Feature Lines for Illustrating Medical Surface Models: Mathematical Background and Survey

7.8 MB·English

by  Kai Lawonn

Feature Lines for Illustrating Medical Surface
Models: Mathematical Background and Survey
KaiLawonnandBernhardPreim
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AbstractThispaperprovidesatutorialandsurveyforaspecifickindofillustrative
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visualization technique: feature lines. We examine different feature line methods.
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Forthis,weprovidethedifferentialgeometrybehindtheseconceptsandadaptthis
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mathematicalfieldtothediscretedifferentialgeometry.Alldiscretedifferentialge-

] ometry terms are explained for triangulated surface meshes. These utilities serve
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asbasisforthefeaturelinemethods.Weprovidethereaderwithallknowledgeto
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re-implement every feature line method. Furthermore, we summarize the methods
. and suggest a guideline for which kind of surface which feature line algorithm is
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c bestsuited.Ourworkismotivatedby,butnotrestrictedto,medicalandbiological
[ surfacemodels.

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0 1 Introduction
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0 Theapplicationofillustrativevisualizationhasincreasedinrecentyears.Theprinci-
. plegoalbehindtheconceptofillustrativevisualizationisameaningful,expressive,
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andsimplifieddepictionofaproblem,asceneorasituation.Asanexample,running
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5 peoplearerepresentedrunningstickmans,whichcanbeseenintheOlympicgames,
1 andotherobjectsbecomesimplifiedlinedrawings,seeFigure1.Morecomplexex-
v: amplescanbefoundinmedicalatlases.Mostanatomicalstructuresarepaintedand
i illustratedwithpencilsandpens.Gray’sanatomyisoneofthefamoustextbooksfor
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medical teaching. Most other textbooks in this area orient to depict anatomy with
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artdrawing,too.
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KaiLawonn
Otto-von-GuerickeUniversityMagdeburg,FacultyofComputerScience,DepartmentofSimula-
tionandGraphicse-mail:[email protected]
BernhardPreim
Otto-von-GuerickeUniversityMagdeburg,FacultyofComputerScience,DepartmentofSimula-
tionandGraphicse-mail:[email protected]
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2 KaiLawonnandBernhardPreim
Other than simplified representa-
tion, illustrative visualization is not
restricted to these fields. Illustrative
techniques are essential for focus-
and-context visualizations. Consider a (a) (b) (c) (d)
scene with anatomical structures and
one specific (important) structure. The Fig. 1 Visual abstraction of the four Olympic
specific structure may be strongly re- disciplines: archery, basketball, football and
handball in the style of the pictograms of the
latedtothesurroundingobjects.There-
OlympicGames2012inLondon.
fore, hiding the other objects is not a
viableoption.Incontrast,depictingall
structuresleadstovisualclutterandopticaldistractionofthemostimportantstruc-
tures. Focus-and-context visualization is characterized by a few local regions that
aredisplayedindetailandwithemphasistechniques,suchasasaturatedcolor.Sur-
roundingcontextualobjectsaredisplayedinalessprominentmannertoavoiddis-
tractionfromfocalregions.Medicalexamplesarevesselswithinteriorbloodflow,
livers with inner structures including vascular trees and possible tumors, proteins
withsurfacerepresentationandinteriorribbonvisualization.Focus-and-contextvi-
sualizationisnotrestrictedtomedicaldata.Anexampleisthevehiclebodyandthe
interiordevices.Theuserorengineerneedstheopportunitytoillustratealldevices
inthesamecontext.
Therearenumerousmethodsfordifferentillustrationtechniques.Thissurveyis
focusedonaspecificillustrativevisualizationcategory:featurelines.Featurelines
areaspecialgroupoflinedrawingtechniques.Anotherclassoflinedrawingmeth-
ods is hatching. Hatching tries to convey the shape by drawing a bunch of lines.
Here,thespatialimpressionofthesurfaceisevenmoreimproved.Severalmethods
existtohatchthesurfacemesh,see[15,20,28,37,51,53].Incontrast,featurelines
try to generate lines at salient regions only. Not only for illustrative visualization,
featurelinescanalsobeusedforrigidregistrationsofanatomicalsurfaces[44]or
forimageanddataanalysisinmedicalapplications[12].Thegoalofthissurveyis
toconveythereadertothedifferentfeaturelinemethodsandofferatutorialwithall
theknowledgetobeabletoimplementeachofthemethods.
Organization.
We first give an overview of the mathematical background. In Section 2, we in-
troducethenecessaryfundamentalsofdifferentialgeometry.Afterwards,weadapt
thesefundamentalstotriangulatedsurfacemeshesinSection3.Section4discusses
generalaspectsandrequirementsforfeaturelines.Next,wepresentdifferentfeature
linemethodsinSection5andcomparetheminSection6.Finally,Section7holds
theconclusionofthissurvey.
FeatureLinesforIllustratingMedicalSurfaceModels 3
Fig.2 Thebasicelementsfordifferentialgeometry.Aparametricsurfaceisgivenandthepartial
derivativescreatethetangentspace.
2 DifferentialGeometryBackground
Thissectionpresentsthefundamentalsofdifferentialgeometryforfeaturelinegen-
eration, which will be crucial for the further sections. We present the basic terms
andproperties.Thissectionisinspiredbydifferentialgeometrybooks[5,6,26].
2.1 BasicPrerequisites
Asurface f: I⊂R2→R3iscalledaparametricsurfaceif f isanimmersion.Anim-
mersionmeansthatallpartialderivatives ∂f areinjectiveateachpoint.Thefurther
∂xi
calculationsaremostlybasedonthetangentspaceofasurface.Thetangentspace
T f of f isdefinedasthelinearcombinationofthepartialderivativesof f:
p
T f (cid:66)span(cid:110)∂f (cid:12)(cid:12)(cid:12) , ∂f (cid:12)(cid:12)(cid:12) (cid:111).
p ∂x1(cid:12)x=u ∂x2(cid:12)x=u
Here,spanisthespaceofalllinearcombinations.Formally:span{v ,v }(cid:66){αv +
1 2 1
βv |α,β∈R}.Withthetangentspace,wecandefineanormalizednormalvectorn.
2
The(normalized)normalvectorn(u)atp= f(u)isdefinedsuchthatforallelements
v∈T f theequation(cid:104)v,n(u)(cid:105)=0holds,where(cid:104).,.(cid:105)denotesthecanonicalEuclidean
p
dotproduct.Therefore,n(u)isdefinedas:
(cid:12) (cid:12)
∂f (cid:12)(cid:12) × ∂f (cid:12)(cid:12)
n(u)= (cid:13)(cid:13)(cid:13)∂∂∂xxf11(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)xx==uu×∂∂∂xxf22(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)xx==uu(cid:13)(cid:13)(cid:13).
4 KaiLawonnandBernhardPreim
ThismapisalsocalledtheGaussmap.Figure2depictsthedomainofaparametric
surfaceaswellasthetangentspaceT f andthenormaln.
p
2.2 Curvature
Thecurvatureisafundamentalpropertytoidentifysalientregionsofasurfacethat
shouldbeconveyedbyfeaturelines.Colloquiallyspoken,itisameasureofhowfar
the surface bends at a certain point. If we consider ourselves to stand on a sphere
ataspecificpoint,itdoesnotmatterinwhichdirectionwego,thebendingwillbe
the same. If we imagine we stand on a plane at a specific point, we can go in any
direction,therewillbenobending.Withoutknowinganymeasureofthecurvature,
we can state that a plane has zero curvature and that a sphere with a small radius
hasahighercurvaturethanaspherewithahigherradius.Thisisduetothefactthat
a sphere with an increasing radius becomes locally more a plane. Intuitively, the
curvaturedependsalsoonthedirectioninwhichwedecidetogo.Onacylinder,we
haveabendinginonedirectionbutnotintheother.Paintingthetraceofawalkon
thesurfaceandviewitin3Dspace,wecouldtreatthisasa3Dcurve.Thedefinition
ofthecurvatureofacurvemaybeadaptedtothecurvatureofasurface.Theadaption
ofthisconceptsisexplainedinthefollowing.Letc: I⊂R→R3bea3Dparametric
curvewith(cid:107)dc(cid:107)=1.Thepropertyofconstantlengthofthederivativeiscalledarc
dt
lengthornaturalparametrization.Onecanshowthatsuchaparametrizationexists
for each continuous, differentiable curve that is an immersion. So, if we want to
measure the size of bending, we can use the norm of the second derivative of the
curve.Therefore,the(absolute)curvatureκ(t)atatimepointtisdefinedas:
(cid:13) (cid:13)
κ(t)=(cid:13)(cid:13)c(cid:48)(cid:48)(t)(cid:13)(cid:13).
To determine the curvature on a cer-
tain point of the surface in a specific
direction, we can employ a curve and
calculate its curvature. This approach
is imperfect because curves that lie in
a plane can have non-vanishing curva-
ture,e.g.,acircle,whereasweclaimed
tohavezerocurvatureonaplanarsur-
face.Therefore,wehavetodistinguish
which part of the second derivative of
the curve contributes to the tangent
spaceandwhichcontributestothenor-
mal part of the surface. Decomposing Fig. 3 The curve’s second derivative is decom-
posedintothetangentialandnormalpart.
the second derivative of the curve into
tangential and normal part of the sur-
faceyields:
FeatureLinesforIllustratingMedicalSurfaceModels 5
c(cid:48)(cid:48)(t)=proj c(cid:48)(cid:48)(t)+(cid:104)c(cid:48)(cid:48)(t),n(cid:105)n,
(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)T(cid:32)pf(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)
(cid:124) (cid:123)(cid:122) (cid:125)
normalpart
tangentialpart
wherec(t)=pandproj xmeanstheprojectionofthepointxontothespaceE,see
E
Figure3.Thecurvatureκ (p)ofthesurfaceatpalongthecurvecisdefinedasthe
c
coefficientofthenormalpart:
κ (p)=(cid:104)c(cid:48)(cid:48)(t),n(cid:105). (1)
c
Hence, we know that c(cid:48)(t)∈T f and (cid:104)c(cid:48)(t),n(cid:105)=0. Deriving the last equation
p
yields:
d
(cid:104)c(cid:48)(t),n(cid:105)=0
dt
d ∂n
(cid:104)c(cid:48)(t),n(cid:105)=(cid:104)c(cid:48)(cid:48)(t),n(cid:105)+(cid:104)c(cid:48)(t), (cid:105).
dt ∂t
Weobtain
∂n
(cid:104)c(cid:48)(cid:48)(t),n(cid:105)=−(cid:104)c(cid:48)(t), (cid:105).
∂t
CombiningthisequationwithEquation1yields
∂n
κc(cid:48)(t)(p)=−(cid:104)c(cid:48)(t), (cid:105). (2)
∂t
Thus, the curvature of a surface at a specific point in a certain direction can be
calculated by a theorem by Meusnier. We call the vectors v,w at p the maximal/
minimal principle curvature directions of the maximal and minimal curvature, if
κv(p)≥κv(cid:48)(p), κw(p)≤κv(cid:48)(p) for all directions v(cid:48) ∈Tpf. If such a minimum and
maximumexists,thenvandwareperpendicular,seeSection2.5foraproof.Ifwe
wanttodeterminethecurvatureindirectionu,wefirstneedtonormalizeu,v,wand
canthendetermineκ (p)by:
u
κ (p)=(cid:104)u,v(cid:105)2κ (p)+(cid:104)u,w(cid:105)2κ (p). (3)
u v w
The coefficients of the curvature are the decomposition of the principle curvature
directionswiththevectoru.
2.3 CovariantDerivative
The essence of the feature line generation is the analysis of local variations in a
specificdirection,i.e.,thecovariantderivative.Therefore,thecovariantderivativeis
acrucialconceptforfeaturelinemethods.Weconsiderascalarfieldonaparametric
surfaceϕ: f(I)→R.Onecanimaginethisscalarfieldasaheatorpressure(aswell
as a curvature) distribution. The directional derivative of ϕ in direction v can be
6 KaiLawonnandBernhardPreim
Fig.4 Given:ascalarfieldinthedomain.Determiningthegradientandusingitascoefficientfor
thebasistangentvectorsleadstothewrongresult(grey).Balancingthedistortionwiththeinverse
ofthemetrictensoryieldsthecorrectgradientonthesurface(black).
writtenasD ϕandisdefinedby:
v
ϕ(x+hv)−ϕ(x)
D ϕ(x)=lim .
v
h→0 h
Ifϕisdifferentiableatx,thedirectionalderivativecanbesimplified:
D ϕ(x)=(cid:104)∇ϕ(x),v(cid:105),
v
where ∇ denotes the gradient. The gradient is an operator applied to a scalar field
and results in a vector field. When we want to extend the definition of the deriva-
tive to an arbitrary surface, we first need to define the gradient of surfaces. In the
following,wemakeuseofthecovariantderivative.Thestandarddirectionalderiva-
tiveresultsinavectorwhichliessomewhereinthe3Dspace,whereasthecovariant
derivativeisrestrictedtostayinthetangentspaceofthesurface.Thegradientisa
two-dimensionalvector.Actually,weneedathree-dimensionalvectorinthetangent
spaceofthesurface.Here,weemploythegradientanduseitascoefficientsofthe
tangentialbasis.Unfortunately,thisleadstowrongresultsbecauseofthedistortions
ofthebasisofthetangentspace,seeFigure4.Thebasisisnotnecessarilyanorthog-
onalnormalizedbasisasinthedomainspaceandcanthereforeleadtodistortions
ofthegradientonthesurface.
One way to calculate this vector is to use the plain scalar field ϕ: R3→R. Af-
terwards,weareabletoattainthegradientinthree-dimensionalspaceandprojectit
onthetangentspace.However,wewanttousethegradientofϕ: R2→Rinthedo-
mainofaparametricsurfaceandcompensatethelengthdistortionsuchthatwecan
useitascoordinateswiththebasisinthetangentspace.Oneimportantfactiswhen
multiplyingthegradientwiththei-thbasisvector,oneobtainsthepartialderivative
ofϕwithx.Hence,weknowthatthethree-dimensionalgradient∇ϕliesinthetan-
i
gentspace.Therefore,itcanberepresentedasalinearcombinationof ∂f , ∂f with
coefficientsα,β: ∂x1 ∂x2
FeatureLinesforIllustratingMedicalSurfaceModels 7
∂f ∂f
∇ϕ=α· +β· .
∂x ∂x
1 2
Multiplyingbothsideswiththebasisvectorsandusingtherelation ∂ϕ =(cid:104)∇ϕ, ∂f (cid:105),
∂xi ∂xi
weobtainanequationsystem:
∂∂∂xϕϕ1 = αα··(cid:104)(cid:104)∂∂∂xff1,,∂∂∂xff1(cid:105)(cid:105)++ββ··(cid:104)(cid:104)∂∂∂xff1,,∂∂∂xff2(cid:105)(cid:105) = (cid:104)(cid:104)∂∂∂xff1,,∂∂∂xff1(cid:105)(cid:105)(cid:104)(cid:104)∂∂∂xff1,,∂∂∂xff2(cid:105)(cid:105)(cid:32)αβ(cid:33).
∂x2 ∂x1 ∂x2 ∂x2 ∂x2 (cid:124)(cid:32)(cid:32)∂(cid:32)(cid:32)(cid:32)x(cid:32)(cid:32)(cid:32)1(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)∂(cid:32)(cid:32)(cid:32)x(cid:32)(cid:32)2(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)∂(cid:32)(cid:32)(cid:32)x(cid:32)(cid:32)2(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)∂(cid:32)(cid:32)(cid:32)x(cid:32)(cid:32)(cid:32)2(cid:32)(cid:32)(cid:32)(cid:125)
g(cid:66)
The matrix g is called the metric tensor. This tensor describes the length and area
distortionfromR2tothesurface.Thelastequationyieldsthecoefficientsα,βwhen
multipliedwiththeinverseofg:
(cid:32) (cid:33) ∂ϕ
αβ =g−1∂∂xϕ1.
∂x2
Thisleadstoageneralexpressionofthegradientforascalarfieldϕ: Rn→R:
(cid:88)n (cid:32) ∂ϕ(cid:33) ∂
∇ϕ= gij , (4)
∂x ∂x
i,j=1 j i
where gij is the i,j-th matrix entry from the inverse of g and ∂ means the basis.
∂xi
Now, we are able to determine the covariant derivative of a scalar field by first
determiningitsgradientandafterwardsusingthedotproduct:
D ϕ=(cid:104)∇ϕ,w(cid:105).
w
2.4 Laplace-BeltramiOperator
TheLaplace-Beltramioperatorisneededforaspecificfeaturelinemethodandwill
therefore be introduced. The Laplace operator is defined as a composition of the
gradient andthedivergence.Wheninterpretingthevectorfieldasaflowfield,the
divergenceisameasureofhowmuchmoreflowleavesaspecificregionthanflow
enters.IntheEuclideanspace,thedivergencedivΦofavectorfieldΦ: Rn→Rnis
thesumofthepartialderivativesofthecomponentsΦ:
i
(cid:88)n ∂
divΦ= Φ.
i
∂x
i=1 i
ThecomputationofthedivergenceforavectorfieldΦ: Rn→RninEuclideanspace
is straightforward. However, for computing the divergence to an arbitrary surface
wehavetobeawareofthelengthandareadistortions.Withoutgivingaderivation
8 KaiLawonnandBernhardPreim
of the divergence, the components Φ of the vector field have to be weighted by
(cid:112)i
the square root of the determinant |g| of the metric tensor g before taking the
derivative.Thesquarerootofthedeterminantofgdescribesthedistortionchange
fromtheEuclideanspacetothesurface.Formally,thedivergenceofavectorfield
Φ: Rn→Rnwithagivenmetrictensorgisgivenby:
1 (cid:88)n ∂ (cid:32)(cid:112) (cid:33)
divΦ= (cid:112) |g|Φi . (5)
|g| i=1 ∂xi
Given the definition of the gradient and the divergence, we can compose both op-
erators to obtain the Laplace-Beltrami operator ∆ϕ of a scalar field ϕ: Rn →R on
surfaces:
1 (cid:88)n ∂ (cid:32)(cid:112) ∂ϕ(cid:33)
∆ϕ=div∇ϕ= (cid:112) |g|gij . (6)
|g|i,j=1∂xi ∂xj
2.5 ShapeOperator
In Section 2.2, we noticed that the curvature of a parametric surface at a specific
pointpinacertaindirectioncanbedeterminedbyEquation2:
∂n
κc(cid:48)(t)(p)=−(cid:104)c(cid:48)(t), (cid:105).
∂t
Actually, this means that the curvature in the direction c(cid:48)(t) is a measure of how
muchthenormalchangesinthisdirection,too.Givenisv∈T f withp= f(u)and
p
|v|=1.Then,wedeterminethecoefficientsα,βofvwiththebasis ∂f , ∂f :
∂x1 ∂x2
(cid:32)αβ(cid:33)=g−1(cid:104)(cid:104)vv,,∂∂∂xff1(cid:105)(cid:105).
∂x2
Weuse(α,β)todeterminethederivativeofnalongvbyusingthetwo-dimensional
(cid:16) (cid:17)
curvec˜(t)=u+t α andcalculate:
β
d
D n(cid:66) n(c˜(t)).
v
dt
WedefineS(v)(cid:66)−D n.ThislinearoperatoriscalledShapeOperator(alsoWein-
v
garten Map or Second Fundamental Tensor). One can see that S(∂f )= ∂n holds.
∂xi ∂xi
Notethatthisoperatorcandirectlyoperateonthe3Dspacewithathree-dimensional
vectorinthetangentspace,aswellasthe2Dspacewiththecoefficientsofthebasis.
Therefore,itcanberepresentedbyamatrixS.RecallEquation2,wesubstitutec(cid:48)
withvand ∂n bySv:
∂t
FeatureLinesforIllustratingMedicalSurfaceModels 9
κ (p)=(cid:104)v,Sv(cid:105).
v
We want to show that the principle curvature directions are the eigenvectors of S.
Assumingv ,v ∈R2arethenormalizedeigenvectorswiththeeigenvaluesλ ≥λ .
1 2 1 2
Every normalized vector w can be written as a linear combination of v ,v : w=
1 2
αv +βv with(cid:107)w(cid:107)=(cid:107)αv +βv (cid:107)=α2+β2+2αβ(cid:104)v ,v (cid:105)=1.Therefore,weobtain:
1 2 1 2 1 2
1
κ (p)=(cid:104)w,Sw(cid:105)= [(α2−β2)(λ −λ )+λ +λ ]. (7)
w 1 2 1 2
2
OnecanseefromEquation7thatκ (p)reachesamaximumifβ=0,α=1,anda
w
minimumisreachedifα=0,β=1.Iftheeigenvalues(curvatures)arenotequal,we
canshowthattheprinciplecurvaturedirectionsareperpendicular.Forthis,weneed
toshowthatS isaself-adjointoperator.Thus,theequation(cid:104)Sv,w(cid:105)=(cid:104)v,Sw(cid:105)holds.
Weshowthisbyusingtheproperty(cid:104)n, ∂f (cid:105)=0andderivethiswithx :
∂xi j
∂n ∂f ∂2f
(cid:104) , (cid:105)+(cid:104)n, (cid:105)=0.
∂x ∂x ∂x∂x
j i i j
WedemonstratethatS isaself-adjointoperatorwiththebasis ∂f :
∂xi
∂f ∂f ∂n ∂f ∂2f ∂n ∂f ∂f ∂f
(cid:104)S( ), (cid:105)=(cid:104)− , (cid:105)=(cid:104)n, (cid:105)=(cid:104)− , (cid:105)=(cid:104)S( ), (cid:105).
∂x ∂x ∂x ∂x ∂x∂x ∂x ∂x ∂x ∂x
i j i j i j j i j i
Now,weshowthattheeigenvectors(principlecurvaturedirections)areperpendic-
ulariftheeigenvalues(curvatures)aredifferent:
λ (cid:104)v ,v (cid:105)=(cid:104)Sv ,v (cid:105)=(cid:104)v ,Sv (cid:105)=λ (cid:104)v ,v (cid:105).
1 1 2 1 2 1 2 2 1 2
Theequationisonlytrueifv ,v areperpendicular(andλ (cid:44)λ holds).
1 2 1 2
3 DiscreteDifferentialGeometry
Thissectionadaptsthecontinuousdifferentialgeometrytodiscretedifferentialge-
ometry,theareaofpolygonalmeshesthatapproximatecontinuousgeometries.The
following notation is used in the remainder of this paper. Let M ⊂R3 be a trian-
gulatedsurfacemesh.Themeshconsistsofverticesi∈V withassociatedpositions
p ∈R3,edgesE={(i,j)|i,j∈V},andtrianglesT={(i,j,k)|(i,j),(j,k),(k,i)∈E}.We
i
writen asthenormalizednormalvectoratvertexi.Ifnothingelseismentioned,we
i
refertonormalvectorsatvertices.Furthermore,N(i)denotestheneighborsofi.So,
forevery j∈N(i),(i,j)∈Eholds.Furthermore,ifweuseatriangleforcalculation,
wealwaysusethisnotation:givenatriangle(cid:52)=(i,j,k)withverticesp,p ,p ,and
i j k
theedgesaredefinedase =p −p ,e =p −p ,e =p −p.
1 i j 2 j k 3 k i
10 KaiLawonnandBernhardPreim
(a) Pointsin2D (b) Points on a surface
mesh
Fig.5 TheVoronoidiagramofdifferentsettings.In(a)aVoronoidiagramofasetofpointsis
determined.In(b)theVoronoiareaiscalculated.Ifoneofthetrianglesisobtuse,thearealeaves
thetriangle.
3.1 VoronoiArea
WeneedtointroducethetermVoronoiarea,asitisimportantforthedetermination
of the curvature. So, given are points in a 2D space. Every point is spread out in
equal speed. If two fronts collide, they stop to spread out further at this region.
After all fronts stopped, every point lies in a region that is surrounded by a front.
ThisregioniscalledaVoronoiregion.Formally,givendistinctpointsx ∈R2 inthe
i
plane,theVoronoiregionforthepointx isdefinedasthesetofpointsV(x )with
k k
V(x )={x∈R2 : (cid:107)x−x (cid:107)≤(cid:107)x−x (cid:107), j(cid:44)k}.
k k j
SeeFigure5(a)foranexampleofaVoronoidiagram.ToobtaintheVoronoiareaofa
vertexonasurfacemesh,theVoronoiareaofeachincidenttriangleisaccumulated.
The Voronoi area calculation is based on the method by Meyer et al. [31]. In case
ofanon-obtusetriangle,theVoronoiareaatp isdeterminedbytheperpendicular
i
bisector of the edges incident to p. The point of intersection, the midpoint of the
i
incident edges and the point itself define the endpoints of the Voronoi area. The
triangleareaoftheVoronoiregionequals:
1(cid:16) (cid:17)
A (p)= (cid:107)e (cid:107)2·cot(e ,e )+(cid:107)e (cid:107)2·cot(e ,e ) .
(cid:52) i 1 2 3 3 1 2
8
Incaseofanobtusetriangle,theVoronoiareaisequalhalfofthetriangleareaifthe
angleatp isobtuse.Otherwiseitisaquarterofthetrianglearea,seeFigure5(b).
i
3.2 DiscreteCurvature
The calculation of the curvatures as well as the principle curvature directions are
important for a number of feature line techniques. Several approaches exist to ap-