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A generalized preimage for the digital analytical hyperplane recognition
Martine Dexet, Eric Andres
To cite this version:
Martine Dexet, Eric Andres. A generalized preimage for the digital analytical hyperplane recognition.
Discrete Applied Mathematics, Elsevier, 2009, 157 (3), pp.476-489. �10.1016/j.dam.2008.05.030�. �hal- 00354684�
hyperplane reognition
M. Dexet
a,
∗
a
LIRMM- CNRS,161 rue Ada,34392 Montpellier Cedex 5, Frane
E. Andres
b
b
SIC,Bât. SP2MI, bvd Marie et Pierre Curie,BP 30179, 86962 Futurosope
Chasseneuil Cedex, Frane
Abstrat
Anewdigitalhyperplanereognitionmethodispresented.Thisalgorithmallowsthe
reognitionofdigitalanalytialhyperplanes,suhasNaive,StandardandSuperover
ones. The priniple is to inrementally ompute in a dual spae the generalized
preimage of the ball set orresponding to a given hypervoxel set aording to the
hosendigitization model. Eah point inthis preimage orrespondsto a Eulidean
hyperplane the digitization of whih ontains all given hypervoxels. An advantage
of thegeneralized preimage isthatit doesnot dependon thehypervoxel loations.
Moreover, the proposed reognition algorithm does not require the hypervoxels to
be onneted or ordered inanyway.
Key words: Digital geometry,hyperplane reognition,generalized preimage
1 Introdution
In digital geometry, objets are usually onsidered as digital point or hyper-
voxel (pixels in 2D and voxels in 3D) sets. Indeed, this is the strutural de-
ompositionmostly usedtostoredigitalinformation.A drawbakof thiskind
of representation is that it does not provide any information on the shape
or topology of digital objets. Another way of obtaining the desription of
∗
Correspondingauthor.
Email addresses: Martine.Dexetlirmm.fr (M.Dexet),
andressi.univ-poitiers.fr (E.Andres).
hyperplane reognition, onsists of determining if a digital point set forms a
hyperplane segment, that is ahyperplanebounded region.
Thereognitionproblemhas sofarmainlybeen studiedindimensions2and3
(see[1℄foranoverviewon2Dreognitionalgorithms),withvariousapproahes
suh as linear programming tehniques [2,3℄, omputational geometry meth-
ods [46℄ or preimage omputation based algorithms [7,8℄. Very few papers
handle the problem in arbitrary dimensions [9,10℄. Computational and ef-
ieny aspets of digital hyperplane reognition problems are investigated
in[11℄.
The present paper is an extension of [8℄ in whih we propose a generalized
approah for the reognition of digital analytial hyperplanes suh as Naive,
StandardandSuperoverhyperplanesusinggeneralizedpreimages.Informally,
the preimage [12℄ of a hypervoxel set onsists of all Eulidean hyperplanes
the digitization of whih ontains the given hypervoxels. More preisely, the
preimage of a hypervoxel set is omputed in a dual spae where eah point
is mapped onto a Eulidean hyperplane. Preimage omputation algorithms
depending on the hypervoxel loations have been proposed in dimensions 2
and 3 [7,13℄.
In this work, we perform the reognition of digital analytial hyperplanes
by omputing the set of Eulidean hyperplanes whih interset the ball set
assoiatedtoagivenhypervoxelsetaordingtothehosendigitizationmodel.
In order to do that, we inrementally ompute the generalized preimage of
the balls orresponding to the hypervoxels. This preimage is dened in any
dimension and is independent of the hypervoxel onnetivity and loation.
More preisely,itisomputed fromthe dualof the ballorrespondingtoeah
hypervoxel. Indeed,eah pointinthis dual objet orresponds toaEulidean
hyperplane whih uts the ball orresponding to the hypervoxel. Hene, a
major part of this paper is devoted to determining the formulas desribing
the dual of a polytope in order to ompute the one orresponding to the
balls assoiated to an analytial digitization model. First, a positive and a
negative extrusion are dened. Then, we show that the dual of a polytope
an be omputed from the extrusions of the dual of its verties. Finally, the
intersetion of allball duals forms the generalized preimage. The reognition
proess onsists therefore simply in omputing the generalized preimage of
a ball set orresponding to a hypervoxel set (i.e. omputing the dual of a
ballset orrespondingto a hypervoxel set). More preisely, we start with the
dual of a ball orresponding to a hypervoxel and add the duals of the balls
orrespondingto the otherhypervoxels aslong as the generalizedpreimage is
not empty.
InSetion2,weintroduesomenotationsanddenitionsaswellastheNaive,
we determine the dual of a polytope and introdue the notion of generalized
preimage of a polytope set. Then, we explain in Setion 4 how our digital
analytial hyperplane reognition algorithm works. We espeially fous on
the Naive,Standard and Superover hyperplaneases. Conlusionand future
works are proposed inSetion 5.
2 Preliminaries
In this setion, we rst propose some notations and give the denitions of a
hypervoxel and a ball. Then, we present four digitization analytial models
onsidered in this work: the Naive and losed Naive models, the Standard
modeland the Superover model.
2.1 Notations and denitions
Let n ∈ Z, n > 0. In the following, we will denote by En the lassial
n-dimensional Eulidean spae, and by J1, kK the subset of integer values {1, . . . , k} ⊂ Z. Moreover, a point with integer-valued oordinates p ∈ Zn
willbe alled adigital point.
We dene anα-hyperube,α ∈R, as follows:
Denition 1 (Hypervoxel) The hypervoxel (or n-dimensional ube) en- tered on the digital point (c1, . . . , cn) ∈ Zn, is the set of points (x1, . . . , xn) ∈ Rn verifying
∀i∈J1, nK, ci− 1
2 ≤xi ≤ci+1 2
Hypervoxels indimensions 2 and 3 are respetively alledpixels and voxels.
Denition 2 (Ball) Let d be a distane in Rn. Then, the ball Bd(c, r) with
enter c∈Rn and radius r ∈R is dened by
Bd(c, r) ={x∈Rn|d(c, x)≤r}
2.2 Disrete analytial models
Inthis work,westudy fourdigitalanalytialmodels:the Naivemodel [14,15℄,
the losed Naive model [16℄, the Standard model [17℄ and the Superover
tizationof Eulidean objets. Moreover, adistane and a ballis assoiatedto
eahmodel.
Inthis setion,we givefor eah modelthe denition ofthe digitalhyperplane
(orn-dimensionalplanes)anddesribepreiselythedigitizationofaEulidean hyperplane aording tothe distane and the ballassoiated to the model.
2.2.1 The Naive models[14,16℄
NaiveandlosedNaivehyperplanesaredenedanalytiallyasfollows(seeFig-
ure 1):
Denition 3 (Naive Hyperplane [14℄) TheNaivehyperplanewithparam-
eters (c0, . . . , cn)∈Rn+1 is the set of points (x1, . . . , xn)∈Zn verifying
−max1≤i≤n|ci|
2 ≤c0+
n
X
i=1
cixi < max1≤i≤n|ci| 2
where c1 ≥ 0, or c1 = 0 and c2 ≥ 0, or ..., or c1 =c2 =. . .=cn−1 = 0
and cn ≥0.
Denition 4 (Closed Naive Hyperplane [16℄) The losed Naive hyper-
planewithparameters(c0, . . . , cn)∈Rn+1 isthesetofpoints(x1, . . . , xn)∈Zn
verifying
−max1≤i≤n|ci|
2 ≤c0+
n
X
i=1
cixi ≤ max1≤i≤n|ci| 2
(a) (b)
Fig. 1.Examples of Naive and losed Naive hyperplanes in dimension 2: (a)Naive
line,(b)Closed Naiveline.
Remark 5 Let p = (x1, . . . , xn) ∈ Rn and p′ = (x′1, . . . , x′n) ∈ Rn. The dis- tane assoiated to the Naive modelsis the distane d1 dened by
d1(p, p′) =
n
X
i=1
|xi−x′i|
and the orresponding ball is Bd1(c,12), c∈ Zn. For instane in dimension 2, the ballBd1(c,12) is a regular rhombus.
theentersofallballswhihareintersetedbythehyperplane(seeFigure2b),
whereastheNaiveoneonsistsoftheentersofallballsutbythehyperplane
exept when a ball vertex is interseted (see Figure 2a). In this ase, several
hypervoxels adjaent to the orresponding hypervoxel do not belong to the
Naive digitization.This is due tothe fat that one inequality in Denition 4
is strit.
(a) (b)
Fig.2. Illustrationof theballsassoiated to the Naive models: (a)Balls assoiated
to aNaive line,(b)Balls assoiatedto a losed Naive line.
Proposition 6 Let B be a ball Bd1(c′,12), c′ ∈ Zn, and let H be a Eulidean
hyperplane with equation c0 +Pni=1cixi = 0 that passes through a vertex v = (v1, . . . , vn) of B. Moreover, we assume that the rst ci 6= 0 veries ci >0.
Let j ∈ J1, nK suh that |cj| = maxni=1|ci|. Then, if cj > 0 (resp. cj < 0),
the digital point (v1, . . . , vj−1, vj − 12, vj+1, . . . , vn) (resp. (v1, . . . , vj−1, vj +
1
2, vj+1, . . . , vn)) belongs to the Naive digitization of H.
PROOF. Bydenition,adigitalpointp= (x1, . . . , xn)belongingtoaNaive
hyperplane veries the followinginequalities:
−max1≤i≤n|ci|
2 ≤c0+
n
X
i=1
cixi < max1≤i≤n|ci| 2
Sine |cj|= maxni=1|ci|, we wantto determine k∈ {−1,1} suh that
−|cj|
2 =c0+
n
X
i=1,i6=j
civi+cj(vj +1 2k)
Then, sine c0+Pni=1civi = 0,we have
−|cj| 2 = 1
2kcj, k ∈ {−1,1}
Hene, if cj >0,we have
−cj
2 = 1
2kcj, k ∈ {−1,1}
and sowededue that k =−1. Else, if cj >0,we have cj
2 = 1
2kcj, k ∈ {−1,1}
and then we dedue that k = 1.
Proposition6 is illustrated inFigure3.
Fig.3.DigitalpointsbelongingtotheNaivedigitizationofaEulideanlineaording
to theslope of the line(in darkgrey).
2.2.2 The Standard [17℄ and Superover[18,19℄ models
Standard and Superover hyperplanes are dened analytially as follows
(see Figure4):
Denition 7 (Standard Hyperplane [17℄) The Standard hyperplane with
parameters (c0, . . . , cn)∈Rn+1 is the set of points (x1, . . . , xn)∈Zn verifying
−
Pn i=1|ci|
2 ≤c0+
n
X
i=1
cixi <
Pn i=1|ci|
2
where c1 ≥ 0, or c1 = 0 and c2 ≥ 0, or ..., or c1 =c2 =. . .=cn−1 = 0
and cn ≥0.
Denition 8 (Superover Hyperplane [19℄) The Superover hyperplane
with parameters (c0, . . . , cn) ∈ Rn+1 is the set of points (x1, . . . , xn) ∈ Zn
−
Pn i=1|ci|
2 ≤c0+
n
X
i=1
cixi ≤
Pn i=1|ci|
2
(a) (b)
Fig.4.Examples ofStandardandSuperoverhyperplanesindimension2:(a)Stan-
dard line,(b)Superoverline.
Remark 9 Let p = (x1, . . . , xn) ∈ Rn and p′ = (x′1, . . . , x′n) ∈ Rn. The dis- tane assoiated to the Standard and Superover models is the distane d∞
dened by
d∞(p, p′) = sup
i∈J1,nK
|xi−x′i|
and the orresponding ball is Bd∞(c,1), c∈ Zn. For instane in dimension 2, the ballBd∞(c,1) is a pixel.
Hene, the Superover digitization of a Eulidean hyperplane also onsists
of the enters of all hypervoxels whih are interseted by the hyperplane(see
Figure4b),whereas theStandardoneonsistsoftheentersofallhypervoxels
ut by the hyperplane exept when a hypervoxel vertex is interseted (see
Figure 4a). In this ase, several hypervoxels adjaent to this vertex do not
belongtothe Standarddigitization.This isduetothefatthatoneinequality
inDenition 7 isstrit.
Proposition 10 LetB be aballBd∞(c′,1), c′ ∈Zn, andlet H be a Eulidean
hyperplan with equation c0 +Pni=1cixi = 0 that passes through a vertex v = (v1, . . . , vn) of B. Moreover, we assume that the rst ci 6= 0 veries ci >0.
Then, eah digital point (x1, . . . , xn) belonging to the standard digitization of
H veries for alli∈J1, nK:
• xi =vi+12 if ci<0,
• xi =vi− 12 if ci >0,
• xi =vi+12 or xi =vi− 12 if ci = 0.
PROOF. Bydenition,a digitalpointp= (x1, . . . , xn)belongingtoa Stan-
dard hyperplaneveries the following inequalities:
−
Pn i=1|ci|
2 ≤c0+
n
X
i=1
cixi <
Pn i=1|ci|
2
We want to determineki ∈ {−1,1},i∈J1, nK, suhthat
−
Pn i=1|ci|
2 =c0+
n
X
i=1
ci(vi+1 2ki)
that is, sinec0+Pni=1civi = 0,
−
Pn i=1|ci|
2 =
n
X
i=1
1 2kici
Hene, we have
n
X
i=1
(|ci| −kici) = 0
and then
n
X
i=1
(k′ici −kici) =
n
X
i=1
(ki′−ki)ci = 0
with ki′ ∈ −1,1and k′ici ≥0
However, sine ∀i∈J1, nK, (k′i−ki)ci ≥0 we dedue that i∈J1, nK, ki =ki′
that is
• if ci >0then ki =−1,
• if ci <0then ki = 1,
• if ci = 0 then ki =−1 orki = 1.
Proposition10 isillustrated in Figure11.
to theslope of the line(in darkgrey).
3 Dual of a polytope
Inordertodenethedual ofapolytope,weuse adualtransformationsimilar
to the well known Hough transform whih is an eient tool usually used in
image proessing to reognize parametri shapes in an image. A review on
existing variationsof this method ispresented in[20℄.
In the two following setions, we rst dene the parameter spae in whih
our dual transformation is performed as well as the positive and negative
extrusionsofa point.Then,wedesribe thedual ofapolytopeanddenethe
notionofgeneralizedpreimage,whihisthebasisofthe reognitionalgorithm
presented inSetion4.
3.1 Denitions and properties
In this work, we use the n-dimensional parameter spae Pn ⊂Rn, and dene the two funtions DE :En→ Pn and DP :Pn → En by:
DE(x1, . . . , xn) =
(
(y1, . . . , yn)∈ Pn|yn=−
n−1
X
i=1
xiyi+xn
)
DP(y1, . . . , yn) =
(
(x1, . . . , xn)∈ En|xn =
n−1
X
i=1
yixi+yn
)
Informally,eahpointinEn (resp.Pn)istransformed byDE (resp.DP)intoa
hyperplaneinPn(resp.En).Inthe restof thispaper,wewillgeneriallywrite Dual forDE orDP.
Denition 11 (Dual objet) Let O be a subset of Rn. Then,
Dual(O) = [
p∈O
Dual(p)
is alled the dual of O.
Proposition 12 Let O1 and O2 be two subsets of Rn suh that O1 ⊆ O2.
Then
Dual(O1)⊆Dual(O2)
PROOF. Sine O1 ⊆ O2, we dedue that Dual(O2) = Sp∈O2Dual(p) =
hS
p∈O1Dual(p)i∪hSp∈O2\O1Dual(p)i. Then,Dual(O1)⊆Dual(O2).
Moreover, the following properties an be dedued from our denition of the
duality.
Proposition 13 Let O1 and O2 be two subsets of Rn. Then,
Dual(O1∪O2) = Dual(O1)∪Dual(O2)
PROOF. Dual(O1 ∪ O2) = Sp∈O1∪O2Dual(p) =
hS
p∈O1Dual(p)i∪hSp∈O2Dual(p)i=Dual(O1)∪Dual(O2).
Proposition 14 Let O1 and O2 be two subsets of Rn. Then,
Dual(O1∩O2)⊆Dual(O1)∩Dual(O2)
PROOF. Sine O1 ∩ O2 ⊆ O1 and O1 ∩ O2 ⊆ O2, we dedue
that Dual(O1∩O2)⊆Dual(O1) and Dual(O1 ∩ O2) ⊆ Dual(O2). Thus,
Dual(O1∩O2)⊆Dual(O1)∩Dual(O2).
Remark 15 Let p ∈ Rn be a point. The dual of eah point whih lies in
Dual(p) is a hyperplane whih passesthrough p.
Moreover, in order to desribe the dual of a polytope, we need to dene the
positive and negativeextrusions of apointas follows:
Denition 16 (Positive and Negative Extrusions) Let
p= (x1, . . . , xn)∈Rn be a point. The positive extrusion of p is dened
by:
p+={p′ = (x′1, . . . , x′n)∈Rn|∀i∈J1, n−1K, xi =x′i and xn ≤x′n}
In the same way, the negative extrusion of p is dened by:
p−={p′ = (x′1, . . . , x′n)∈Rn|∀i∈J1, n−1K, xi =x′i and xn ≥x′n}
Let O1 and O2 be two subsets of Rn suh that O1 ⊆ O2. Then, O+1 ⊆ O+2
and O−1 ⊆O−2. Moreover, the following properties an be dedued from De-
nition 16.
Proposition 17 Let O1 and O2 be two subsets of Rn. Then,
(O1∪O2)+ =O1+∪O+2
In the same way, (O1∪O2)− =O−1 ∪O2−.
PROOF. (O1∪O2)+ =Sp∈O1∪O2p+ =hSp∈O1p+i∪hSp∈O2p+i=O+1 ∪O+2.
The proof of (O1∪O2)− =O1−∪O−2 isobtained in the same way.
Proposition 18 Let p∈Rn be a point. Then,
Dual(p)+ =Dual(p+)
In the same way, Dual(p)− =Dual(p−).
PROOF. Letusonsiderp= (x1, . . . , xn)∈ En.Then,Dual(p+) =DE(p+) =
[
p′∈p+
Dual(p′) = [
p′=(x′1,...,x′n)∈p+
{(y1, . . . , yn)∈ Pn|yn=−
n−1
X
i=1
x′iyi+x′n}=
{(y1, . . . , yn)∈ Pn|yn ≥ −
n−1
X
i=1
xiyi+xn}= [
p′∈DE(p)
p′+ =DE(p)+ =Dual(p)+.
The proof of Dual(p)− =Dual(p−)an beobtained inthe same way.
Proposition18 isillustrated in Figure6.
3.2 Polytope dual representation
In this work, we need todene the dual of apolytope. An n-polytope, n ∈Z, is dened as follows:
p
Dual(p)
(a)
p Dual(p)
(b)
Fig.6.Positiveandnegativeextrusionsofapointp(half-lines)andtheirdualobjet:
ahalf-spae,(a) Positive extrusionof p,(b)Negative extrusion.
Denition 19 (n-polytope) Let P be a polytope in dimension n, or n-
polytope. Then, there exists a nite set of k half-spaes H = {H1, . . . , Hk}
suh that P = Tki=1Hi, and suh that if Hi is the hyperplane forming
the boundary of the half-spae Hi (or boundary hyperplan of Hi), then
∀i∈J1, kK, Hi∩P 6=∅.
Notations: Let P be an n-polytope, and let H be the orresponding half- spae set. Wedene three subsets of H,denoted H0, H+ and H−,as follows:
• H0 is the half-spae set in H dened by an equation similar to cn+Pn−1i=1 ciXi ≥0orsimilartocn+Pn−1i=1 ciXi ≤0,with(c1, . . . , cn)∈ En.
• H+ is the half-spae set in H dened by an equation similar to Xn≥cn+Pn−1i=1 ciXi, (c1, . . . , cn)∈ En.
• H− is the half-spae set in H dened by an equation similar to Xn≤cn+Pn−1i=1 ciXi, (c1, . . . , cn)∈ En.
Moreover, we denote H0, H+ and H− the three boundary hyperplane sets
orresponding respetively to the half-spae sets H0, H+ and H−.
Proposition 20 Let P be an n-polytope. Then, P =P+∩P−
with
P+= \
H∈(H0∪H+)
H
and
P− = \
H∈(H0∪H−)
H
