Image Deformation Using Moving Least Squares

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1、ImageDeformationUsingMovingLeastSquaresScottSchaefer∗TravisMcPhail†JoeWarren‡TexasA&MUniversityRiceUniversityRiceUniversity(a)(b)(c)(d)Figure1:DeformationusingMovingLeastSquares.Originalimagewithcontrolpointsshowninblue(a).MovingLeastSquaresdeforma-tionsusingaffinetransformations(b),similarity

2、transformations(c)andrigidtransformations(d).Abstractthepositionandorientationofthesehandles,theimageshouldde-forminanintuitivefashion.WeprovideanimagedeformationmethodbasedonMovingLeastWeviewthisdeformationasafunctionfthatmapspointsintheSquaresusingvariousclassesoflinearfunctionsincludingaffi

3、ne,undeformedimagetothedeformedimage.Applyingthefunctionsimilarityandrigidtransformations.Thesedeformationsarereal-ftoeachpointvintheundeformedimagecreatesthedeformedisticandgivetheusertheimpressionofmanipulatingreal-worldimage.Nowconsideranimagewithasetofhandlespthattheuserobjects.Wealsoallo

4、wtheusertospecifythedeformationsusingmovestonewpositionsq.Forftobeusefulfordeformationsiteithersetsofpointsorlinesegments,thelaterusefulforcontrol-mustsatisfythefollowingproperties:lingcurvesandprofilespresentintheimage.Foreachofthesetechniques,weprovidesimpleclosed-formsolutionsthatyieldfast•

5、Interpolation:Thehandlespshouldmapdirectlytoqunderdeformations,whichcanbeperformedinreal-time.deformation.(i.e;f(pi)=qi).CRCategories:I.3.5[ComputerGraphics]:ComputationalGe-•Smoothness:fshouldproducesmoothdeformationsometryandObjectModeling—Boundaryrepresentations;Curve,surface,solid,andobje

6、ctrepresentations;Geometricalgorithms,languages,andsystems•Identity:Ifthedeformedhandlesqarethesameasthep,thenfshouldbetheidentityfunction.(i.e;qi=pi⇒f(v)=v).Keywords:Deformations,movingleastsquares,rigidtransforma-tionsThesepropertiesareverysimilartothoseusedinscattereddatainterpolation.Thefi

7、rsttwopropertiessimplystatethatthefunc-tionfinterpolatesthescattereddatavaluesandissmooth.Thelast1Introductionpropertyissometimesreferredtoaslinearprecisionintheapproxi-mationfield.Itstatesthatifdataissampledfromalinearfunction,Imagedeformatio