基于最小二乘的曲面拟合算法研究

292Vol.29,No.2200904JournalofHangzhouDianziUniversityApr.2009,,(,310018):2008-06-28:(C24002):(1981-),,,,.:,,,,:;;;:TP301.6:A:1001-9146(2009)02-0048-040(Computer-AidedGeometricDesign,CAGD),[1][2],:,;,,,,,,,1CAGD,[3],,,,,,[4]:f(x,y)=a1+a2x+a3y+a4x2+a5xy+a6y2+a7x3+a8x2y+a9xy2+a10y3(1):z=f(x,y),1i=(xi,yi),1i(xi,yi),i=1,2,,N,[b1(i),b2(1),bn(1)],n=N1i:f(l)=nj=1ajbj(1)(2),bj(1)=xpyq,0p+q0,p0,q0,j=1,2,,nbj(1),j=1,2,,n.n,a1,a2,,an,BA=Z,BnN,B=b1(11)b1(12)b1(1N)b2(11)b2(12)b2(1N)bn(11)bn(12)bn(1N),AT=(a1,a2,,an),ZT=(z1,z2,,zN)A,Znn=N,,,2,[5],,,:E(f)=Ni=1(f(li)-zi)2=Ni=1(Nj=1ajbj(1i)-zi)2(3),E(f),E(f),:9E9aj=0j=1,2,,n(4)24:9E9aj=Ni=12[f(li)-zi]9f(li)9aj=2Ni=1bji[f(li)-zi]=2Ni=1[bji(a1b1i+a2b2i++anbni)-bjizi]=2[a1(Ni=1bjib1i)+a2(Ni=1bjib2i)++an(Ni=1bjibni)-(Ni=1bjizi)]=0(5)5,:a1Ni=1b1ib1i+a2Ni=1b1ib2i++anNi=1b1ibni=Ni=1b1izia1Ni=1b2ib1i+a2Ni=1b2ib2i++anNi=1b2ibni=Ni=1b2izia1Ni=1bnib1i+a2Ni=1bnib2i++anNi=1bnibni=Ni=1bnizi(6)6:BBTA=BZ(7)B=b1(11)b1(12)b1(1N)b2(11)b2(12)b2(1N)bn(11)bn(12)bn(1N),AT=(a1,a2,,an),ZT=(z1,z2,,zN)942:BnN,A,Zna1,a2,,an3VC++.NET2005,OpenGL,,4,U,V,,,,49(77),4916161491[0,1],,1,111ixiyizii=0-0.0002360.982348-0.483352i=1-0.0349761.198324-0.678115i=20.7147290.652642-0.177686i=3-0.0238690.669794-0.158547i=40.4521430.368296-0.156738i=50.1465400.589337-0.346574i=60.6671850.720721-0.599163i=70.3389520.328451-0.261953i=80.7148530.548667-0.541611i=90.484634-0.027283-0.070892i=100.7154530.614536-0.888670i=110.4790370.589764-0.866682i=121.0315940.135872-0.651269i=130.2780910.279388-0.748587i=141.031761-0.184888-0.335382i=151.008976-0.012538-0.4903144,,,,,,0520091[1].B[M].:,2001:45-72.[2],.[J].,2004,(1):84-88.[3],,,.[J].,1999,20(3):41-46.[4]FloaterM.Parameterizationandsmoothapproximationofsurfacetriangulations[J].ComputerAidedGeometricDesign,1997,14(3):231-250.[5],.[J].,1990,(4):40-41.AlgorithmofSurfaceFittingResearchBasedonLeast-squaresMethodsLiEr2tao,ZHANGGuo2xuan,ZENGHong(InstituteofComputerApplicationTechnology,HangzhouDianziUniversity,HangzhouZhejiang310018,China)Abstract:Thispaperintroducesanalgorithmresearchontheleast-squaresmethod(LSM)basedsurfacefitting,andelaboratesonbasicprincipleandapproachofLSM.Byusingthisalgorithmforsurfacefitting,thefittingofthesurfacehastheadvantagesofhighdegreeofaccuracyandsmoothness.Atlast,throughestablishingoptimalfittingparameters,itisshapedthemodelofFergusonbicubiccurvedsurfacetosurfacefitting.Theresultsshowthatusingthismethodtofitsurfaceissuperiorandeffective.Nowadays,dealingwithfittingproblembasedonLSMhasbeenusedwidely.Keywords:surfacefitting;least-squares;parameter;bicubiccurvedsurface152: