An Introduction to Ricci Flow and Volumetric Approximation with Applications to Shape Modeling

118 Pages·2014·13.01 MB·English

SIGGRAPH!Asia!2014!–!Course!“An!Introduction!to!Ricci!Flow!and!Volumetric!
Approximation!with!Applications!to!Shape!Modeling”!
G.!Patanè!(CNR-IMATI,!Italy),!X.S.!Li!(Lousiana!State!Univ.,!USA),!X.D.!Gu!(Stony!Brook!Univ.,!USA)!
!
!
!
!
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!
Table&of&Contents!
! Course'Overview!
! Course'Notes!
! Course'Slides!
SIGGRAPH!Asia!2014!–!Course!“An!Introduction!to!Ricci!Flow!and!Volumetric!
Approximation!with!Applications!to!Shape!Modeling”!
G.!Patanè!(CNR-IMATI,!Italy),!X.S.!Li!(Lousiana!State!Univ.,!USA),!X.D.!Gu!(Stony!Brook!Univ.,!USA)!
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!
Course'Overview!
SIGGRAPH Asia 2014 Course Notes
�
An Introduction to Ricci Flow and Volumetric Approximation with
Applications to Shape Modeling
GiuseppePatane´ XinShaneLi† DavidXianfengGu‡
⇤
Abstract essential to address a wide range of problems. For instance, vol-
umetric Laplacian eigenfunctions are suited to define volumetric
Extending a shape-driven map to the interior of the input shape descriptors,whichareconsistentwiththeirsurface-basedcounter-
and to the surrounding volume is a difficult problem since it parts. Inasimilarway, harmonicvolumetricfunctionshavebeen
typically relies on the integration of shape-based and volumetric appliedtovolumetricparameterizationandtothedefinitionofpoly-
information, together with smoothness conditions, interpolating cubesplines.
constraints, preservation of feature values at both a local and
Thissurveydiscussesthemainvolumetricapproximationschemes
global level. This survey discusses the main volumetric approxi-
forboth3Dshapesandd-dimensionaldata,andprovidesaunified
mation schemes for both 3D shapes and d-dimensional data, and
discussion on the integration of surface-based and volume-based
provides a unified discussion on the integration of surface-based
shapeinformation. Italsodescribestheapplicationofshape-based
and volume-based shape information. Then, it describes the
andvolumetrictechniquestoshapeprocessingwithvolumetricpa-
application of shape-based and volumetric techniques to shape
rameterizationandtothefeature-drivenapproximationwithmov-
modeling through volumetric parameterization and polycube
ingleast-squarestechniquesandradialbasisfunctions. Whilepre-
splines; feature-driven approximation through kernels and radial
viousworkhasaddressedtheprocessingandanalysisof3Dshapes
basisfunctions. WealsodiscusstheHamilton’sRicciflow,which
throughmethodsthatexploiteithertheirsurface-basedorvolumet-
isapowerfultooltocomputetheconformalshapestructureandto
ricrepresentations,thissurveypresentsaunifiedoverviewonthese
designRiemannianmetricsofmanifoldsbyprescribedcurvatures.
works through volumetric approximations of surface-based scalar
We conclude the presentation by discussing applications to shape
functions. Thisunifiedschemealsoprovidesabasisforgeneraliz-
analysisandmedicine.
ingthosemethodsthathavebeenprimarilydefinedonsurfacesbut
areopentoandbenefitoftheintegrationwithvolumetricinforma-
Keywords: Riemannian surface and metric; Ricci flow; con-
tion. Furthermore,itsystematicallypresentsthetheory,algorithm,
formal structure; Laplace-Beltrami operator; heat diffusion
andapplicationsofdiscreteRicciflow.Inthefollowing,weprovide
equation;implicitapproximation;volumeparameterization;shape
adetaileddescriptionofthemainpartsofourcontribution.
modeling;medicine
PartI Introduction
1 Course description �
Wepresenttheoutlineandthemainaimsofthiscourseonspectral
Shapemodelingtypicallyhandlesa3Dshapeasatwo-dimensional surface-basedandvolume-basedtechniques,anddiscretecurvature
surface,whichdescribestheshapeboundaryandisrepresentedas flowmethodsforshapemodelingandanalysis.
a triangular mesh or a point cloud. However, in several applica-
tionsavolumetricrepresentationismoresuitedtohandlethecom- Part II Differential operators and spaces for
plexity of the input shape. For instance, volumetric representa- shapem�odeling
tionsaccuratelymodelthebehaviorofnon-rigiddeformationsand
volumeconstraintsareimposedtoavoiddeformationartifacts. In
We start with an introduction to the spectral surface-based and
shape matching, volumetric descriptors, such as Laplacian eigen-
volume-basedtechniques,anddiscretecurvatureflowmethodsfor
functions,heatkernels,anddiffusiondistances,aredefinedstarting
shapemodeling,togetherwithapresentationofthebackgroundun-
fromtheirsurface-basedcounterparts.
derlyingthemainspectralandcurvatureflowtechniquesforshape
modeling. Keyconceptsfromsmoothgeometry,suchasRieman-
In the aforementioned applications, the underlying problem re-
nian metric, Gaussian curvature, Laplace-Beltrami operator, heat
quirestheprolongationofthesurface-basedinformation,whichis
diffusion equation, and Ricci flow are systematically introduced.
typicallyrepresentedasashape-drivenmap, totheinteriorofthe
Wealsopresentthemainresultsontheconvergenceandtheunique-
inputshapeor,moregenerally,tothesurroundingvolume.Extend-
nessofthesolutiontoRicciflowandthegeometricapproximation
ingasurface-basedscalarfunctiontoavolumetricmapisadifficult
theorem. Starting from this background on the main differential
problemsinceittypicallyreliesontheintegrationofshape-based
propertiesofmanifolds,wedefineanddiscussthepropertiesofthe
andvolumetricinformation,togetherwithsmoothnessconditions,
harmonicmaps,theLaplacianeigenfunctions,andthesolutionsto
interpolating constraints, preservation of features at both a local
theheatequation.
andgloballevel.Besidestheunderlyingcomplexityanddegreesof
freedominthedefinitionofvolumetricapproximationsofsurface-
based maps, volumetric approximations (e.g., the extension of a Part III From surface-based to volume-based
�
surface-basedscalarfunctiontoavolume-basedapproximation)are shapemodeling
⇤ConsiglioNazionaledelleRicerche,IstitutodiMatematicaApplicatae UsingtheconceptsintroducedinPartII,weaddressthevolumetric
TecnologieInformatiche,Genova,Italy,[email protected] approximationproblem. Afteranoverviewontheaimsofthevol-
†LouisianaStateUniversity,SchoolofElectricalEngineering&Com- umetricapproximationinthecontextofshapemodelingandanal-
puterScience,USA,[email protected] ysis, weclassifythemainapproachesproposedbypreviouswork
‡StateUniversityofNewYorkatStonyBrook,DepartmentofComputer and detail the following approximation schemes: (i) linear preci-
Science,NewYork,USA,[email protected] sionmethodsthroughgeneralizedbarycentriccoordinates;(ii)im-
plicit methods with radial basis functions; (iii) surface-based and Part III From surface-based to volume-based
�
cross-volumeparameterization; (iv)polycubesplines; (v)moving shapemodeling(70minutes)
least squares techniques. More precisely, we introduce the com-
putationoftheinter-surfaceharmonicmap,extendittovolumetric
1. The volumetric approximation problem (5 minutes: G.
harmonicmap,andconstructthepolycubeshapeparameterization
Patane`)
andsplines. Then,wediscussvolumetricapproximationsthrough
radial basis function with constraints on the approximation error Definition
•
andthepreservationofthecriticalpoints.
Aimsandmotivations
•
PartIV Applications&Conclusions 2. Mainapproaches(25minutes:G.Patane`)
�
Linear precision methods and generalized barycentric
Oncethecontinuousanddiscretesettingshavebeenintroduced,we •
coordinates
focusonthemainapplicationsofthevolumetricapproximationto
shapemodelingandmedicine. Inthe context ofshapemodeling, Functionapproximationwithradialbasisfunctions
we outline how the Laplacian eigenvectors of a given surface are •
extended into the shape interior, thus providing the basis for the Movingleast-squaresapproximation
•
definitionofshape-awarebarycentriccoordinatesandofvolumet-
[Break(15minutes)]
ricdescriptors,suchasthevolumetricglobalpointsignature,bihar-
monic and diffusion embeddings, which have been primarily de- 3. Fromcross-surfacetocross-volumemapping(40minutes:X.
finedforthesurfacesetting. Wealsopresenttemplate-basedshape Li)
descriptorsandthecomputationofharmonicvolumetricmappings
between solid objects with the same topology for volumetric pa- Cross-surfaceandcross-volumemapping
•
rameterization, solid texture mapping, and hexahedral remeshing.
Volumetricharmonicmapping
In the context of medicine, we discuss applications to respiratory •
motion modeling, medical and forensic skull modeling and facial
Polycubeparameterization
reconstruction. Finally, weconcludethecoursewithadiscussion •
ofopenproblemsandfutureperspectives,alsoaddressingquestions
PartIV Applications&Conclusions(50minutes)
andanswerswithallpresenters. �
1. Applicationstoshapemodelingandanalysis(20minutes: D.
2 Schedule Gu,G.Patane`)
Surface-basedandvolume-baseddescriptorsforshape
PartI Introduction(10minutes) •
� correspondenceandcomparison
1. Outlineandmotivations(10minutes:D.Gu,G.Patane`) Volumepreservingmappingsbetweensurfacesandim-
•
agerestoration
Part II Differential operators and spaces for 2. Applicationstomedicine(20minutes:D.Gu,X.Li)
�
shapemodeling(80minutes)
Motionmodelingforradiotherapyplanning
•
1. MappingsonRiemannsurfaces(20minutes:D.Gu) Skullandfacialmodelingandrestoration
•
Riemannianmetric,isothermalcoordinates Conformalbrainmappingandbraincortexanalysis
• •
Virtualcolonoscopy
Holomorphicdifferentials •
•
3. Conclusions,Questions&Answers(10minutes: G.Patane`,
Quasi-conformalmappingandBeltramiequation
• X.Li,D.Gu)
2. Ricciflow(30minutes:D.Gu)
3 Targeted audience and background
Yamabe equation and convergence theorem of Ricci
•
flow
Intendedaudience Thetargetaudienceofthistutorialincludes
graduatestudentsandresearchersinterestedinRiemanniangeome-
DiscreteRicciflow,convergence,uniqueness
• try,spectralgeometryprocessing,andimplicitmodeling.Ourgoals
arethreefold:(i)toshowthepossibilityofintegratingshape-based
Discreteconformalmappingandmetricdeformation
• and volume-based information; (ii) to introduce and discuss the
3. Laplacian operator and spectral processing (30 minutes: G. fundamentalresultsanditsapplicationsthatarerelevanttoshape
Patane`) modelingand,moregenerally,computergraphics;(iii)toidentify
open problems and future work. The main topics cover volumet-
Laplace-Beltramioperatoron3Dshapes ricparameterizationandpolycubesplines;implicitmodelingwith
• radialbasisfunctionsandkernelmethods; spectralshapeanalysis
Harmonicequation,Laplacianeigenproblem,andheat throughdescriptorsanddistances;discreteRicciflow;applications
•
diffusionequation to medicine. Several topics are of interest for a wider audience;
among them, we mention shape correspondence, descriptors and
Spectral distances and kernels: commute-time, bi- comparison; shape driven scalar functions for shape and volume
•
harmonic,anddiffusiondistances analysis.
Prerequisites Knowledge about differential geometry, mesh (T7) SIGGRAPHAsia’2010“SpectralGeometryProcessing”(B.
processing,functionapproximation. Levy,R.H.Zhang);
(T8) Eurographics’2010 State if the Art Reports “A Survey on
Levelofdifficulty: Advancedcourse. Shape Correspondence (O. van Kaick, R. H. Zhang, G.
Hamarneh,D.Cohen-Or).
4 Course Rationale CourseT6focusedonthediscreteexteriorcalculusanditsrelation
withdigitalgeometryprocessinganddiscretedifferentialgeometry.
Tutorialoriginality Whileprevioustutorialshaveaddressedthe TutorialT7presentedthemainconceptsbehindspectralmeshpro-
processing and analysis of 3D shapes through methods that ex- cessingon3Dshapesanditsapplicationstofiltering,shapematch-
ploit either their surface-based or volumetric representations, we ing, remeshing, segmentation, and parameterization. Tutorial T8
will present a unified overview on these works through volumet- reviewedthemainmethodsforthecomputationofthecorrespon-
ricapproximationsofsurface-basedscalarfunctions. Thisunified dencesbetweengeometricshapes.
schemewillalsoprovideabasisforgeneralizingthosemethodsthat
havebeenprimarilydefinedonsurfacesbutareopentoandbenefit 5 Lecturers biographies
oftheintegrationwithvolumetricinformation. Furthermore, itis
thefirsttutorialthatsystematicallypresentsthetheory,algorithm, DavidXiangfengGu
andapplicationsofdiscreteRicciflow.Inthefollowing,weprovide
alistofpreviousworkonthetopicsthatisrelatedtothistutorial.
Affiliation StateUniv.ofNewYorkatStonyBrook
e-mail [email protected]
Relatedtutorialsorganizedbythelecturers URL http://www.cs.sunysb.edu/ gu/
David Gu is an associate professor in Computer Science de-
(T1) SIGGRAPHAsia2013Course“Surface-BasedandVolume- partment, Stony Brook University. He received a Ph.D. from
Based Techniques for Shape Modeling and Analysis” (G. Harvard university (2003), supervised by a Fields medalist, Prof.
Patane`,X.S.Li,X.D.Gu); Shing-TungYau.Hisresearchfocusesoncomputationalconformal
geometry, and its applications in graphics, vision, geometric
(T2) Shape Modeling International’2012 Tutorial “Spectral, Cur- modelingnetworksandmedicalimaging.
vatureFlowSurface-BasedandVolume-BasedTechniquesfor
ShapeModelingandAnalysis”(G.Patane`,X.D.Gu,X.S.Li,
XinShaneLi
M.Spagnuolo);
Affiliation LouisianaStateUniversity
(T3) Eurographics’2007 Tutorial “3D shape description and
e-mail [email protected]
matchingbasedonpropertiesofrealfunctions”(S.Biasotti,
URL http://www.ece.lsu.edu/xinli
B.Falcidieno, P.Frosini, D.Giorgi, C.Landi, S.Marini, G.
Patane`,M.Spagnuolo); Xin Li is an assistant professor in School of Electrical En-
gineering and Computer Science, Louisiana State University.
(T4) ICIAM’2007 Mini-Symposium “Geometric-Topological He received his Ph.D. in Computer Science from Stony Brook
Methods for 3D Shape Classification and Matching” (M. University (SUNY) in 2008. His research focus is on geometric
Spagnuolo,G.Patane`); modelingandcomputing,andtheirapplicationsingraphics,vision,
medicalimaging,andcomputationalforensics.
(T5) SMI’2008 Mini-Symposium on “Shape Understanding via
SpectralAnalysisTechniques”(B.Levy,R.Zhang,M.Retuer,
GiuseppePatane`
G.Patane`,M.Spagnuolo).
Affiliation CNR-IMATI,Genova,Italy
ThiscourseproposalrevisesandextendsourT1SIGGRAPHAsia
e-mail [email protected]
2013 Course “Surface-Based and Volume-Based Techniques for
URL http://www.ge.imati.cnr.it
ShapeModelingandAnalysis”. Accordingtorecentresultsofthe
authorsandthefeedbacktothepreviouscourse, additionalmate- Giuseppe Patane` is researcher at CNR-IMATI (2001-today).
rialon(i)spectrum-freecomputationoftheheatkernelanddiffu- HereceivedaPh.D.in”MathematicsandApplications”fromthe
siondistances;(ii)applicationstomedicinehavebeenincludedin University of Genova (2005) and a Post Lauream Degree Master
the notes and slides of this new course proposal. Tutorial T2 ad- fromthe”F.SeveriNationalInstituteforAdvancedMathematics”
dressed the main volumetric approximation schemes for both 3D (2000).From2001,hisresearchactivitieshavebeenfocusedonthe
shapes and n-dimensional data, and provides a unified discussion definitionofparadigmsandalgorithmsformodelingandanalyzing
ontheintegrationofsurface-basedandvolume-basedshapeinfor- digitalshapesandmultidimensionaldata.
mation. TutorialsT3andT4coveredavarietyofmethodsfor3D
shapematchingandretrieval,whicharecharacterizedbytheuseof
areal-valuedfunctiondefinedontheshapetoderiveitssignature.
TutorialT5addressedspectralanalysisforshapeunderstandingand
severalapplications,whichincludesurfaceparameterization,defor-
mation,compression,andnon-rigidshaperetrieval.
Relatedtutorials
(T6) SIGGRAPH’2013 Course “Geometry Processing with Dis-
creteExteriorCalculus”(F.deGoes,K.Crane,M.Desbrun,
P.Schroeder);
SIGGRAPH!Asia!2014!–!Course!“An!Introduction!to!Ricci!Flow!and!Volumetric!
Approximation!with!Applications!to!Shape!Modeling”!
G.!Patanè!(CNR-IMATI,!Italy),!X.S.!Li!(Lousiana!State!Univ.,!USA),!X.D.!Gu!(Stony!Brook!Univ.,!USA)!
!
!
!
!
!
Course'Notes!
Course'Notes!–!Index!
!
Contents
1 Introduction 2
2 RiemannsurfacesandRicciflow 3
2.1 MappingsonRiemannSurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 SurfaceRicciflow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Differentialoperatorsandspacesforshapemodeling 9
3.1 Laplace-Beltramioperatoron3Dshapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Laplacianmatrixandequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4 Fromsurface-basedtovolume-basedshapemodeling 14
4.1 Linearprecisionapproximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.2 FunctionapproximationwithRBFs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.3 Fromsurface-tocross-volumeparameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.4 Polycubeparameterizationandpolycubesplines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.5 Movingleast-squaresandlocalapproximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.6 Topology-drivenapproximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.7 Computationalcost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5 Applications 23
5.1 Shapemodelingandanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.2 Medicalapplications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
6 Conclusionsandfuturework 26 !
SIGGRAPH Asia 2014 Course Notes
�
An Introduction to Ricci Flow and Volumetric Approximation
with Applications to Shape Modeling
GiuseppePatane´ DavidXianfengGu† XinShaneLi‡
⇤
Abstract
Extendingashape-drivenmaptotheinterioroftheinputshapeandtothesurroundingvolumeisadifficultproblemsince
ittypicallyreliesontheintegrationofshape-basedandvolumetricinformation,togetherwithsmoothnessconditions,interpo-
latingconstraints, preservationoffeaturevaluesatbothalocalandgloballevel. Thissurveydiscussesthemainvolumetric
approximation schemes for both 3D shapes and d-dimensional data, and provides a unified discussion on the integration of
surface-basedandvolumetricshapeinformation.Then,itdescribestheapplicationofsurface-basedandvolumetrictechniques
to shape modeling through volumetric parameterization and polycube splines; feature-driven approximation through kernels
andradialbasisfunctions.WealsodiscusstheHamilton’sRicciflow,whichisapowerfultooltocomputetheconformalshape
structureandtodesignRiemannianmetricsofmanifoldsbyprescribedcurvatures.Weconcludethepresentationbydiscussing
applicationstoshapeanalysisandmedicine.
Keywords:Riemanniansurfaceandmetric;Ricciflow;conformalstructure;Laplace-Beltramioperator;heatdiffusionequation;
implicitapproximation;volumeparameterization;shapemodeling;medicine
1 Introduction
Shape modeling typically handles a 3D shape as a two-dimensional surface, which describes the shape boundary and is rep-
resentedasatriangularmeshorapointcloud. However,inseveralapplicationsavolumetricrepresentationismoresuitedto
handlethecomplexityoftheinputshape. Forinstance,volumetricrepresentationsaccuratelymodelthebehaviorofnon-rigid
deformations and volume constraints are imposed to avoid deformation artifacts. In shape matching, volumetric descriptors,
suchasLaplacianeigenfunctions,heatkernels,anddiffusiondistances,aredefinedstartingfromtheirsurface-basedcounter-
parts.
Intheaforementionedapplications,theunderlyingproblemrequirestheprolongationofthesurface-basedinformation,which
istypicallyrepresentedasashape-drivenmap,totheinterioroftheinputshapeor,moregenerally,tothesurroundingvolume.
Extendingasurface-basedscalarfunctiontoavolumetricmapisadifficultproblemsinceittypicallyreliesontheintegration
ofshape-basedandvolumetricinformation,togetherwithsmoothnessconditions,interpolatingconstraints,preservationoffea-
turesatbothalocalandgloballevel. Besidestheunderlyingcomplexityanddegreesoffreedominthedefinitionofvolumetric
approximationsofsurface-basedmaps, volumetricapproximations(e.g., theextensionofasurface-basedscalarfunctiontoa
volume-basedapproximation)areessentialtoaddressawiderangeofproblems. Forinstance,volumetricLaplacianeigenfunc-
tionsaresuitedtodefinevolumetricdescriptors,whichareconsistentwiththeirsurface-basedcounterparts. Inasimilarway,
harmonicvolumetricfunctionshavebeenappliedtovolumetricparameterizationandtothedefinitionofpolycubesplines.
Thissurveydiscussesthemainvolumetricapproximationschemesforboth3Dshapesandd-dimensionaldata,andprovidesa
unified discussion on the integration of surface-based and volumetric information. It also describes the application of shape-
basedandvolumetrictechniquestoshapeprocessingwithvolumetricparameterizationandtothefeature-drivenapproximation
withmovingleast-squarestechniquesandradialbasisfunctions.Whilepreviousworkhasaddressedtheprocessingandanalysis
of 3D shapes through methods that exploit either their surface-based or volumetric representations, this survey presents a
unified overview on these works through volumetric approximations of surface-based scalar functions. This unified scheme
alsoprovidesabasisforgeneralizingthosemethodsthathavebeenprimarilydefinedonsurfacesbutareopentoandbenefitof
theintegrationwithvolumetricinformation. Furthermore,itsystematicallypresentsthetheory,algorithm,andapplicationsof
discreteRicciflow. Inthefollowing,weprovideadetaileddescriptionofthemainpartsofourcontribution.
⇤ConsiglioNazionaledelleRicerche,IstitutodiMatematicaApplicataeTecnologieInformatiche,Genova,Italy,[email protected]
†StateUniversityofNewYorkatStonyBrook,DepartmentofComputerScience,NewYork,USA,[email protected]
‡LouisianaStateUniversity,SchoolofElectricalEngineering&ComputerScience,USA,[email protected]
1+ µ
| |
U↵ U� 1 µ
�| |
✓
�
�
�
↵
K = 11+|µµ|
� �| |
↵� ✓= 1argµ
2
z↵ z�
(a) (b)
Figure1: (a)Riemannsurface. Alltransitionsf arebiholomorphic. (b)Beltramicoefficient.
ab
Outline and contribution We start with an introduction to the spectral surface-based and volumetric techniques, and
discrete curvature flow methods for shape modeling, together with a presentation of the background underlying the main
spectralandcurvatureflowtechniquesforshapemodeling(Sect.2). Keyconceptsfromsmoothgeometry,suchasRiemannian
metric, Gaussian curvature, Laplace-Beltrami operator, heat diffusion equation, and Ricci flow are systematically introduced
(Sect.3).WealsopresentthemainresultsontheconvergenceandtheuniquenessofthesolutiontoRicciflowandthegeometric
approximationtheorem. Startingfromthisbackgroundonthemaindifferentialpropertiesofmanifolds,wedefineanddiscuss
thepropertiesoftheharmonicmaps,theLaplacianeigenfunctions,andthesolutionstotheheatequation.
After an overview on the aims of the volumetric approximation in the context of shape modeling and analysis, we classify
themainapproachesproposedbypreviousworkanddetailthefollowingapproximationschemes(Sect.4): (i)linearprecision
methodsthroughgeneralizedbarycentriccoordinates;(ii)implicitmethodswithradialbasisfunctions;(iii)surface-basedand
cross-volume parameterization; (iv) polycube splines; (v) moving least squares techniques; (vi) and topology-driven approx-
imation. More precisely, we introduce the computation of the inter-surface harmonic map, extend it to volumetric harmonic
map, and construct the polycube shape parameterization and splines. Then, we discuss volumetric approximations through
radialbasisfunctionwithconstraintsontheapproximationerrorandthepreservationofthecriticalpoints.
Oncethecontinuousanddiscretesettingshavebeenintroduced,wefocusonthemainapplicationsofthevolumetricapproxi-
mationtoshapemodelingandmedicine(Sect.5). Inthecontextofshapemodeling,weoutlinehowtheLaplacianeigenvectors
of a given surface are extended into the shape interior, thus providing the basis for the definition of shape-aware barycentric
coordinates and of volumetric descriptors, such as the volumetric global point signature, biharmonic and diffusion embed-
dings, which have been primarily defined for the surface setting. We also present template-based shape descriptors and the
computationofharmonicvolumetricmappingsbetweensolidobjectswiththesametopologyforvolumetricparameterization,
solid texture mapping, and hexahedral remeshing. In the context of medicine, we discuss applications to respiratory motion
modeling, medical and forensic skull modeling and facial reconstruction. Finally (Sect. 6), we conclude the presentation by
discussingopenproblemsandfutureperspectives.
2 Riemann surfaces and Ricci flow
Firstly,weintroducemappingsonRiemannsurfaces,andquasi-conformalmappingandTeichmullerspaces(Sect.2.1). Then,
wediscussthesurfaceRicciflowanditsdiscretization(Sect.2.2).
2.1 Mappings on Riemann Surfaces
SupposeN beadifferentialmanifoldofdimensionn. ARiemannianmetriconN isafamilyofinnerproductsg :T N
p p ⇥
TpN !R, p2N , such that, for all differentiable vector fields X,Y on N , p!gp(X(p),Y(p)) defines a smooth surface
N !R. Selectingasetoflocalcoordinates(x1,x2,···,xn),themetrictensorcanbewrittenasg=Âi,jgijdxidxj. Considering
the differential map f :(M,g) (N ,h) between two Riemannian manifolds, the pull back metric on M induced by f is
!
givenby f⇤h=JThJ,whereJ=(∂∂xyij)istheJacobianmatrixof f. Surfacesareexamplesof2dimensionalmanifolds.
Figure2: Holomorphic1-formbasisonagenustwosurface.
SupposeN isanorientablesurfaceembeddedinE3 andgtheinducedEuclideanmetric;let(x,y)bethelocalparametersof
the metric surface (N ,g). If the Riemannian metric has the local representation g=e2l(x,y)(dx2+dy2), then (x,y) is called
isothermalcoordinatesofthesurface. Inparticular,l :N Riscalledtheconformalfactor. Thefollowingtheoremshows
!
theexistenceofisothermalcoordinates[Chern1955].
Theorem2.1 Suppose(N ,g)isasmoothorientedmetricsurface,thenforeachpointpthereexistsaneighborhoodU(p)ofp
suchthatlocalcoordinatesexistonU(p).
Throughtheisothermalcoordinates,weintroducetheGaussiancoordinatesasfollows. Let(N ,g)beanorientedsurfacewith
aRiemannianmetricand(u,v)anisothermalcoordinates. Then,theGaussiancurvatureofthesurfaceisgivenby
∂2 ∂2
K(u,v)=�Dgl, Dg=e�2l(u,v) ∂u2 +∂v2 ,
✓ ◆
where D is the Laplace-Beltrami operator induced by g. The Gauss curvature is intrinsic to the Riemannian metric and the
g
totalcurvatureisatopologicalinvariant. AccordingtotheGauss-Bonnettheorem[SchoenandYau1994;DoCarmo1976],the
totalGaussiancurvatureisgivenby
KdA+ k ds=2pc(N ),
g
ZN Z∂N
wherec(N )istheEulernumberofthesurface,k isthegeodesiccurvatureontheboundary,and∂N istheboundaryofthe
g
surface.
HolomorphicDifferentialsWenowintroduceholomorphic1-formsonReimanniansurfaces.Suppose f :Cˆ Cˆ beacomplex
function,whereCˆ =C • istheextendedcomplexplane. Definingthecomplexdifferentialoperators, !
[{ }
∂ 1 ∂ ∂ ∂ 1 ∂ ∂
= i , = +i ,
∂z 2 ∂x� ∂y ∂z¯ 2 ∂x ∂y
✓ ◆ ✓ ◆
thefunction f iscalledholomorphicif ∂f =0everywhere. If f isinvertibleand f 1 isalsoholomorphic,then f iscalledbi-
∂z¯ �
holomorphic. If f istreatedasamappingbetweencomplexplanes,thenholomorphicfunctionsareangle-preserving,namely,
conformal.
SupposeN isatopologicalsurface,withanatlas (Uk,fk) ,wherefk:Uk Cisalocalcomplexcoordinatechart. Ifallthe
{ } !
localcoordinatetransitions(Fig.1(a))
fij=fj�fi�1:fi(Ui\Uj)!fj(Ui\Uj),
arebi-holomorphic,thentheatlasiscalledaconformalatlasandthesurfaceN iscalledaRiemannsurface. Werecallthatall
orientedmetricsurfacesareRiemannsurfaces.
Suppose (N ,g) is an oriented surface with a Riemannian metric, the atlas formed by local isothermal coordinate charts is a
conformal structure. Hence, all oriented metric surfaces are Riemann surfaces, their Riemannian metrics induce conformal
structures. A holomorphic 1-form on a Riemann surface N is an assignment of a function f(z) on each chart z such that
i i i
if z is another local coordinate, then f(z)=f (z ) dzj . All holomorphic 1-forms form a group with 2g real dimension,
j i i j j dzi
denoted as W(N ), where g is the genus of N . Fig.⇣2 sh⌘ows the basis of holomorphic 1-forms on a genus two surface. A
holomorphic 1-form can be decomposed to two real harmonic 1-forms. According to Hodge theory [Schoen and Yau 1997],