Shape analysis and structuring

305 Pages·2008·8.254 MB·English

by  Leila de Floriani, Michela Spagnuolo

Mathematics and Visualization
SeriesEditors
GeraldFarin
Hans-ChristianHege
DavidHoffman
ChristopherR.Johnson
KonradPolthier
MartinRumpf
Leila De Floriani
Michela Spagnuolo
Editors
Shape Analysis
and Structuring
With 156 Figures, 29 in Color and 7 Tables
ABC
LeilaDeFloriani MichelaSpagnuolo
DepartmentofComputer IstitutoperlaMatematicaApplicata
andInformationSciences(DISI) eleTecnologieInformatiche
UniversityofGenova ConsiglioNazionaledelleRicerche
ViaDodecaneso,35 ViaDeMarini,6
16146Genova,Italy 16149Genova, Italy
defl[email protected] [email protected]
and
DepartmentofComputerScience
UniversityofMaryland
CollegePark,MD20740,USA
ISBN978-3-540-33264-0 e-ISBN978-3-540-33265-7
SpringerSeriesinMathematics and Visualization ISSN1612-3786
LibraryofCongressControlNumber:2007935447
MathematicsSubjectClassificationNumbers(2000):68U05,68U07,65D18,65D17
(cid:1)c 2008Springer-VerlagBerlinHeidelberg
Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis
concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,
reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication
orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,
1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare
liabletoprosecutionundertheGermanCopyrightLaw.
Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply,
evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws
andregulationsandthereforefreeforgeneraluse.
Coverimage:GershonElber,TECHNION,Israel
Coverdesign:WMXDesignGmbH.Heidelberg
Printedonacid-freepaper.
9 8 7 6 5 4 3 2 1
springer.com
1 Introduction
The idea of composing this book arose from the desire to enrich and systematise
the extensive state-of-the-art studies that were carried out within the framework of
AIM@SHAPE, a Network of Excellence funded by European Commission under
the FP6-IST 1. The main goal of the network is to develop new methodologies for
modellingandprocessingknowledgeembeddedindigitalshapes.
Current approaches to modelling are focused on the geometry of shapes, while
theirsemantics,e.g.,meaningorfunctionalityinagivencontext,isstilloverlooked.
This is partly due to the lack of methods for the automatic extraction of the se-
manticcontentfromdigitalshapes,knownastheprocessofsemanticannotationin
research areas related to the development of the Semantic Web, and partly to the
evolutionofresearchonshapemodellingwhichinthepastyearswashighlyfocused
onthegeometricaspectsofshapes.Theshiftfromapurelygeometrictoasemantic-
awarelevelofrepresentationofdigitalshapesistheultimatescientificobjectiveof
AIM@SHAPE.
Inthisscenario,acrucialroleisplayedbygeometryprocessingmethodsthatare
aimed to preserve and enhance shape information as well as to effectively capture
thestructureofashapebyidentifyingrelevantshapecomponentsandtheirmutual
relationships. Each chapter of the book provides a detailed state-of-the-art review
on a specific topic, which is crucial for shape analysis and structuring, contains a
classificationofthetechniquesdevelopedinthearea,anddiscussesopenproblems.
Structuralanalysisplayafundamentalroleintheautomaticextractionofseman-
tic information. While shapes are fully characterized by a specific geometry, shape
information is treated differently by the human brain with respect to several other
formsofinformation.Atageometriclevel,adigitalshapeisacomputationalstruc-
turewhichdefinesageometricrepresentation.Differenttypesofgeometricmodels
can be used to describe the same object. Examples are polygons, surface models
(e.g.,splines,NURBSsurfaces),orsolidmodels(e.g.,triangleortetrahedralmeshes,
boundary representations, constructive volumetric representations). The structural
levelintherepresentationisreachedbyorganizinggeometricinformationtoreflect,
or make explicit, the association between the various components of the shape. A
structural description is the basis for developing semantic-based shape representa-
tions,sinceitabstractsfromthelow-level,detailed,descriptionprovidedbyageo-
metricmodel.
Severaltechniqueshavebeendevelopedintheliteratureforprocessingdifferent
aspectsofthegeometryofshapes,inparticularshapeinterrogationandre-meshing
techniques enhance a shape description with information which can be effectively
usedtoattachsemanticstotheshape.Shapeinterrogationistheprocessofextract-
ing information from a geometric model. Geometric models need to be analyzed
with respect to different aspects, such as visual pleasantness, technical smoothing,
geometric constraints, or surface intrinsic properties. The various methods devel-
1AIM@SHAPE,http://www.aimatshape.net
VI Introduction
opedintheliteratureareusedtodetectsurfaceimperfections,toanalyzeshapes,or
tovisualizedifferentforms.SuchmethodsarereviewedinChapterone.
Re-meshing is often used for efficient shape approximation, and it consists of
repartitioningasetofprimitivessothattheybestfittheoriginalshape.Re-meshing
preservestheshape,inthesensethatitstillapproximatestheshapeafterre-meshing,
anditcanbedesignedtoenhancetheshape.Everyshapefeatureislocallyfitwitha
primitivethatminimallycharacterizestheshape.Inaddition,somerecentre-meshing
techniquesoperatethroughacarefulanalysisbasedonmulti-scalediscretedifferen-
tialgeometrysoastoestimatethemain(anddetailed)axisofsymmetryoftheshape.
Theshapeis,thus,locallyclassifiedasspherical,parabolic,elliptic,andhyperbolic
in order to drive the re-meshing process. Such classification may be also used for
shapeenhancement.Chaptertwopresentsasurveyonre-meshingtechniques.
Afirstwayofstructuringshapeinformationisprovidedbythosetechniquesthat
organizeageometricshapedescriptiondefinedbyafunction,byamesh,orbyaset
of points into a representation of the shape at different levels of resolutions, from
which concise and adaptive shape descriptions can be extracted. This topic has re-
ceived considerable attention in recent years in many fields of computer graphics,
geometricmodellingandvisualization,andnumerousresearcheffortshavebeende-
votedtoit.Thisbookcontainstwochaptersonfocusedonmulti-resolutionanalysis,
andtheotheronsubdivisionsurfaces.
Multi-resolution analysis provides a powerful tool for efficiently representing
functionsatmultiplelevelsofdetail.Herein,acomplexfunctionisdecomposedinto
a coarser low-resolution part, together with a collection of detail coefficients, nec-
essarytorecovertheoriginalfunction.Multi-resolutionanalysishasmanyinherent
advantages,includingcompression,progressivetransmission,visualizationandedit-
ingatdifferentlevelsofdetail.Anoverviewofmethodsformulti-resolutionanalysis
ispresentedinChapterthree.
Subdivision surfaces define the basis for generating a smooth surface from a
coarse mesh, and, thus, they have bees extensively used in geometric modeling for
creating,editingandtransmittingashape.Thesurfaceisdefinedbytheinitialcoarse
mesh plus a subdivision scheme to progressively subdivide the mesh by inserting
new vertices and connecting them to the edges and faces until a smooth surface is
obtainedinthelimit.Chapterfourcontainsareviewofsurfacesubdivisionschemes
andtheirapplicationingeometricmodeling.
The third part of the book is devoted to structural shape representations. In the
aboveframework,manyresearcheffortshavebeendevotedtostudyconcise,struc-
turalrepresentationsofashapebasedonskeletalstructures,suchasthemedialaxis,
or the Reeb graph. Skeletal structures provide an abstract shape representation by
idealizedlinesthatretaintheconnectivityoftheoriginalshape.Inadvancedfields,
such as virtual human modeling, available modeling tools to represent structured
geometryfocusonaddingaskeletontothe3Dgeometryinordertoanimateitand
providedifferentdegreesofrealism.Asurveyofdifferentskeletalstructuresispre-
sentedinChapterfive.
Anotherclassofstructuralrepresentationsisprovidedbymorphology-basedde-
scriptionsforscalarandvectorfields.Therehasbeenaconsiderableamountofwork
Introduction VII
in the literature on extracting critical features (point, integral lines, etc.) from two-
dimensional scalar fields describing grey-level images and terrains, and, more re-
cently,someworkhasbeendoneonvolumedataonextractingcriticalfeaturesand
forrepresentingthetopologicalstructureofthefieldiso-surfaces.Asurveyofmor-
phological representations for two-dimensional and three-dimensional scalar fields
ispresentedinChaptersix.
Topological methods based on features, like critical points or separatrix lines,
havealsobeenappliedfortheanalysisofvectorfields.Thebasicideaistousesuch
featuresforsegmentingtheflowintoareasofdifferentflowbehaviour,andusethis
asatoolforunderstandingcomplexphenomenadescribedbythevectorfields.After
introducing topological features for two-dimensional and three-dimensional vector
fields, Chapter seven presents a survey of methods for extracting topological fea-
turesfromvectorfieldsandusingthemasvisualizationtoolsforcomplexflowphe-
nomena, represented both as static and dynamic fields. Applications of topological
methodsforcompressing,simplifying,comparing,andconstructingvectorfieldsare
alsodiscussed.
The last part of the book, namely Chapter eight, provides a review on the use
of structural data for modelling shapes with a high semantic characterization, e.g.,
virtual humans. In this case, the structural model, called the control skeleton, has
by itself a specific role in the evaluation of the many different shapes associated
with all the possible postures that the body model can reach. The first part of the
Chapterdiscussesthecontrolarticulatedskeletonstructureanddifferentapproaches
tobuildskeletonsandbindittotheshapegeometry.Thesecondpartaddressesthe
generationoflevel-of-detailmodelsforvirtualhumans,intermsofthegeometryand
ofthearticulatedskeleton.
Wewouldliketoacknowledgetheworkofalltheauthorsofthevariouschapters
inthisbook,thatcontributedwiththeirexpertiseandenergytoassembleasubstantial
partofstate-of-the-artreportsonavarietyofinterestingtopics.Ourspecialthanksgo
toBiancaFalcidieno,thecoordinatoroftheAIM@SHAPENetworkofExcellence,
for her enthusiasm and support in this work and for having inspired much of the
motivations of this collection, and to all partners in AIM@SHAPE. Many people
contributed to the preparation of the book, and we would like to thank specifically
EmanueleDanovaroforhisinvaluablehelpineditingandpreparation.
Finally,wewouldliketoacknowledgethesupportoftheEuropeanNetworkof
ExcellenceAIM@SHAPE,contractnumber506766.
LeilaDeFloriani MichelaSpagnuolo
Contents
1 Introduction .................................................... V
ShapeInterrogation
Stefanie Hahmann, Alexander Belyaev, Laurent Buse´, Gershon Elber,
BernardMourrain,ChristianRo¨ssl ................................... 1
1 Introduction .................................................... 1
2 DifferentialGeometryofcurvesandsurfaces......................... 2
2.1 Curves.................................................... 2
2.2 Surfaces................................................... 3
3 Interrogationofdiscreteshapes .................................... 5
3.1 SurfaceNormalEstimation................................... 5
3.2 CurvatureTensorEstimation ................................. 6
3.3 ApplicationstoDiscreteShapeAnalysis........................ 8
4 First-OrderShapeAnalysis........................................ 9
4.1 Reflectionlines............................................. 9
4.2 Highlightlines ............................................. 10
4.3 Isophotes.................................................. 11
4.4 Detectionofinflections ...................................... 12
4.5 Geodesicpathsonsurfacesandmeshes ........................ 13
5 Second-ordershapeanalysis....................................... 13
5.1 LocalshapeanalysiswithGaussiancurvature ................... 14
5.2 FocalSurfaceandCorrespondingSurfaceFeatures............... 15
5.3 Hedgehogdiagramsandcurvatureplots ........................ 17
5.4 Generalizedfocalsurfaces ................................... 18
5.5 Colormappings ............................................ 19
6 Characteristiclines .............................................. 20
6.1 Isolines ................................................... 21
6.2 Linesofcurvature,umbilics .................................. 22
6.3 CurvatureExtremaforShapeInterrogation ..................... 22
6.4 SpecialSurfacePoints....................................... 24
7 RobustSymbolicbasedShapeInterrogationandAnalysis .............. 25
X Contents
7.1 CurvatureAnalysis ......................................... 26
7.2 Silhouette,Isoclines/IsophotesandReflectionlines............... 28
7.3 SurfaceRecognition ........................................ 28
8 Interrogationofalgebraiccurvesandsurfaces ........................ 30
8.1 Subdivisionmethods ........................................ 32
Univariatepolynomials ...................................... 32
Multivariatepolynomials .................................... 33
UnivariateRootSolver ...................................... 34
Multivariaterootfinding ..................................... 34
8.2 Algebraicmethods.......................................... 35
Resultant-basedmethods..................................... 35
Normalforms.............................................. 39
9 Conclusion ..................................................... 41
References ......................................................... 42
RecentAdvancesinRemeshingofSurfaces
PierreAlliez,GiulianaUcelli,CraigGotsman,MarcoAttene............... 53
1 Introduction .................................................... 53
1.1 Remeshing ................................................ 54
1.2 Applications ............................................... 54
1.3 MainIssues................................................ 54
2 StateoftheArt.................................................. 56
2.1 StructuredRemeshing ....................................... 57
Semi-Regular .............................................. 57
CompletelyRegular......................................... 61
HighlyRegular............................................. 62
2.2 CompatibleRemeshing...................................... 63
2.3 HighQualityRemeshing..................................... 66
2.4 FeatureRemeshing ......................................... 71
Feature-preserving.......................................... 72
Feature-enhancing .......................................... 73
2.5 Error-drivenRemeshing ..................................... 74
References ......................................................... 78
MultiresolutionAnalysis
Georges-PierreBonneau,GershonElber,StefanieHahmann,BasileSauvage.. 83
1 Introduction .................................................... 83
2 HierarchicalFreeformRepresentations.............................. 84
3 MultiresolutionMethodsforFreeformRepresentations ................ 85
3.1 WaveletDecompositionofB-splineFunctions................... 86
3.2 DirectFreeformCurveandSurfaceManipulation ................ 88
3.3 LinearConstraints .......................................... 91
3.4 Bi-LinearandNon-LinearConstraints ......................... 92
3.5 IntrinsicMultiresolutionDecompositionofFreeformShapes ...... 96
3.6 MultiresolutionMorphing.................................... 98
Contents XI
3.7 VariationalMultiresolutionMethodsforFreeformSurface ........ 100
VariationalShapeDesign .................................... 101
VariationalMultiresolutionModeling .......................... 102
4 MultiresolutionAnalysisforIrregularMesh-basedRepresentations...... 102
4.1 IrregularTriangulations...................................... 103
4.2 SurfaceMeshes ............................................ 108
5 ConclusionsandOpenIssues...................................... 109
References ......................................................... 110
SubdivisionSurfacesandApplications
ChiaraEvaCatalano,IoannisIvrissimtzis,AhmadNasri..................115
1 Introduction .................................................... 115
2 Subdivisionbasics ............................................... 116
2.1 Subdivisionschemes ........................................ 116
2.2 Subdivisionanalysis ........................................ 118
Laurentpolynomials ........................................ 118
Spectralanalysisofthesubdivisionmatrix...................... 119
Thecontinuitydegreeofsubdivisionsurfaces ................... 120
3 Artifactsinsubdivisionsurfaces ................................... 120
3.1 Firstorderartifacts.......................................... 121
3.2 Secondorderartifacts ....................................... 123
3.3 Higherlevelartifacts ........................................ 126
4 Implementationandusabilityissues ................................ 128
4.1 Directevaluationofsubdivisionsurfaces ....................... 128
4.2 Visualqualityandshapepredictability ......................... 129
4.3 Multi-resolutionsubdivisionsurfaces .......................... 131
5 Constraint-basedsubdivisionsurfaces............................... 131
5.1 Pointinterpolation .......................................... 132
5.2 Interpolationwithnormalconstraints .......................... 132
5.3 Interpolationofisolatedcurves ............................... 132
5.4 Interpolationofisolatedcurveswithcrossderivative ............. 135
5.5 Loftedsubdivisionsurfaces .................................. 135
5.6 Interpolationofanetofcurves................................ 137
5.7 Non-uniformsubdivisionsurfaces ............................. 138
6 Conclusions .................................................... 138
References ......................................................... 140
SkeletalStructures
Silvia Biasotti, Dominique Attali, Jean-Daniel Boissonnat, Herbert
Edelsbrunner,GershonElber,MichelaMortara,GabriellaSannitidiBaja,
MichelaSpagnuolo,MirelaTanase,RemcoVeltkamp .....................145
1 Introduction .................................................... 145
1.1 Overview.................................................. 146
2 Definitionsofgeometricmedialstructures ........................... 148
3 Exactrepresentationofmedialstructures ............................ 150