthinning steps,provided that ea h step doesnot alterthe topologi al hara teristi s. Insequential thinning algo-rithms,ea hstep onsistsofdete tingand hoosingaso- alledsimplevoxel,that maybe hara terizedlo ally(see[1,2℄),andremovingit.Su hapro essusually involves many arbitrary hoi es, and the nal result may depend, sometimes heavily, on any of these hoi es. This is why parallel thinning algorithms are generallypreferredtosequentialones.However,removingasetofsimplevoxels atea hthinningstep,inparallel,mayaltertopology.Theframeworkof riti al kernels,introdu edbyoneoftheauthorsin[3℄,providesa onditionunderwhi h wehavetheguaranteethatasubsetofvoxels anberemovedwithout hanging topology.This onditionis, toourknowledge,themostgeneraloneamongthe related works.Furthermore, riti alkernelsindeed provideamethod to design newparallelthinningalgorithms,inwhi hthepropertyoftopologypreservation isbuilt-in, andin whi h anykindof onstraintmaybeimposed(see[4,5℄).

Among the dierent parallel thinning algorithms that have been proposed in the literature, we an distinguish symmetri from asymmetri algorithms. Symmetri algorithms(seee.g.[6,7,8℄)(alsoknownasfullyparallelalgorithms) produ eskeletonsthatareinvariantunder90degreesrotations.They onsistof theiterationofthinningstepsthataremadeof1)theidenti ationandsele tion

⋆⋆

position in spa e, and 2) the removal, in parallel, of all sele ted voxels from the obje t. Symmetri algorithms, on the positive side, produ e a result that is uniquely dened: no arbitrary hoi e is needed. On the negative side, they generallyprodu ethi kskeletons,seeFig.1.

(a) (b) ( ) (d)

Fig.1. Dierent types of skeletons. (a): Curvilinear skeleton, symmetri . (b): Curvilinearskeleton,asymmetri .( ):Surfa eskeleton,symmetri .(d):Surfa e skeleton,asymmetri .

Asymmetri skeletons,ontheopposite,arepreferredwhenthinnerskeletons arerequired.Thepri etopayisa ertainamountofarbitrary hoi estobemade. In all existing asymmetri parallel thinning algorithms, ea h thinning step is dividedintoa ertainnumberofsubsteps.Intheso- alleddire tionalalgorithms (seee.g.[9,10,11℄),ea hsubstepisdevotedto thedete tionandthedeletion of voxelsbelongingtooneside oftheobje t:allthevoxels onsideredduringthe substephave,forexample,theirsouthneighborinsidetheobje tandtheirnorth neighboroutside theobje t.The order in whi h thesedire tional substeps are exe utedissetbeforehand,arbitrarily.Subgrid(orsubeld)algorithms(seee.g. [12,13℄) form the se ond ategory of asymmetri parallel thinning algorithms. There, ea h substepis devoted tothedete tionand thedeletion ofvoxelsthat belongto a ertain subgrid,forexample,allvoxelsthathaveeven oordinates. Consideredsubgridsmustformapartitionofthegrid.Again,theorderinwhi h subgridsare onsideredisarbitrary.

Subgridalgorithmsare notoftenusedin pra ti ebe ausetheyprodu e ar-tifa ts,that is,wavingskeletonbran heswheretheoriginalobje tissmoothor straight. Dire tional algorithms are the most popular ones. Most of them are implementedthroughsetsofmasks,onepersubstep. Aset ofmasksisused to hara terizevoxelsthatmustbekeptduringagivensubstep,inorderto1) pre-servetopology,and2)prevent urvesorsurfa estodisappear.Thus,topologi al onditions and geometri al onditions annot be easily distinguished, and the slightestmodi ationofanymaskinvolvestheneedtomakeanewproofofthe

onsider liques.A liqueisasetofmutuallyadja entvoxels.Then,weidentify the riti alkerneloftheobje t,a ordingtosomedenitions, whi hisaunion of liques.Themaintheorem ofthe riti alkernelsframework [3,5℄ statesthat we anremoveinparallelanysubsetoftheobje t,providedthatwekeepatleast onevoxelofevery liquethat onstitutesthe riti alkernel,andthisguarantees topology preservation. Here, aswetry to obtain thin skeletons, our goal is to keep,wheneverpossible,exa tlyonevoxelineverysu h lique.Thisleadsusto proposeageneri parallelasymmetri thinnings heme,thatmaybeenri hedby addinganysortofgeometri al onstraint.Forexample,wedenethenotionsof 1D and 2D isthmuses. A 1D (resp. 2D) isthmus is a voxel that is lo ally like apie e of urve (resp.surfa e).Fromourgeneri s heme, weeasily derive,by adding the onstraint to preserve isthmuses, spe i algorithms that produ e urvilinearorsurfa eskeletons.

2 Voxel Complexes

Inthisse tion,wegivesomebasi denitionsforvoxel omplexes,seealso[14,1℄. Let

Z

bethesetofintegers.We onsiderthefamiliesofsets

F

1

0

,

F

1

1

,su hthat

F

1

0

= {{a} | a ∈ Z}

,

F

1

1

= {{a, a + 1} | a ∈ Z}

.A subset

f

of

Z

n

,

n

≥ 2

, that istheCartesianprodu tofexa tly

d

elementsof

F

1

1

and

(n − d)

elementsof

F

1

0

is alled afa e oran

d

-fa e of

Z

n

,

d

isthedimension of

f

.Intheillustrations ofthispaperex eptFig.6,a3-fa e(resp.2-fa e,1-fa e,0-fa e)isdepi tedbya ube(resp.square,segment,dot), seee.g.Fig.4.

A3-fa e of

Z

3

isalso alleda voxel. A niteset that is omposed solely of voxelsis alleda(voxel) omplex (seeFig.2).Wedenoteby

V

3

the olle tionof allvoxel omplexes.

Wesaythattwovoxels

x, y

areadja ent if

x

∩ y 6= ∅

.Wewrite

N (x)

forthe set of allvoxelsthat are adja entto avoxel

x

,

N (x)

isthe neighborhood of

x

. Notethat,forea hvoxel

x

,wehave

x

∈ N (x)

.Weset

N

∗

_{(x) = N (x) \ {x}}

. Let

d

∈ {0, 1, 2}

.We say that twovoxels

x, y

are

d

-neighbors if

x

∩ y

is a

d

-fa e. Thus, two distin t voxels

x

and

y

are adja ent if and only if they are

d

-neighborsforsome

d

∈ {0, 1, 2}

. Let

X

∈ V

3

.Wesaythat

X

is onne ted if,forany

x, y

∈ X

,there existsa sequen e

hx

0

, ..., x

k

i

ofvoxelsin

X

su hthat

x

0

= x

,

x

k

= y

,and

x

i

isadja ent to

x

i−1

,

i

= 1, ..., k

.

3 Simple Voxels

Intuitivelyavoxel

x

ofa omplex

X

is alledasimplevoxelifitsremovalfrom

X

doesnot hangethetopologyof

X

.Thisnotionmaybeformalizedwiththe help of the followingre ursivedenition introdu ed in [5℄, see also [15,16℄ for

b

a

c

d e

f

h

g

b

f

h

d

(a) (b)

Fig.2.(a) A omplex

X

whi h is made of 8 voxels, (b) A omplex

Y

⊆ X

, whi hisathinningof

X

.

Denition1. Let

X

∈ V

3

.

Wesaythat

X

isredu ible ifeither: i)

X

is omposedofasinglevoxel;or ii)thereexists

x

∈ X

su hthat

N

∗

_{(x) ∩ X}

isredu ibleand

X

\ {x}

isredu ible. Denition 2. Let

X

∈ V

3

. A voxel

x

∈ X

is simple for

X

if

N

∗

_{(x) ∩ X}

is redu ible. If

x

∈ X

is simple for

X

, we say that

X

\ {x}

is an elementary thinningof

X

.

Thus,a omplex

X

∈ V

3

isredu ibleifandonlyifitispossibletoredu e

X

toasinglevoxelbyiterativelyremovingsimplevoxels.Observethataredu ible omplexisne essarilynon-emptyand onne ted.

In Fig.2 (a), the voxel

a

is simple for

X

(

N

∗

_{(a) ∩ X}

is made of a single voxel),thevoxel

d

isnotsimplefor

X

(

N

∗

_{(d) ∩ X}

isnot onne ted),thevoxel

h

issimplefor

X

(

N

∗

_{(h) ∩ X}

ismadeoftwovoxelsthatare

2

-neighborsandis redu ible).

In[5℄,itwasshownthattheabovedenitionofasimplevoxelisequivalentto lassi al hara terizationsbasedon onne tivitypropertiesofthevoxel's neigh-borhood [17,18,19,20,2℄. An equivalen e wasalso establishedwith a denition basedon theoperationof ollapse [21℄,this operationis adis reteanalogueof a ontinuousdeformation(ahomotopy),seealso[15,3,2℄.

Thenotionofasimplevoxelallowsonetodenethinningsofa omplex,see anillustrationFig.2(b).

Let

X, Y

∈ V

3

. Wesaythat

Y

is athinning of

X

orthat

X

isredu ible to

Y

,ifthereexistsasequen e

hX

0

, ..., X

k

i

su hthat

X

0

= X

,

X

k

= Y

,and

X

i

is anelementarythinningof

X

i−1

,

i

= 1, ..., k

.

Thus,a omplex

X

isredu ibleifandonlyifitisredu ibletoasinglevoxel.

4 Criti al Kernels Let

X

bea omplexin

V

3

.It iswellknownthat, ifweremovesimultaneously (inparallel)simplevoxelsfrom

X

,wemay hangethetopologyoftheoriginal obje t

X

. Forexample,thetwovoxels

f

and

g

aresimplefortheobje t

X

de-pi tedFig.2(a).Nevertheless

X

\ {f, g}

hastwo onne ted omponentswhereas

withthewarrantythatwedonotalterthetopologyoftheseobje ts[3,4,5℄.This method isvalidfor omplexes ofarbitrarydimension.

Let

d

∈ {0, 1, 2, 3}

andlet

C

∈ V

3

.Wesaythat

C

isa

d

- lique ora lique if

∩{x ∈ C}

isa

d

-fa e.If

C

isa

d

- lique,

d

istherankof

C

.

If

C

ismadeofsolelytwodistin tvoxels

x

and

y

,wenotethat

C

isa

d

- lique ifandonlyif

x

and

y

are

d

-neighbors,with

d

∈ {0, 1, 2}

.

Let

X

∈ V

3

and let

C

⊆ X

bea lique.Wesaythat

C

isessential for

X

if wehave

C

= D

whenever

D

isa liquesu hthat:

i)

C

⊆ D ⊆ X

;and ii)

∩{x ∈ C} = ∩{x ∈ D}

.

Observe that any omplex

C

that is made of a single voxel is a lique (a 3- lique). Furthermore any voxel of a omplex

X

onstitutes a lique that is essentialfor

X

.

InFig.2(a),

{f, g}

is a2- lique that isessentialfor

X

,

{b, d}

is a0- lique that is not essential for

X

,

{b, c, d}

is a0- lique essential for

X

,

{e, f, g}

is a 1- liqueessentialfor

X

.

Denition 3. Let

S

∈ V

3

. The

K

-neighborhood of

S

, written

K(S)

, isthe set madeofallvoxelsthatareadja enttoea hvoxelin

S

.Weset

K

∗

_{(S) = K(S)\ S}

. Wenotethat wehave

K(S) = N (x)

whenever

S

ismadeofasinglevoxel

x

. Wealsoobservethatwehave

S

⊆ K(S)

whenever

S

isa lique.

Denition4. Let

X

∈ V

3

andlet

C

bea liquethatisessentialfor

X

.Wesay that the lique

C

isregular for

X

if

K

∗

_{(C) ∩ X}

isredu ible. Wesay that

C

is riti al for

X

if

C

isnotregularfor

X

.

Thus, if

C

isa liquethat ismadeofasinglevoxel

x

,then

C

isregularfor

X

ifandonlyif

x

issimplefor

X

.

InFig. 2 (a), the liques

C

1

= {b, c, d}

,

C

2

= {f, g}

, and

C

3

= {f, h}

are essentialfor

X

.Wehave

K

∗

_(C

₁

_{)∩X = ∅}

,

K

∗

_(C

₂

_{)∩X = {e, h}}

,and

K

∗

_(C

₃

_{)∩X =}

{g}

.Thus,

C

1

and

C

2

are riti alfor

X

,while

C

3

isregularfor

X

.

The following result is a onsequen e of a general theorem that holds for omplexes of arbitrary dimensions [3,5℄, see an illustration Fig. 2 (a) and (b) wherethe omplexes

X

and

Y

satisfythe onditionofTh.5.

Theorem5. Let

X

∈ V

3

andlet

Y

⊆ X

.

The omplex

Y

isathinningof

X

ifany liquethatis riti al for

X

ontainsat leastonevoxel of

Y

.

5 A generi 3D parallel and asymmetri thinning s heme Our goal is to dene a subset

Y

of a voxel omplex

X

that is guaranteed to in lude at least onevoxel of ea h lique that is riti al for

X

. ByTh. 5, this subset

Y

willbeathinningof

X

.

Letus onsiderthe omplex

X

depi tedFig.3(a).Therearepre iselythree liquesthatare riti alfor

X

:

-the0- lique

C

1

= {b, c}

(wehave

K

∗

_(C

1

) ∩ X = ∅

); -the2- lique

C

2

= {a, b}

(wehave

K

∗

_(C

2

) ∩ X = ∅

); -the3- lique

C

3

= {b}

(thevoxel

b

isnotsimple).

Suppose that,in order to build a omplex

Y

that fullls the ondition of Th. 5,wesele tarbitrarilyonevoxelofea h liquethatis riti alfor

X

.Following su hastrategy,we ouldsele t

c

for

C

1

,

a

for

C

2

,and

b

for

C

3

.Thus,wewould have

Y

= X

, no voxelwould be removed from

X

. Now, we observethat the omplex

Y

′

_{= {b}}

satisesthe onditionofTh. 5.This omplexisobtainedby onsideringrstthe3- liquesbeforesele tingavoxelin the

2

-,

1

-,or

0

liques. The omplex

X

ofFig.3(b) providesanother exampleofsu hasituation. Therearepre iselythree liquesthat are riti alfor

X

:

-the1- lique

C

1

= {e, f, g, h}

(wehave

K

∗

_(C

₁

_{) ∩ X = ∅}

); -the1- lique

C

2

= {e, d, g}

(wehave

K

∗

_(C

₂

_{) ∩ X = ∅}

); -the2- lique

C

3

= {e, g}

(

K

∗

_(C

₃

_{) ∩ X}

isnot onne ted).

If we sele t arbitrarily one voxel of ea h riti al lique, we ould obtain the omplex

Y

= {f, d, g}

. On the other hand, if we onsider the 2- liques before the1- liques,weobtaineither

Y

′

_{= {e}}

or

Y

′′

_{= {g}}

.Inboth asestheresultis betterinthesensethat weremovemorevoxelsfrom

X

.

This dis ussionmotivates the introdu tionof the following 3D asymmetri and parallel thinning s heme AsymThinningS heme(see also [4,5℄). The main featuresofthiss hemearethefollowing:

-Takingintoa ounttheobservationsmadethroughthetwopreviousexamples, riti al liquesare onsidereda ordingtotheirde reasingranks(step4).Thus, ea hiterationis madeof foursub-iterations (steps 4-8).Voxelsthat havebeen previouslysele ted arestored in aset

Y

(step 8). Atagivensub-iteration, we onsidervoxelsonlyin riti al liquesin ludedin

X

\ Y

(step6).

-

Select

isafun tionfrom

V

3

to

V

3

,thesetofallvoxels.Morepre isely,

Select

asso iates, to ea h set

S

of voxels,a unique voxel

x

of

S

. We refer to su h a fun tion as asele tion fun tion. This fun tion allowsus to sele t a voxel in a given riti al lique(step7).Apossible hoi eistotakefor

Select(S)

,therst pixelof

S

in thelexi ographi orderofthevoxels oordinates.

- In order to ompute urvilinear or surfa eskeletons, we have to keep other voxelsthanthe ones that arene essaryforthe preservation ofthe topologyof the obje t

X

. In the s heme, the set

K

orresponds to a set of features that we want to be preserved by a thinning algorithm (thus, we have

K

⊆ X

). This set

K

, alled onstraint set, is updateddynami ally at step10.

Skel

X

is afun tionfrom

X

on

{T rue, F alse}

that allowsustokeepsomeskeletal voxels of

X

,e.g.,somevoxelsbelongingtoparts of

X

thataresurfa es or urves.For example,ifwewanttoobtain urvilinearskeletons,apopularsolutionisto set

Skel

X

(x) = T rue

whenever

x

isa so- alled endvoxel of

X

: anend voxelis a voxelthathasexa tlyoneneighborinside

X

;seeFig.7(a)askeletonobtainedin thisway.However,thissolutionislimitedanddoesnotpermittoobtainsurfa e

c

b

a

e

g

f

h

d

(a) (b) Fig.3.Two omplexes.

By onstru tion, at ea h iteration, the omplex

Y

at step 9 satises the onditionof Th. 5. Thus, theresultof thes heme isa thinningofthe original omplex

X

. Observealso that,ex ept step 4,ea hstep of thes hememay be omputedin parallel.

Algorithm1:AsymThinningS heme

(X, Skel

X

)

Data:

X

∈ V

3

,

Skel

X

isafun tionfrom

X

on

{T rue, F alse}

Result:

X

K

:=

∅

; 1 repeat 2

Y

:=

K

; 3 for

d

← 3

to

0

do 4

Z

:=

∅

; 5

forea h

d

- lique

C

⊆ X \ Y

thatis riti alfor

X

do 6

Z

:=

Z

∪ {Select(C)}

; 7

Y

:=

Y

∪ Z

; 8

X

:=

Y

; 9

forea hvoxel

x

∈ X \ K

su h that

Skel

X

(x) = T rue

do

K

:=

K

∪ {x}

; 10

untilstability ; 11

Fig.4 provides an illustration of the s heme AsymThinningS heme. Let us onsider the omplex

X

depi ted in (a). We suppose in this examplethat we do notkeep any skeletal voxel,i.e., for any

x

∈ X

, we set

Skel

X

(x) = F alse

. Thetra esofthe liquesthat are riti alfor

X

arerepresentedin(b),thetra e of a lique

C

is thefa e

f

= ∩{x ∈ C}

. Thus, the set of the liques that are riti alfor

X

ispre isely omposedofsix0- liques,two1- liques,three2- liques, andone3- lique.In( )thedierentsub-iterationsofthes hemeareillustrated (steps 4-8):

- when

d

= 3

, only one lique is onsidered, the dark grey voxel is sele ted whateverthesele tionfun tion;

- when

d

= 2

,all thethree 2- liques are onsidered sin enone ofthese liques ontainsthe abovevoxel. Voxelsthat ouldbesele ted byasele tionfun tion

(d) (e) (f)

(g) (h)

Fig.4. (a): A omplex

X

made of pre isely 12 voxels. (b): The tra es of the liquesthat are riti alfor

X

. ( ): Voxelsthat havebeen sele tedbythe algo-rithm. (d): The result

Y

of the rstiteration. (e): Thetra es of the 4 liques that are riti alfor

Y

.(f):Theresultofthese ond iteration.(g)and(h): Two otherpossiblesele tionsattherstiteration.

- when

d

= 1

, only one lique is onsidered, avoxel that ould be sele ted is depi tedinlightgrey;

-when

d

= 0

,no liqueis onsideredsin eea hofthe0- liques ontainsatleast onevoxelthathasbeenpreviouslysele ted.

Afterthesesub-iterations,weobtainthe omplexdepi tedin(d).Thegures(e) and (f) illustratethe se onditeration, at theend ofthis iterationthe omplex is redu edto asinglevoxel.In(g)and (h)twoother possiblesele tionsat the rstiterationaregiven.

Of ourse,theresultofthes hememaydependonthe hoi eofthesele tion fun tion. This is the pri e to be paid if we try to obtain thin skeletons. For example,somearbitrary hoi eshavetobemadeforredu ingatwovoxelswide ribbontoasimple urve.

Inthesequelofthepaper,wetakefor

Select(S)

,the rstpixelof

S

in the lexi ographi orderofthevoxels oordinates.

Fig.5showsanotherillustration,onbiggerobje ts,of AsymThinningS heme. Here also, for any

x

∈ X

, we have

Skel

X

(x) = F alse

(noskeletalvoxel).The

6 Isthmus-based asymmetri thinning

Inthisse tion,weshowhowtouseourgeneri s hemeAsymThinningS hemein order to get apro edure that omputeseither urvilinearorsurfa eskeletons. Thisthinningpro edurepreservesa onstraintset

K

thatismadeofisthmuses. Intuitively, a voxel

x

of an obje t

X

is said to be a

1

-isthmus (resp. a

2

-isthmus)iftheneighborhoodof

x

orresponds-uptoathinning-totheoneof apointbelongingtoa urve(resp.asurfa e)[5℄.

Wesaythat

X

∈ V

3

isa

0

-surfa e if

X

ispre iselymadeoftwovoxels

x

and

y

su h that

x

∩ y = ∅

. Wesaythat

X

∈ V

3

isa

1

-surfa e (orasimple losed urve)if: i)

X

is onne ted;andii)Forea h

x

∈ X

,

N

∗

_{(x) ∩ X}

isa

0

-surfa e. Denition6. Let

X

∈ V

3

,let

x

∈ X

. Wesaythat

x

isa

1

-isthmusfor

X

if

N

∗

_{(x) ∩ X}

isredu ibletoa

0

-surfa e. Wesaythat

x

isa

2

-isthmusfor

X

if

N

∗

_{(x) ∩ X}

isredu ibletoa

1

-surfa e. Wesaythat

x

isa

2

+

-isthmusfor

X

if

x

isa

1

-isthmusora

2

-isthmusfor

X

. Our aim is to thin an obje t, while preserving a onstraint set

K

that is made of voxels that are dete ted as

k

-isthmuses during the thinning pro ess. We obtain urvilinearskeletonswith

k

= 1

,surfa e skeletons with

k

= 2

, and surfa e/ urvilinearskeletonswith

k

= 2

+

. These three kindsof skeletons may beobtainedbyusingAsymThinningS heme,withthefun tion

Skel

X

dened as follows:

Skel

X

(x) =

 T rue

if

x

isa

k

-isthmus,

F alse

otherwise, with

k

∈ {1, 2, 2

+

_}

.

x

x

x

(a)

(b)

(c)

Fig.6.Inthisgure,avoxelis representedbyits entralpoint.(a):A voxel

x

andtheset

N (x)∩X

(bla kpoints).(b):Aset

S

whi hisa

1

-surfa e,

N

∗

_(x)∩X

isredu ibleto

S

,thus

x

isa2-isthmus.for

X

.( ):Avoxel

x

andtheset

N (x)∩X

(bla kpoints).Thevoxel

x

isa1-isthmus for

X

.

Observethereisthepossibilitythatavoxelbelongstoa

k

-isthmusatagiven stepofthealgorithm,but notat furthersteps.Thisiswhypreviouslydete ted isthmusesarestored(seeline10of AsymThinningS heme).

InFig.7(b-f),weshowa urvilinearskeleton, asurfa eskeletonand a sur-fa e/ urvilinearskeletonobtainedbyourmethod fromthesameobje t.

7 Con lusion

We introdu ed an original generi s heme for asymmetri parallel topology-preserving thinning of 3D obje ts made of voxels, in the framework of riti- al kernels. We sawthat from this s heme, one an easily deriveseveral thin-ning operatorshavingspe i behaviours,simply by hangingthedenition of skeletalpoints.Inparti ular,weshowedthat ultimate, urvilinear,surfa e,and surfa e/ urvilinear skeletons an be obtained, based on the notion of 1D/2D isthmuses.

Akeypoint,intheimplementationofthealgorithmsproposedinthispaper, isthedete tionof riti al liquesandisthmusvoxels.In[5℄,weshowedthatitis possibletodete t riti al liquesthankstoasetofmasks,inlineartime.Wealso showedthatthe ongurationsof1Dand2Disthmusesmaybepre- omputedby alinear-timealgorithmandstoredinlookuptables.Finally,basedona breadth-rststrategy,thewholemethod anbeimplementedtorunin

O(n)

time,where

n

isthenumberofvoxelsoftheinput3Dimage.

In an extended paper, in preparation, we will show how to deal with the robustness to noiseissue thanks to the notion of isthmus persisten e. We will also ompare ourmethod withall existing asymmetri parallel skeletonization algorithmsa tinginthe3D ubi grid.

( ) (d)

(e) (f)

Fig.7.Asymmetri skeletonsobtainedbyusingAsymThinningS heme.(a):the fun tion

Skel

X

isbasedonend voxels.(b, ,d): thefun tion

Skel

X

isbasedon

k

-isthmuses,with

k

= 1, 2

and

2

+

1. Kong, T.Y., Rosenfeld, A.: Digital topology: introdu tion and survey. Comp. Vision,Graphi sandImagePro .48(1989)357393

2. Couprie, M., Bertrand, G.: New hara terizations of simple points in 2D, 3D and 4D dis rete spa es. IEEE Transa tions on Pattern Analysis and Ma hine Intelligen e31(4)(April2009)637648

3. Bertrand, G.: On riti al kernels. Comptes Rendusdel'A adémie des S ien es, SérieMath.I(345)(2007)363367

4. Bertrand,G.,Couprie,M.: Two-dimensionalthinningalgorithmsbasedon riti al kernels. JournalofMathemati alImagingandVision31(1) (May2008)3556 5. Bertrand,G.,Couprie,M.:PowerfulParallelandSymmetri 3DThinningS hemes

Based on Criti al Kernels. Journal of Mathemati al Imaging and Vision 48(1) (2014)134148

6. Manzanera,A.,Bernard,T.,Prêteux,F.,Longuet,B.: n-dimensional skeletoniza-tion: a unied mathemati al framework. Journal of Ele troni Imaging 11(1) (2002)2537

7. Lohou, C., Bertrand, G.: Two symmetri al thinning algorithms for 3D binary images. PatternRe ognition 40(2007)23012314

8. Palágyi,K.:A3Dfullyparallelsurfa e-thinningalgorithm. Theoreti alComputer S ien e406(1-2)(2008)119135

9. Tsao, Y., Fu, K.: A parallel thinning algorithm for 3d pi tures. CGIP 17(4) (De ember1981)315331

10. Palágyi,K.,Kuba,A.:Aparallel3d12-subiterationthinningalgorithm.Graphi al ModelsandImagePro essing61(4) (1999)199221

11. Lohou,C., Bertrand,G.: A 3d6-subiteration urvethinningalgorithmbasedon p-simplepoints. Dis reteAppliedMathemati s151(2005)198228

12. Bertrand,G.,Aktouf,Z.: Athree-dimensionalthinningalgorithmusingsubelds. In:VisionGeometryIII.Volume2356.,SPIE(1996)113124

13. Németh,G.,Kardos,P.,Palágyi,K.: Topologypreserving3Dthinningalgorithms using fourandeightsubelds. InCampilho,A.,Kamel,M.,eds.:ImageAnalysis and Re ognition. Volume 6111 of Le tureNotes inComputer S ien e. Springer Berlin/Heidelberg(2010)316325

14. Kovalevsky,V.: Finite topologyas appliedto imageanalysis. ComputerVision, Graphi sandImagePro essing46(1989)141161

15. Kong, T.Y.: Topology-preserving deletion of 1's from 2-, 3- and 4-dimensional binary images. In: Dis rete Geometry for Computer Imagery. Volume 1347 of LNCS.,Springer(1997)318

16. Bertrand,G.: Newnotionsfordis retetopology. In:Dis reteGeometryfor Com-puterImagery.Volume1568ofLNCS.,Springer(1999)218228

17. Bertrand,G.,Malandain,G.: Anew hara terizationofthree-dimensionalsimple points. PatternRe ognitionLetters15(2)(1994)169175

18. Bertrand, G.: Simplepoints,topologi al numbersandgeodesi neighborhoodsin ubi grids. PatternRe ognitionLetters15(1994)10031011

19. Saha,P.,Chaudhuri,B.,Chanda,B.,DuttaMajumder,D.: Topologypreservation in3D digitalspa e. PatternRe ognition 27(1994)295300

20. Kong, T.Y.: Ontopology preservationin 2-D and 3-D thinning. International JournalonPatternRe ognitionandArti ialIntelligen e9(1995)813844 21. Whitehead,J.:Simpli ialspa es,nu leiand

m

-groups. Pro eedingsoftheLondon