Mesh Editing with Curvature Flow Laplacian

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EurographicsSymposiumonGeometryProcessing(2005)M.Desbrun,H.Pottmann(Editors)

MeshEditingwithCurvatureFlowLaplacian

OscarKin-ChungAu1†Chiew-LanTai1HongboFu1LigangLiu21HongKongUniversityofScience&Technology2ZhejiangUniversity

1.IntroductionDifferentialcoordinatesareessentiallyvectorsencodedintheglobalcoordinatesystem.Sincethelocalfeaturesonamesharedeformedandrotatedduringediting,thediffer-entialcoordinatesmustsomehowbetransformedtomatchthedesiredneworientations,otherwisedistortionlikeshear-ingandstretchingwilloccur.Thistransformationproblemisbasicallyachicken-and-eggproblem:thereconstructionofthedeformedsurfacerequiresproperlyorienteddiffer-entialcoordinates,whilethereorientationofthesecoordi-natesdependontheunknowndeformedmesh.WepresentaniterativeLaplacian-basededitingframeworktosolvethistransformationproblem.Theonlyuserinputrequiredarethepositionsofthehandles,nottheirlocalframes.Thusoursys-temsupportssimplepointhandleediting.Ouriterativeup-datingprocessfindsthebestorientationsoflocalfeatures,includingtheorientationsatthepointhandles.

2.LaplacianEditingLetV=(v1,v2,...,vn)bethemeshvertexpositions,and

i∗betheindexsetofverticesadjacenttovi.TheLaplacianCoordinate(LC)ofavertexviisli=∑j∈i∗wij(vj−vi),

wherewijistheweightoftheedge(i,j)correspondingtovertexvi.Inmatrixform,itisl=LV,whereLisann×nmatrixwithelementsderivedfromwij.Werefertotheseele-mentsastheLaplaciancoefficients.ThebasicideaofLapla-cianeditingistofindthepositionsV󰀄ofthedeformedmesh

byminimization,argminV󰀂󰀁󰀁LV󰀄−l󰀁󰀁2,constrainedbythepositionsofsomeselectedverticesasthehandlesofthemodel[S∗04,L∗04].ThisisequivalenttosolvingasparselinearsystemAV󰀄=binleastsquaressense.ThusV󰀄canbesolvedfromthenormalequationsATAV󰀄=ATb.

Previousrelatedmethodscannotproducegoodresultswhenthehandlesinvolvelargeanglerotationoraretrans-lateddistantlyfromtheiroriginallocations.Lipmanetal.[L∗04]usedanintermediatereconstructedsurfacetoguesstheneworientationsoftheLCs.Sorkineetal.[S∗04]employedimplicitlydefinedtransformationsontotheLCs.However,itisuntenableforlargeanglerotationandanisotropicscaling.Yuetal.[Y∗04]solvedthetransforma-tionproblembypropagatingthetransformationsofhandlestoallvertices.Lipmanetal.[L∗05]encodethevertexdiffer-encesinlocalframesandminimizetheleastsquareserrorofthechangesinthelocalframes.SincebothapproachesneedFigure1:Resultofpreviousmethodsifahandleisonlytranslated,notrotated(middle).Ourmethoddeformsthelo-calfeaturesnaturallybasedononlytheposition(nottrans-formation)ofthehandle(right).Figure2:(left)Inputmodelwithhandlesatthefeet,nosetip,andtailend.(right)Editingbymovingthepointhandlesatthenosetipandtail.thetransformations(orlocalframes)ofthehandlesasinput,ifthehandlesonlyundergotranslation,thereisnotransfor-mationchangetobepropagatedorminimized(seeFigure1);thustheseapproachescannotavoidshearingandstretchingdistortioncausedbyhandletranslation.3.CurvatureFlowLaplacianEditingWeobservethat,toreducedistortion,theeditedmeshshouldretain(1)parameterizationinformation(shapesoftriangles);(2)geometryinformation(sizesoflocalfeatures).Tosep-aratethetwotypesofinformation,weadoptthecurva-tureflowLaplaceoperator:theedge(i,j)hasweightwij=cotαij+cotβij,whereαijandβijarethetwoanglesop-positetheedge.NowtheLCisanapproximationofthein-tegratedmeancurvaturenormalatvi:li=∑j∈i∗wij(vj−vi)=4Areaiκini,whereAreaiistheoneringtrianglesareaofvertexvi,andκiandniarethemeancurvatureandunitnormalvectoratvi,respectively.Weconsiderthelocalparameterizationinformationas

c󰀂TheEurographicsAssociation2005.OscarKin-ChungAuetal./MeshEditingwithCurvatureFlowLaplaciancapturedbytheLaplaciancoefficients,andthegeometryin-formationasencodedbythemagnitudesoftheLCs.AstheLCsareinthedirectionsofthevertexnormals,weregardthenormalsasnotcontaininganylocalinformation(computableonthefly).Thus,ourapproachtriestokeepthemagnitudesofLCsandtheLaplaciancoefficientssimilarbeforeandaf-terediting.Sincebothsetsofinformationgenerallydependonthevertexpositionsnonlinearly,weproposeaniterativeupdatingmethodthatimprovesthevertexpositionsviandtheLCliineachiteration,minimizingparameterizationandgeometrydistortionsprogressively.

Algorithm.LetvtiandltibethevertexpositionsandtheLCs

attimet,respectively,andletv0i=viandl0i=li.

Step1.UpdatethevertexpositionsWeusethecurrentltitocomputethevertexpositionsvt+1i;thatis,wefixtheLCsandsolvethenormalequationswiththecurrenthandlepositionsasconstraints.

Step2.UpdatetheLaplaciancoordinatesWeupdatetheLCstomatchthecurrentdeformedsurface;thatis,wefixthevertexpositionsvt+1iandcomputethenewLCslt+1i.Weusethemeancurvaturenormalscomputedfromthecurrentvertexpositionsvt+1iasthevaluesoflt+1i,butscalethemtohavethemagnitudesoftheoriginall0i,inordertokeeptheoriginalfeaturesizes.