Powerful Parallel Symmetric 3D Thinning Schemes Based on Critical Kernels

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Powerful Parallel Symmetric 3D Thinning Schemes Based on Critical Kernels

Gilles Bertrand, Michel Couprie

To cite this version:

Gilles Bertrand, Michel Couprie. Powerful Parallel Symmetric 3D Thinning Schemes Based on Critical

Kernels. 2012. �hal-00731083�

(2)

Critial Kernels

∗

GillesBertrand and MihelCouprie

UniversitéParis-Est, Laboratoired'InformatiqueGaspard-Monge, ESIEE Paris

Cité Desartes, BP 99,93162 Noisy-le-GrandCedex Frane

g.bertrandesiee.fr,m.ouprieesiee.fr

September 12, 2012

Abstrat

Themain ontributionof thepresentartileonsists of

new3Dparallelandsymmetrithinningshemeswhih

havethefollowingqualities:

- They are eetive and sound, in the sense that they

are guaranteedto preservetopology. Thisguarantee is

obtainedthankstoatheoremonritialkernels;

- They are powerful, in the sense that they remove

morepoints,inoneiteration,thananyothersymmetri

parallelthinningsheme;

- They are versatile, as onditionsfor thepreservation

of geometrial features (e.g., urve extremities or

surfae borders) are independent of those aounting

fortopologypreservation;

- They are eient: we provide in this artile a small

set ofmasks,atingin thegrid

Z ³

,that issuient,in addition to the lassial simple point test, to straight-

forwardlyimplementthem.

Keywords: Thinningalgorithm,skeleton,parallelal-

gorithm, ritial kernel, ubial omplex, simple point,

ollapse.

1 Introdution

Computingtheskeletonofa3Dshapeisafundamental

stepin severalappliationsdealingwithshapeanalysis,

shapereognition,registration,visualization,animation,

et. A fundamental property of skeletons is topology

preservation: askeletonmusthavethesametopologial

harateristisastheoriginalshape.

∗

Thiswork has been partially supported bythe ANR-2010-

BLAN-0205KIDICOprojet.

Indisrete grids (

Z ²

,

Z ³

,

Z ⁴

), a topology-preserving transformation an be dened thanks to the notionof

simple point [25℄: intuitively, a point of an objet (a

subset of

Z ^d

) is alled simpleif it anbe deleted from this objetwithout alteringtopology. Let us illustrate

this notion by Fig. 1, whih displays a samesubset of

Z ²

under twousual representations: asa set of points (left),andasasetofpixels(right). Inthisexample,the

pixels(or points)

a, b, c

^are ^simple^but

x, y, z, t

^are ^not.

Thisnotion,pioneeredbyDuda,Hart,Munson[18℄,Go-

lay [20℄ and Rosenfeld [43℄, hassine been the subjet

of an abundant literature. In partiular, loal hara-

terizationsofsimplepointshavebeenproposed(seee.g.

[13, 17℄), on whih eientimplementationof thinning

proeduresarebased.

y a b

c z

t x

Figure1: Illustrationof2Dsimplepoints/pixels. Theset

X

^is^made^of^the^pointsrepresentedasblakdissonthe left,and bygraypixelsonthe right. Thepoints/pixels

a, b, c

^are^simple^while

x, y, z, t

^are^not: ^deleting

x

^would

reate a hole in

X

^, ^deleting

y

^would ^suppress ^a ^hole,

deleting

z

^would^splitâônnetedômponent,ând^delet-

ing

t

^would ^suppressâônnetedômponent.

Themostnatural wayto thin anobjetonsistsof

removingsomeofitsborderpointsinparallel. Byparal-

lel,wemeanthat thesameoperationisexeutedsimul-

(3)

Figure2: Dierentkindsofskeletons: (a)surfaeskeleton,(b)urvilinearskeleton,()minimalskeleton.

taneously and independentlyfor eah image point. By

repeating suh a proedure until stability, onean ob-

tainawell-enteredskeletonoftheoriginalobjet(see

Fig.2). Furthermore,parallelthinningalgorithms tend

toprodueskeletonswhiharemorerobusttosmallvari-

ationsofshapeontours,in omparisonwithsequential

algorithmswhihmustmakearbitraryhoiesregarding

theorderoftheproessingofpoints.

However,paralleldeletionofsimplepointsdoesnot,in

general,guaranteetopologypreservation: seeforexam-

pleFig.1wherethepoints

a

^and

b

^are^both^simple,^and

removingthese twopoints simultaneouslywould merge

two bakground omponents. In fat, suh a guaran-

tee is not obvious to obtain, evenfor the 2D ase (see

[16℄,wherefteenpublishedparallelthinningalgorithms

areanalyzed,andounter-examplesareshownforveof

them).

For the 2D ase, A. Rosenfeld introdued in [44℄ a

methodthatonsistsofdividingeahthinningstepinto

four substeps. Eahof these substepsonsiders asan-

didate for deletion, only the simple points that have

no neighborbelonging to the objet in one of the four

main diretions (north, south, east, west) and haveat

leasttwo8-neighborsbelongingtotheobjet. However,

thisso-alleddiretionalstrategyannotbestraightfor-

wardly extended to 3D. In this ase, the six main di-

retions arenorth, south, east, west, upand down. In

Fig. 3, the voxels

x, y

^are ^simple ^voxels ^that ^have ^no

neighbor belonging to the objet in the diretion up,

butifweremovethemin parallel,theobjetsplits.

Someauthors(see e.g. [9,33, 39,38℄)haveproposed

thinningalgorithmsbasedontheso-alledsubeldstrat-

egy, ageneralstrategywhih permits theparallel dele-

tionofertain simplepoints. It onsistsofonsidering,

ineahsubstep,onlysimplepointsthatbelongtoagiven

subgrid(alsoalled subeld). Forexamplein 2D(resp.

3D),four(resp. eight)disjointsubeldsmaybedened

by saying that two points belong to the same subeld

if the parity of eah of their oordinates is the same.

Variants with four or even two subelds, in 3D, have

also beenproposed; but additionalonditions must be

hekedto ensuretopologypreservation.

Thediretionalandthesubeldstrategyshareaom-

mondrawbak: dependingontheorderoftheonsidered

diretions or subelds, one an obtain dierent skele-

tons. Analternativetothesestrategiesonsistsofdelet-

ing points in a symmetri manner. By symmetri, we

mean that this operation is invariant by any isometry

(anisometry, in

Z ^d

, isabijetionwhihpreservesadja- eny relations). For topology preservation, additional

onditionsmustbeveriedwhendeleting simplepoints

in this way. Suh onditionsare diultto design: in-

deed, veryfew symmetri3D thinningalgorithms have

beenpublished[31,32,37,29,40℄,andamongthese,[31℄

and[32℄donotpreservetopology(see[27,28℄).

Reently, one of the authors introdued a general

framework, alled ritial kernels [8℄, that permits to

x y

Figure 3: All voxelsare simple, thevoxels

x

^and

y

^are

bothup voxels.

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hekthe topologial soundness of parallel thinningal-

gorithmsin anydimension,andalsotodesignnewones

thatpreservetopologybyonstrution.

As proven in [12℄, ritial kernels onstitute a non-

trivial generalization of all previously proposed frame-

works with similar aims, namely minimal non-simple

sets[42℄andP-simplepoints[6℄. Thankstoritialker-

nels, wewere abletoproposein [11℄nine new2Dthin-

ning algorithms whih respond to spei needs (sym-

metry, entering, thinness, geometrial riterions, et.)

andwhihhadnoequivalentamongpreviouslypublished

works. Thelearseparationoftopologialandgeometri-

alonstraints,whihisakeyfeatureofthisframework,

makeseasythedesignofsuhalgorithms.

Themainontributionofthepresentartileonsistsof

new3Dparallelandsymmetrithinningshemeswhih

havethefollowingqualities:

-Theyare eetiveand sound,asthemain theorem of

ritialkernelsandadditional properties proven in this

artileprovidetheguaranteeoftopologypreservation;

-Theyarepowerful,inthesensethattheyremovemore

points,in oneiteration, thananyother symmetripar-

allelthinningsheme. Inpartiular,theyanbeusedto

omputeminimalskeletons;

- They are versatile, as onditionsfor thepreservation

ofgeometrialfeatures(e.g.,urveextremitiesorsurfae

borders)areindependentofthose aountingfortopol-

ogy preservation. Wegive in this artileexamples and

illustrations of minimal, urvilinear and surfae skele-

tonsproduedusingtheseshemes;

- They are eient: we provide in this artile a small

set ofmasks,atingin thegrid

Z ³

,that issuient,in additiontothelassialsimplepointtest,tostraightfor-

wardlyimplementthem.

Alltheproofsofpropertiesstatedbelowareintheap-

pendix. Somepreliminaryresultsoftheworkpresented

inthis paperappearin[10℄.

2 Cubial and Xel Complexes

Inthis setion, wegivesome basidenitions forubi-

al omplexes, seealso [26, 3, 2℄. Weonsider herethe

three-dimensional ase. Note that most of the notions

introduedin therst setionsmakesense in arbitrary

n

-dimensionalubialspaes.

Let

Z

bethesetofintegers. Weonsider thefamilies 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 ³

whih is the Cartesianprodutofexatly

d

^elements^of

F ¹ ₁

and

(n−d)

elementsof

F ¹ ₀

isalledafae ora

d

^-fae^of

Z ³

,

d

^is^the

dimension of

f

^,^we^write

dim(f ) = d

^.

We denote by

F ³

the set omposed of all

d

^-faes ^of

Z ³

, with

d ∈ {0, 1, 2, 3}

^. ^A

d

^-fae ^of

Z ³

isalled apoint if

d = 0

^, ^a ^(unit) ^segment ^if

d = 1

^, ^a ^(unit) ^square ^if

d = 2

^,â^(unit) ûbe îf

d = 3

^.

If

X

îsâ^nite^setôf^faesⁱⁿ

F ³

,wewrite

X ⁻ = {y ∈ F ³ | y ⊆ x

^for^some

x ∈ X }

^,

X ⁻

^is^the^losure ^of

X

^. ^A

niteset

X

^of^faesⁱⁿ

F ³

isaubial omplex(in

F ³

)if

X = X ⁻

^. ^We ^denote^by

C ³

the olletionomposedof allsuhomplexes.

Let

X

^be ^a^nite ^set ^of ^faes ⁱⁿ

F ³

. Wesay that

X

is a xel omplex (in

F ³

) if, for any

x, y ∈ X

^, ^we ^have

y = x

^whenever

y ⊆ x

^. ^We^denote^by

X ³

theolletion

omposedofallsuhomplexes. Observethat,if

X ∈ X ³

and

Y ⊆ X

^,^then^we^have^neessarily

Y ∈ X ³

.

If

X

îsâ^nite^setôf^faesⁱⁿ

F ³

,wedenoteby

X ⁺

^the

setoffaes in

X

^whih^are^maximal ^for^inlusionⁱⁿ

X

^.

Thus, if

X ∈ C ³

,wehave

X ⁺ ∈ X ³

and

(X ⁺ ) ⁻ = X

^. ^If

X ∈ X ³

,wehave

X ⁻ ∈ C ³

and

(X ⁻ ) ⁺ = X

^.

Therefore,itisequivalent,withtheaboveorrespon-

denes, to speify aubial omplex ora xel omplex.

SeeanillustrationFig. 4.

x y

z t

(a) (b) ()

(d) (e)

Figure4: (a):Fourpoints

x, y, z, t

^. ^(b):^A^graphial^rep-

resentation of the set of faes

{{x, y, z, t}, {x, y}, {z}}

^.

(): A set of faes

X

^, ^whih îs ^neither â ûbial ôm-

plexnoraxelomplex. (d):Theset

X ⁺

^, ^whih^is^a^xel

omplexomposedof4segments,1square, and1ube.

(e):Theset

X ⁻

^,^whih îsâûbialômplex.

3 Simple Faes

Intuitively afae

x

ôf â ^xel ômplex

X

îs ^simple îf îts

removal from

X

^does ^not ^hange ^the ^topology ^of

X

^.

Inthissetion, weproposeadenition ofasimplefae

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(a) (b) () (d)

Figure 5: The ubialomplexof Fig. 4(e) andthree

stepsof elementaryollapses.

based on the operation of ollapse [47, 19℄. This op-

eration, whih is a disrete analogue of a ontinuous

deformation (a homotopy), is dened hereafter for an

arbitraryubialomplex

X ∈ C ³

.

Let

X ∈ C ³

and

x, y ∈ X

^suh ^that

x ⊂ y

^. ^If

y

^is

the onlyfae of

X

^distint ^from

x

^that ^ontains

x

^, ^we

saythat

(x, y)

^is^a^free ^pair ^for

X

^,^and^that^the^ubial

omplex

X \ {x, y}

îsânêlementary ôllapseôf

X

^.

Let

X, Y ∈ C ³

. We say that

X

^ollapses ^onto

Y

^if

there exists asequene

hX 0 , ..., X ^k i

^suh ^that

X 0 = X

^,

X ^k = Y

^,^and

X ⁱ

îsânêlementaryôllapseôf

X ⁱ ₋ 1

^,

i = 1, ..., k

^. ^SeeillustrationFig. 5.

Now,wegivethedenitionof asimplefae in anar-

bitraryxelomplex

X ∈ X ³

,see[8℄. Itmaybeseenasa disreteanalogueof theonegivenbyT.Y.Kongin [23℄

whihlieson ontinuous deformationsinthe Eulidean

spae. SeetheillustrationgivenFig. 6.

Denition1. Let

X ∈ X ³

andlet

x ∈ X

^. ^We^say^that

x

^is^simple^for

X

^if

X ⁻

^ollapses^onto

(X \ {x}) ⁻

^. ^If

x

is simple for

X

^, ^we ^say ^that

X \ {x}

îs ânêlementary

thinningof

X

^.

Let

X, Y ∈ X ³

. Wesaythat

Y

îs â^thinning ôf

X

^if

there exists asequene

hX 0 , ..., X ^k i

^suh ^that

X 0 = X

^,

X ^k = Y

^, ^and

X ⁱ

îs ân êlementary ^thinning ôf

X ⁱ ₋ 1

^,

i = 1, ..., k

^.

Observethat, if

Y

îs â^thinning ôf

X

^, ^then

X ⁻

^ol-

lapsesonto

Y ⁻

^.

4 Critial Kernels

Let

X

^be^a^xel^omplexⁱⁿ

F ³

. Asseenintheintrodu- tion, if we remove simultaneously (in parallel) simple

faesfrom

X

^,^we^may^hange^the^topology^of^the^orig-

inal objet

X

^. ^More ^preisely, ^we ^may ^obtain ^a^set

Y

suhthat

X ⁻

^does^not^ollapse^onto

Y ⁻

^.

Thus,itisnotpossibletousediretlythenotionofsim-

ple fae for thinning disrete objets in asymmetrial

manner.

In this setion, we reall a framework for thinning

disreteobjetsinparallelwiththewarrantythatwedo

x

y z

X Y Z T

X ⁻ Y ⁻ Z ⁻ T ⁻

Figure 6: Four xel omplexes

X

^,

Y = X \ {x}

^,

Z = Y \ {y}

^,

T = Z \ {z}

⁽

X

îs ^the^xel ômplex ôf ^Fig. ⁴

(d)). Theubial omplexes

X ⁻

^,

Y ⁻

^,

Z ⁻

^,

T ⁻

^are^also

given. Thefae

x

^is^simple^for

X

^,

y

^is^simple^for

Y

^,^but

z

^is^not^simple^for

Z

^,^for

Z ⁻

^does^not^ollapse^on

T ⁻

^.

(a) (b)

Figure7: (a) Axelomplex

X

^whih ^is^made^of ³^seg-

ments, 3squares, and 4 ubes, (b) the faes whih are

essentialfor

X

ând ^whih âre ^not^faes ôf

X

^are ^high-

lightedindark.

notalterthetopologyoftheseobjets[8℄. Thismethod

holds for omplexes of arbitrary dimension. As far as

weknow,thisistherstgeneralmethodwhihpermits

tothin arbitraryomplexesin asymmetriway.

Let

C ∈ X ³

. Wesaythat

C

^is^a

d

^-lique^,^or^a^lique,

if

∩{x ∈ C}

^is^a

d

^-fae.

Denition 2. Let

X ∈ X ³

and let

x ∈ X ⁻

^. ^We ^say

that

x

îs ân êssential ^fae ^for

X

^if

x

^is ^preisely ^the

intersetion of all faes of

X

^whih ^ontain

x

^, ^i.e., ^if

x = ∩{y ∈ X | x ⊆ y}

^. ^If

x

îsânêssential^fae^for

X

^,^we

write

x ⁺ X = {y ∈ X | x ⊆ y}

^,^and^we^say^that^the^lique

x ⁺ X

^is^essential^for

X

^.

Let

x

^be ^any ^fae ^of

X ∈ X ³

. We observethat

x

^is

an essential fae for

X

^and ^we ^have

x ⁺ X = {x}

^. ^The

essentialfaesforthexelomplex

X

^of^Fig. ⁷^(a)^whih

arenotfaesof

X

^arehighlightedFig. 7(b).

Denition3. Let

X ∈ X ³

andlet

x

^be^an^essential^fae

for

X

^. ^We^say^that

x

^is^regular^for

X

^if

x

^is ^simple^for

(X \ x ⁺ X )∪ {x}

^. ^We^say^that

x

^is^ritial^for

X

^if

x

^is^not

regularfor

X

^. ^If

x

^is^ritial ^(resp. ^regular)^for

X

^,^we

saythat thelique

x ⁺ X

^is^ritial^(resp. ^regular)^for

X

^.

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Observethat,inthepreviousdenition,

(X \x ⁺ X )∪{x}

isaxelomplex. If

x ∈ X

^,^we^have

(X \ x ⁺ X ) ∪ {x} = X

^.

Thus, a fae

x ∈ X

^is ^regular ^for

X

îf ând ônly îf ît

is simple for

X

^. Ôbserve âlso ^that â

0

^-lique ^whih ^is

essential for

X

^is ^neessarily ^ritial ^for

X

^. ^See ^Fig.

8 and 9 whih illustrate the notion of a ritial fae.

Note that an alternativeand equivalent denition of a

regular/ritialfaeisgivenin [8℄.

x y z

t

(a)

x

y

(b) ()

z

t

(d) (e)

Figure 8: (a): The xel omplex

X

^of ^Fig. ⁷ ^and ^four

essential faes

x

^,

y

^,

z

^,

t

(highlighted). (b): The xel omplex

(X \ x ⁺ X ) ∪ {x}

^:

x

^is^regular^for

X

^. ⁽⁾^The^xel

omplex

(X \ y ⁺ X ) ∪ {y}

^:

y

^is^ritial^for

X

^. ^(d): ^The^xel

omplex

(X \ z X ⁺ ) ∪ {z}

^:

z

^is^regular^for

X

^. ^(e)^The^xel

omplex

(X \ t ⁺ X ) ∪ {t}

^:

t

^is^ritial^for

X

^.

Remark 4. Let

X ∈ X ³

, let

x

^be^an ^essential^fae ^for

X

^, ^and ^let

C

^be ^the ^lique

x ⁺ X

^. ^If

C

^is ^regular ^for

X

^,

andif

x ∈ X

^,^then^(as^mentioned^above)

x

^is^simple^for

X

^,^and ^we^have

C = {x}

^. ^Thus

X \ C

îs â^thinning ôf

X

^: ^weân^remove^suhâ^regular^lique^from^theôbjet

withoutaltering the topology. Now letus onsider the

asewhere

C

^is^regular^but

x 6∈ X

^. ^For^that^purpose,^let

(a) (b)

Figure9: (a): Thexelomplex

X

^of^Fig. ^7: ^the^faes^of

X

^whih^are^ritial^for

X

^(not^simple) ^arehighlighted.

(b): The faes whih are ritial for

X

^and ^whih ^are

notfaes of

X

^arehighlighted.

usonsiderthexelomplex

X

^of^Fig. ⁸^(a)^and^the^faes

x

^and

z

^. ^Let

C

^be^the^lique^(made^of²^squares)^suh

that

C = x ⁺ X

^,

C

^is^a ^regular^lique ^for

X

^. ^We^observe

that

X \ C

îs â ^thinning ôf

X

^(see ^Fig. ⁸^(b)). ^Now

let

C ^′ = z X ⁺

^(a ^liqueômposed ôf ^twoûbes) ^whihîs

alsoaregularliquefor

X

^. ^We^note^that

X \ C ^′

^has^not

thesametopologyas

X

⁽

X

^has^two^tunnels^and

X \ C ^′

has only one tunnel, see Fig. 8 (a) and (d)). Thus

X \ C ^′

ânnot^beâ^thinningôf

X

^. ^In^fat,^the^dierene

betweenthesetwosituationsisthatthetwofaesof

x ⁺ X

areregular (i.e. simple) for

X

^, ^while^there îs â^fae ôf

z ⁺ X

^whih ^is ^not ^regular ^for

X

^(the ^ube ^above

z

^). ^In

thesequel of this setion, we will give someonditions

whih,intheontextofritialfaesandritialliques,

ensurethat agivensubset

Y ⊆ X

îsâ^thinningôf

X

^.

Thefollowingresultisaonsequeneofageneralthe-

oremwhihholdsforomplexesofarbitrarydimensions

(see[8℄).

If

X ∈ X ³

,theritialkernelof

X

îs^theûbialômplex

omposedofallfaesthatareritialfor

X

^and^all^faes

thatareinludedinthese faes.

Theorem5. Let

X ∈ X ³

andlet

Y ⊆ X

^.

Thexelomplex

Y

îsâ^thinningôf

X

^if

Y ⁻

^ontains^the

ritial kernelof

X

^.

Inother words,thexelomplex

Y

îsâ^thinningôf

X

if

Y ⁻

ôntainsâll^faes ^thatâre^ritial^for

X

^. ^See^Fig.

10 whih provides two examples of a omplex

Y

^that

satisestheaboveproperty.

AsadiretonsequeneofTh. 5,weobtainthefollow-

ingpropertywhihwill beourguidelineforthesequel.

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Corollary 6. Let

X ∈ X ³

andlet

Y ⊆ X

^.

The xelomplex

Y

îsâ^thinningôf

X

^if^any ^lique ^that

isritial for

X

ôntainsât^leastône^faeôf

Y

^.

Weonludethis setionbygivingaharaterization

oftheomplexeswhih satisfytheonditionof Th.5.

Theorem 7. Let

X ∈ X ³

andlet

Y ⊆ X

^. ^The ^ubial

omplex

Y ⁻

^ontains^the ^ritial^kernel^of

X

îfândônly

if any

Z

^suh^that

Y ⊆ Z ⊆ X

îsâ^thinningôf

X

^.

(a) (b)

() (d)

Figure10: (a): Axelomplex

X

^made^of¹²^ubes. ^(b):

Thefaesthatareritialfor

X

^arehighlighted. ()and (d) : twoxel omplexes

Y ^′ ⊆ X

^and

Y ^′′ ⊆ X

^. ^By ^Th.

5,

Y ^′

^and

Y ^′′

^are^both^thinnings^of

X

^.

5 Charaterization of ritial

liques in voxel omplexes

Inthispaper,weinvestigateamethodologyforthinning

objets whih are made of voxels (i.e., unit ubes).

For that purpose, wepropose, in thefollowing, several

haraterizations of

d

^-liques^(with

d = 3, 2, 1, 0

⁾^whih

are ritial for suh objets. We rst give a few basi

denitionsforvoxelomplexes.

We denote by

V ³

the olletionof all xel omplexes whih are omposed solely of unit ubes. A unit ube

isalso alledavoxel,anelementof

V ³

isalled avoxel omplex.

Forexample,thexelomplexofFig. 10(a)isavoxel

omplex,whiletheoneofFig. 7(a)is not.

Figure 11: Dierent types of neighborhoods:

N 2 ^∗ (x)

(squares),

N 1 ^∗ (x)

^(squares^and ^irles),

N 0 ^∗ (x)

^(squares,

irles, and triangles). The voxel

x

^orresponds ^to ^the

entralpoint.

Let

d ∈ {0, 1, 2}

^. ^We ^say ^that ^two ^voxels

x, y

^are

d

^-adjaent ^if

x ∩ y

^is ^a

k

^-fae, ^with

k ≥ d

^. ^If

x

^is ^a

voxel, we write

N d (x)

^for ^the ^set ^of ^all ^voxels ^whih

are

d

^-adjaent ^to

x

^,

N d (x)

^is ^the

d

-neighborhood of

x

^.

Notethat, foreahvoxel

x

^,^we^have

x ∈ N d (x)

^. ^We^set

N d ^∗ (x) = N d (x) \ {x}

^. ^See^anillustrationFig. 11where

thevoxel

x

^isrepresentedbyapoint.

Let

X, Y ∈ V ³

, with

Y ⊆ X

^. ^We ^say ^that

Y

^is

d

^-onneted ⁱⁿ

X

^if, ^for ^any

x, y ∈ Y

^, ^there ^exists ^a

sequene

hx 0 , ..., x ^k i

^of ^voxels ⁱⁿ

X

^suh ^that

x 0 = x

^,

x ^k = y

^,^and

x ⁱ

^is

d

^-adjaent^to

x ⁱ ₋ 1

^,

i = 1, ..., k

^.

Wesaythat

X ∈ V ³

is

d

^-onneted ^if

X

^is

d

^-onneted

in

X

^.

A3-lique whih isritial for

X ∈ V ³

is aset om- posedsolelyofonevoxelwhihisnotsimplefor

X

^. ^Thus,

anyharaterizationofsimplevoxelsissuienttohar-

aterizesuhliques.

Thefollowingpropositionshowsthat,whenonsider-

ingvoxelomplexes, Denition 1leadsto aharateri-

zation of simple voxelswhih is equivalent to previous

ones [5,13,46, 22, 17℄. If

X ∈ V ³

, wewrite

X

^for ^the

setofvoxelswhiharenotin

X

^.

Proposition 8. Let

X ∈ V ³

andlet

x ∈ X

^.

Thevoxel

x

^is^simple^for

X

îfândônly îf:

1)Theset

N 0 ^∗ (x)∩X

^is^non-empty^and

0

^-onneted;^and

2)The set

N 2 ^∗ (x) ∩ X

^is ^non-empty^and

2

^-onnetedⁱⁿ

N 1 ^∗ (x) ∩ X

^.

Let

d ∈ {0, 1, 2}

^. ^The^voxels^whih^belong^to

d

^-liques

thatareritialfor

X ∈ V ³

maybedetetedby:

1)Detetingall

d

^-faesⁱⁿ

X ⁻

^whih^are^essential^for

X

(8)

(Def. 2);

2)Detetingallessential

d

^-faes

x

^whih^are^not^simple

for

(X \ x ⁺ X ) ∪ {x}

^(Def. ^3);

3)Labelingallthevoxelsof

X

^whih^ontain^suh^faes.

In the following, we propose to haraterize ritial

liques in a way suh that the omputation of

X ⁻

^is

notneessary.

Werst observethat, upto

π/2

^rotations,^the^three

ongurations

C 2

^,

C 1

^, ^and

C 0

^given ⁱⁿ ^Fig. ¹² ^may

be used for the detetion of an arbitrary (regular or

ritial) lique whih is essential for a given voxel

omplex

X

ⁱⁿ

V ³

(in this gure a voxel is represented byapoint). Infat,itmaybeseenthat:

-

C 2

^may ^be ^used ^for ^deteting ^a ^2-lique

C

^whih ^is

essentialfor

X

^: ^there îs ^suh â^lique îf^both ^voxels

A

and

B

^areⁱⁿ

X

^. ^In^this^ase,^we^have

C = {A, B}

^.

-

C 1

C

^whih ^is

essentialfor

X

^: ^there îs ^suh â^liqueîf ^both

A

^and

D

are in

X

^or ^both

B

^and

C

^are ⁱⁿ

X

^. ^In ^this ^ase, ^we

have

C = {A, B, C, D} ∩ X

^.

-

C 0

C

^whih ^is

essentialfor

X

^: ^there îs^suhâ^liqueîf

A

^and

H

^,^or

B

and

G

^,^or

C

^and

F

^,^or

D

^and

E

^areⁱⁿ

X

^. ^In^this^ase,

wehave

C = {A, B, C, D, E, F, G, H} ∩ X

^.

Wenowintrodueanotionofneighborhood whih is

fundamentalforourpurpose.

Denition 9. Let

S ∈ V ³

. The

K

-neighborhood of

S

^,

written

K(S)

^, îs ^the ^set ^made ôf âll ^voxels ^whih âre

0

^-adjaent^to^eah^voxelⁱⁿ

S

^. ^We^set

K ^∗ (S) = K(S) \ S

^.

We note that we have

K(S) = N 0 (x)

^whenever

S

^is

madeofasinglevoxel

x

^. ^We^also^observe^that:

-wehave

K(T ) ⊆ K(S)

^whenever

S ⊆ T

^;

-wehave

S ⊆ K(S)

^whenever

S

^is^a^lique;

A B

A B D C

A E

B F

C

D H

G

C 2 C 1 C 0

Figure 12: Masksfor

2

^-liques⁽

C 2

^),

1

^-liques⁽

C 1

^), ^and

0

^-liques⁽

C 0

^). ^Here,^a^voxel^isrepresentedbyitsentral point.

A B

X Y

X Y X

Y X

Y Y

X Y X

X

6

6 Y

Y

7 7 0

0 1 2 1

2 3

3

4

5

5 K 2

A B

Y Y Y

Y D

C X X

X X

3

2 0 3

0 1 1

2 A E

B F

C

D H

G

K 1 K 0

Figure13:

K

-neighborhoodsfor

2

^-liques⁽

K 2

^),

1

^-liques

(

K 1

^), ^and

0

^-liques⁽

K 0

^). ^A ^voxel^is representedby its entralpoint.

- we have

K(S) = K(T )

^whenever

S

^and

T

^are ^two

liquessuhthat

∩{x ∈ S} = ∩{x ∈ T }

^.

The

K

-neighborhoodsoftheongurations

C 2

^,

C 1

^,^and

C 0

^are^given^Fig. ^13. ^Observe^that ^we^have

K ^∗ (S ) = ∅

fortheonguration

C 0

^.

Let

X ∈ V ³

. Asmentionedearlier,a

0

^-lique^whih^is

essentialfor

X

^is^neessarily^ritial. ^With^the^following

twopropositions, wegivesomeharaterizationsfor

2

^-

and

1

^-liques^whih ^are^regular ^for

X

^. ^Reall^that ^a

2

^-

lique whih is essentialfor

X

^is ^neessarily ^omposed

oftwovoxelswhihare

2

^-adjaent(onguration

C 2

^).

Proposition 10. Let

X ∈ V ³

, let

C = {x, y}

^be ^a

2

^-

liquewhih isessential for

X

^. ^The ^lique

C

^is^regular

for

X

îfândônly îf:

1) The set of voxels

K ^∗ (C) ∩ X

^is ^non-empty ^and

0

^-

onneted; and

2)Thereexiststwo voxels

x ^′ , y ^′ ∈ K ^∗ (C) ∩ X

^suh^that

x ^′ ∈ N 2 ^∗ (x)

^,

y ^′ ∈ N 2 ^∗ (y)

^,^and

x ^′ ∈ N 2 ^∗ (y ^′ )

^.

Proposition11. Let

X ∈ V ³

,let

C

^be^a

1

^-lique^whih

isessential for

X

^. ^The ^lique

C

^is^regular^for

X

^if ^and

only if the set of voxels

K ^∗ (C) ∩ X

^is ^non-empty ^and

0

^-onneted.

We are now in position to propose some masks for

detetingritial liques. These masks

K 2

^,

K 1

^,

K 0

^are

(9)

y x

(a)

B A

(b)

Figure 14: (a): Thexel omplex

X

^whih îs^the ôneôf

Fig. 10 (a). Here, eah voxel of

X

^is represented by a blak disk. (b): The mask

K 2

^, ^with

A, B

^mathing

voxels

x, y

^of

X

^. ^Conditionⁱⁱ⁾ ^of^Def. ¹² ^for

K 2

^is^not

satisedbutonditioni)isfullledsinethesetofvoxels

{X 0 , ..., X 7 , Y 0 , ..., Y 7 } ∩ X

^is^not

0

^-onneted. ^Thus,^by

Prop. 13,thevoxels

x, y

^onstitute^a

2

^-lique^of

S

^whih

isritial for

X

^. ^See^also^Fig.¹⁰ ^(b)^where ^the^ritial

fae

z = x ∩ y

^ishighlighted,wehave

z X ⁺ = {x, y}

^.

desribedusingFig.13. Foreahofthesemasks,wealso

onsider allthemasksobtainedfrom them byapplying

π/2

^rotationsâboutêahâxis. ^We^get⁷^masks⁽³^for

K 2

^,

3for

K 1

^, ^and ¹^for

K 0

^). ^See^Fig. ¹⁴ ^for^anillustration oftheuseofthemask

K 2

^.

Denition 12. Let

X ∈ V ³

, and let

S

^be ^a ^set ^of

voxelsof

X

^. ^We^say^that:

1)

S

^mathes

K 2

ⁱⁿ

X

^if

S = {A, B}

^;^and

i)thesetofvoxels

{X 0 , ..., X 7 , Y 0 , ..., Y 7 } ∩X

^is^either

emptyornot

0

^-onneted;^or

ii)foreah

i ∈ {0, 2, 4, 6}

^,

X i

^or

Y i

^belongs^to

X

^.

2)

S

^mathes

K 1

ⁱⁿ

X

^if

S = {A, B, C, D} ∩ X

^;^and

i)at leastoneof thesets

{A, D}

^,

{B, C }

^is^a^subset

of

X

^;^and

ii) we haveeither

[ U ∩ X 6= ∅

^and

V ∩ X 6= ∅ ]

^or

[ U ∩ X = ∅

^and

V ∩ X = ∅ ]

^, ^with

U = {X 0 , ..., X 3 }

and

V = {Y 0 , ..., Y 3 }

^.

3)

S

^mathes

K 0

ⁱⁿ

X

^if

S = {A, B, C, D, E, F, G, H}∩X

and at least one of the sets

{A, H}, {B, G}, {C, F }

^,

{D, E}

îsâ^subsetôf

X

^.

Prop. 13is adiretonsequeneofProp. 10and11.

Proposition 13. Let

X ∈ V ³

, let

S

^be ^a^set ^of ^voxels

in

X

^,^and^let

d ∈ {2, 1, 0}

^. ^The^set

S

^is^a

d

^-lique^whih

isritial for

X

îfândônly îf

S

^mathes

K d

ⁱⁿ

X

^.

Weonludethis setionbygivingaharaterization

ofsimplevoxelsandregularliquesthat isbasedonthe

denedreursivelyasfollows.

Denition14. Let

X ∈ V ³

. Wesaythat

X

îs^reduible îfêither:

i)

X

îsômposed ôfâ^single^voxel;ôr

ii)thereexists

x ∈ X

^suh^that

N ₀ ^∗ (x) ∩ X

^is^reduible

and

X \ {x}

^is^reduible.

Thefollowingtheorem allowsus to haraterizesim-

ple voxels withreduible sets, see also[23, 7℄ forother

reursiveapproahesforsimpliity.

Theorem15. Let

X ∈ V ³

andlet

x ∈ X

^. ^The^voxel

x

issimplefor

X

îf ândônlyîf

N ₀ ^∗ (x) ∩ X

^is^reduible.

Thus, aomplex

X ∈ V ³

isreduibleifand onlyifit is possibleto redue

X

^to ^a ^single^voxel ^byiteratively removingsimplevoxels.

Morepreisely,

X ∈ V ³

isreduibleif andonlyifthere existsasequene

hx 0 , ..., x ^k i

^suh^that

X = {x 0 , ..., x ^k }

^,

and

x ⁱ

^is^simple^for

{x ⁱ , ..., x ^k }

^,

i ∈ [0, k − 1]

^.

The following theorem is an extension of Th. 15 to

arbitraryregularliques.

Theorem 16. Let

X ∈ V ³

and let

C

^be ^a ^lique ^that

isessential for

X

^. ^The ^lique

C

^is^regular^for

X

^if ^and

onlyif

K ^∗ (C) ∩ X

^is^reduible.

Thus, Th. 16 makes it possible to haraterize, in

aunied way, regular

d

^-liques, ^with

d = 3, 2, 1, 0

^. ^In

partiular, for

d = 3

^, ^we ^get ^the haraterization of simplevoxelsgivenTh. 15. In this ase, the lique

C

ismadeofasinglevoxel

x

^,^and^we^have

K ^∗ (C) = N 0 ^∗ (x)

^.

Let

X ∈ V ³

be a reduible omplex whih is not omposed of a single voxel. By the very denition of

suh a omplex, and by Th. 15, there exists a simple

voxel for

X

^suh ^that

X \ {x}

^is ^reduible. ^But ^if

x

îs ân ârbitrary ^simple ^voxel ^for

X

^, ^then

X \ {x}

is not neessarily reduible. Suh a situation ours

when

X \ {x}

îs ân ôbjet^suh âs ^the ^so-alled^dune

hat [48℄ orhouse with two rooms [14℄, see also [34℄ for

algorithmiissues.

The following result shows that there is not enough

spaeforsuhobjetstobeinthe

K

-neighborhoodofa lique.

Theorem17. Let

C ∈ V ³

suhthat

C

îs â^lique, ând

let

S ⊆ K ^∗ (C)

^. ^If

S

^is^reduible,^then

S \{x}

^is^reduible

whenever

x

^is^a^simple^voxel ^for

S

^.

(10)

Let

X ∈ V ³

andlet

C

^beâ^lique^thatîsêssential^for

X

^. Âs â ônsequene ôf ^Th. ¹⁶ ând ^17, determining

whether

C

^is ^regular^or ^ritial^for

X

^may ^be^done ^by

thefollowinggreedyalgorithmRegularClique.

Algorithm1: RegularClique

Data:

X ∈ V ³

,alique

C

^whih^is^essential^for

X

Result:

Regular S = K ^∗ (C) ∩ X

^;

1

repeat 2

arbitrarilyseletavoxel

x

^that^is^simple^for

S

^;

3

S = S \ {x}

^;

4

until stability ; 5

If

Card(S ) = 1

^,^then

Regular = T rue

^;

6

Else

Regular = F alse

^;

7

6 Cruial Kernels and Minimal

Skeletons

Ourgoalistodeneasubsetofavoxelomplex

X

^that

isguaranteedtoinludeatleastonevoxelofeahlique

that is ritial for

X

^. ^By^Cor. ^6, ^this^subset ^will ^be^a

thinningof

X

^.

We want this subset to be as small as possible in

ordertoobtainaneientthinningproedure. Wealso

wantourmethodtobeindependentofarbitraryhoies,

in partiular of a hoie of spei voxels in a given

ritiallique. For thatpurpose thefollowingnotionof

amaximalritialfae wasintrodued[11,12℄.

Let

X ∈ V ³

andlet

x

^be^a^ritial^fae^for

X

^. ^We^say

that

x

^is

M

^-ritial ^for

X

^if

x

îs ^not â^proper^fae ôf

afae whihis ritial for

X

^. ^If

x

^is

M

^-ritial ^for

X

^,

we saythat thelique

x ⁺ X

^is

M

^-ruial ^for

X

^. ^We^say

thatavoxel

x ∈ X

^is

M

^-ruial^for

X

^if

x

^belongs^to^a

liquewhihis

M

^-ruial ^for

X

^.

If

X ∈ V ³

, we denote by

M(X)

^the ^set ^omposed

of all voxels that are

M

^-ruial ^for

X

^,

M(X )

^is

the

M

^-ruial ^kernel ^of

X

^. ^Thus,

M(X )

^is ^the^set ^of

voxelsof

X

^thatôntainâ^fae^whihîs

M

^-ritial^for

X

^.

InFig.15(a),the

M

^-ritial^faesôfâômplex

X

^are

highlighted(seealsoFig. 10(b)wheretheritialfaes

of thesame omplexare given). The

M

^-ruial ^kernel

of

X

^is ^given^Fig. ¹⁵^(b).

Remark 18. Let

X ∈ V ³

and let

C ⊆ X

^. ^It ^may ^be

seenthat

C

^is ^an

M

^-ruial ^lique^for

X

îf ândônly îf

C

^is^a^ritial^lique^for

X

^and^no^proper^subset^of

C

^is

aliquewhihisritialfor

X

^.

Remark 19. Let

X ∈ V ³

andlet

C ⊆ X

^. ^It^has^been

proved that

C

^is ^non-simple ^for

X

^whenever

C

^is ^an

M

^-ruial^lique^for

X

^,^whih^means^that^the^set

X \ C

is not a thinning of

X

^. ^In ^fat, ^it ^was ^shown ⁱⁿ ^[12℄

(Th. 28), thatasubsetof

X

^is^an

M

^-ruial^lique ^for

X

îf ând ônlyîf ît îs â^minimal ^non-simple ^set ^for

X

^,

see [42, 21, 30, 24℄ forother propertiesof the so-alled

MNS's.

Bytheverydenition of an

M

^-ruial ^voxel,

M(X )

whih is ritial for

X

^, ^thus

M(X )

îs â ^thinning ôf

X

^. Nevertheless, throughthe followingobservation, it may be seen that it is possible to obtain a subset of

voxels of

X

^whih ^fullls ^the ^onditions ^given ⁱⁿ ^the

very beginning of this setion, and whih ontains less

voxelsthan

M(X )

^.

LetusonsideragainFig.15(a). Thevoxel

x

^ontains

an

M

^-ritial^1-fae^and^thus^it^belongs^to

M(X)

^. ^But

this1-faeisalsoinludedinthevoxel

y

^,^whih^ontains

a 2-fae whih is also

M

^-ritial. ^This ^motivates ^the

followingreursivedenition ofaruial voxelwhihis

basedondimension.

Denition20. Let

X ∈ V ³

and

C

^be^a

d

^-lique^whih

isritialfor

X

^. ^We^say^that

C

^is

D

^-ruial ^for

X

^if:

i)

d = 3

^;^or

ii)

d ∈ {2, 1, 0}

^and

C

^does ^not ^ontain ^any ^voxel ^be-

longingto a

d ^′

^-lique ^that ^is

D

^-ruial ^for

X

^and ^suh

that

d ^′ > d

^.

Wesaythatavoxel

x ∈ X

^is

D

^-ruial^for

X

^if

x

^belongs

toaliquethat is

D

^-ruial ^for

X

^.

Note that a voxel that is not simple neessarily

onstitutesa

3

^-lique ^whih ^is

D

^-ruial. ^Observe^also

that, if

d ^′ 6= d

^, ^a ^voxel

x

^that ^belongs ^to ^a

d

^-lique

whih is

D

^-ruial^annot^belong^to^a

d ^′

^-lique^whih^is

also

D

^-ruial.

If

X ∈ V ³

,wedenoteby

D(X)

^the^setômposedôfâll

voxelswhihare

D

^-ruial^for

X

^,

D(X )

^is^the

D

^-ruial

kernelof

X

^.

Again,bytheverydenitionofa

D

^-ruial^fae,

D(X )

whih is ritial for

X

^. ^Thus, ^by ^Cor. ^6,

D(X)

^is ^a

thinningof

X

^.

(11)

thinning,the

D

^-ruial ^kernelôrresponds^to ânôpera-

tionwhihismorepowerfulthanthe

M

^-ruial^kernel.

Theorem21. Let

X ∈ V ³

. The

D

^-ruial ^kernel^of

X

isasubsetof its

M

^-ruial ^kernel.

Observe that the example of Fig. 15 (a) shows

that the above inlusion may be strit: the voxel

x

^is

M

^-ruial^for

X

^but^not

D

^-ruial^for

X

^.

Let

X ∈ V ³

. Let

hX 0 , ..., X ^k i

^be ^the^sequene^of^dis-

tint elements suh that

X 0 = X

^,

X ^k = D(X k )

^, ^and

X ⁱ = D(X i − 1 )

^,^for

i = 1, ..., k

^. ^The^set

X ^k

^is ^the^mini-

mal

D

^-skeleton^of

X

^.

InFig. 15(), theomplex

Z = D(X )

^ishighlighted, theomplex

X

^being^theôneôf^Fig.¹⁵^(a). ^Theômplex

Z ^′ = D(Z)

^is^givenⁱⁿ^(d)^and^(e). ^We^have

Z ^′ = D(Z ^′ )

^,

thus

Z ^′

^is^the^minimal

D

^-skeleton^of

X

^.

Twootherexamplesofminimal

D

^-skeletons^are^given

Fig. 16. Wewill seein thenextsetion thataminimal

D

^-skeleton^may^beôbtained^byânâlgorithm^whihîsâ

speialinstaneofageneriparallelthinningsheme.

7 Three Generi Symmetri Thin-

ning Shemes

In this setion, we propose three generi thinning

shemes whih permit to ompute a wide variety of

skeletons.

Forthatpurpose,werstintroduethenotionofa

D

^-

ruialkernelwhihisonstrainedtopreserveagivenset

K

^(Def. ^23),^and^whihgeneralizesthedenitionofa

D

^-

ruial kernelpresentedin Se. 6. Infat, forthinning

objets, we often want to keep other voxels than the

onesthatareruial. Intuitively,theset

K

^orresponds

to a set of features that we wantto be preserved bya

thinningalgorithm(likeextremitiesofurves,ifwewant

toobtainaurvilinearskeleton).

Allthethreeproposedthinningshemesarebasedon

suhonstrained

D

^-ruial ^kernels.

Denition 22. Let

X ∈ V ³

,

K ∈ V ³

, and let

C

^be^a

d

^-lique^whih^is^ritial^for

X

^and^suh^that

C ⊆ X \ K

^.

Wesaythat

C

^is

D

^-ruial^for

hX, Ki

^if:

i)

d = 3

^;^or

ii)

d ∈ {2, 1, 0}

^and

C

^does ^not ^ontain ^any ^voxel^be-

longingto a

d ^′

^-lique^whih^is

D

^-ruial ^for

hX, Ki

^and

suhthat

d ^′ > d

^.

x y

(a)

(b) ()

(d) (e)

Figure 15: (a): A voxelomplex

X

^and ^its

M

^-ritial

faes (highlighted). (b): The omplex

Y = M(X )

^is

highlighted. (): Theomplex

Z = D(X )

^ishighlighted.

(d): The omplex

Z ^′ = D(Z)

^is highlighted. (e): We have

Z ^′ = D(Z ^′ )

^:

Z ^′

^is^the^minimal

D

^-skeleton^of

X

^.

We say that a voxel

x ∈ X

^is

D

^-ruial ^for

hX, K i

^if

x

^is ⁱⁿ

K

^or^if

x

^belongs ^to ^a^lique^whih ^is

D

^-ruial

for

hX, K i

^.

Denition 23. Let

X ∈ V ³

,

K ∈ V ³

. Wedenote by

D(X, K )

^the ^set ômposed ôf âll ^voxels ^whih âre

D

^-

ruialfor

hX, Ki

^,

D(X, K )

^is^the

D

^-ruial^kernel^of

X

onstrainedby

K

^.

From the previous denitions and from Cor. 6, we

immediatelydeduethefollowingpropositionwhihen-

suresthatanyonstrained

D

^-ruial^kernelôfânôbjet

preservesthetopologyofthisobjet.

Proposition 24. Let

X ∈ V ³

,

K ∈ V ³

. The

D

^-ruial

kernelof

X

^onstrained^by

K

îsâ^thinningôf

X

^.

By onstrution, the following proedure D-ruial

omputes the

D

^-ruial ^kernel ôf ân ôbjet

X ∈ V ³

onstrained by

K

^. Ît ônsists ôf ⁵ ^steps, êah ^step

may be done in parallel. Voxels that are not simple

and ritial liques may be deteted with the hara-

terizations given Prop. 8 and Prop. 13, or with the

(12)

(a) (b)

Figure 16: Twovoxel omplexes and their minimal

D

^-

skeleton(in red).

unied haraterization given Th. 16 whih an be

implementedusingalgorithm RegularClique.

Algorithm2: D-ruial

Data:

X ∈ V ³

,

K ∈ V ³

Result:

X

R 3

^:=^setôfâll^voxelsôf

X

^whih^are^not^simple^for

1

X

^or^whih ^areⁱⁿ

K

^;

R 2

^:=^setôfâll^voxels^belonging^to âny

2

^-lique

2

whih isritialfor

X

^and^inludedⁱⁿ

X \ R 3

^;

R 1

1

^-lique

3

whih isritialfor

X

^and^inludedⁱⁿ

X \ (R 3 ∪ R 2 )

^;

R 0

0

^-lique

4

whih isritialfor

X

^and^inludedⁱⁿ

X \ (R 3 ∪ R 2 ∪ R 1 )

^;

X

^:=

R 3 ∪ R 2 ∪ R 1 ∪ R 0

^;

5

Wepresentnowtherstthinningshemewhihon-

sistsinomputingiteratively,startingfrom

X

^,

D

^-ruial

kernelsonstrainedbyagivenset

K

^,^this^onstraint^set

isxedfromthebeginning. ByProp. 24,theresultisa

thinningof

X

^. ^Furthermore,theresultontains

K ∩ X

^.

Denition 25. Let

X ∈ V ³

,

K ∈ V ³

. Let

hX 0 , ..., X k i

bethesequeneofdistintelementssuhthat

X 0 = X

^,

X ⁱ = D(X ⁱ ₋ 1 , K )

^for

i = 1, ..., k

^, ^and

X ^k = D(X ^k , K )

^.

Theset

X ^k

^is^the

D

^-skeleton ^of

X

^onstrained^by

K

^.

Observethat theminimal

D

^-skeleton ôf ânôbjet

S

isa

D

^-skeleton^of

S

^onstrained^by

K

^,^with

K = ∅

^.

Notealsothat the

D

^-skeleton^of

X

^onstrained^by

K

maybeeasilyobtainedbyrepeating, untilstability,the

The seond thinning sheme is based on adynami

onstraint set. This onstraint set is dened thanks

to afuntion

Ψ

^from

V ³

to

V ³

whih is xed from the beginning. Thisfuntionallowsonetodene,ateahit-

eration,theverysubsetoftheobjetwhihmustbepre-

servedduring thethinning proedure. Again,byProp.

24,theresultisathinningof

X

^.

Denition26. Let

Ψ

^be^a^funtion^from

V ³

to

V ³

. Let

X ∈ V ³

and let

hX 0 , ..., X ^k i

^be^the^sequene^of^distint

elementssuhthat

X 0 = X

^,

X ⁱ = D(X i − 1 , Ψ(X ⁱ ₋ 1 ))

^for

i = 1, ..., k

^,^and

X k = D(X k , Ψ(X k ))

^. ^The^set

X k

^is^the

D

^-skeleton^of

X

^onstrained^by

Ψ

^.

The third thinning sheme is based, as above, on a

dynamionstraintsetandamap

Ψ

^from

V ³

to

V ³

whih is xed from the beginning. Thedierene is that the

onstraint set is built iterativelyfrom

Ψ

^and ^from ^the

onstraint obtained at the previous iteration step. By

Prop. 24,theresultisathinningof

X

^.

Denition27. Let

Ψ

^be^a^funtion^from

V ³

to

V ³

. Let

X ∈ V ³

,

K ∈ V ³

, and let

hX 0 , ..., X ^k i

^be ^the ^sequene

ofdistint elementssuh that

X 0 = X

^,

K 0 = K

^,

X ⁱ = D(X i − 1 , K ⁱ )

^with

K ⁱ = K ⁱ ₋ 1 ∪ Ψ(X ⁱ ₋ 1 )

^for

i = 1, ..., k

^,

and

X ^k = D(X k , K ^k ∪ Ψ(X ^k ))

^. ^The ^set

S ^k

^is ^the

D

^-

skeleton of

X

inrementallyonstrainedby

Ψ

^and

K

^.

Again,it may beseenthat the

D

^-skeleton^of

X

^on-

strainedby

Ψ

^,^or^the

D

^-skeleton^of

X

inrementallyonstrainedby

Ψ

^and

K

^,^may ^beêasily ôbtained^by îtera-

tivelyapplyingtheproedureD-ruial.

8 Examples

In this setion, we give several examples of spei

instanesofthethree abovethinningshemes.

A rstbasi exampleof a

D

^-skeleton ^onstrained ^by

aset of voxels

K

^is ^given ^Fig. ^17. ^Here

K

^is ^made^of

5 points, thus the

D

^-skeleton ôf ^the ôriginal ôbjet

X

onstrained by

K

îs â ûrvilinear ^shape. ^Note ^that â

D

^-skeleton^may^ontain^some^simple^points^that^do^not

belong to the onstrained set. In other words suh a

skeleton may be thik, whih is the prie to pay for

symmetry.

Aseond exampleofsuh askeleton isgivenFig. 18

where

X

îs â^solid ûbe ând

K

îs â^subset ôf

X

^whih

Powerful Parallel Symmetric 3D Thinning Schemes Based on Critical Kernels

HAL Id: hal-00731083

https://hal-upec-upem.archives-ouvertes.fr/hal-00731083

Preprint submitted on 12 Sep 2012

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Powerful Parallel Symmetric 3D Thinning Schemes Based on Critical Kernels

Gilles Bertrand, Michel Couprie

To cite this version:

Gilles Bertrand, Michel Couprie. Powerful Parallel Symmetric 3D Thinning Schemes Based on Critical

Kernels. 2012. �hal-00731083�

∗

Z 3

∗

Z 2

Z 3

Z 4

Z d

Z 2

a, b, c

x, y, z, t

y a b

c z

t x

X

a, b, c

x, y, z, t

x

X

y

z

t

a

b

x, y

Z d

x y

x

y

Z 3

n

Z

F 1 0

F 1 1

F 1 0 = {{a} | a ∈ Z }

F 1 1 = {{a, a + 1} | a ∈ Z }

f

Z 3

d

F 1 1

(n−d)

F 1 0

d

Z 3

d

f

dim(f ) = d

F 3

d

Z 3

d ∈ {0, 1, 2, 3}

d

Z 3

d = 0

d = 1

d = 2

d = 3

X

F 3

X − = {y ∈ F 3 | y ⊆ x

x ∈ X }

X −

X

X

F 3

F 3

X = X −

C 3

X

F 3

Z ³

Z ²

Z ³

Z ⁴

Z ^d

Z ²

Z ^d

Z ³

F ¹ ₀

F ¹ ₁

F ¹ ₀ = {{a} | a ∈ Z }

F ¹ ₁ = {{a, a + 1} | a ∈ Z }

Z ³

F ¹ ₁

F ¹ ₀

Z ³

F ³

Z ³

Z ³

F ³

X ⁻ = {y ∈ F ³ | y ⊆ x

X ⁻

F ³

F ³

X = X ⁻

C ³

F ³

F ³

X ³

X ∈ X ³

Y ∈ X ³

F ³

X ⁺

X ∈ C ³

X ⁺ ∈ X ³

(X ⁺ ) ⁻ = X

X ∈ X ³

X ⁻ ∈ C ³

(X ⁻ ) ⁺ = X

X ⁺

X ⁻

X ∈ C ³

X ∈ C ³

X, Y ∈ C ³

hX 0 , ..., X ^k i

X ^k = Y

X ⁱ

X ⁱ ₋ 1

X ∈ X ³

X ∈ X ³

X ⁻

(X \ {x}) ⁻

X, Y ∈ X ³

hX 0 , ..., X ^k i

X ^k = Y

X ⁱ