New characterizations of simple points, minimal non-simple sets and P-simple points in 2D, 3D and 4D discrete spaces

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New characterizations of simple points, minimal non-simple sets and P-simple points in 2D, 3D and 4D

discrete spaces

Michel Couprie, Gilles Bertrand

To cite this version:

Michel Couprie, Gilles Bertrand. New characterizations of simple points, minimal non-simple sets and

P-simple points in 2D, 3D and 4D discrete spaces. Discrete Geometry for Computer Imagery, 2008,

France. pp.105-116. �hal-00622026�

non-simple sets and P-simple points in 2D, 3D

and 4D disrete spaes

MihelCouprieandGillesBertrand

UniversitéParis-Est,LABINFO-IGM,UMRCNRS8049,A2SI-ESIEE,Frane

(m.ouprie,g.bertrand)esiee.f r

Abstrat. Inthisartile,wepresentnewresultsonsimplepoints,min-

imalnon-simplesets (MNS)andP-simplepoints.Inpartiular,wepro-

pose new haraterizations whih hold in dimensions 2, 3 and 4, and

whihleadtoeientalgorithmsfordetetingsuhpointsorsets.This

workissettledintheframeworkofubial omplexes,andsomeofthe

mainresultsarebasedonthepropertiesofritial kernels.

Introdution

Topology-preservingoperators,likehomotopiskeletonization,areusedinmany

appliationsofimageanalysistotransformanobjetwhileleavingunhangedits

topologialharateristis.Indisretegrids(

Z ²

,

Z ³

,

Z ⁴

),suhatransformation anbedened thanksto thenotionof simplepoint[20℄:intuitively,apointof

anobjetisalled simpleifit anbedeleted from thisobjetwithoutaltering

topology.

Themost natural wayto thin an objet onsists in removing someof its

borderpointsinparallel,inasymmetrialmanner.However,paralleldeletionof

simplepointsdoesnotguaranteetopologypreservationin general.Infat,suh

aguaranteeis notobviousto obtain,even forthe2D ase(see[10℄). C. Ronse

introduedtheminimalnon-simplesets[28℄tostudytheonditionsunderwhih

pointsmayberemovedsimultaneouslywhilepreservingtopologyof2Dobjets.

Thisleadstoveriationmethodsforthetopologialsoundnessofparallelthin-

ning algorithms. Suh methods have been proposed for 2D algorithms by C.

Ronse [28℄and R.Hall[15℄,theyhavebeendevelopedforthe3D asebyT.Y.

Kong [21,16℄ and C.M. Ma [25℄,as well as for the 4D ase by C-J. Gau and

T.Y. Kong [13,19℄. For the 3Dase, G. Bertrand [1℄ introdued thenotion of

P-simple point as a veriation method but also as a methodology to design

parallelthinningalgorithms[2,9,23,24℄.

IntroduedreentlybyG.Bertrand,ritialkernels[3,4℄onstituteageneral

frameworksettledintheategoryofabstratomplexesforthestudyofparallel

thinninginanydimension.Itallowseasydesignofparallelthinningalgorithms

whihproduenewtypesofskeletons,withspeigeometrialproperties,while

guaranteeing their topologial soundness [7,5,6℄. A new denition of asimple

thenotionsofanessentialfaeandofaoreofafaeallowtodenetheritial

kernel

K

^ofaomplex

X

^.Themostfundamental resultprovedin [3,4℄isthat, ifasubset

Y

^of

X

^ontains

K

^{, then}

X

^ollapsesonto

Y

^,hene

X

^and

Y

^have

thesametopology.

Inthis artile,wepresentnewresultsonsimplepoints,minimalnon-simple

sets (MNS), P-simple points and ritial kernels. Let us summarize the main

onesamong theseresults.

First of all, we state some onuene properties of the ollapse operation

(Th.5,Th.6),whih playafundamentalrolein theproofofforthomingtheo-

rems.Thesepropertiesdonotholdingeneralduetotheexisteneoftopologial

monsterssuhasBing'shouse([8℄,seealso[27℄);weshowthattheyareindeed

trueinsomedisretespaeswhiharenotlargeenoughtoontainsuhounter-

examples.

Based on these onuene properties, we derive a new haraterization of

2D,3Dand4Dsimplepoints(Th.7)whihleadstoasimple,greedylineartime

algorithmforsimpliityheking.

Then,weshowtheequivalene(up to 4D)betweenthenotionofMNS and

thenotionofruiallique,derivedfromtheframeworkofritialkernels.This

equivalene (Th. 21) leads to the rst haraterization of MNS whih an be

veriedusingapolynomialmethod.

Finally, we showthe equivalenebetween thenotionof P-simplepoint and

the notionof weakly ruial point, also derivedfrom the framework of ritial

kernels. This equivalene (Th. 23) leads to the rst loal haraterization of

P-simplepointsin4D.

Thispaperis self-ontained, howevertheproofsannot beinluded due to

spaelimitation.Theyanbefoundin [12,11℄,togetherwithsomeillustrations

anddevelopments.

1 Cubial Complexes

Abstratomplexes havebeenpromotedin partiularbyV. Kovalevsky[22℄in

order to provide asound topologial basis for image analysis.Forinstane, in

thisframework,weretrievethemainnotionsandresultsofdigitaltopology,suh

asthenotionofsimplepoint.

Intuitively,aubialomplexmaybethoughtofasasetofelementshaving

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

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

omplexes, see also [7,5,6℄ for more details. We onsider here

n

-dimensional omplexes,with

0 ≤ n ≤ 4

^.

Let

S

^beaset.If

T

^isasubsetof

S

^{, wewrite}

T ⊆ S

^{.Wedenote by}

|S|

^the

numberofelementsof

S

^.

Let

Z

bethesetofintegers.Weonsiderthefamiliesofsets

F ¹ ₀

,

F ¹ ₁

,suhthat

F ¹ ₀ = {{a} | a ∈ Z }

^,

F ¹ ₁ = {{a, a + 1} | a ∈ Z }

^{. Asubset}

f

^of

Z ⁿ

,

n ≥ 2

^,whih

is theCartesianprodut of exatly

m

^elementsof

F ¹ ₁

and

(n − m)

^elementsof

F ¹ ₀

is alled a fae or an

m

^{-fae of}

Z ⁿ

,

m

^{is the dimension of}

f

^{, we write}

dim(f ) = m

^.

Observethatanynon-emptyintersetionoffaesisafae.Forexample,the

intersetionoftwo

2

^-faes

A

^and

B

^{maybeeither a}

2

^-fae(if

A = B

^),a

1

^-fae,

a

0

^{-fae,ortheemptyset.}

(a) (b) () (d) (e)

Fig.1.Graphialrepresentationsof:(a)a

0

-fae,(b)a

1

-fae,()a

2

-fae,(d)a

3

-fae, (e)a

4

-fae.

Wedenoteby

F ⁿ

thesetomposedofall

m

^-faesof

Z ⁿ

,with

0 ≤ m ≤ n

^.An

m

^{-fae of}

Z ⁿ

isalled a point if

m = 0

^{, a (unit) interval if}

m = 1

^{, a (unit)}

square if

m = 2

^{, a (unit) ube if}

m = 3

^{, a (unit) hyperube if}

m = 4

^(see

Fig.1).

Let

f

^beafaein

F ⁿ

.Weset

f ˆ = {g ∈ F ⁿ | g ⊆ f }

^and

f ˆ ^∗ = ˆ f \ {f }

^.

Any

g ∈ f ˆ

^{isa faeof}

f

^,andany

g ∈ f ˆ ^∗

^{isa proper faeof}

f

^.

If

X

^{is a nite set of faes in}

F ⁿ

, we write

X ⁻ = ∪{ f ˆ | f ∈ X }

^,

X ⁻

^{is the}

losureof

X

^.

Aset

X

^offaesin

F ⁿ

isa ell oran

m

^{-ell ifthereexistsan}

m

^-fae

f ∈ X

^,

suhthat

X = ˆ f

^{.The boundaryof aell}

f ˆ

^istheset

f ˆ ^∗

^.

Aniteset

X

^offaesin

F ⁿ

isa omplex(in

F ⁿ

)if

X = X ⁻

^.Anysubset

Y

^of

aomplex

X

^{whihisalsoaomplexisa subomplexof}

X

^.If

Y

^isasubomplex

of

X

^,wewrite

Y X

^.If

X

^isaomplexin

F ⁿ

,wealsowrite

X F ⁿ

.InFig.2 andFig.3,someomplexesarerepresented.Notiethat anyellisaomplex.

Let

X ⊆ F ⁿ

beanon-emptysetoffaes.Asequene

(f i ) ^ℓ _i=0

^offaesof

X

^is

apath in

X

^(from

f 0

^to

f ℓ

^{) ifeither}

f i

^{is afae of}

f i+1

^or

f i+1

^{isafae of}

f i

^,

forall

i ∈ [0, ℓ − 1]

^.Wesaythat

X

^{isonneted if,for anytwofaes}

f, g

ⁱⁿ

X

^,

there isapathfrom

f

^to

g

ⁱⁿ

X

^{; otherwisewesaythat}

X

^{is disonneted.We}

say that

Y

^{is aonnetedomponent of}

X

^if

Y ⊆ X

^,

Y

^{isonneted andif}

Y

ismaximalforthesetwoproperties(i.e., wehave

Z = Y

^whenever

Y ⊆ Z ⊆ X

and

Z

^isonneted).

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

^.

Let

X F ⁿ

,

X 6= ∅

^{,the number}

dim(X) = max{dim(f ) | f ∈ X ⁺ }

^{is the}

dimension of

X

^.Wesaythat

X

^isan

m

^-omplexif

dim(X ) = m

^.

Wesaythat

X

^{is pure if,foreah}

f ∈ X ⁺

^,wehave

dim(f ) = dim(X)

^.

Intuitivelyasubomplexof aomplex

X

^{is simpleifitsremovalfrom}

X

^does

nothangethetopologyof

X

^{. Inthissetion werealladenition ofasimple}

subomplexbasedontheoperationofollapse[14℄,whihisadisreteanalogue

ofaontinuousdeformation(ahomotopy).

Let

X

^beaomplexin

F ⁿ

andlet

f ∈ X

^{.Ifthereexistsonefae}

g ∈ f ˆ ^∗

^suh

that

f

^{istheonlyfae of}

X

^{whihstritly inludes}

g

^,then

g

^{issaidto be free}

for

X

^andthepair

(f, g)

^{issaidtobea freepair for}

X

^{.Notiethat,if}

(f, g)

^is

afreepair,thenwehaveneessarily

f ∈ X ⁺

^and

dim(g) = dim(f ) − 1

^.

Let

X

^{beaomplex,andlet}

(f, g)

^{beafreepairfor}

X

^{. Theomplex}

X \{f, g}

isan elementary ollapseof

X

^.

Let

X

^,

Y

^{betwoomplexes.Wesaythat}

X

^ollapsesonto

Y

^if

Y = X

^orifthere

existsaollapsesequenefrom

X

^to

Y

^{,i.e.,asequeneofomplexes}

hX 0 , ..., X ℓ i

suhthat

X 0 = X

^,

X ℓ = Y

^,and

X i

^{isanelementaryollapseof}

X i−1

^,

i = 1, ..., ℓ

^.

If

X

^ollapsesonto

Y

^and

Y

^{isaomplex madeof asinglepoint,wesay that}

ollapses ontoapoint.

Fig. 2 illustrates a ollapse sequene. Observe that, if

X

^{is a ell of any}

dimension,then

X

^{ollapsesontoapoint.Itmayeasilybeseenthattheollapse}

operationpreservesthenumberofonnetedomponents.

(a)

f

(b) ()

(d) (e) (f)

Fig.2.(a):apure

3

-omplex

X F ³

,anda

3

-fae

f ∈ X ⁺

^.(f):aomplex

Y

^whihis

thedetahmentof

f ˆ

^from

X

^{.(a-f):aollapsesequenefrom}

X

^to

Y

^.

Let

X, Y

^{be two omplexes. Let}

Z

^{suh that}

X ∩ Y Z Y

^{, and let}

f, g ∈ Z \ X

^{. It may be seen that the pair}

(f, g)

^{is a free pair for}

X ∪ Z

^if

andonlyif

(f, g)

^{isafreepairfor}

Z

^{.Thus,byindution,wehavethefollowing}

property.

Proposition1([3,4℄). Let

X, Y F ⁿ

.Theomplex

X ∪ Y

^{ollapses onto}

X

^if

andonly if

Y

^ollapsesonto

X ∩ Y

^.

Theoperationofdetahmentallowstoremoveasubsetfromaomplex,while

guaranteeingthattheresultisstill aomplex.

Denition2([3,4℄). Let

Y ⊆ X F ⁿ

.Weset

X ⊘ Y = (X ⁺ \ Y ⁺ ) ⁻

^{.The set}

X ⊘ Y

^{isaomplex whih isthe detahmentof}

Y

^from

X

^.

Inthe following,wewill be interested in the ase where

Y

^{is a singleell.}

ForexampleinFig.2,weseeaomplex

X

^{(a)ontaininga}

3

^-ell

f ˆ

^,and

X ⊘ f ˆ

isdepitedin(f).

Let us now reall here a denition of simpliity based on the ollapse op-

eration, whih anbe seenasa disrete ounterpart of theone given by T.Y.

Kong [17℄.

Denition3([3,4℄). Let

Y ⊆ X

^;wesaythat

Y

^{is simplefor}

X

^if

X

^ollapses

onto

X ⊘ Y

^.

TheollapsesequenedisplayedinFig.2(a-f)showsthattheell

f ˆ

^issimple

fortheomplexdepitedin(a).

Thenotion of attahment, as introdued byT.Y. Kong [16,17℄, leadsto a

loalharaterizationofsimplesets, whihfollowseasily fromProp.1.

Let

Y X F ⁿ

. The attahment of

Y

^for

X

^{is the omplex dened by}

Att

(Y, X) = Y ∩ (X ⊘ Y )

^.

Proposition4([3,4℄). Let

Y X F ⁿ

.Theomplex

Y

^issimplefor

X

^ifand

only if

Y

^{ollapsesontoAtt}

(Y, X)

^.

LetusintrodueinformallytheShlegel diagrams asagraphialrepresenta-

tionforvisualizingtheattahmentofaell.InFig.3a,theboundaryofa

3

^-ell

f ˆ

anditsShlegel diagramaredepited.Theinterestofthisrepresentationliesin

thefatthatastruturelike

f ˆ ^∗

^{lyinginthe3Dspaemaybe}representedinthe 2Dplane. Notiethat one

2

^{-fae oftheboundary,herethesquare}

ef hg

^,isnot

representedby a losedpolygon in the shlegel diagram,but wemay onsider

that itisrepresentedbytheoutsidespae.

AsanillustrationofProp.4,Fig.3bshows(bothdiretlyandbyitsShlegel

diagram) theattahmentof

f ˆ

^{for theomplex}

X

^{ofFig.2a, andweaneasily}

verifythat

f ˆ

^ollapsesonto

Att( ˆ f , X)

^.

(a)

a b

c d

e f

g h

e

g h

f a

c d

b

(b)

Fig.3.(a):Theboundaryofa

3

-ellanditsShlegeldiagram.(b):Theattahmentof

f ˆ

^for

X

^(seeFig.2a).

Representing4D objetsis not easy. To start with, let us onsider Fig. 4a

where arepresentationofthe3Domplex

X

^{ofFig.2aisgivenunder theform}

oftwohorizontal ross-setions,eahblakdotrepresentinga

3

^-ell.

Inasimilarway,wemay representa4Dobjetbyits3D setions,asthe

objet

Y

^{in Fig.4b.Suh anobjetmaybethought ofasatime seriesof 3D}

objets.InFig.4b,eahblakdotrepresentsa

4

^{-ellofthewhole4Domplex}

Y

^.

Shlegel diagramsarepartiularly usefulforrepresentingtheattahmentof

a 4D ell

f ˆ

^{, whenever this attahment if not equal to}

f ˆ ^∗

^{. Fig. 5a shows the}

(a)

f

(b)

h n m

g i

k l

j

Fig.4.(a):Analternativerepresentationofthe3Domplex

X

^{ofFig.2a.(b):Asimilar}

representationofa4Domplex

Y

^.

Shlegel diagramof the boundary ofa

4

^{-ell(see Fig.1e),where oneof the}

3

^-

faes is represented by the outside spae. Fig. 5b showsthe Shlegel diagram

of theattahmentof the

4

^-ell

g

ⁱⁿ

Y

^{(seeFig.4b). Forexample,the}

3

^-ell

H

representedintheenterofthediagramistheintersetionbetweenthe

4

^-ell

g

andthe

4

^-ell

h

^.Also,the

2

^-ell

I

^(resp.the

1

^-ell

J

^,the

1

^-ell

K

^,the

0

^-ell

L

⁾

is

g ∩ i

^(resp.

g ∩ j

^,

g ∩ k

^,

g ∩ l

^).Thetwo

2

^{-ellswhiharethe}intersetionsof

g

with,respetively,

m

^and

n

^{,arebothinludedin the}

3

^-ell

H

^{.Observethatthe}

ell

g

^{isnotsimple(itsattahmentisnotonneted).}

(a) (b)

H I J

K L

Fig.5.(a):TheShlegeldiagramoftheboundaryofa

4

-ell.(b):TheShlegeldiagram oftheattahmentofthe

4

-ell

g

^{ofFig.4b,whihisnotsimple.}

3 Conuenes

Let

X F ⁿ

. If

f

^{is a faet of}

X

^{, thenby Def. 3,}

f ˆ

^{is simpleifand only if}

X

ollapses onto

X ⊘ f ˆ

^{. From Prop. 4,we see that heking the simpliity of a}

ell

f

^{reduestothesearhforaollapsesequenefrom}

f ˆ

^toAtt

( ˆ f , X)

^.Wewill

showinSe.4thatthehugenumber(espeiallyin4D)ofpossiblesuhollapse

sequenesneednotbeexhaustivelyexplored,thankstotheonueneproperties

(Th.5andTh.6)introduedinthissetion.

Consider three omplexes

A, B, C

^{. If}

A

^{ollapses onto}

C

^and

A

^ollapses

onto

B

^{,thenweknowthat}

A, B

^and

C

^{havethesametopology.If}furthermore wehave

C B A

^{, itis temptingtoonjeturethat}

B

^ollapsesonto

C

^.

Inthe two-dimensional disrete plane

F ²

, the aboveonjeture is true, we allitaonueneproperty.Butquitesurprisinglyitdoesnotholdin

F ³

(more generally in

F ⁿ , n ≥ 3

^{), and this fat onstitutes indeed one of the prinipal}

diultieswhendealingwithertaintopologialproperties,suhasthePoinaré

onjeture for example. A lassialounter-exampleto this assertionis Bing's

Intheboundaryofan

n

^-faewith

n ≤ 4

^{,thereisnotenoughroomtobuild}

suhounter-examples,andthussomekindsofonuenepropertieshold.

Theorem5 (Conuene1). Let

f

^{be a}

d

^{-fae with}

d ∈ {2, 3, 4}

^,let

A, B f ˆ ^∗

suhthat

B A

^,and

A

^{ollapsesontoa point. Then,}

B

^{ollapses ontoa point}

if andonlyif

A

^{ollapses onto}

B

^.

Theseond onuene theorem may be easily derived from Th. 5 and the

fat that

f ˆ

^{ollapsesontoapoint.}

Theorem6(Conuene2). Let

f

^bea

d

^-faewith

d ∈ {2, 3, 4}

^,andlet

C, D f ˆ ^∗

^suhthat

D C

^,and

f ˆ

^{ollapses onto}

D

^.Then,

f ˆ

^{ollapses onto}

C

^{if and}

only if

C

^{ollapses onto}

D

^.

4 New haraterization of simple ells

In theimage proessing literature,a(binary)digital image isoften onsidered

asaset ofpixels in 2Dorvoxelsin 3D. Apixel isanelementarysquare anda

voxelisanelementaryube,thusaneasyorrespondeneanbemadebetween

thislassialviewandtheframeworkof ubialomplexes.

If

X F ⁿ

and if

X

^{is a pure}

n

^{-omplex,then wewrite}

X ⊑ F ⁿ

. In other words,

X ⊑ F ⁿ

meansthat

X ⁺

^{isaset omposed of}

n

^{-faes (e.g.,pixelsin 2D}

orvoxelsin3D).If

X, Y ⊑ F ⁿ

and

Y X

^,thenwewrite

Y ⊑ X

^.

Notiethat,if

X ⊑ F ⁿ

andif

f ˆ

^isan

n

^-ellof

X

^,then

X ⊘ f ˆ ⊑ F ⁿ

. There is indeed an equivalene between the operation on omplexes whih onsists

of removing (by detahment) a simple

n

^{-ell, and the removal of an 8-simple}

(resp.26-simple,80-simple)pointintheframeworkof2D(resp.3D,4D)digital

topology(see[16,17,7,6℄).

FromProp.4and Th.6,wehavethefollowingharaterizationof asimple

ell,whih doesonlydependonthestatus ofthefaes whiharein theell.

Theorem 7. Let

X ⊑ F ^d

, with

d ∈ {2, 3, 4}

^.Let

f

^{be a faet of}

X

^{, and let}

A =

^Att

( ˆ f , X)

^{.The twofollowing statementshold:}

i)The ell

f ˆ

^issimplefor

X

^{if andonlyif}

f ˆ

^{ollapses onto}

A

^.

ii)Supposethat

f ˆ

^issimplefor

X

^.Forany

Z

^suhthat

A Z f ˆ

^,if

f ˆ

^ollapses

onto

Z

^then

Z

^{ollapses onto}

A

^.

Now,thanksto Th.7,ifwewanttohekwhetheraell

f ˆ

^{issimpleornot,}

itissuienttoapplythefollowinggreedyalgorithm:

Set

Z = ˆ f

^;

Seletanyfreepair

(f, g)

ⁱⁿ

Z \ A

^,andset

Z

^to

Z \ {f, g}

^;

Continueuntileither

Z = A

^{(answeryes)ornosuhpairisfound(answerno).}

Ifthisalgorithm returnsyes,thenobviously

f ˆ

^ollapsesonto

A

^{and byi),}

f ˆ

^{issimple.Intheotherase,wehavefoundasubomplex}

Z

^of

A

^suhthat

f ˆ

ollapsesonto

Z

^and

Z

^{doesnotollapseonto}

A

^{, thus bythenegationof ii),}

f ˆ

isnotsimple.

Thisalgorithmmaybeimplementedtoruninlineartimewithrespettothe

Letusbrieyrealltheframeworkintroduedbyoneoftheauthors(in[3,4℄)for

thinning,inparallel,disreteobjetswiththewarrantythatwedonotalterthe

topologyoftheseobjets.Wefoushereonthetwo-,three-andfour-dimensional

ases, but in fat someof the resultsin this setion are validfor omplexes of

arbitrarydimension.Thisframeworkisbasedsolelyonthreenotions:thenotion

of anessentialfaewhihallowsus todene theoreofafae,andthenotion

ofaritialfae.

Denition 8. Let

X F ⁿ

and let

f ∈ X

^{.We say that}

f

^{is an essential fae}

for

X

^if

f

^{ispreisely the}intersetionof allfaetsof

X

^whihontain

f

^,i.e.,if

f = ∩{g ∈ X ⁺ | f ⊆ g}

^{.Wedenote byEss}

(X)

^{the set omposedof allessential}

faes of

X

^.If

f

^{isanessential faefor}

X

^{,wesay that}

f ˆ

^{isan essentialellfor}

X

^.If

Y X

^andEss

(Y ) ⊆

^Ess

(X)

^,thenwewrite

Y E X

^.

Observethat afaet of

X

^{is neessarilyanessentialfae for}

X

^{, i.e.,}

X ⁺ ⊆

Ess

(X )

^{.Observealsothat,if}

X

^and

Y

^arebothpure

n

^{-omplexes,wehavethat}

Y E X

^whenever

Y

^{isasubomplexof}

X

^.

Denition 9. Let

X F ⁿ

and let

f ∈

^Ess

(X)

^{. The ore of}

f ˆ

^for

X

^{is the}

omplex Core

( ˆ f , X ) = ∪{ˆ g | g ∈

^Ess

(X ) ∩ f ˆ ^∗ }

^.

Proposition10([3℄). Let

X F ⁿ

,andlet

f ∈

^Ess

(X)

^.Let

K = {g ∈ X | f ⊆ g}

^,andlet

Y = X ⊘ K

^{.Wehave: Core}

( ˆ f , X) =

^Att

( ˆ f , Y ∪ f ˆ ) = ˆ f ∩ Y

^.

Corollary 11 ([3,4℄). Let

X F ⁿ

, andlet

f ∈ X ⁺

^{.We have: Core}

( ˆ f , X) =

Att

( ˆ f , X)

^.

Denition 12. Let

X F ⁿ

and let

f ∈ X

^{.We say that}

f

^and

f ˆ

^{are regular}

for

X

^if

f ∈

^Ess

(X )

^andif

f ˆ

^{ollapses onto Core}

( ˆ f , X)

^{.We saythat}

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

^.

A fae

f

ⁱⁿ

X

^{is a maximalritial fae,or an M-ritialfae (for}

X

^{), if}

f

^is

afaet ofCriti

(X )

^.

Inotherwords,

f

^{isanM-ritialfaeifitisritialandnotinludedinany}

otherritialfae.

Proposition 13 ([3,4℄). Let

X F ⁿ

, andlet

f ∈

^Ess

(X )

^.Let

Y = ∪{ˆ g | g ∈ X ⁺

^and

f ⊆ g}

^and

Z = [X ⊘ Y ] ∪ f ˆ

^{.The fae}

f

^isregularfor

X

^{ifand only}

if

f ˆ

^{issimple for}

Z

^.

The following theorem is the most fundamental result onerning ritial

kernels.Weuseit fortheproofsofour mainpropertiesin dimension4orless,

butnotie thatthetheoremholdswhateverthedimension.

Theorem 14 ([3,4℄). Let

n ∈ N

,let

X F ⁿ

. i)The omplex

X

^{ollapses ontoitsritial kernel.}

ii) If

Y E X

^{ontainsthe ritial kernelof}

X

^,then

X

^{ollapses onto}

Y

^.

iii)If

Y E X

^{ontainstheritialkernelof}

X

^,thenany

Z

^suhthat

Y Z E X

ollapses onto

Y

^.

If

X

^{is apure}

n

^{-omplex(e.g., aset of}

3

^{-ells,orvoxels,in}

F ³

),theritial kernelof

X

^{is notneessarily a pure}

n

^{-omplex. The notionof ruial lique,}

introdued in [7℄, allows us to reover apure

n

^-subomplex

Y

^{of an arbitrary}

pure

n

^-omplex

X

^{,under theonstraintthat}

X

^ollapsesonto

Y

^.

Denition 15 ([7℄). Let

X ⊑ F ⁿ

, and let

f

^{be an M-ritial fae for}

X

^{. The}

set

K

^{of all the faets of}

X

^{whih ontain}

f

^{is alled a ruial lique (for}

X

^).

Morepreisely,

K

^{isthe ruialliqueinduedby}

f

^.

(a) (b) () (d) (e) (f)

Fig.6.Cruialliquesin

F ³

(representedinlightgray):(a)induedbyanM-ritial

0

- fae;(b,)induedbyanM-ritial

1

-fae;(d,e,f)induedbyanM-ritial

2

-fae.The onsidered M-ritialfaesare indark gray, theore of theseM-ritial faes(when

non-empty)isrepresentedinblak.

Some3D ruial liquesare illustratedin Fig.6.ByTh. 14 and theabove

denition,ifasubomplex

Y ⊑ X ⊑ F ⁿ

ontainsalltheritialfaetsof

X

^,and

atleastonefaetofeah ruialliquefor

X

^,then

X

^ollapsesonto

Y

^.

Now,letusstatetwopropertiesofruialliqueswhihareessentialforthe

proofofoneofourmain results(Th.21).

Proposition16. Let

X ⊑ F ^d

,with

d ∈ {2, 3, 4}

^,let

f

^{be an M-ritial fae of}

X

^,let

K

^{bethe ruiallique induedby}

f

^,andlet

k

^beanyfaetof

K

^.Let

K ^′

be suhthat

K ^′ ⊆ K \ {k}

^and

K ^′ 6= K \ {k}

^.

Then,

k

^{isasimple faeof the omplex}

[X ⊘ K ^′ ]

^.

Proposition17. Let

X ⊑ F ^d

,with

d ∈ {2, 3, 4}

^,let

f

^{be an M-ritial fae of}

X

^,let

K

^{bethe ruiallique induedby}

f

^,andlet

k

^beanyfaetof

K

^.

Then,

k

^{isnot asimplefae ofthe omplex}

[X ⊘ K] ∪ k ˆ

^.

6 Minimal non-simple sets

C.Ronseintroduedin[28℄theminimalnon-simplesets(MNS)toproposesome

onditionsunderwhihsimplepointsanberemovedinparallelwhilepreserving

evenbeenonsidered in[13,18,19℄.

Themainresultofthissetion(Th.21)statestheequivalenebetweenMNS

andruial liquesindimensions2,3and4.This equivaleneleadstotherst

haraterization of MNS whih an be veried using a polynomial method. In

ontrast,theverydenitionofaMNS(seebelow),aswellastheharaterization

ofTh. 18,involvestheexaminationofallsubsetsofagivenandidateset,e.g.,

asubsetofa

2 × 2 × 2 × 2

^blokin4D.

Let

X ⊑ F ^d

,with

d ∈ {2, 3, 4}

^.Asequene

hk 0 , . . . , k ℓ i

^offaetsof

X

^issaid

tobeasimple sequenefor

X

^if

k 0

^issimplefor

X

^{,andif,forany}

i ∈ {1, . . . , ℓ}

^,

k i

^{is simplefor}

X ⊘ {k j | 0 ≤ j < i}

^{. Let}

K

^{bea set of faetsof}

X

^{. The set}

K

^{is saidto be F-simple (where F stands for faet)for}

X

^if

K

^{is empty, or}

if theelementsof

K

^{anbe orderedasa simplesequene for}

X

^{. Theset}

K

^is

minimalnon-simplefor

X

^{ifitisnotF-simplefor}

X

^{andifallitspropersubsets}

areF-simple. Thefollowingharaterizationwillbeusedin thesequel.

Theorem18 (adaptedfromGauandKong[13℄,theorem3). Let

X ⊑ F ^d

,with

d ∈ {2, 3, 4}

^{, and let}

K ⊆ X ⁺

^{. Then}

K

^{is a minimal non-simple set for}

X

^if

andonly ifthe twofollowing onditions hold:

i)Eah

k

^of

K

^{isnon-simple for}

[X ⊘ K] ∪ ˆ k

^.

ii)Eah

k

^of

K

^issimplefor

[X ⊘ K ^′ ]

^whenever

K ^′ ⊆ K \{k}

^and

K ^′ 6= K \{k}

^.

Forexample, it may beseen that the sets displayed in Fig.6 in light gray

areindeedminimalnon-simplesets.

Th.19isakeyproperty 1

whih isusedtoproveProp.20andTh.23.

Theorem 19. Let

f

^{be a}

d

^{-fae with}

d ∈ {2, 3, 4}

^{, let}

ℓ

^{be an integer stritly}

greaterthan

1

^,let

X 1 , . . . , X ℓ

^be

ℓ

subomplexes of

f ˆ

^{. The twofollowing asser-}

tionsare equivalent:

i)Forall

L ⊆ {1, . . . , ℓ}

^suhthat

L 6= ∅

^,

∪ i∈L X i

^{ollapses ontoa point.}

ii) For all

L ⊆ {1, . . . , ℓ}

^suhthat

L 6= ∅

^,

∩ i∈L X i

^{ollapses ontoapoint.}

Letus nowestablishthelinkbetweenMNSandruialliques.

Proposition20. Let

X ⊑ F ^d

,with

d ∈ {2, 3, 4}

^,let

K

^{beaminimalnon-simple}

set for

X

^{,and let}

f

^{be the} intersetion of allthe elements of

K

^{. Then,}

f

^isan

M-ritial fae for

X

^and

K

^{istheinduedruial lique.}

If

K

^{isaruialliquefor}

X

^{,thenfromTh.18,Prop.16andProp.17,}

K

^is

aminimalnon-simplesetfor

X

^.Conversely,if

K

^{isaminimalnon-simplesetfor}

X

^{,thenbyProp.20,}

K

^{isaruiallique.Thus,wehavethefollowingtheorem.}

Theorem 21. Let

X ⊑ F ^d

,with

d ∈ {2, 3, 4}

^{,and let}

K ⊆ X ⁺

^.Then

K

^isa

minimal non-simpleset for

X

^{if andonlyif itisaruiallique for}

X

^.

1

Notiethatasimilarpropertyholdsin

R ³

,intheframeworkofalgebraitopology,if wereplaethenotionofollapsibilityontoapointbytheoneofontratibility[18,

In the preeding setion, we saw that ritial kernels whih are settled in the

frameworkof abstrat omplexes allowto derivethe notionof aminimal non-

simplesetproposedin theontextofdigitaltopology.Alsoin theframeworkof

digitaltopology,oneoftheauthorsintroduedthenotionofP-simplepoint[2℄,

and provedfor the 3Dase aloal haraterization whih leads to alinearal-

gorithm for testing P-simpliity. In[7℄, westated the equivalene betweenthe

notionof2DP-simplepointsandanotionderivedfromtheoneofruiallique.

Here,weextendthisequivaleneresultupto 4D.

Let

X ⊑ F ⁿ

,andlet

C ⊆ X ⁺

^.Afaet

k ∈ C

^{issaidtobeP-simplefor}

hX, Ci

if

k

^{issimpleforallomplexes}

X ⊘ T

^,suhthat

T ⊆ C \ {k}

^.

Denition 22. Let

X ⊑ F ⁿ

, and let

C

^{be a set of faets of}

X

^{, we set}

D = X ⁺ \ C

^{. Let}

k ∈ C

^{, we saythat}

k

^{is weaklyruial for}

hX, Di

^if

k

^ontainsa

fae

f

^{whih isritialfor}

X

^{,andsuhthatallthefaetsof}

X

^ontaining

f

^are

in

C

^.

Theorem23. Let

X ⊑ F ^d

,with

d ∈ {2, 3, 4}

^,let

C

^{be asetof faets of}

X

^,let

D = X ⊘ C

^.Let

k ∈ C

^{,the faet}

k

^{isP-simplefor}

hX, Ci

^ifandonlyif

k

^isnot

weakly ruial for

hX, Di

^.

Conlusion

We provided in this artile anew haraterization of simplepoints, in dimen-

sions up to 4D, leading to an eient simpliity testing algorithm. Moreover,

we demonstrated that the main onepts previously introdued in order to

study topology-preservingparallel thinning in the framework of digitaltopol-

ogy, namely P-simple points and minimal non-simple sets, may be not only

retrieved in the framework of ritial kernels, but also better understood and

enrihed. Critialkernelsthusappearto onstituteaunifyingframeworkwhih

enompassespreviousworksonparallel thinning.

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