Homotopic thinning in 2D and 3D cubical complexes based on critical kernels

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Homotopic thinning in 2D and 3D cubical complexes based on critical kernels

Michel Couprie, Gilles Bertrand

To cite this version:

Michel Couprie, Gilles Bertrand. Homotopic thinning in 2D and 3D cubical complexes based on

critical kernels. 19th IAPR international conference on Discrete Geometry for Computer Imagery

(DGCI 2016), Apr 2016, Nantes, France. pp.131-142, �10.1007/978-3-319-32360-2_10�. �hal-01360276�

omplexes based on ritial kernels

Mihel CouprieandGillesBertrand

UniversitéParis-Est,LIGM,ÉquipeA3SI,ESIEEParis,Frane

e-mail:mihel.ouprieesiee.fr,gilles.bertrandesiee.fr

Abstrat. Weproposeasymmetrithinningshemeforubialorsim-

pliialomplexes of dimension2 or 3.We showhowto obtain, witha

samegenerithinningsheme, ultimate,urveorsurfaeskeletonsthat

areuniquelydened(noarbitraryhoieisdone).

Introdution

Weproposea symmetrithinningshemeforubialorsimpliial omplexesof

dimension2or 3.Ourmotivationsarelistedbelow:

- Complexes an be used for the representation of disrete geometri objets,

yieldingbetterunderstandingoftheirstruture andtopologialproperties;

- The framework of digital topology does not permit to obtain skeletons that

areprovablythin,however,suha propertyan beprovedin theframework of

omplexes;

-Toourknowledge,theredoesnotyetexistanysymmetrialthinningalgorithm

in theframework ofomplexes. Onlyasymmetrialgorithms, basedontheol-

lapse operationhavebeenproposed. However,asymmetri thinningalgorithms

an produe, forthesame objet,drastially dierentresultsdepending ofthe

orientation of theobjet in spae (see Fig. 8). On the other hand, symmetri

algorithmsguaranteea90degreerotationinvariane.

Inourprevious works onritial kernels,wehaveproposed methods where

theinputandtheoutputwerehomogenousomplexes,thatis,setsofpixelsor

sets ofvoxels(seee.g.[2,3℄).The aseofgeneralomplexes(madeof elements

ofvariousdimensions) hasneverbeenonsidered inthisframework.

Here,weshowhowtoobtain,withasamegenerithinningsheme,ultimate,

urveorsurfaeskeletonsthatareuniquelydened(noarbitraryhoieisdone).

Wealsoshowthat,ifathinskeletonisneeded,itisbettertouseoursymmetri

method rstandnishthethinningwithafewstepsofollapse.

1 Cubial Complexes

Althoughwefousonubialomplexesinthispaper,allthenotionsandmeth-

odsintroduedfromheretosetion5anbereadilytransposedtotheframework

in ordertoprovidea soundtopologialbasisforimageanalysis.

Intuitively,aubialomplexmaybethoughtofasa setofelementshaving

variousdimensions(e.g.,verties,edges,squares, ubes) gluedtogether aord-

ing to ertain rules. In this setion, we reall briey some basi denitions on

omplexes,seealso[2,6℄formoredetails.Weonsiderhere

n

-dimensionalom- plexes,with

0 6 n 6 3

^.

Let

S

^beaset.If

T

^isasubsetof

S

^,wewrite

T ⊆ S

^.Let

Z

^{denotethesetof}

integers.

We onsider the families of sets

F ¹ 0

^,

F ¹ 1

^{, suh that}

F ¹ 0 = {{a} | a ∈ Z}

^,

F ¹ 1 = {{a, a + 1} | a ∈ Z}

^{. A subset}

f

^of

Z ⁿ

^,

n > 2

^{, whih is the Cartesian}

produtofexatly

m

^elementsof

F ¹ 1

^and

(n − m)

^elementsof

F ¹ 0

^{isalleda fae}

oran

m

^{-fae of}

Z ⁿ

^,

m

^{isthedimension of}

f

^,wewrite

dim(f ) = m

^.

Observethatanynon-emptyintersetionoffaesisa fae.For example,the

intersetionoftwo

2

^-faes

A

^and

B

^{maybeeither a}

2

^-fae(if

A = B

^),a

1

^-fae,

a

0

^{-fae,or theemptyset.}

(a) (b) () (d)

Fig.1.Graphialrepresentationsof:(a)a

0

-fae,(b)a

1

-fae,()a

2

-fae,(d)a

3

-fae.

Wedenote by

F ⁿ

^{the set omposed of all}

m

^{-faes of}

Z ⁿ

^{, with}

0 6 m 6 n

^.

An

m

^{-fae of}

Z ⁿ

^{isalled apoint if}

m = 0

^{, a (unit)interval if}

m = 1

^{, a (unit)}

square if

m = 2

^{,a(unit) ube if}

m = 3

^(seeFig.1).

Let

f

^beafaein

F ⁿ

^.Weset

f ˆ = {g ∈ F ⁿ | g ⊆ f }

^and

f ˆ ^∗ = ˆ f \ {f }

^.

Any

g ∈ f ˆ

^isafaeof

f

^.

If

X

^{isanitesetoffaesin}

F ⁿ

^,wewrite

X ⁻ = ∪{ f ˆ | f ∈ X }

^,

X ⁻

^isthelosure

of

X

^.

Aset

X

^offaesin

F ⁿ

^{isa ell or an}

m

^{-ell ifthereexistsan}

m

^-fae

f ∈ X

^,

suhthat

X = ˆ f

^{.Theboundaryofaell}

f ˆ

^istheset

f ˆ ^∗

^.

Aniteset

X

^offaesin

F ⁿ

^isaomplex(in

F ⁿ

^)if

X = X ⁻

^.Anysubset

Y

^of

aomplex

X

^{whihisalsoaomplexisasubomplexof}

X

^.If

Y

^isasubomplex

of

X

^,wewrite

Y X

^.If

X

^isaomplexin

F ⁿ

^,wealsowrite

X F ⁿ

^.InFig.2,

some omplexesarerepresented.Notiethatanyellisa omplex.

Let

X ⊆ F ⁿ

^{. A fae}

f ∈ X

^{is afaet of}

X

^{ifthere is no}

g ∈ X

^{suh that}

f ∈ g ˆ ^∗

^.Wedenoteby

X ⁺

^{thesetomposed ofallfaetsof}

X

^.

If

X

^{is a omplex, observe that in general,}

X ⁺

^{is not a omplex, and that}

[X ⁺ ] ⁻ = X

^.

Inthis setionwerealla denition of theoperationof ollapse[7℄, whih isa

disreteanalogueofa ontinuousdeformation(ahomotopy).

Let

X

^beaomplexin

F ⁿ

^andlet

f ∈ X

^{.Ifthereexistsonefae}

g ∈ f ˆ ^∗

^suh

that

f

^{isthe onlyfae of}

X

^{whih stritly inludes}

g

^{, then}

g

^{is saidto befree}

for

X

^andthepair

(f, g)

^{issaidto bea freepairfor}

X

^{.Notiethat,if}

(f, g)

^is

a freepair,thenwehaveneessarily

f ∈ X ⁺

^and

dim(g) = dim(f ) − 1

^.

Let

X

^{beaomplex,andlet}

(f, g)

^{beafreepairfor}

X

^{. Theomplex}

X \{f, g}

isanelementaryollapseof

X

Let

X

^,

Y

^{betwo omplexes. We saythat}

X

^{ollapses onto}

Y

^if

Y = X

^or

if there exists a ollapse sequene from

X

^to

Y

^{, i.e., a sequene of omplexes}

hX 0 , ..., X ^ℓ i

^{suh that}

X 0 = X

^,

X ^ℓ = Y

^{, and}

X ⁱ

^{is an elementary ollapse of}

X i−1

^,

i = 1, ..., ℓ

^{. Fig.2} illustratesaollapsesequene.

(a) (b) () (d)

Fig.2.(a):aomplex

X F ³

^{.(a-d):aollapsesequenefrom}

X

^.

Remark 1. Let

V

^{be aset of 2-faes (pixels) or aset of 3-faes(voxels), and}

let

x ∈ V

^{.The element}

x

^{issimple,inthe senseofdigital topology (see[8,6℄) if}

the omplex

V ⁻

^{ollapses onto}

(V \ {x}) ⁻

^.

3 Critial kernels

Letusbrieyrealltheframeworkintroduedbyoneoftheauthors(in[1℄)for

thinning, in parallel, disrete objets with the warranty that we do not alter

thetopologyofthese objets.Wefoushere onthetwo-andthree-dimensional

ases,but infat theresultsin thissetionarevalidforomplexesofarbitrary

dimension. This framework is based solely on three notions: the notion of an

essential fae whih allows us to dene the oreof a fae, and thenotion ofa

ritialfae(seeillustrations inFig.3).

Denition 2 ([1℄). Let

X F ⁿ

^andlet

f ∈ X

^{.We say that}

f

^{isan essential}

fae for

X

^if

f

^{is preisely the} intersetion of all faets of

X

^{whih ontain}

f

^,

i.e., if

f = ∩{g ∈ X ⁺ | f ⊆ g}

^{.We denote by Ess}

(X )

^{the set omposed of all}

essentialfaesof

X

^.If

f

^{isanessentialfaefor}

X

^,wesaythat

f ˆ

^{isan essential}

ellfor

X

^.If

Y X

^andEss

(Y ) ⊆

^Ess

(X )

^{,then wewrite}

Y E X

^.

Fig.3. (a): a omplex

X F ²

, the essential faes are shownin gray. (b,,d,e): an essentialfae (ingray)anditsore(inblak).Thefaesin(b,e)areregular,thosein

(,d)areritial.

Observethat a faet of

X

^{is neessarilyanessentialfae for}

X

^{, i.e.,}

X ⁺ ⊆

Ess

(X)

^.

Denition 3([1℄). Let

X F ⁿ

andlet

f ∈

^Ess

(X)

^{.The oreof}

f ˆ

^for

X

^isthe

omplex Core

( ˆ f , X) = ∪{ˆ g | g ∈

^Ess

(X ) ∩ f ˆ ^∗ }

^.

Denition4([1℄). Let

X F ⁿ

andlet

f ∈ X

^.Wesaythat

f

^and

f ˆ

^areregular

for

X

^if

f ∈

^Ess

(X)

^andif

f ˆ

^{ollapses ontoCore}

( ˆ f , X)

^{.We say that}

f

^and

f ˆ

are ritialfor

X

^if

f ∈

^Ess

(X)

^andif

f

^{isnot regularfor}

X

^.

If

X F ⁿ

,wesetCriti

(X ) = ∪{ f ˆ | f

^{isritial for}

X }

^{,wesaythatCriti}

(X)

isthe ritialkernelof

X

^.

If

f

^{isapixel(resp.avoxel),thensayingthat}

f

^{isregularisequivalenttosay}

that

f

^{issimpleinthelassialsense(seeRem.(1)and[6℄).Thus,thenotionof}

regularfae generalizesthe oneof simplepixel (resp.voxel) to arbitraryfaets

andeventofaes thatarenotfaets.

The following theorem is the most fundamental result onerning ritial

kernels.Weuseit here in dimension2 or 3,but notie that thetheorem holds

whateverthedimension.

Theorem 5([1℄). Let

n ∈ N

,let

X F ⁿ

.

i)The omplex

X

^{ollapsesontoitsritial kernel.}

ii) If

Y E X

^{ontainsthe ritialkernel of}

X

^,then

X

^{ollapses onto}

Y

^.

iii)If

Y E X

^{ontainstheritialkernelof}

X

^,thenany

Z

^suhthat

Y Z E X

ollapses onto

Y

^.

Let

n

^{beapositiveinteger,let}

X F ⁿ

.WedeneCriti

n (X )

^{as follows:}

Criti

0 (X ) = X

^{, and Criti}

ⁿ (X ) =

^Criti

(

^Criti

ⁿ⁻¹ (X))

^{, whenever}

n > 0

^{. If}

Criti

n (X) =

^Criti

ⁿ⁺¹ (X )

^{,thenwesaythatCriti}

ⁿ (X )

^{istheultimateskeleton}

of

X

^{andwewrite Criti}

ⁿ (X ) =

^Criti

^∞ (X)

^.

FromTh. 5, wededue immedialtely that forany

X F ⁿ

, theomplex

X

ollapsesontoCriti

∞ (X)

^{.SeeFig.4 foran}illustration.

Inthissetion,weintrodueournewgeneriparallelthinningsheme,seealgo-

rithm1.Itisgeneriinthesensethatanynotionofskeletalelement(introdued

below)maybeused,forobtaining,e.g.,ultimate,urve,orsurfaeskeletons.

Inorderto omputeurveor surfaeskeletons,wehaveto keepother faes

than the ones that are neessary for the preservation of the topology of the

objet

X

^{.Inthesheme,theset}

K

^{orrespondstoasetoffeaturesthatwewant}

to be preserved by a thinning algorithm (thus, we have

K ⊆ X

^{). This set}

K

^,

alled onstraintset,isupdateddynamiallyat line3of thealgorithm. To this

aim, we will dene a funtion Skel

X

^from

X ⁺

^onto

{

^True

,

^False

}

^{, that allows}

ustodetetsomeskeletal faets of

X

^{,e.g.,some faetsbelongingtopartsof}

X

thataresurfaesorurves.Thesedetetedfaetsareprogressivelystoredin

K

^.

Algorithm 1:SymThinningSheme

(X,

^Skel

X )

Data:

X F ⁿ

^,Skel

^X

^{isafuntionfrom}

X ⁺

^on

{

^True

,

^False

}

Result:

X K

^:=

∅

^;

1

repeat 2

K := K ∪ {x ∈ X ⁺

^suhthatSkel

^X (x) =

^True

}

^;

3

X

^:=Criti

(X ) ∪ K ⁻

^;

4

untilstability ; 5

Notie that, before line 4, the omplex

Y =

^Criti

(X) ∪ K ⁻

^{is suh that}

Y E X

^andCriti

(X ) ⊆ Y

^{.Thus,byTh.5(ii),theoriginalomplex}

X

^ollapses

onto theresultof SymThinningSheme, forany

X

^{andanyfuntionSkel}

^X

^.

SeeFig.4 foranillustrationof SymThinningSheme,usingafuntionSkel

X

that yields False for any faet. The result of this operation is, obviously, the

ultimateskeletonoftheinputomplex

X

^.

(a) (b) () (d)

Fig.4. (a): a omplex

X F ³

^{. (b): after one exeution of the main loop of}

SymThinningSheme: Criti

1 (X) =

^Criti

(X )

^{. (): after two exeutions of the main}

loop:Criti

2 (X)

^{.(d):thenalresult:Criti}

³ (X ) =

^Criti

^∞ (X )

^.

gatedoratparts,weusetwokindsofskeletalfaetsalledisthmuses.

Intuitively, a faet

f

^{of a omplex}

X

^{is said to be a}

1

^{-isthmus (resp. a}

2

^-

isthmus) iftheoreof

f ˆ

^for

X

^{orresponds totheone of anelementbelonging}

toa urve(resp.a surfae)[3℄.

Let

X ⊆ F ⁿ

^{bea non-emptyset offaes.Asequene}

(f ⁱ ) ^ℓ i=0

^offaesof

X

^is

a path in

X

^(from

f 0

^to

f ^ℓ

^)if

f ⁱ ∩ f i+1 6= ∅

^,forall

i ∈ [0, ℓ − 1]

^.Wesaythat

X

isonneted if,foranytwofaes

f, g

ⁱⁿ

X

^{,thereisapathfrom}

f

^to

g

ⁱⁿ

X

^.

Wesaythat

X F ⁿ

^{is a}

0

^{-surfae if}

X ⁺

^{is preisely madeof two faets}

f

and

g

^of

X

^suhthat

f ∩ g = ∅

^.

Wesaythat

X F ⁿ

^isa

1

^{-surfae (ora simplelosedurve)if:}

i)

X ⁺

^{isonneted;and}

ii)Foreah

f ∈ X ⁺

^,Core

( ˆ f , X)

^isa

0

^-surfae.

Wesaythat

X F ⁿ

^{isansimple open urve if:}

i)

X ⁺

^{isonneted;and}

ii)Foreah

f ∈ X ⁺

^,Core

( ˆ f , X)

^isa

0

^{-surfaeor asingleell.}

Denition 6. Let

X F ⁿ

^,let

f ∈ X ⁺

^.

Wesaythat

f

^isa

1

^-isthmusfor

X

^{if Core}

( ˆ f , X)

^{is a}

0

^-surfae.

Wesaythat

f

^isa

2

^-isthmusfor

X

^{if Core}

( ˆ f , X)

^{is a}

1

^-surfae.

Wesaythat

f

^isa

2 ⁺

^-isthmusfor

X

^if

f

^isa

1

^{-isthmus ora}

2

^{-isthmus for}

X

^.

Ouraimistothinanobjet,whilepreservingaonstraintset

K

^thatismade

offaesthataredetetedas

k

^{-isthmusesduringthethinningproess.Weobtain}

urveskeletonswith

k = 1

^{,andsurfaeskeletonswith}

k = 2 ⁺

^{.Thesetwokinds}

of skeletons maybe obtained byusing SymThinningSheme, with thefuntion

Skel

X

^{dened asfollows:}

Skel

X (x) =

True if

x

^isa

k

^-isthmusfor

X

^,

False otherwise,

with

k

^beingsetto

1

^or

2 ⁺

^.

Observethata faetmaybea

k

^{-isthmusata givenstepofalgorithm1,but}

notatfurther steps.Thisiswhypreviouslydetetedisthmusesarestoredin

K

^.

Fig.5illustratesurveandsurfaeskeletons.Weobservethattheseskeletons

ontainfaes ofall dimensions:3,2,1,0.This istheounterpartofthehoie

ofhavingasymmetriproess,henea90degreesrotationinvarianeproperty,

asillustratedinFig.6.Wedealwiththethinnessissueinthenextsetion.

Observealsothat,inFig.6,theobtainedskeletonsaresimpleopenurves,as

dened above.Moregenerally,despitethefatthat theyareomposedof faes

ofvariousdimensions,partsofproduedskeletonsanbediretlyinterpretedas

pieesofurvesorsurfaes.

5 Asymmetri thinning sheme

Thinnerskeletonsmaybeobtainedifwegiveupthesymmetry.Tothisaim,the

Fig.5.(a): aomplex

X F ³

^{.(b):urveskeletonof}

X

^{.():surfaeskeletonof}

X

^.

Fig.6.Illustrationof90degreesrotationinvarianewiththesymmetrithinning(al-

gorithmSymThinningSheme).

orrespondstoaspeialaseofamethodintroduedbyLiuetal.in[10℄(seealso

[4℄)for produing families of lteredskeletons. Here, weareinterested in non-

ltered skeletons obtained through parameter-free thinning methods. Besides,

thelteringapproahof[10℄an easilybeadaptedto ourmethod.

Ingeneral,removingfreepairsfroma omplexin paralleldoesnotpreserve

topology.Butunder ertainonditions parallelollapseoffreepairsisfeasible.

First,weneedto dene thediretion ofa freefae. Let

X

^{bea omplexin}

F ⁿ

^{, let}

(f, g)

^{be a free pairfor}

X

^{. Sine}

(f, g)

^{is free, weknowthat}

dim(g) = dim(f )− 1

^{,anditanbeeasilyseenthat}

f = g ∪g ^′

^where

g ^′

^{isthetranslateof}

g

byoneofthe

2n

^vetorsof

Z ⁿ

^{withalloordinatesequalto}

0

^{exeptone,whih}

is either

+1

^or

−1

^.Let

v

^{denotethis vetor,and}

c

^{itsnon-nulloordinate.We}

dene Dir

(f, g)

^{as theindex of}

c

ⁱⁿ

v

^{, itis thediretion ofthefree pair}

(f, g)

^.

Itsorientation isdenedas Orient

(f, g) = 1

^if

c = +1

^{,and asOrient}

(f, g) = 0

Considering two distint free pairs

(f, g)

^and

(i, j)

^{for a omplex}

X

ⁱⁿ

F ⁿ

suhthatDir

(f, g) =

^Dir

(i, j)

^andOrient

(f, g) =

^Orient

(i, j)

^,wehave

f 6= i

^.It

aneasilybeseenthat

(f, g)

^isfreefor

X \ {i, j}

^,and

(i, j)

^isfreefor

X \ {f, g}

^.

Looselyspeaking,

(f, g)

^and

(i, j)

^{mayollapseinanyorderorinparallel.More}

generally, wehavethefollowingproperty.

Proposition 7 ([5℄). Let

X

^{be a omplex in}

F ⁿ

^{, andlet}

(f 1 , g 1 ), . . . , (f ^m , g m )

be

m

^{distint freepairsfor}

X

^{havingallthesamediretion andthe sameorien-}

tation.The omplex

X

^ollapsesonto

X \ {f 1 , g 1 , . . . , f m , g m }

^.

Now,wearereadytointroduealgorithm2.

Algorithm 2:ParDirCollapse

(X,

^Skel

^X )

Data:

X F ⁿ

^,Skel

^X

^{isafuntionfrom}

X ⁺

^on

{

^True

,

^False

}

Result:

X

K

^:=

∅

^;

L = {{f, g} | (f, g)

^isfreefor

X}

^;

1

while

L 6= ∅

^do

2

K := K ∪ {x ∈ X ⁺

^suhthatSkel

^X (x) =

^True

}

^;

3

fordir

= 1 → n

^do

4

fororient

= 0 → 1

^do

5

for

d = n → 1

^do

6

T = ∪{{f, g} ∈ L | (f, g)

^isfreefor

X

^and

f / ∈ K

^,

7

Dir

(f, g) =

^dir,Orient

(f, g) =

^orient,

dim(f) = d}

^;

8

X = X \ T

^;

9

Notiethat oppositeorientations(e.g.,northandsouth)aretreatedonse-

utivelyin a same diretionalsubstep. To obtain urve or surfae skeletons, we

setthefuntionSkel

X

^asfollows:

Skel

X (x) =

True if

dim(x) = 1

^,

False otherwise.

forurveskeletons,and

Skel

X (x) =

True if

dim(x) ∈ {1, 2}

^,

False otherwise.

forsurfaeskeletons.

Fig. 7 shows results of algorithm ParDirCollapse. Notie that the urve

skeletonisonly omposedof 1- and0-faes,and that thesurfaeskeletondoes

notontain any 3-fae.Indeed, thefollowing property guarantees that a urve

skeletonin2D(resp.asurfaeskeletonin3D)doesnotontainany2-fae(resp.

3-fae).

Proposition8 ([5℄). Let

X

^{beanite omplex in}

F ⁿ

^{, with}

n > 0

^{,that has at}

leastone

n

^{-fae. Then}

X

^{hasatleastone free}

(n − 1)

^-fae.

Fig.7.(a): aomplex

X F ³

^{.(b):aurveskeleton by ollapseof}

X

^.():asurfae

skeletonbyollapseof

X

^.

Theprieto payfor gettingthisthinness property isthe loss of90degrees

rotationinvariane.TheexampleofFig.8showsthatdierenesofarbitrarysize

maybeobserved betweenskeletonsof a same shape,dependingon itsposition

in spae. Onthe left, wesee that two parallelskeletonbranhes orrespond to

a single branh of the right image. The length of this split branh may be

arbitrarilybig,dependingonthesize ofthewhole objet.

Fig.8. Illustration of asymmetrithinning (algorithm ParDirCollapse). Theboxed

areaisdetailedinFig.9.

Fig. 9 details the diretional substeps of algorithm ParDirCollapse and

showshowthisalgorithmmaygivebirth todierentskeletonongurationsfor

(d) (e) (f)

Fig.9.Detailofthethinningbyollapse(algorithmParDirCollapse)oftheomplexes

ofFig.8.(a,d):rststep.(b,e):seondstep.(,f):thirdstep.Blakfaesaretheones

thatremainattheendofthestep.Theorderinwhihthefaesofdierentdiretions

and orientations are proessed is the same in all ases: 1. horizontal, left to right

(white);2.horizontal,righttoleft(lightgray);3.vertial,downwards(mediumgray);

4.vertial,upwards(darkgray).Anarrowindiatestheonly1-faethatisaddedtothe

onstraintset

K

^{atthebeginningoftheseonditeration.Atthebeginningofthethird}

step,allthe1-faesinblakarein

K

^{.Weobservethebirthoftwoparallelbranhesin}

(),andthemergingoftwobranhesin(f).

6 Experiments, disussion and onlusion

Skeletonsarenotoriouslysensitivetonoise,andthisismajorproblemformany

appliations.Evenintheontinuousase,theslightestperturbationofasmooth

ontourshapemayprovoketheappearaneofanarbitrarilylongskeletonbranh,

thatwewillrefertoasaspuriousbranh.Adesirablepropertyofdisreteskele-

tonization methods is to generate as few spurious branhes as possible, in re-

sponse totheso-alled disretization(or voxelization) noise that is inherent to

anydisretization proess.

Itwouldmakelittlesensetodiretlyompareresultsof SymThinningSheme

withthoseof ParDirCollapse,as thegoalsofthesetwo methodsaredierent.

On the other hand, we may ompare the results of i) ParDirCollapse with

thoseof ii)SymThinningSheme followed by ParDirCollapse,as botharethin

skeletons.

Firstofall,letustakea lookat Fig.10,where thelattermethod isapplied

tothesameobjetsasinFig.6andFig.8.Weseethatthesplitbranhartifat

ofFig.8isavoided.

Wewill omparethe two methods with respet to their ability to produe

skeletonsthat are freeof spuriousbranhes.Inthe following, weompare how

dierentmethodsbehavewithrespetto thisproperty.

Inordertogetgroundtruthskeletons,wedisretizedsixsimple3Dshapesfor

whihtheskeletonsareknown:abentylinderformingaknot(

X 1

^),aEulidean

byafewasymmetrithinningsteps(algorithmParDirCollapse).

ball (

X 2

^{), a thikened straight segment(}

X 3

^{), a torus (}

X 4

^{), a thikened spiral}

(

X 5

^{,seeFig.11),anellipsoid(}

X 6

^{).Forexample,aurveskeletonofadisretized}

torus should ideallybea simplelosed disrete urve (a 1-surfae).Any extra

branh of the skeleton must undoubtedly be onsidered as spurious. Thus, a

simpleandeetiveriterionforassessingthequalityofaskeletonizationmethod

istoountthenumberofextrabranhes,orequivalentlyinourase,thenumber

of extraurve extremities(free faes). Notiethat,evenif theoriginal objets

are omplexes obtained by taking the losure of sets of voxels (3-faes), the

intermediateandnalresultsareindeedgeneralomplexes,whih mayontain

2-faets and1-faets.

Inorderto omparemethods, weusetheindiator

S(X) = |c(X) − c ⁱ (X )|

^,

where

c(X )

^{stands forthe number of urve}extremities for theresultobtained from

X

^{afterthinning,and}

c ⁱ (X )

^{standsfortheidealnumberofurve}extremities toexpetwiththeobjet

X

^{.Inotherwords,}

S(X)

^{ountsthenumberofspurious}

branhesin theskeletonofobjet

X

^{, aresultof}

0

^{beingthebestone.}

Table 1.

Objet

X 1 X 2 X 3 X 4 X 5 X 6

S(ParDirCollapse(SymThinningSheme(

X i

^{))) 4 0 0 0 0 0}

S(ParDirCollapse(

X ⁱ

^{)) 16 0 0 0 8 1}

Additionally, we performed disrete rotations of the objet

X 4

^{(torus), by}

angles ranging from 1 to 89 degrees by steps of 1 degree, and omputed the

values of

S(X)

^{for all these rotated objets and for both methods. The mean}

Fig.11. Results for objet

X 5

^{. Left:} ParDirCollapse(

X 5

^{). Center:}

SymThinningSheme(

X 5

^).Right:ParDirCollapse(SymThinningSheme(

X 5

^)).

valueof

S(X)

^was131.0forParDirCollapseand69.2forSymThinningSheme followedbyParDirCollapse,whihalwaysgavethebest result.

Toonlude,oursymmetriparallelthinningshemeistherstonethatper-

mitstothingeneral2Dor3Domplexesinasymmetrialmanner,avoidingany

arbitraryhoie.Wealsoshowedexperimentallythatif,however,thinskeletons

arerequired,thenitisbetterto useoursymmetrithinningshemerst.

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