41 3 Vol.41,No.3 016 ActaGeodaeticaetCartographicaSinica Jun.,01 GEXuming,WUJicang.GeneralizedRegularizationtoIl-posedTotalLeastSquaresProblem[J].ActaGeodaeticaetCartographicaSinica, 01,41(3):37-377.(. [J].,01,41(3):37-377.) 1 1, 1. 0009;. 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 :1001-1595(01)03-037-06 : (41074019); (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
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
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, 10 33 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
3 : 375 1 3: [19] Δx x-x * 1, Tab.1 Biasandrelativebiasnorm 3 1 R-LS R-TLS R-GTLS Δx 6.0706 6.07 0.4335 foxgood 10 5 x-x * 0.8398 0.8400 0.0600 : 1 1 3 Fig.3 Comparison ofdiferent methods andinflectionpointselection 3 Tab.3 Biasandrelativebiasnorm R-LS R-TLS R-GTLS Δx 0.736 0.7451 0.604 x-x * 0.1646 0.1666 0.058 3 Fig. Comparisonofdiferentmethodsandinflection pointselection R-LS R-TLS R- GTLS Tab. Biasandrelativebiasnorm R-TLS R-LS R-LS R-TLS R-GTLS 1 Δx 5.9390 5.8880 0.1991 R-LS R-TLS x-x * 0.816 0.8146 0.075 ;R-GTLS
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, -11.01m) [0] A, R-GTLS 89543 4 4 Tab.4 Resultsbydiferentmethodsandcorrespondingbiasandrelativebiasnorm LS R-LS TLS R-TLS R-GTLS 0 0.0531 0.053-0.4874 0.0405 0.018 0-0.0846-0.0699-0.1587-0.0117-0.007 0-0.8545-0.6798-0.713-0.0463 0.006 68 68.0396 68.0406 66.5933 68.091 68.0153-6 -6.0953-5.905-387.4079-5.9351-5.9851 9 8.819 8.948-164.0349 8.9654 8.9837 14 14.0074 14.0078 14.0788 14.0089 14.006 41 40.9064 40.9468 384.3145 40.9677 40.990-11 -11.696-11.133 1075.660-11.010-11.0078 Δx 1.1610 0.7079 1758.0810 0.1068 0.0351 x-x * 0.0135 0.008 0.4663 0.001 0.0004 4 4,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
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