3 : 373 R-LSR-TLS TSVD Tikhonov Tikhonov Ax b, A R m n,b R n,m n (1) min Ax-b Lx δ (5),A ;b ;x,δ ;L 1 b [9] A Lagrange min Ax-b = Δb Ax=b+Δb () L ( x,

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1 41 3 Vol.41,No ActaGeodaeticaetCartographicaSinica Jun.,01 GEXuming,WUJicang.GeneralizedRegularizationtoIl-posedTotalLeastSquaresProblem[J].ActaGeodaeticaetCartographicaSinica, 01,41(3): (. [J].,01,41(3): ) 1 1, ;. GeneralizedRegularizationtoIl-posedTotalLeastSquaresProblem GEXuming 1,WUJicang 1, 1.DepartmentofSurveying and Geo-Informatics,TongjiUniversity,Shanghai0009,China;.Key Laboratory of Modern EngineeringSurveying,SBSM,Shanghai0009,China Abstract:Totalleastsquaresmethodisaderegularizingprocedure,sotheil-posedproblemswilbemoreserious. Thatmeanserorsinthedata are morelikelyto afectthetotalleastsquaressolutionthantheleastsquares solution.itisproposedusinggeneralizedregularizationtosolveil-posedproblemsintotalleastsquares,soasto improvestabilityoftheresults.finaly,numericalexperimentsarecariedouttodemonstratetheperformanceand eficiencyofthegeneralizedregularizationmethodwhichhavesignificantadvantagesinsolvingil-posedproblems. Keywords:il-posedtotalleastsquares;EIV;generalizedregularization;L-curve : (TLS), : ;EIV; ;L :P07 :A : (01) : ( ); (010DFB0190) 1 [1] EIV TLS TLS [8] (GSVD) TLS (T-TLS); [9] (R-TLS) ; [] [10 1] R- [1], TLS,TLS [13] [3] TLS ; [14] ; [4] (WTLS) ; [5] [13 14] TLS [15] [16] EIV L [16] (LS), L [9] R-TLS [10 11] [16] L [6] [7] (TSVD) (R-GTLS) LS

2 3 : 373 R-LSR-TLS TSVD Tikhonov Tikhonov Ax b, A R m n,b R n,m n (1) min Ax-b Lx δ (5),A ;b ;x,δ ;L 1 b [9] A Lagrange min Ax-b = Δb Ax=b+Δb () L ( x, λ) = Ax-b +λ ( Lx -δ )(6),λLagrange x (5) δ x EIV min { Ax-b +λ Lx } (7),λ min (A,b)-(A #,b # ) F A # x=b # (3) A # b # min (A,b)-(A #,b # ) F, A # x=b #, Lx δ (8) [1-] x LS=(A T A) -1 A T b (4a) Lagrange L ( A #,x, μ) = ( A, b) - ( A #,A # x ) F+ x TLS=(A T A-σ n+1i) -1 A T b (4b) μ ( Lx -δ ) (9),σn+1 [A b] n+1, μ Lagrange δ ;I, (7) A 0 min Ax-b { +ζ Lx 1+ x } (10), ζ R-TLS, [9] (9),, (R-TLS)x R-TLS ( A T A+λII+λLL T L ) x=a T b (11), 1 λi=- b-ax (11a) (σn+1i) 1+ x A T A (8)L=I (11) [17] Tikhonov A ( A T A+λIL I) x=a T b (1) [18] 1,λIL =λi+λl(1) [16] TSVD δ< x LS Tikhonov T-TLS TSVD A, [10],T-TLS [A b], [8],T-TLS Tikhonov [13] [9], Lx LS Lx TLS δ

3 374 June01Vol.41No.3AGCS htp: xb.sinomaps.com δ < Lx LS, (8) x λl =x R-GTLS 3 min b-ax δ = Lx 1+ x (13) [19] RegularizationTool007 (13) L=I (1) 3,1+ x 1+δ δ R-LS,R-TLS R-GTLS3 R- GTLS [18] L 1: [19] Regularization Tool 007 i laplace L, δ (7) δ L b-ax λ, Lx λ λ 1(a) x( ),R- (10), δ LS ( * ),R-TLS ( λl λi )R-GTLS ( ) 1(b) (11) R-GTLS L ( ), x R-TLS [16] L,, : w 1= b-axλ L,w = Lx 1+ x λl λl b-ax λl 1+ x λl [a,b] λl R-TLS x λl L L Lx λ δ [16,18]L, L x λl x R-TLS : (1) λl L () λl, (11a)λI (3) (11) x λl (11a) λi (4) λi x λl (3) (4) (4) (5) (5) () (3) (4) 1λL λi x λl (6) (6) b-axλ L Lx 1+ x λl λl L 1 (7) λl,pointselection Fig.1 Comparisonofdiferentmethodsandinflection

4 3 : : [19] Δx x-x * 1, Tab.1 Biasandrelativebiasnorm 3 1 R-LS R-TLS R-GTLS Δx foxgood 10 5 x-x * : Fig.3 Comparison ofdiferent methods andinflectionpointselection 3 Tab.3 Biasandrelativebiasnorm R-LS R-TLS R-GTLS Δx x-x * Fig. Comparisonofdiferentmethodsandinflection pointselection R-LS R-TLS R- GTLS Tab. Biasandrelativebiasnorm R-TLS R-LS R-LS R-TLS R-GTLS 1 Δx R-LS R-TLS x-x * ;R-GTLS

5 376 June01Vol.41No.3AGCS htp: xb.sinomaps.com,r-gtls 4: [0] R-LS R-TLS P 1 P P 10 10, 10 3 P 11 P 1, R-GTLSR-LS R-TLS 1 3-6,9)(14,41,-11)) 3 R-LS, ±0.01m R-TLS 33 3 R-GTLS,3, (0.01m,-0.01m,0.00m) (68.01m, -5.99m,8.98 m)(14.0 m,40.99 m, 3 P 13 ( (0,0,0) (68, m) [0] A, R-GTLS Tab.4 Resultsbydiferentmethodsandcorrespondingbiasandrelativebiasnorm LS R-LS TLS R-TLS R-GTLS Δx x-x * ,R-TLS R-LS R- GTLS 3 : R-TLS (1) R-GTLS, (R-TLS), 4 (R-LS) 5, (),R-LSR-TLS 4 R-GTLS R- LSR-TLS

6 3 : 377 (3)R-GTLS R-LS R-TLS [1] GOLUB G H,LOAN F C VAN.AnalysisoftheTotal LeastSquaresProblem[J].SIAM Journalon Numerical Analysis,1980,17(6): [] HUFFELSVAN,VANDEWALLE J.AnalysisandProperties ofthe Generalized TotalLeastSquaresProblem AX=B whensomeoralcolumnsofaaresubjecttoerrors[j]. SIAMJournalon MatrixAnalysisandApplications,1989, 10(3): [3] LU Tieding,ZHOUShijian,ZHANGLiting,etal.Sphere TargetFixingofPointCloudDataBasedonTLS [J].Journal ofgeodesyandgeodynamics,009,(4): ( Regularized Total Least Squares Solution to Il-posed Error-in-variable Model [J].Journal of Geodesy and Geodynamics,009,9(): (,. [J].,009,9(): ) [14] WANGLeyang,XU Caijun,LU Tieding.RidgeEstima- tion MethodinIl-posed Weighted TotalLeastSquares Adjustment[J].GeomaticsandInformation Sciencesof. WuhanUniversity,010,35(11): (, [J].,009,. (4): ) [4] CHEN Weixian,CHEN Yi,YUAN Qing,etal.Applica- tionofweightedtotalleastsquarestotargetfitingof Three-dimensionalLaserScanning[J].JournalofGeodesy andgeodynamics,010,(30):90-96.( [J]. :,010,35(11): ) [15] SCHAFFIN B,WIESER A.On Weighted TotalLeast- squaresadjustmentforlinearregression[j].journalof Geodesy,008,8(7): [16] HANSENPC.AnalysisofDiscreteIl-posedProblemsby [J].,010,(30):90-96.) [5] YUAN Qing,LOU Lizhi,CHEN Weixian,etal.The Applicationofthe WeightedTotalLeast-squarestoThree Dimensional-datum Transformation[J].ActaGeodaeticaet CartographicaSinica,011,40(S): ( [17] MeansoftheL-curve[J].SIAM Review,199,34(4): MARKOVSKYI,HUFFEL Svan.Overview oftotal LeastSquares Methods[J].Signal Processing,007 (87): [18] HANSEN P C.Rank-deficientand DiscreteIl-posed [J].,011,40(S): ) [6] TIKHONOV A N.SolutionofIncorrectlyFormulated ProblemsandtheRegularization Method[J].SovietMath- ematicsdoklady,1963,4: Problems[M].Philadelphia:SIAM,1998. [19] HANSENPC.RegularizationTools:A MatlabPackage foranalysisandsolutionofdiscreteil-posedproblems [J].NumericalAlgorithms,1994,6(1):1-35. [7] HANSEN P C.TruncatedSingularValueDecomposition SolutionstoDiscreteIl-posedProblemswithIl-determined NumericalRank[J].SIAMJournalonScientificandStatis- ticalcomputing,1990,11(3): [0] WANG Zhenjie.Regularized Methodin SurveyingIl- posedproblem[m].beijing:sciencepress,006:15. (. [M]. :,006:15.) [8] FIERRO R D,GOLUBG H,HANSENPC,etal.Regu- ( : ) larizationby Truncated TotalLeastSquares[J].SIAM JournalonScientificandStatisticalComputing,1997,18 (1): [9] GOLUBG H,HANSENPC,O LEARY DP.Tikhonov RegularizationandTotalLeastSquares[J].SIAM Journal on Matrix Analysis and Applications,1999,1 (1): [10] SIMA D M,HUFFELSvan,GOLUBG H.Regularized TotalLeastSquaresBasedon QuadraticEigenvalueProblem Solvers[J].BIT Numerical Mathematics,004,44: [11] GUO H,RENAUT R.A Regularized TotalLeast SquaresAlgorithm[C] TotalLeastSquaresandErrors- in-variables Modeling. Amsterdan: Kluwer Academic Publishers,00: [1] BECK A,BEN-TAL A.OntheSolutionoftheTikhonov Regularization ofthe Regularized Total Least Squares Problem[J].SIAM Journalon Optimization,006,17 (1): [13] YUANZhenchao,SHEN Yunzhong,ZHOUZebo,etal. : : : (1985-) Firstauthor:GEXumming(1985-),male,postgradu- ate,majorsingeodeticdata processing and precise surveyingengineering. ganyu099@163.com

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