Description of the symmetric convex random closed sets as zonotopes from their Feret diameters

0.51 MB·

by  Saïd Rahmani

ModernStochastics:TheoryandApplications3(2016)325–364
DOI:10.15559/16-VMSTA70
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Description of the symmetric convex random closed

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sets as zonotopes from their Feret diameters

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SaïdRahmani ,Jean-CharlesPinoli,JohanDebayle
. ∗
h
t
a ÉcoleNationaleSupérieuredesMinesdeSaint-Etienne,SPIN/LGFUMR
m CNRS5307,158CoursFauriel,42023Saint-Etienne,France
[
[email protected](S.Rahmani),[email protected](J.-C.Pinoli),[email protected](J.Debayle)

1
v Received:24October2016,Revised:7December2016,Accepted:14December2016,
4 Publishedonline:3January2017
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2 Abstract Inthispaper,the2-Drandomclosedsets(RACS)arestudiedbymeansoftheFeret
1
diameter,alsoknownasthecaliperdiameter.Morespecifically,itisshownthata2-Dsymmet-
0
ricconvex RACS can be approximated asprecisely as wewant by some random zonotopes
.
1 (polytopesformedbytheMinkowskisumoflinesegments)intermsoftheHausdorffdistance.
0 Such an approximation is fully defined from the Feret diameter of the 2-D convex RACS.
7 Particularly,themomentsoftherandomvectorrepresentingthefacelengthsofthezonotope
1
approximationarerelatedtothemomentsoftheFeretdiameterrandomprocessoftheRACS.
:
v
Keywords Zonotopes,randomclosedset,theFeretdiameter,polygonalapproximation
i
X
2010MSC 60Dxx,52A22
r
a
1 Introduction
1.1 Contextandobjectives
Thegeometricalcharacterizationofgranularmedia(grains,pores,fibers,etc.)is an
importantissueinmaterialsandprocesssciences.Indeed,severalgranularmediacan
be modeled as randomsets where the heterogeneityof the particles is studied with
a probabilistic approach [12, 24]. In this context, the random closed sets (RACSs)
havebeenparticularlystudied[29,21,8,2]togetgeometricalcharacteristicsofsuch
∗Correspondingauthor.
©2016TheAuthor(s).PublishedbyVTeX.OpenaccessarticleundertheCCBYlicense.
www.i-journals.org/vmsta
326 S.Rahmanietal.
granularmedia. A RACS denotesa randomvariable defined on a probabilityspace
(Ω,A,P)valuedin(F,F),thefamilyofclosedsubsetsofRd providedwiththeσ-
algebraF:=σ F F F X = X K ,whereKdenotestheclassofcompact
subsetsonRd.{I{nap∈roba|bi∩listic6po∅in}tof∈view},thedistributionofaconvexRACSis
uniquelydeterminedfromthe Choquetcapacity functional[23, 16]. However,such
a descriptionis not suitable for explicitly determiningthe geometricalshape of the
RACS. An alternativeway is to describe a RACS by the probabilitydistribution of
real-valuedgeometricalcharacteristics(area,perimeter,diameters,etc.).
1.2 Originalcontribution
Theaimofthispaperistoshowhowsuchcharacteristicscanbeusedtodescribethe
geometricalshape of a convexrandomclosed set in R2. It has already been shown
[25]thatthemomentsoftheFeretdiameterofaconvexrandomclosedsetinR2can
beobtainedbytheareameasuresonmorphologicaltransformsofit.AFeretdiameter
(alsoknownascaliperdiameter)isameasureofasetsizealongaspecifieddirection.
It can be defined as the distance between the two parallel planes restricting the set
perpendiculartothatdirection.
A set X R2 is said to be central symmetric or, more simply, symmetric if it
isequaltothe∈setX˘ := X.NotethattheFeretdiameterisnotsensitivetosucha
centralsymmetrization[2−2]. Indeed,for a nonemptycompactconvexset X R2,
itssymmetrizedset 1(X X˘)(see[21,17])hasthesameFeretdiameterasX⊂.Con-
2 ⊕
sequently,theFeretdiameterofaconvexsetX isnotenoughtofullyreconstructX
(butonlyitssymmetrizedset).However,theFeretdiameterisstillusefultodescribe
theshapeofconvexsetsfortworeasons.Firstly,aconvexsetX anditssymmetrized
set 1(X X˘)sharealotofcommongeometricaldescriptors(perimeter,eccentricity,
2 ⊕
etc.).Secondly,therearemanyapplicationsinwhichsymmetricconvexparticlesare
considered.In this way,the reportedwork is focusedon the symmetricconvexsets
(i.e.,X = 1(X X˘)).Byabusingthenotation,theconditions“nonemptyandcom-
2 ⊕
pact”willbeoftenomittedinthispaper.Inotherwords,withoutexplicitmentioning
ofthecontrary,aconvexsetwillrefertoanonemptycompactconvexset.
Inthispaper,weshowthattheFeretdiameterofarandomsymmetricconvexset
can be used to define some approximationsof it as random zonotopes. The polyg-
onal approximation of a deterministic convex set has already been studied several
times[18,14,4,7].However,inmostcases,theapproximationismadebyusingthe
supportfunction,whichisnotavailableinmostofthegeometricstochasticmodels.
Randompolygonshavealreadybeenstudiedseveraltimes[10,3,20].However,they
aredefinedindifferentwaysandforotherobjectives,andtheyarenotcharacterized
fromtheirFeretdiameters.Inourpointofview,a zonotope(whichis a Minkowski
sum of line segments) is described by its faces (direction and length) and can be
characterizedby its Feretdiameter. We will show thatthe Feretdiameter ofa sym-
metric convexset evaluatedon a finite numberof directionsN > 1 can be used to
define some approximations of it as zonotope. Such zonotope approximations will
begeneralizedtotherandomsymmetricconvexsets.Therefore,arandomsymmetric
convexsetwillbeapproximatedbyarandomzonotope,andsuchapproximationswill
be characterizedfrom the Feret diameterof the randomsymmetricconvexset. The
consideredrandomzonotopewillbeuniquelydeterminedbythelengthsofitsfaces,
Descriptionofarandomsymmetricconvexsetbyarandomzonotope 327
andtheirdirectionswillbeassumedtobeknown.Theapproximationsconsideredare
consistentasN withrespecttotheHausdorffdistance.
→∞
Thisworkis a preliminarystudyin orderto describethegeometricalcharacter-
istics of a population of convex particles in the context of image analysis. Indeed,
suchimagesofpopulationofconvexparticlescanbemodeledbystochasticgeomet-
ric models. In such a model, the projection of a particle represented by a random
convexset. Consequently,thisworkcan be usedto getinformationon suchconvex
particles.In addition,whentheparticlesaresupposedto be symmetric,theyhavea
symmetric2-DprojectionthatcanbefullycharacterizedbytheFeretdiameter.Such
a symmetrichypothesisissuitablein severalindustrialapplicationsinchemicalen-
gineering(gasabsorption,distillation,liquid–liquidextraction,petroleumprocesses,
crystallization,etc.).
An area of applicationis the gas–liquidreactions.Indeedin a such process, the
gasbubblescanbemodeledasanellipsoidthe2-Dprojectionsofwhichareellipses
(see[32,30,5]).Themainareaofapplicationiscrystalsmanufacturing.Indeed,many
crystalsare3-Dzonohedrons,andtheir2-Dprojectionsarezonotopes.Forexample,
the crystalsof oxalateammonium[26, 1], the crystalsof calciumoxalate dihydrate
[31],orthe(L)-glutamicacid[6].Insuchapplications,theconsideredapproximations
coincidewiththerealdata.
1.3 Outlineofthepaper
Thepaperisorganizedasfollows.Thefirstpartisdevotedtothecaseofadetermin-
isticsymmetricconvexsetX.SomepropertiesoftheFeretdiameterarefirstrecalled,
andthenforanyintegerN > 1,anapproximationX(N) ofX asazonotope[11]is
0
described.ItisshownthatthisapproximationisconsistentasN withrespect
→ ∞
totheHausdorffdistance[27].AmoreaccuratezonotopeapproximationX˜(N)ofX
0
that is invariant up to a rotation is also defined with the consistency also satisfied.
ThisapproximationisparticularlyinterestingtodescribethegeometricalshapeofX.
Thesecondpartisdevotedto acharacterizationoftherandomzonotopes.First,
weexploresomepropertiesoftherandomprocessassociatedwiththeFeretdiameter.
Then we study the random zonotopes, define some their classes, and discuss their
descriptions by their faces. Finally, we study the characterization of some random
zonotopesfromtheirFeretdiametersrandomprocess.
Inthelastpart,westudyarandomsymmetricconvexsetX.Weshowthatitcan
still be describedas precisely as we want by a randomzonotopeX(N) and up to a
0
rotationbyarandomzonotopeX(N)withrespecttotheHausdorffdistance.Wegive
∞
thepropertiesoftheseapproximatesandshowthattheycanbecharacterizedfromthe
FeretdiameterrandomprocessofX.Inparticular,themeanandautocovarianceofthe
FeretdiameterrandomprocessofX canbeusedtogetthemeanandthevariancesof
therandomvectorscomposedbythefacelengthsoftheirzonotopeapproximations.
2 DescriptionofasymmetricconvexsetasazonotopefromitsFeretdiameter
TheaimofthissectionistodiscusshowaconvexsetX canbedescribedasazono-
tope. We will show that X can always be approximated as precisely as we want
328 S.Rahmanietal.
by zonotopes and how such zonotopes can be characterized from the Feret diame-
ter of X. First, we need to recall the definition of the Feret diameter and some its
properties.
2.1 Feretdiameterandthesupportfunction
Definition1(Supportfunction). LetX R2 beaconvexset.Thesupportfunction
⊂
ofX isdefinedas
R2 R
fX : x −→ sup x,s =max x,s ,
(cid:12)(cid:12) 7−→ s∈Xh i s∈Xh i
(cid:12)
where , denotesth(cid:12)eEuclideandotproduct.
h· ·i (cid:12)
The support function allows us to fully characterize a convex set. Indeed, any
positivehomogeneousconvexreal-valuedfunctiononR2isthesupportfunctionofa
convexset [27]. Inthe following,wegivesome importantpropertiesofthe support
function.Theproofsareomittedsincetheycanbefoundintheliterature[13,27].
Proposition1(Propertiesofthesupportfunction). LetX R2 beaconvexset.Its
⊂
supportfunctionsatisfiesthefollowingproperties:
1. Positivehomogeneity: r 0, f (rx)=rf (x).
X X
∀ ≥
2. Subadditivity:f (x+y) f (x)+f (y).
X X X
≤
3. f =f +f ,where denotestheMinkowskiaddition.
X Y X Y
⊕ ⊕
4. Ifsisavectorialsimilarityandb R2,thenf (x)=f (s(x))+ x,b .
s(X)+b X
∈ h i
5. Reconstruction:
X = y R2 y,x f (x) . (1)
X
∈ h i≤
x R2
\∈ (cid:8) (cid:12) (cid:9)
(cid:12)
6. If,inaddition,0 X,thenf 0.
X
∈ ≥
7. d (X,Y) = f f , where d denotes the Hausdorff distance, and
H X Y H
k − k∞
istheuniformnormontheunitsphere.
k·k∞
Items 1 and 2 relate the convexity of the support function, and expression (1)
allows the reconstruction of a convex set from its support function. Note that the
positivehomogeneityof the supportfunctioninvolvesthatit canbe completelyde-
terminedontheEuclideanunitsphere.Weadoptthefollowingrepresentationforthe
supportfunctionofX:
R R
hX : θ −→ h (θ)=f (t( sin(θ),cos(θ))),
(cid:12) X X
(cid:12) 7−→ −
(cid:12)
whichisacontinuou(cid:12)sand2π-periodicfunction.
(cid:12)
NotethattheFeretdiameterofaconvexsetX,denotedH ,canbeexpressedby
X
thesupportfunctionas
θ R, H (θ)=h (θ)+h (θ), (2)
∀ ∈ X X X˘
Descriptionofarandomsymmetricconvexsetbyarandomzonotope 329
where X˘ is the usual notation for the symmetric set X. It is easy to see that the
FeretdiameterofX coincideswiththesupportfunctio−nofX X˘,where denotes
⊕ ⊕
theMinkowskisum.Therefore,thefunctionalH issufficienttofullycharacterize
X
thesymmetrizedbody 1(X X˘).NotethatifXisalreadysymmetric,thenH fully
2 ⊕ X
characterizesX.WerecallsomeimportantpropertiesoftheFeretdiameter.
Proposition 2 (Properties of the Feret diameter). Let X be a convex set. Then its
FeretdiameterH satisfiesthefollowingproperties:
X
1. FortowconvexsetsX andY,H =H +H .
X Y X Y
⊕
2. r R, H = rH .
rX X
∀ ∈ | |
3. IfR isarotationandb R2,then θ R, H (θ)=H (θ+η).
η ∈ ∀ ∈ Rη(X)+b X
4. π-periodicity: θ R, H (θ+π)=H (θ).
X X
∀ ∈
5. FortwosymmetricbodiesX andY,H H X Y
X Y
≤ ⇔ ⊆
6. Foranyθ,β [0,2π],
∈
β β+π
H (θ+β) H (θ)+2 sin H θ+ . (3)
X X X
≤ 2 2
(cid:12) (cid:18) (cid:19)(cid:12) (cid:18) (cid:19)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Proof.
1,2,3. AccordingtoEq.(2),thefirstthreeitemscomedirectlyfromProposition1.
4. Theπ-periodicityfollowsfromh (θ)=h (θ+π),θ R.
X˘ X ∈
5. BecauseofthesymmetryofX andY,ifH H ,thenh h .There-
X Y X Y
fore,foranyx R2,f (x) f (x),so y ≤R2 y,x f ≤(x) y
X Y X
R2 y,x f ∈(x) ,andthu≤sX Y byP{ro∈positi|ohn1.5i.≤ }⊆{ ∈
Y
|h i≤ } ⊂
SupposethatX Y.Then x R2, s,x s X s,x s Y
⊂ ∀ ∈ {h i| ∈ }⊂ {h i| ∈ } ⇒
f (x) f (x) h h H H .
X Y X Y X Y
≤ ⇒ ≤ ⇒ ≤
6. Forany(θ,β) R2,letα=β+π,x=t ( sin(θ),cos(θ)),z =t ( sin(θ+
∈ − −
α),cos(θ+α))andy =z+x,sothat
f (y x) f ( x)+f (y),
X X X
− ≤ −
h (θ+α) h (θ+π)+f (y),
X X X
≤
and
2 2
y = sin(θ)+sin(θ+α) + cos(θ)+cos(θ+α)
k k
q
= (cid:0)(2+2 sin(θ)sin(θ+(cid:1)α)+(cid:0)cos(θ)cos(θ+α) (cid:1)
=√q2 1+(cid:0)cos(α) (cid:1)
p
330 S.Rahmanietal.
α
=√2 2cos2
s 2
(cid:18) (cid:19)
α
=2 cos
2
(cid:12) (cid:18) (cid:19)(cid:12)
(cid:12) β (cid:12)
(cid:12) (cid:12)
=2(cid:12)sin (cid:12).
2
(cid:12) (cid:18) (cid:19)(cid:12)
(cid:12) (cid:12)
Usingtheformu(cid:12)las (cid:12)
(cid:12) (cid:12)
α α
sin(θ)+sin(θ+α)=2sin θ+ cos ,
2 2
(cid:18) (cid:19) (cid:18) (cid:19)
α α
cos(θ)+cos(θ+α)=2cos θ+ cos
2 2
(cid:18) (cid:19) (cid:18) (cid:19)
andtakingη Rsuchthaty = y t ( sin(η),cos(η)),wehave
∈ k k −
2sin(θ+ α)cos(α)
sin(η)= 2 2 ,
y
k k
2cos(θ+ α)cos(α)
cos(η)= 2 2 .
y
k k
Let s be the sign of cos(α). Then sin(η) = ssin(θ + α) and cos(η) =
2 2
scos(θ+ α).
2
Finally,η θ+ β+π,θ+ β+π +π ,anditcanbeexpressedas
∈{ 2 2 }
β
h (θ+β π) h (θ+π)+2 sin h (η).
X X X
− ≤ 2
(cid:12) (cid:18) (cid:19)(cid:12)
(cid:12) (cid:12)
ThisresultistrueforanyconvexsetX,in pa(cid:12)(cid:12)rticular,fo(cid:12)(cid:12)rY = 1(X X˘).
2 ⊕
However,h = H ,andthenbytheπ-periodicityoftheFeretdiameterwe
y X
have
β β+π
θ β [0,2π], H (θ+β) H (θ)+2 sin H θ+ .
X X X
∀ ∈ ≤ 2 2
(cid:12) (cid:18) (cid:19)(cid:12) (cid:18) (cid:19)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
The Feret diameter can also be related to the mixed area [27] by using a line
segment as a structural element. Indeed, using the Steiner formula [27] with two
convexsetsX andY,wehave
A(X Y)=A(X)+2W(X,Y)+A(Y),
⊕
whereW(X,Y)denotesthemixedareabetweenXandY.Themixedareafunctional
W(, )isasymmetricmapping,whichisalsohomogeneousinitstwovariables(see
· ·
[19, 27]fordetails). Itisoftenused todescribesome morphologicalcharacteristics
of a convex set X by using different structuring elements. For instance, if X is a
boundedconvexsetandB istheunitdisk,thenW(X,B) = 1U(X),whereU(X)
2
Descriptionofarandomsymmetricconvexsetbyarandomzonotope 331
denotes the perimeter of X. Let X be a boundedconvexset, and S be a unit line
θ
segmentdirectedbyθ.Then
1
W(X,S )= H (θ). (4)
θ X
2
Theproofisomittedsinceitconsistsinasimpledrawingandcanbefoundinthe
literature[25,19].
Remark 1. This relation is very importantbecause it involves an interpretation of
themixedareaofaconvexsetwiththeMinkowskiadditionoflinesegmentsfromits
Feretdiameter.Indeed,foranyθ ,θ [0,π]andα ,α R ,
1 2 1 2 +
∈ ∈
A(X α S α S )=A(X α S )+2W(X α S ,α S )
⊕ 1 θ1 ⊕ 2 θ2 ⊕ 1 θ1 ⊕ 1 θ1 2 θ2
=A(X)+α H (θ )+α H (θ )
1 X 1 2 X⊕α1Sθ1 2
=A(X)+α H (θ )+α H (θ )+α H (θ ).
1 X 1 2 X 2 2 α1Sθ1 2
However,α H (θ )=W(α S ,S )=A(α S α S ).Then,
2 α1Sθ1 2 1 θ1 θ2 1 θ1 ⊕ 2 θ2
1
W(X,α S α S )= α H (θ )+α H (θ ) .
1 θ1 ⊕ 2 θ2 2 1 X 1 2 X 2
This result can be easily generalized by i(cid:0)nduction to any Minkow(cid:1)ski sum of line
segments: n 1, i=1,...,n, α R ,θ R,wehave
i + i
∀ ≥ ∀ ∈ ∈
n n
1
W X, α S = α H (θ ). (5)
i θi 2 i X i
!
i=1 i=1
M X
Relation (5) has an important kind of linearity. Indeed, it implies formulae for the
computation of the mixed area between a convex set and a symmetric body from
theirFeretdiameter(seeRemark3).
2.2 Approximationofasymmetricconvexsetbya0-regularzonotope
NowwegivesomepropertiesoftheFeretdiameterofaconvexsetanditsconnection
with the mixed area. Here the zonotope will be defined and particularly the class
of the 0-regular zonotopes, some properties of the zonotopes will be discussed. In
particular, we will show how a symmetric convex set can be approximatedby a 0-
regularzonotopeaspreciselyaswewant.
Let denotetheclassofallsymmetricconvexsetsofR2,wherethesymmetryis
giveninCthesenseofMinkowski:X = 1(X X˘).LetS betheunitlinesegment
2 ⊕ 0
[ 1,1],andS its rotationbytheanglet [0,π[. Considernowthe convexsetX
−2 2 t ∈
suchthat
n
X = α S , n N , i=1,...n, α R , θ [0,π[. (6)
i θi ∈ ∗ ∀ i ∈ + i ∈
i=1
M
NotethatXisacompactconvexsymmetricpolygonwithatmost2nfaces,wherefor
alli=1,...,n,α isthelengthofthetwofacesofX orientedbyθ .Itiseasytosee
i i
thateverycompactconvexsymmetricpolygonhasanevennumberoffacesandcan
be representedas (6) up to a translation. Furthermore,note that X has a nonempty
interiorifandonlyifn>1.
332 S.Rahmanietal.
Definition 2 (Zonotopes). Any compact convex symmetric polygon such as (6) is
called a zonotope.ForN N , (N) denotesthe set of allzonotopeswith at most
∗
∈ C
2N faces:
N
(N) = α S α RN, θ [0,π[N ,
C ( i θi ∈ + ∈ )
i=1
M (cid:12)
whereα=t (α ,...α )andθ =t (θ ,.(cid:12)..θ ).
1 N 1 N
Several geometric characteristics and properties of zonotopes can be easily ex-
pressedfromrepresentation(6).
Proposition3(Geometricalcharacterizationofzonotopes). LetN N ,andX =
∗
N α S beanelementof (N).LetH beitsFeretdiameterfun∈ction,U(X)its
i=1 i θi C X
perimeter,andA(X)itsarea.Then
L
N
η R, H (η)= α sin(η θ ) , (7)
X i i
∀ ∈ −
i=1
X (cid:12) (cid:12)
(cid:12) (cid:12)
N
U(X)=2 α , (8)
i
i=1
X
N N
1
A(X)= α α sin(θ θ ) . (9)
i j i j
2 −
i=1j=1
XX (cid:12) (cid:12)
(cid:12) (cid:12)
Proof.
(6) Forany(β,η) R2,thesupportfunctionofthelinesegmentS inthedirec-
β
∈
tionηis
h (η)= max t cos(β)sin(η)+sin(β)cos(η)
Sβ t [ 1,1] −
∈−2 2
(cid:8) (cid:0) (cid:1)(cid:9)
= max tsin(β η)
t [ 1,1] −
∈−2 2
1 (cid:8) (cid:9)
= sin(β η)
2 −
H (η)= si(cid:12)n(β η) .(cid:12)
⇒ Sβ (cid:12) − (cid:12)
Thenrelation(7)follows(cid:12)fromPropo(cid:12)sitions2.1and2.2.
(cid:12) (cid:12)
(7) IfX isapolygonof2N facesoflengthα ,i=1,...,N,theperimetercanbe
i
obtainedbyaddingupthefacelengths.
(8) Forthearea,theresult(9)isprovedbyinductiononN:forN = 1,X = S
θ1
andA(X) = 0,sothat(9)issatisfied.Supposethat(9)istrueforn N and
≤
letusshowthatitistrueforN +1.SinceX =( N α S ) α S ,
i=1 i θi ⊕ N+1 θN+1
bytheSteinerformulawehave
L
N N
A(X)=A α S +2W α S ,α S .
i θi i θi N+1 θN+1
! !
i=1 i=1
M M
Descriptionofarandomsymmetricconvexsetbyarandomzonotope 333
Then,by(4),
N
2W α S ,α S =α H (θ ),
i θi N+1 θN+1! N+1 LNi=1αiSθi N+1
i=1
M
andfinally,bytheheredityassumptionand(7),
N N
1
A(X)= α α sin(θ θ ) +α H (θ )
2 i j i− j N+1 LNi=1αiSθi N+1
i=1j=1
XX (cid:12) (cid:12)
N N (cid:12) (cid:12) N
1
= α α sin(θ θ ) +α α sin(θ θ )
i j i j N+1 i N+1 i
2 − −
i=1j=1 i=1
XX (cid:12) (cid:12) X (cid:12) (cid:12)
N+1N+1 (cid:12) (cid:12) (cid:12) (cid:12)
1
= α α sin(θ θ ) ,
i j i j
2 −
i=1 j=1
X X (cid:12) (cid:12)
(cid:12) (cid:12)
whichproves(9).
Inthefollowing,weusearegularsubdivisionθ.Wewillshowthatifthesubdi-
visionstepissufficientlysmall,thenanysymmetricconvexsetcanbeapproximated
byazonotopeaspreciselyaswewant.
Definition 3 (0-regular zonotopes). For N N , let (N) denote the class of all
∈ ∗ C0
zonotopeswithatmost2N facesorientedbytheregularsubdivisionof[0,π[ byN
elements:
N
(i 1)π
(N) = α S α RN withθ = − , i=1,...,N.
C0 i θi ∈ + i N
( )
i=1
M (cid:12)
(cid:12)
Suchzonotopesarecalled0-regularzonotopes.
Notethat (N) (N) and (N1) (N2) ifandonlyifN isasplitterofN .
C0 ⊂ C C0 ⊂ C0 1 2
Inaddition, (N) canbeidentifiedtoRN bythemappingα X = ( N α S ),
C0 + → i=1 i θi
which is an isomorphism between the semigroups(RN,+) and ( (N), ). That is,
+ C0 L⊕
thismappingisabijection,and
N N N
α,α RN RN, α +α S = α S α S .
∀ ′ ∈ + × + i ′i θi i θi ⊕ ′i θi
! ! !
i=1 i=1 i=1
(cid:0) (cid:1) M(cid:0) (cid:1) M M
Theorem1(Approximationin (N)). LetX .
C0 ∈C
1. For all N > 1, let F(N) denote the squared matrix ( sin(θ θ ))
i j 1 i,j N
andH(N) =t (H (θ ),...,H (θ )).Then | − | ≤ ≤
X X 1 X N
N
X(N) = F(N)−1H(N) S (10)
0 X i θi
i=1
M(cid:0) (cid:1)
334 S.Rahmanietal.
belongsto (N)andsatisfies
C0
π
N >1, d X,X(N) (6+2√2)sin diam(X), (11)
∀ H 0 ≤ 2N
(cid:18) (cid:19)
(cid:0) (cid:1)
where diam(X) = sups R(HX(s)) denotesthe maximaldiameterofX and
d theHausdorffdistanc∈e.
H
Consequently,thesequenceof0-regularzonotopes(X(N)) approximates
0 N>1
X inthefollowingsense:
d X,X(N) 0 asN . (12)
H 0 −→ −→∞
WecallX(N)the (N)-(cid:0)approxima(cid:1)tionofX.
0 C0
2. Inaddition,foranyN >1,thesetX(N)istheuniqueelementof (N)satisfy-
0 C0
ing
H (θ )=H (θ ), i=1,...,N. (13)
X(N) i X i
0
3. Furthermore,X(N)containsX andcanbeexpressedas
0
N
1
X(N) = x R2, x,t sin(θ ),cos(θ ) H (θ ) . (14)
0 ∈ − i i ≤ 2 X i
i=1(cid:26) (cid:27)
\ (cid:12)(cid:10) (cid:0) (cid:1)(cid:11)(cid:12)
(cid:12) (cid:12)
Proof.
2. For integer N > 1, it is easy to see that the matrix F(N) is invertible since
F(N) isacirculantmatrix[15]anditseigenvaluesareexactlythecoefficients
ofthe discreteFouriertransform[28] ofthe signal sin() (thesecoefficients
| · |
areallstrictlypositive).Letα=F(N)−1H(N)besuchthat
X
N
X(N) = α S .
0 i θi
i=1
M
LetusshowthatX(N) istheuniqueelementof (N) satisfyingH (θ ) =
0 C0 X0(N) i
H (θ ), i = 1,...,N. Suppose that there exists X (N) satisfying
X i ′ ∈ C0
HX′(θi) = HX(θi), i = 1,...,N. Then X′ can be written as X′ =
N α S , and then H(N) = F(N)α. The invertibility of F(N) implies
i=1 ′i θi X ′
α=α,whichmeansthatX(N) =X .
L ′ 0 ′
1. LetusfindanupperboundfortheHausdorffdistance.
Forallη R,thereexistsi 1,...,N suchthatη =θ +δwith δ π .
∈ ∈{ } i | |≤ 2N
Usinginequality(3)withθ =θ andβ =δforX(N),wehave
i 0
δ δ+π
H (η) H (θ )+2 sin H θ + .
X0(N) ≤ X0(N) i (cid:12) (cid:18)2(cid:19)(cid:12) X0(N)(cid:18) i 2 (cid:19)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)