Discrete linear objects in dimension n: the standard model

HAL Id: hal-00354729

https://hal.archives-ouvertes.fr/hal-00354729

Submitted on 21 Jan 2009

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires

Discrete linear objects in dimension n: the standard model

Eric Andres

To cite this version:

Eric Andres. Discrete linear objects in dimension n: the standard model. Graphical Models, Elsevier, 2003, 65 (1-3), pp.92 - 111. �10.1016/S1524-0703(03)00004-3�. �hal-00354729�

Standard Model

Eri ANDRES

a

a

IRCOM-SIC, SP2MI, BP30179, F-86962 Futurosope Cedex, Frane

Abstrat

Anewanalytialdesriptionmodel,alledthestandardmodel,forthedisretization

of Eulideanlinear objets (point,m-at, m-simplex)in dimension n is proposed.

The objets aredened analytiallybyinequalities.Thisallows a globaldenition

independentofthenumberofdisretepoints.Amethodisprovidedtoomputethe

analytialdesriptionforagiven linearobjet.Adisretestandardmodelhasmany

properties in ommon with the superover model from whih it derives. However,

ontrary to superover objets, a standard objet does not have bubbles.A stan-

dardobjet is (n 1) -onneted, tunnel-freeand bubble-free. The standard model

is geometrially onsistent. The standard model is wellsuited formodellingappli-

ations.

Key words: disretegeometry, digitization,dimensionn,simplexe, m-at

Email address: andressi.univ-poitiers.fr(Eri ANDRES).

When working in disrete geometry, aside from onsidering an objet simply

asaset ofdisretepoints,theproblemofdeningdisretegeometrialobjets

arises.A disrete2D linesegmentan bedened as8-onneted, 4-onneted

or even disonneted as a dotted line. There is not a unique way of dening

a disrete objet or of digitizing a Eulidean objet. This problem has been

aroundforfortyyearsandmanydierentdisreteobjetdenitionshavebeen

proposed. One an say that authors have followed three main approahes to

denedisretegeometrialobjets:analgorithmiapproah,atopologialap-

proah and a more reent analytial approah followed in this paper. In the

algorithmi approah [1,10,13,16,21{24,34℄a disrete objet is the result of a

generation algorithm. Historially, the rst approah that has been used, it

has shown a number of limitations.It is oftendiÆult to ontrol the proper-

ties ofthe sodened disreteobjets. Forinstane, the disreteobjetsmight

not be geometrially onsistent : the edge of a 3D triangle is typially not

neessary a 3D line segment or the 3D triangle is not a piee of 3D plane

[21,22℄.It isalsodiÆulttopropose generationalgorithmsfordisreteobjets

in dimension higher than three. Exept for n-dimensional lines [34℄, to the

best authors knowledge, no disrete objet, in dimensions higher than three,

has been algorithmially dened. In the topologial approah, a disrete ob-

jet is typially dened as a lass of objets verifying loal properties, often

topologialin nature[18{20,25,28℄.While itis,by denition, easier toobtain

the desired properties, it is diÆult to be sure with suh an approah, that

thelass ofobjetsdened byagivenset ofpropertiesisnot largerthan what

is initiallyexpeted. A third, more reent approah, denes a disreteobjet

manyadvantages suhas providinga ompat denition(independentonthe

numberof pointsformingthe disreteobjet), aglobal ontrolof the disrete

objet.Ithasalsoanadvantagethatisnotimmediatlyvisiblewhenone isnot

familiar with this approah. It allows a good ontrol of the loal topologial

properties of the objet. The many links with mathematial morphologyare

also an interesting property of some analytially dened models suh as the

superover model [7,19,26,29,31,33℄. One of the main advantages is that it is

relativelyeasytodenedisreteobjetsinanarbitrarydimension[3,4,7,30,32℄.

The standard modelintroduedin the following pagesis analytiallydened.

A new analytialdesription modelfor all linear objets indimension n (dis-

rete points, m-ats and geometrial simplies) is presented in this paper.

The analytial model is alled the standard model. The names derives from

the name given by J. Franon [18℄ to (n 1)-onneted analytial disrete

3D planes (see also[4℄for general detailson disreteanalytial hyperplanes).

To the best authors knowledge, it is the rst time that a disrete model is

proposedthatdenes alargelass ofdisreteobjetsinarbitrarydimensions.

The standard modelisalled adisreteanalytialmodelbeausethe disrete

objets(points,m-ats,simplies)aredenedanalytiallybyinequalities.The

analytialdenitionisindependentofthe numberofdisretepointsoftheob-

jet. Forinstane, a3D standard triangle isdened by 17or less inequalities

independently of itssize.

Themodel wepropose hasmanyinterestingproperties.Themodelisgeomet-

rially onsistent: for instane, the verties of a 3D standard polygon are 3D

standardpoints,theedgesofa3Dstandard polygonare3D standardlineseg-

mentsand the 3D standard polygon is a piee of a3D standard plane. It has

Andres and Barneva [12℄ and therefore is (n 1)-onneted and tunnel-free.

In 3D, (n 1)-onnetivity in our notations orresponds to the lassial 6-

onnetivity. Contrary to the superover model, from whih it derives, the

standardobjetsarebubble-free.Oneoftheproblemsofthe superovermodel

isthatitisnot topologiallyonsistent.Asuperoverm-atisalways(n 1)-

onneted but sometimes it has simple points (loated on so-alled bubbles

on the objet). This makes the model diÆult to use in pratie [14,15℄. For

instane, a superover of a Eulidean nD point an be omposed of any 2 i

disrete points, 0 i n. A standard m-at is almost idential to the su-

perover m-at, it remains (n 1)-onneted and tunnel-free, exept for the

simple points in the bubbles that are removed. The standard digitization of

a nD Eulidean point is always omposed only of one disretepoint.Finally,

the standard model has a very important property in the framework of dis-

rete modelling: St(F [G)=St(F)[St(G). This means that, for instane,

the denition of the standard 3D polygon is suÆient todene the standard

model of an arbitraryEulidean polygonal 3D objet.

The denition of the standard model is derived from the superover model

[2,5,6,14,15,31,7℄.Astandard objetisobtained by asimplerewritingproess

of the inequalities dening analytially asuperover objet [7℄. A superover

linear objet is dened by a set of inequalities"

P

n

i=1 a

i X

i a

0

". The simple

points in the bubbles are points that verify "

P

n

i=1 a

i X

i

= a

0

". In order to

remove the simple points, and thus bubbles, some of the inequalities need

simply to be rewritten into "

P

n

i=1 a

i X

i

< a

0

". The seletion of inequalities

that are modied is based on an orientation onvention. Depending on the

orientation of the half-spae, the orresponding inequality is modied ornot.

superovermodelonwhihthestandardmodelisbased.Insetion3the stan-

dard model is introdued and dened. We start, in setion3.1, by explaining

why suh a \heavy" mathematial mahinery is neessary to dene (n 1)-

onneteddisreteobjets. We show inpartiularwhy alassial,misleading,

approah does not work. In setion 3.2, we explain the basi ideas behind

the standard model. In setion 3.3, we introdue the orientation onvention

that forms the basis of the denition of the standard model. The standard

model is dened for all linear primitives in dimension n in setion 3.4. The

properties of the standard primitives, espeially the tunnel-freeness and the

(n 1)-onnetivity,arepresented insetion3.5.Insetion4,weexaminethe

dierent lasses of standard linear objets to see how the denition is trans-

lated in pratie and how the dierent inequalities dening the objets are

established. Conlusion and several perpetives are presented insetion 5.

2 Preliminaries

2.1 Basi notations in disrete geometry

Mostofthefollowingnotationsorrespond tothosegiven byCohenandKauf-

manin[14,15℄andthosegiven byAndresin[7℄.Weprovideonlyashortreall

of these notions.

Let Z n

be the subset of the nD Eulidean spae R n

that onsists of all the

integer oordinate points.A disrete (resp. Eulidean)point is an elementof

Z n

(resp. R n

). A disrete (resp. Eulidean) objet is a set of disrete (resp.

Eulidean) points.A disrete inequality isaninequalitywith oeÆients inR

Fig. 1. Triangle T = S 2

P 0

;P 1

;P 2

, edge S 1

P 0

;P 1

, straight line

A 1

P 0

;P 1

andhalf-spaeE

P 0

;P 1

;P 2

;P 2

fromwhih weretain onlythe integeroordinatesolutions.A disrete analyt-

ial objet is a disrete objet dened by a nite set of disrete inequalities.

An m-atis aEulidean aÆnesubspae of dimension m.

Let us onsider a set P of m + 1 linearly independent Eulidean points

P 0

;:::;P m

. We denote A m

(P) the m-at indued by P (i.e. the m-aton-

taining P). We denote S m

(P) the geometrial simplex of dimension m in

R n

indued by P (i.e. the onvex hull of P). For S = S m

(P) a geometrial

simplex, we denote S = A m

(P) the orresponding m-at. For a n-simplex

S = S n

(P), we denote E(S;P i

) the half-spae of boundary A n 1

(P nP i

)

that ontains P i

(see gure 1).

We denote p

i

the i-th oordinate of a point or vetor p. Two disrete points

p and q are k-neighbours, with 0 k n, if jp

i q

i

j 1 for 1 i n, and

k n P

n

i=1 jp

i q

i

j.Thevoxel V(p)R n

ofadisretenDpointpisdened

by V(p) = h

p

1 1

2

;p

1 +

1

2 i

h

p

n 1

2

;p

n +

1

2 i

. For a disrete objet F,

V(F)= S

p2F

V(p).Wedenote n

the setof allthe permutationsoff1;:::;ng.

LetusdenoteJ n

m

the set ofallthe stritlygrowingsequenes ofm integersall

between 1 and n: J n

m

=fj 2Z m

j1j

1

<j

2

<:::<j

m

ng. This denes a

Let us onsider an objet F in the n-dimensional Eulidean spae R n

, with

n>1.

The orthogonal projetion is dened by:

i

(F)=f(q

1

;:::;q

i 1

;q

i+1

;:::;q

n

)jq 2R n

g; for 1in;

j

(F)=(

j

1 Æ

j

2

ÆÆ

j

m

)(F); for j 2J n

m :

The orthogonal extrusionis dened by:

"

j

(F)= 1

j (

j

(F)),for j 2J n

m :

Example:LetusonsiderthesetofpointsP =fP 0

(0;0;0);P 1

(9;1;1);P 2

(3;8;4)g.

The orresponding simplex T =S 2

(P) is a3D triangle.The orthogonal pro-

jetion

2

(T)=S 2

(

2

(P))=S 2

(f(0;0);(9;1);(3;4)g) is a2D triangle.The

orthogonal extrusion "

2

(T) = f(0;t;0);(9;t;1);(3;t;4)jt 2Rg is a 3D Eu-

lideanobjet dened by 3 half-spaes.

We dene anaxis arrangement appliation r

j

, for j 2J n

m , by:

r

j :R

n

!R n

x7!

x

j ( 1)

;x

j ( 2)

;:::;x

j (n)

where the permutation

j 2

n

isdened by:

j

= 8

>

>

>

>

>

>

<

>

>

>

>

>

>

:

for 1im;

j (j

i )=i

else, for m<in;

j (k

r )=i

r r+1 r s

s m. The axis arrangement appliation has been speially designed so

that it veries the two following properties:

j

(F)=

( 1;2;:::;m)

r 1

j (F)

and

"

j

(F)=r

j

"

(1;2;:::;m)

r 1

j (F)

forall F in R n

and j 2J n

m .

Example: Let us onsider the 5D point P (1;2;3;4;5) and j = (2;4) 2 J 5

2 .

The orresponding axis arrangement appliation is dened by r

( 2;4)

: x 7!

(x

3

;x

1

;x

4

;x

2

;x

5

)andr 1

(2;4)

:x7!(x

2

;x

4

;x

1

;x

3

;x

5

).Theorthogonalprojetion

veries

( 2;4)

(P)=

( 1;2)

r 1

( 2;4) (P)

=

( 1;2)

(2;4;1;3;5)=(1;3;5):

The orthogonal extrusionveries "

(2;4)

(1;3;5)=r

( 2;4)

"

(1;2)

r 1

(2;4) (P)

=r

(2;4)

"

(1;2)

(2;4;1;3;5)

=r

(2;4)

1

( 1;2)

(1;3;5)

and therefore

"

(2;4)

(1;3;5)=r

( 2;4)

(f(t;u;1;3;5)j(t;u)2R 2

g)=f(1;t;3;u;5)j(t;u)2R 2

g.

2.2 Geometri properties of the Superover

A disrete objet G is a over of a Eulidean objet F if F V(G) and

8p2G;V(p)\F 6=?.ThesuperoverS(F)ofaEulideanobjetF isdened

by S(F) =fp2Z n

jV(p)\F 6=?g (see Figure 2a). S(F) is by denition a

over of F. It is easy to see that if G is a over of F then G S(F). The

superoverofF anbedenedindierentways:S(F)=

F B 1

1

2

\Z n

=

n

p2Z n

d 1

(p;F) 1

2 o

(see Figure 2b) where B 1

(r) if the ball entered

on the origin, of radius r for the distane d 1

. This links the superover to

mathematialmorphology[29,31,7,26℄.

Thesuperoverhas manyproperties.Letusonsider twoEulideanobjetsF

and G, and a multi-index j 2 J n

m

, then: S(F) = S

2F

S( ), S(F [G) =

S(F)[ S(G), if F G, then S(F) S(G). These properties are well

known [14,15℄. The following properties are morereentand are useful inthe

framework of this paper: S(F G) = S(F)S(G), r

j

(S(F)) = S(r

j (F)),

j

(S(F))=S(

j

(F))and "

j

(S(F))=S("

j

(F))=r

j (Z

m

S(

j

(F)))[7℄.

Denition 1 (Bubble)

A k-bubble, with 1 k n, is the superover of a Eulidean point that has

exatly k half-integeroordinates.

Ahalf-integerisa reall+ 1

2

,withl aninteger. Ak-bubbleisformedof 2 k

dis-

retepoints.A 2-bubblean beseen ingure2a(markedby the blak irle).

The two white dots are what we all here "simple" points.This orresponds

to anextension of the notion of simple pointsthat ts a superover simplex.

A point P belongingto the superover simplex S is said to bea simple point

if it isa simplepointfor S with the lassial denition given insetion 2.1.

Denition 2 (Bubble-free)

Theoverofanm-atissaidtobebubble-freeifithasnok-bubblesfork>m.

The over of a simplexS is said to be bubble-free ifS is bubble-free.

forkm;are disretepointsthatarepart ofalltheoversof F.Ifweremove

any of these points, the disrete objet is not a over anymore. In the k-

bubbles, for k > m; there are disrete points that are \simple" points. The

aimof thispaperistopropose disreteanalytialobjetsthat arebubble-free

and(n 1)-onneted by removingsomeofthe simplepoints.Ingure2a,by

removing one of the two simple points, we obtain a bubble-free, 1-onneted

disrete2D line segment.

Lemma 1 A disrete point p belongs to a k-bubble, k >m, of the superover

of an m-at F if and only if there exists a point 2 F with k half-integer

oordinates suh that p2S( ).

The proof of this lemma isobvious.

3 Standard Model

The aim of this paper is to propose a new over lass, alled the standard

over. The standard over is so far only dened for linear objets in all di-

mensions. The disrete analytial model has been designed to onserve most

of the properties of the superover, to be bubble-free and (n 1)-onneted.

Thesuperovermodelhas almostallthepropertiesweare lookingfor:tunnel-

freeness, (n 1)-onnetivity, stability forunion, et. The onlyproperty that

ismissingisthe bubble-freeness. Some superoverobjets have simplepoints.

The model is therefore not topologially onsistent and this is a problem for

several appliations suh as, for instane, polygonalization. For this reason

several attempts have been made to modify the superover disretization by

tion thatsuhattempts an'twork.In ourapproah,presented insetion3.2,

we explain how, by studying the analytial desription of linear objets, it is

possibletoremoveseletivelythe simplepointsinthe superovermodelwhile

preserving the modeling properties. In the setion that follow the standard

model and its properties are introdued.

3.1 What does not work with the lassial approah

Several unsuessful attempts havebeen made todene disreteobjetsthat

have superover type modeling properties with bubble-freeness and (n 1)-

onnetivity properties[27,14,15℄. All these ideas basiallymodify, invarious

ways, the denition of a voxel in order to avoid bubbles. We give here a

simple suh example and show why it does not work that way (see [14℄ for

some other examples). In Figure 3, the pixel denition has been hanged. A

pixelisnowformedoftheSWvertex(blakdisk),thetwoorrespondingedges

(bold edges) and its' interior. The three other verties and two other edges

donot belong tothe pixel.This denition derivesthat S

p2Z

nV(p)=R n

with

V(p)\V(q)=? for p6=q:The disretisation of adisreteline isneessarily

bubble-free. However, aswe see in gure 3,the disretised line x

1 x

2

=0 is

not 1-onneted. In fat, it has been shown as early as in 1970 [27℄, that no

hangeinthedenitionofthepixelorvoxelanleadtoaorretsolution.This

means that a simple pixel denition modiation avoids bubbles but reates

primitivesthataren'ttopologiallyonsistent.Thismakessuhamodeluseless

forappliationssuhaspolygonalization.Tunnel-freenessproperty isalsolost

with suhan approah.

1 2 1 2

modied pixeldenition.

Fig.4.Standardand Superoverstraight line.Theblakpointsbelongto bothline.

The whitepoint belongsonlytothe superover.

3.2 Standard modelapproah : a modiation of the superover denition

The disrete analytialdesription of the superover of a linear onvex is de-

nedasintersetion ofhalf-spaesdened by disreteinequalities P

n

i=1 a

i x

i

a

0

[2,5{7℄. A linear onave objet is simply onsidered asunion of onvexes.

The orientation of eah half-spae is heked with an orientation onvention

and dependingonit, itsinequality \ P

n

i=1 a

i x

i a

0

"remainsunhanged oris

replaed by \ P

n

i=1 a

i x

i

<a

0

".