基于总体最小二乘法的三维坐标转换参数解算

©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.(2009)3620359203:2009208213:(19762),,,,,610061(19632),,,,615000(19772),,,,610212:,,,,,,:,(TLS),:P226.3:A0[1],,,[2],3137,,(LeastSquares,LS),(Gauss2Markov,G2M)G2M,,,,(TotalLeastSquares,TLS)LS13D3D,x,y,zx,y,z,:xyz=xyz+Rxyz(1),x,yz;;R:R=r11r12r13r21r22r23r31r32r33r11=cos()cos(),r12=-cos()sin(),r13=-sin,r21=cos()sin()-sin()sincos(),r22=cos()cos()+sin()sinsin(),r23=-sin()cos(),r31=sin()sin()+cos()sincos(),r32=sin()cos()-cos()sinsin(),r33=cos()cos(),,,X,YZx,y,z,,,7,(1),,7(1),,(1),===0,=1,(1),:x-xy-yz-z=100x0-z-y010y-z0x001zyx0dxdydzdddd(2)n(n3)(2),:l=A(3),lR3n1;AR3n7;R71,rank(A)=73nl,0,:l+e=A,eN(0,20P-1)(4),20;PR3n3n,:minee2p,l+e=A(5)(5)LS,2p,:=(ATPA)-1ATPl(6),2TSL3DTLSGolubVanLoan[3],LS,TLS,,(3),A,,A,A=[A1;A2],A1R3n3,A2R3n4;=[T1;T2]T,1R31,2R41;:9533536200912SHANXIARCHITECTUREVol.35No.36Dec.2009©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.=100010001100010001,A2=x10z1-y1y1-z10x1z1y1-x10xn0zn-ynyn-zn0xnznyn-xn0,1=dxdydz,2=dddd:l+el=A1;A2+EA212(7),elR3n1;EA2R3n4,03n3nDA255C,:min[EA2;el]D[EA2;el]CF,:l+el=[A1;A2+EA2]12(8)(8)TLS,FFrobenius[5](8)[3,6],3:1)D[A1;A2;l]QR,:QTD[A1;A2;l]=R11R12R1b0R22R2b(9),R11R33;R12R34;R22R(3n-3)4;R1bR31;R2bR(3n-3)12)(9),:[R22;R2b]CC-12-10(10)(10)[R22;R2b]C(SingularValueDecomposition,SVD),:[R22;R2b]C=UVT(11),U=[u1,1,,u1,3n-3,,u3n-3,1,,u3n-3,3n-3]R(3n-3)(3n-3)V=[v1,1,,v1,5,,v5,1,,v5,5]R55;=diag(1,,5)(11),2:2=-1c5v5,5C14[v1,5,v2,5,,v4,5]T(12),C14=diag(c1,c2,,c4)C4();c5C,A23)22(9),:R111+R122-R1b=0(13)(13),1:1=R-111(R1b-R122)(14)3,110(),(1)2(),0.01,0,LSTLS,21m(x-y-z)(x-y-z)1994.8196982.8943105.1253994.8245982.9042105.11993553983.0716396746.194832.83362995.0016982.8721100.7669995.0015982.8743100.75363553983.2397396746.141128.47433993.0188983.0819104.6561993.0160983.0845104.66683553981.3694396746.792532.39614993.2046983.0621101.5637993.2174983.0742101.55663553981.5427396746.761229.28295992.1702982.7885104.3935992.1888982.7858104.39343553980.4782396746.732732.10116991.3230982.8562103.6719991.3178982.8549103.67193553979.6694396747.011531.41677990.2440980.6089103.0562990.2450980.5962103.05373553978.0757396745.103630.78198988.9804980.6839102.4040988.9723980.6673102.40803553976.8742396745.495930.12269987.7439981.4384101.4658987.7507981.4314101.46323553975.8631396746.515329.212610995.0033982.8629102.7767994.9797982.8657102.76013553983.2426396746.125230.49992x/my/mz/m/()/()/()3552779.5815396037.4850-72.00100.99869230710.33204-0.20293-14.13354LS3552778.9681396041.1634-74.71840.99788174400.32013-0.06223-14.28133TLS3552778.9681396041.1634-74.71840.99788174400.32013-0.06226-14.28133,LSTLS(4);2,,(2),TLSLS4,(TLS),LSTLSLSl,A,,:[1],,,.[J].,2008,38(23):1212128.[2].GPS[J].,2009,2(90):90292.0633536200912©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.(2009)3620361202GPS:2009207213:(19662),,,,315040(19812),,,,,315040:GPS,GPS,,,:,,:TU198:AGPS,,,,WGS2841GPS,,,GPSX,P1P2,WGS284X1X2,:X2=X1+X12(1),X12P1,P2P1,X1,P2:X2=X1+X12(2),X12[1]:ýk12(t)=1/[k2(t)-j2(t)-k1(t)+j1(t)]-ýNk(3),ji(t)PiSj,ji(t)=[(xj(t)-xi)2+(yj(t)-yi)2+(zj(t)-zi)2]1/2;xj(t),yj(t),zj(t)Sj;xi,yi,ziPi;ýNk;ýk12(t)tP1,P2Sj,SkWGS284Xio,Xi,:ji(t)=jio(t)-[lji(t)mji(t)nji(t)]Xi=jio(t)-rji(t)Xi(4),rji(t)PiSj,rji(t)=[lji(t)mji(t)nji(t)];Xi=[xiyizi]Trjki=rki(t)-rji(t),:(3):V=-1/[1/rjk2x2+1/(ljk-Crjk1x1)](5)ljk=-[k20(t)-j20(t)-k10(t)+j10(t)]+[ýk12(t)+ýNjk12](6)n,ýNjk12,1,:V=-1/[1/r2x2+1/(1-r1x1)](7),V=[v12v13vln]T;ri=[r12ir13irlni]T;l=[l12l13lln]T(7),:x2=x1-r-12rx1(8),r=r2-r1,x12x1GPS,:x12=x2-x1=Qx1(9)[3]O.Akyilmaz.Totalleastsquaressolutionofcoordinatetransfor2mation[J].SurveyReview,2007,39(303):68280.[4],,.[J].,2008,17(6):628.[5]IvanMarkovsky,SabineVanHuffel.Overviewoftotalleast2squaresmethods[J].SignalProcessing,2007,87(10):228322302.[6].[M].:,2004.Calculatingparametersof3DcoordinatestransformationbasedonthemethodofTotalLeastSquaresLUZhongGUIYou2longLIUYongAbstract:Itpointsoutthatthetraditionalmethodusedforcalculatingparametersof3DcoordinatestransformationistheLeastSquares(LS)method,thismethodonlytakescoordinatesareerroneousofonecoordinatesystemintoaccount,butitisnotagreewiththerealityapparently.Inordertoresolvethisproblem,amethodofcalculatingparametersof3Dcoordinatestransformationwasproposedandtheprogramwasimple2mentedandasimulatedexperimentationtotestthenewmethodalsodesignedbythispaper.TheresultshowsthattheTotalLeastSquares(TLS)methodcanbeusedtocalculateparametersof3Dcoordinatestransformationeffectively.Keywords:3Dcoordinatestransformation,TotalLeastSquares(TLS),calculatingparameters1633536200912SHANXIARCHITECTUREVol.35No.36Dec.2009