Geometric Curve Evolution and Image Processing

187 Pages·2003·2.219 MB·English

by  Frédéric Cao (auth.)

1805
Lecture Notes in Mathematics
Editors:
J.–M.Morel,Cachan
F.Takens,Groningen
B.Teissier,Paris
3
Berlin
Heidelberg
NewYork
HongKong
London
Milan
Paris
Tokyo
Fre´de´ric Cao
Geometric Curve Evolution
and Image Processing
1 3
Author
Fre´de´ricCao
IRISA/INRIA
CampusUniversitairedeBeaulieu
35042RennesCedex
France
e-mail:[e-mail]
http://www.irisa.fr/prive/fcao
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MathematicsSubjectClassification(2000):35K65,35D05,53A04,68U10
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Preface
These lectures intend to give a self-contained exposure of some techniques for
computingtheevolutionofplanecurves.Themotionsofinterestaretheso-called
motionsbycurvature.Theymeanthat,atanyinstant,eachpointofthecurvemoves
withanormalvelocityequaltoafunctionofthecurvatureatthispoint.Thiskindof
evolutionisofsomeinterestindifferentialgeometry,forinstanceintheproblemof
minimalsurfaces.Theinterestisnotonlytheoreticalsincethemotionsbycurvature
appearinthemodelingofvariousphenomenaascrystalgrowth,flamepropagation
andinterfacesbetweenphases.Morerecently,theseequationshavealsoappearedin
theyoungfieldofimageprocessingwheretheyprovideanefficientwaytosmooth
curvesrepresentingthecontoursoftheobjects.Thissmoothingisanecessarystepfor
imageanalysisassoonastheanalysisusessomelocalcharacteristicsofthecontours.
Indeed, natural images are very noisy and differential features are unreliable if
one is not careful before computing them. A solution consists in smoothing the
curvestoeliminatethesmalloscillationswithoutchangingtheglobalshapeofthe
contours.Whatkindofsmoothingissuitableforsuchatask?Theanswershallbe
givenbyanaxiomaticapproachwhoseconclusionsarethattheclassofadmissible
motionsisreducedtothemotionsbycurvature.Oncethisisestablished,thewell-
posedness of these equations has to be examined. For certain particular motions,
thisturnstobetruebutnocompleteresultsareavailableforthegeneralexistenceof
thesemotions.Thisproblemshallbeturnedaroundbyintroducingaweaknotion
of solution using the theory of viscosity solutions of partial differential equations
(PDE).Acompletetheoryofexistenceanduniquenessofthoseequationswillbe
presented,asself-containedaspossible.(Onlyatechnical,thoughimportant,lemma
willbeskipped.)Thenumericalresolutionofthemotionsbycurvatureisthenext
topic of interest. After a rapid review of the most commonly used algorithms, a
completelydifferentnumericalschemeispresented.Itsoriginalityisthatitsatisfies
exactlythesameinvariancepropertiesastheequationsofmotionbycurvature.It
is also inconditionally stable and its convergence can be proved in the sense of
viscositysolutions.Moreover,itallowstopreciselycomputemotionsbycurvature,
when the normal velocity is a power of the curvature more than 3, or even 10 in
VI Preface
some cases, which seems a priori nearly impossible in a numerical point of view.
Manynumericalexperimentsarepresented.
Whothisvolumeisaddressedto?
We hope that these notes shall interest people from both communities of applied
mathematics and image processing. We tried to make them as self-contained as
possible.Nevertheless,weskippedthemostdifficultresultssincetheirproofuses
techniquesthatwouldhaveledustoofarfromourmainway.Indeed,theselectures
areaddressedtoresearchersdiscoveringthecommonfieldofmathematicsandim-
ageprocessingbutalsotograduateandPhDstudentswantingtospanatheoryfrom
AtoZ:fromthebasicaxioms,tomathematicalresultsandnumericalapplications.
ThechaptersaremostlyindependentexceptChap.6thatusesresultsfromChap.4.
Thebibliographyoneverysubjectwetackleishuge,andwecannotpretendtogive
exhaustive references on differential geometry, viscosity solutions, mathematical
morphologyorscalespacetheory.Attheendofmostchapters,wegiveshortbibli-
ographicalnotesdetailinginafewwordsthemainstepsthatproducedsignificant
advancesinthetheory.
Acknowledgements
MyfirstthanksgotoJean-MichelMorelwhowasmyadvisorduringmythesisand
also proposed me to write these notes. It has been a constant pleasure to work in
histeaminwhichcollaborationsareasnumerousasfruitful.Iwouldliketothank
himforhisscientificsupportbutalsoforalltheotherqualitieshedevelopsinhis
relations.
IwouldalsoliketoexpressmygratitudetoYvesMeyer,withwhomanydiscussion
isalwaysanopportunitytolearnsomethingandtorealizewehavestillsomuchto
learn.Ihavealsobeenalwaysimpressedbyhisenthusiasm,andthankhimsincerely
forhisencouragements.
IowealsomuchtoLionelMoisan,whocontributedtotheredactionofChap.6and
whosehelp,speciallyforthenumericalpart,wasveryprecious.
ManythanksalsotoJean-PierreLeCadreforhiscarefulreadingofthistext.
IwouldalsoliketothankAndre´sAlmansa,VicentCaselles,FrancineCatte´,Agne`s
Desolneux,Franc¸oiseDibos,JacquesFroment,YannGousseau,Fre´de´ricGuichard,
Georges Koepfler, Franc¸ois Malgouyres, Simon Masnou, Pascal Monasse, Pablo
Muse´,DenisPasquignon,Fre´de´ricSurforallthediscussionswehadforthelastfew
years.
Rennes,October2002 Fre´de´ricCao
Contents
Preface ....................................................... V
PartI Thecurvesmoothingproblem
1 Curveevolutionandimageprocessing ......................... 3
1.1 Shaperecognition ......................................... 5
1.1.1 Axiomsforshaperecognition... ....................... 7
1.1.2 ...andtheirconsequences ............................ 8
1.2 Curvesmoothing.......................................... 8
1.2.1 Thelinearcurvescalespace .......................... 9
1.2.2 Towardsanintrinsicheatequation ..................... 10
1.3 Anaxiomaticapproachofcurveevolution ..................... 12
1.3.1 Basicrequirements.................................. 12
1.3.2 Firstconclusionsandfirstmodels...................... 13
1.4 Imageandcontoursmoothing ............................... 15
1.5 Applications.............................................. 15
1.5.1 Activecontours..................................... 15
1.5.2 Principlesofashaperecognitionalgorithm.............. 17
1.5.3 Opticalcharacterrecognition ......................... 18
1.6 Organizationofthevolume ................................. 19
1.7 Bibliographicalnotes ...................................... 20
2 Rudimentarybasesofcurvegeometry ......................... 23
2.1 Jordancurves............................................. 23
2.2 Lengthofacurve.......................................... 24
2.3 Euclideanparameterization ................................. 25
2.4 Motionofgraphs.......................................... 27
VIII Contents
PartII Theoreticalcurveevolution
3 Geometriccurveshorteningflow.............................. 31
3.1 Whatkindofequationsforcurvesmoothing? .................. 32
3.1.1 Invariantflows ..................................... 32
3.1.2 Symmetrygroupofflow ............................. 32
3.2 Differentialinvariants ...................................... 40
3.2.1 Generalformofinvariantflows ....................... 40
3.2.2 ThemeancurvatureflowistheEuclideanintrinsicheatflow 42
3.2.3 Theaffineinvariantflow:thesimplestaffineinvariant
curveflow ......................................... 42
3.3 Generalpropertiesofsecondorderparabolicflows:adigest ...... 46
3.3.1 Existenceanduniquenessformeancurvatureandaffine
flows.............................................. 46
3.3.2 Short-timeexistenceinthegeneralcase ................ 48
3.3.3 Evolutionofconvexcurves ........................... 48
3.3.4 Evolutionofthelength .............................. 49
3.3.5 Evolutionofthearea ................................ 49
3.4 Smoothingstaircases....................................... 50
3.5 Bibliographicalnotes ...................................... 53
4 Curveevolutionandlevelsets ................................ 55
4.1 Fromcurveoperatorstofunctionoperatorsandviceversa........ 57
4.1.1 Signeddistancefunctionandsupportingfunction ........ 57
4.1.2 Monotoneandtranslationinvariantoperators ............ 57
4.1.3 Levelsetsandtheirproperties......................... 58
4.1.4 Extensionofsetsoperatorstofunctionsoperators ........ 60
4.1.5 Characterizationofmonotone,contrastinvariantoperator . 62
4.1.6 Asymptoticbehaviorofmorphologicaloperators......... 64
4.1.7 MorphologicaloperatorsyieldPDEs ................... 69
4.2 CurveevolutionandScaleSpacetheory....................... 70
4.2.1 MultiscaleanalysisaregivenbyPDEs ................. 70
4.2.2 Morphologicalscalespace ........................... 73
4.3 Viscositysolutions......................................... 74
4.3.1 Definitionofviscositysolution........................ 76
4.3.2 Proofofuniqueness:themaximumprinciple ............ 80
4.3.3 ExistenceofsolutionbyPerron’sMethod............... 85
4.3.4 Contrastinvarianceoflevelsetsflow................... 88
4.3.5 Viscositysolutionsshortenlevellines .................. 89
4.4 Morphologicaloperatorsandviscositysolution................. 90
4.4.1 Medianfilterandmeancurvaturemotion ............... 90
4.4.2 Affineinvariantschemes ............................. 92
4.5 Conclusions .............................................. 93
4.6 Curvaturethresholding ..................................... 93
Contents IX
4.6.1 Viscosityapproach .................................. 93
4.6.2 Openingandclosing................................. 95
4.7 Alternativeweaksolutionsofcurveevolution .................. 100
4.7.1 Brakke’svarifoldsolution ............................ 100
4.7.2 Reactiondiffusionapproximation...................... 100
4.7.3 Minimalbarriers.................................... 101
4.8 Bibliographicalnotes ...................................... 101
PartIII Numericalcurveevolution
5 Classicalnumericalmethodsforcurveevolution ................ 107
5.1 Parametricmethods........................................ 107
5.1.1 Finitedifferencemethods ............................ 107
5.1.2 Finiteelementschemes .............................. 108
5.2 Nonparametricmethods.................................... 108
5.2.1 Sethian’slevelsetsmethods .......................... 108
5.2.2 AlvarezandGuichard’sfinitedifferencesscheme ........ 109
5.2.3 Amonotoneandconvergentfinitedifferenceschemes..... 109
5.2.4 Bence,MerrimanandOsherschemeformeancurvature
motion ............................................ 109
5.2.5 Ellipticregularization................................ 110
6 Ageometricalschemeforcurveevolution ...................... 111
6.1 Preliminarydefinitions ..................................... 112
6.2 Erosion .................................................. 114
6.3 Propertiesoftheerosion.................................... 115
6.4 Erosionandlevelsets ...................................... 119
6.4.1 Consistency........................................ 119
6.4.2 Convergence ....................................... 124
6.5 Evolutionbyapowerofthecurvature ........................ 124
6.5.1 Consistency........................................ 125
6.5.2 Convergence ....................................... 129
6.6 Anumericalimplementationoftheerosion .................... 133
6.6.1 Scalecovariance.................................... 133
6.6.2 Generalalgorithm................................... 135
6.6.3 Erodedsetandenvelope ............................. 136
6.6.4 Swallowtails....................................... 138
6.7 Numericalexperiments..................................... 139
6.7.1 Evolvingcircles .................................... 140
6.7.2 Convexpolygons ................................... 141
6.7.3 Unclosedcurve..................................... 142
6.7.4 Evolvingnonconvexcurves .......................... 142
6.7.5 Invariance ......................................... 150
6.7.6 Numericalmaximumprinciple ........................ 152
X Contents
6.7.7 Imagefiltering...................................... 156
6.8 Bibliographicalnotes ...................................... 166
Conclusionandperspectives ..................................... 167
Discussion .................................................... 167
Openproblems ................................................ 169
A ProofofThm.4.34.......................................... 171
References .................................................... 177
Index......................................................... 185