generalized total least squares prediction algorithm for

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Advances in Space Research 59 (2017) 815–823

Generalized total least squares prediction algorithm for universal3D similarity transformation

Bin Wang a,b,⇑, Jiancheng Li a, Chao Liu c, Jie Yu d

aSchool of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, 430079 Wuhan, Chinab Institute of Geodesy, University of Stuttgart, Geschwister-Scholl str. 24D, 70174 Stuttgart, Germany

cSchool of Geodesy and Geomatics, Anhui University of Science and Technology, 168 Middle Shungeng Road, 232001 Huainan, ChinadCollege of Surveying and Geo-Informatics, Tongji University, 1239 Siping Road, 200092 Shanghai, China

Received 6 June 2016; received in revised form 8 September 2016; accepted 15 September 2016Available online 23 September 2016

Abstract

Three–dimensional (3D) similarity datum transformation is extensively applied to transform coordinates from GNSS-based datum toa local coordinate system. Recently, some total least squares (TLS) algorithms have been successfully developed to solve the universal 3Dsimilarity transformation problem (probably with big rotation angles and an arbitrary scale ratio). However, their procedures of theparameter estimation and new point (non-common point) transformation were implemented separately, and the statistical correlationwhich often exists between the common and new points in the original coordinate system was not considered. In this contribution, ageneralized total least squares prediction (GTLSP) algorithm, which implements the parameter estimation and new point transformationsynthetically, is proposed. All of the random errors in the original and target coordinates, and their variance–covariance information willbe considered. The 3D transformation model in this case is abstracted as a kind of generalized errors–in–variables (EIV) model and theequation for new point transformation is incorporated into the functional model as well. Then the iterative solution is derived based onthe Gauss–Newton approach of nonlinear least squares. The performance of GTLSP algorithm is verified in terms of a simulated exper-iment, and the results show that GTLSP algorithm can improve the statistical accuracy of the transformed coordinates compared withthe existing TLS algorithms for 3D similarity transformation.� 2016 COSPAR. Published by Elsevier Ltd. All rights reserved.

Keywords: 3D similarity transformation; Errors–in–variables model; Total least squares prediction; Gauss–Newton approach

1. Introduction

Nowadays, Global Navigation Satellite System (GNSS)has been extensively applied in the fields of geodesy andmany other areas. The GNSS techniques are based on spec-ified coordinate systems, such as WGS-84, ITRF 2008, andCGCS 2000 etc. As a result, the coordinates of the pointsobtained from GNSS observations are defined in these sys-tems. However, many geodetic and surveying engineering

http://dx.doi.org/10.1016/j.asr.2016.09.018

0273-1177/� 2016 COSPAR. Published by Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: School of Geodesy and Geomatics, WuhanUniversity, 129 Luoyu Road, 430079 Wuhan, China.

E-mail address: [email protected] (B. Wang).

applications are based on local coordinate systems. Thusthe coordinate transformation problem must be solved inthe GNSS applications.

Three–dimensional (3D) similarity datum transforma-tion is the most commonly used model for geodetic trans-formation and has attracted extensive attention. Thepurpose of 3D transformation is to predict the coordinatesof new points in the target system by using their originalcoordinates and the coordinates of common points in bothoriginal and target coordinate systems. In order to achievethis goal, two necessary procedures, including the parame-ter estimation and new point transformation, should beimplemented in practice.


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816 B. Wang et al. / Advances in Space Research 59 (2017) 815–823

Parameter estimation is an important part for the 3Dsimilarity transformation problem, in which there aretotally seven transformation parameters to be estimated,including three translation parameters, three rotationangles and a scale ratio. In geodetic field, many algorithmshave been developed to solve these parameters. If the rota-tion angles are small enough and the scale ratio is close to1, the transformation equation can be simplified into thewell-known Bursa model (Seeber, 1993, p. 2; Yang,1999), which is normally processed as a linear Gauss–Markov (GM) model. However, the rotation angles couldbe large in practice, therefore many least squaresalgorithms have been proposed to solve the nonlinearGM model, such as the methods based on the Procrustesanalysis (Grafarend and Awange, 2003, 2012; Grafarend,2006) and the unit quaternion (Shen et al., 2006).

It is well known that the random errors in the coefficientmatrix are not considered in the traditional GM model.Nevertheless, the coordinates in the original system areusually observed as well, which means that parts of the ele-ments in the coefficient matrix are affected by randomerrors. Thus the transformation model cannot always beabstracted as a GM model. Then the errors–in–variables(EIV) model should be introduced to replace the GMmodel in the datum transformation problems. The methodto adjust EIV model was named total least squares (TLS)and was firstly introduced by Golub and Van Loan(1980) in the field of statistics. It is generally acknowledgedthat Teunissen (1988) was the first person who formulatedand solved the TLS problem in an exact form in geodeticliterature. Since then a large number of algorithms, espe-cially the weighted TLS (WTLS) algorithms which cansolve EIV models in heteroscedastic case were developed(e.g., Schaffrin and Wieser, 2008; Neitzel, 2010; Shenet al., 2011; Amiri-Simkooei and Jazaeri, 2012;Mahboub, 2012; Xu et al., 2012; Fang, 2013; Zhou andFang, 2015). In addition, some extended TLS algorithmswere also investigated, for instance, the TLS algorithmwith equality (Schaffrin and Felus, 2009; Zhang andZhang, 2014; Fang, 2015; Fang and Wu, 2016; Fanget al., 2015) and inequality constraints (Zhang et al.,2013; Fang, 2014a; Fang and Wu, 2016; Zeng et al.,2015), the TLS solution to multivariate EIV model(Schaffrin and Felus, 2008), the gross error processing forTLS (Amiri-Simkooei and Jazaeri, 2013; Wang et al.,2016) and the variance component estimation method inEIV models (Amiri-Simkooei, 2013; Xu and Liu, 2014;Mahboub, 2014). Many of these algorithms can beemployed to estimate parameters in the 2D or 3D datumtransformation problem.

The TLS solution to 3D transformation under universalconditions (probably with big rotation angles and an arbi-trary scale ratio) has become a hot research issue in recentyears. Felus and Burtch (2009) derived an algorithm bysolving the multivariate EIV model. Fang (2014b) provideda TLS solution by introducing the quasi indirect errors

adjustment (QIEA). Chang (2015) generated a closed-form least squares solution to 3D similarity transformationproblem under Gauss–Helmert model in the equallyweighted case. Additionally, Fang (2015) proposed aWTLS algorithm with universal constraints for fullyweighted 3D datum transformations.

Although a large amount of TLS algorithms for 3Dtransformation have been proposed, a limitation existscommonly: they only focused on how to estimate the trans-formation parameters, whereas their procedures of theparameter estimation and new point transformation wereimplemented separately. The statistical correlation betweenthe common and new points in the original system, whichoften exists in practice particularly when the coordinatesare obtained by GNSS observation networks, was ignored.In order to overcome this limitation, Li et al. (2012) pro-posed a seamless 3D datum transformation model whichintegrated the processes of the parameter estimation andnew point transformation. The experimental resultsshowed that the accuracy of the transformed coordinatescan be improved especially when the correlation mentionedabove is strong. However, this transformation model wasderived based on the Bursa model which is simplifiedunder the conditions with small rotation angles and a scaleratio close to 1, implying that it is inapplicable under theconditions with big rotation angles and an arbitrary scaleratio.

In this paper, we propose a generalized total leastsquares prediction (GTLSP) algorithm for the universal3D similarity transformation, in which the parameter cal-culation and new point transformation can be implementedsynthetically. All of the random errors in the original andtarget coordinates, and their variance–covariance informa-tion are taken into account in this algorithm. The 3D trans-formation model is described as a kind of generalized EIVmodel. In addition, the equation for new point transforma-tion is also included in the functional model. Then theGauss–Newton approach of nonlinear least squares isemployed to derive the iterative solution.

The remaining part of the article is organized as follows:the background of 3D similarity transformation model isintroduced in Section 2. In Section 3, the abstract func-tional model and the corresponding stochastic model arepresented, and then the iterative formulae of the GTLSPalgorithm are derived based on the Gauss–Newtonapproach. In Section 4, we make a detailed discussionabout the application of this new algorithm for 3D similar-ity transformation. A simulated experiment is presented toverify the performance of GTLSP algorithms in Section 5.Finally, some conclusions from theoretical and experimen-tal aspects are given in Section 6.

2. Background of the 3D similarity transformation model

The 3D similarity transformation model is (Wolf andGhilani, 1997; Felus and Burtch, 2009):

B. Wang et al. / Advances in Space Research 59 (2017) 815–823 817

x2iy2iz2i

264

375�

ex2iey2iez2i

264

375 ¼ lM

x1iy1iz1i

264

375�

ex1iey1iez1i

264

375

0B@

1CAþ

Dx

Dy

Dz

264

375

ð1Þwhere ½ x1i y1i z1i �T and ½ x2i y2i z2i �T represent thecoordinates of the ith point in the original and target coor-

dinate systems, ½ ex1i ey1i ez1i �T and ½ ex2i ey2i ez2i �Tare their corresponding random error vectors, respectively.M is the rotation matrix, and it is expressed as below:

M ¼ M3 �M2 �M1 ð2Þ

M1 ¼1 0 0

0 cos a sin a

0 � sin a cos a

264

375;

M2 ¼cos b 0 � sin b

0 1 0

sin b 0 cos b

264

375;

M3 ¼cos c sin c 0

� sin c cos c 0

0 0 1

264

375 ð3Þ

The parameter vector to be estimated is as following:

n ¼ ½Dx Dy Dz l a b c �T ð4Þwhich includes three translation parameters (Dx, Dy, Dz),three rotation angles (a, b, c) and a scale ratio parameter (l).

By using the estimated transformation parameters, theoriginal coordinates of the new points can be convertedinto the target coordinate system. Most previous studieson TLS algorithms for 3D datum transformation onlyfocused on the parameter estimation. Meanwhile the proce-dures of the parameter calculation and coordinates trans-formation of new points were implemented separately. Toovercome this shortcoming, Li et al. (2012) proposed aseamless 3D transformation model which integrated theprocesses of parameter estimation and new point transfor-mation. It can actually be regarded as a kind of TLS pre-diction method. However, this model was derived basedon the Bursa model which is simplified under the condi-tions with small rotation angles and a scale ratio close to1. Thus it is inapplicable for more universal 3D transfor-mation problems with large rotation angles and an arbi-trary scale ratio. The GTLSP algorithm derived in nextsection can effectively solve this problem.

3. Derivation of the GTLSP algorithm

For common points, the transformation model can beabstracted as a kind of generalized EIV model:

L1 � eL1¼ u1ðe1; nÞ ¼ f 1ða1 � e1; nÞ ð5Þ

where f 1 and u1 are both abstract vector functions withdimensions m � 1, a1 and L1 denote the n � 1 and m � 1observation vectors in original and target systems, e1 and

eL1are their corresponding random error vectors, respec-

tively. n is the t � 1 parameter vector to be estimated.For new points, the transformation equation is

expressed as below:

�L2 ¼ u2ðe2; nÞ ¼ f 2ða2 � e2; nÞ ð6Þwhere f 2 and u2 are abstract vector functions as well, andboth of their dimensions are p � 1. a2 represents the q � 1observation vector in the original system, e2 is the corre-

sponding random error vector. �L2 denotes the p � 1 targetvector which needs to be predicted.

The corresponding stochastic model is:

e � Nð0; r20QeÞ ð7Þ

where

e ¼eL1

e1

e2

264

375; Qe ¼

QL1L1QL1a1

QL1a2

Qa1L1Qa1a1

Qa1a2

Qa2L1Qa2a1

Qa2a2

264

375 ð8Þ

Qe is a positive definite cofactor matrix of the total errorvector e, without loss of generality, it is assumed as a fullycorrelated matrix. QL1L1

, Qa1a1and Qa2a2

are the cofactor

matrices of eL1, e1 and e2, respectively. QL1a1

, QL1a2and

Qa1a2are the corresponding cofactor matrices between

eL1, e1 and e2. The estimation criterion of the model above

is presented as following:

eTPe ¼ min ð9Þwhere P denotes the weight matrix, and P ¼ Q�1

e .Since Eqs. (5) and (6) are essentially nonlinear models,

the Gauss–Newton approach of nonlinear least squares isemployed to derive the solution. We assume that the

approximate values of n, e1 and e2 are n0, e01 and e02, respec-

tively. The right-hand members of Eqs. (5) and (6) are

expressed through Taylor series expansion at (n0; e01, e02):

L1 � eL1¼ u1ðe01; n0Þ þ A1dnþ B1ðe1 � e01Þ ð10Þ

�L2 ¼ u2ðe02; n0Þ þ A2dnþ B2ðe2 � e02Þ ð11Þwhere dn is the correction vector of n0, dn ¼ n� n0, and:

A1 ¼ @u1ðe1; nÞ@nT

��ðe01; n0Þ; B1 ¼ @u1ðe1; nÞ@eT1

��ðe01; n0Þ ð12Þ

A2 ¼ @u2ðe2; nÞ@nT

��ðe02; n0Þ; B2 ¼ @u2ðe2; nÞ@eT2

��ðe02; n0Þ ð13Þ

The Lagrange objective function of GTLSP is con-structed as below:

U ¼ eTPeþ 2kTðL1 � eL1� u1ðe01; n0Þ � A1dn

� B1ðe1 � e01ÞÞ þ 2KTð�L2 � u2ðe02; n0Þ � A2dn

� B2ðe2 � e02ÞÞ ð14Þwhere k and K are the m � 1 and p � 1 vectors of‘‘Lagrange multipliers”, respectively. The solution of thistarget function can be derived via the Euler–Lagrange nec-essary conditions, namely,


1

2

@U@e

��ð~e; dn; k; K ; ~L2Þ ¼ P~e�k

BT1 k

BT2 K

264

375 ¼ 0 ð15aÞ

1

2

@U@dn

��ð~e; dn; k; K ; ~L2Þ ¼ �AT1 k� AT

2 K ¼ 0 ð15bÞ

1

2

@U@k

��ð~e; dn; k; K ; ~L2Þ ¼ L1 � ~eL1� u1ðe01; n0Þ

� A1dn� B1ð~e1 � e01Þ ¼ 0 ð15cÞ1

2

@U@K

��ð~e; dn; k; K ; ~L2Þ ¼ ~L2 � u2ðe02; n0Þ � A2dn

� B2ð~e2 � e02Þ ¼ 0 ð15dÞ1

2

@U

@�L2

��ð~e; dn; k; K ; ~L2Þ ¼ K ¼ 0 ð15eÞ

where ‘‘�” and ‘‘^” represent predicted and estimated ones,respectively.

From Eqs. (15a) and (15e), we can readily obtain theerror vector:

~e¼~eL1

~e1

~e2

264

375¼Qe

k

BT1 k

BT2 K

264

375¼

QL1L1QL1a1

QL1a2

Qa1L1Qa1a1

Qa1a2

Qa2L1Qa2a1

Qa2a2

264

375

k

BT1 k

0

264

375

¼ðQL1L1

þQL1a1BT

1 ÞkðQa1L1

þQa1a1BT

1 ÞkðQa2L1

þQa2a1BT

1 Þk

2664

3775 ð16Þ

Thus, we can derive the expression for each residual vector

in the form of k as below:

~eL1¼ ðQL1L1

þQL1a1BT

1 Þk ð17aÞ~e1 ¼ ðQa1L1

þQa1a1BT

1 Þk ð17bÞ~e2 ¼ ðQa2L1

þQa2a1BT

1 Þk ð17cÞBy inserting Eqs. (17a) and (17b) into Eq. (15c), we

obtain the following equation:

L1 � ðQL1L1þQL1a1

BT1 Þk� u1ðe01; n0Þ � A1dn

� B1ðQa1L1þQa1a1

BT1 Þkþ B1e

01 ¼ 0 ð18Þ

Then the expression of k can be derived as:

k ¼ ðQL1L1þQL1a1

BT1 þ B1Qa1L1

þ B1Qa1a1BT

1 Þ�1

� ðL1 � u1ðe01; n0Þ � A1dnþ B1e01Þ

¼ Q�1l ðL1 � u1ðe01; n0Þ � A1dnþ B1e

01Þ ð19Þ

with

Ql ¼ QL1L1þQL1a1

BT1 þ B1Qa1L1

þ B1Qa1a1BT

1 ð20ÞBy substituting Eqs. (15e) and (19) into Eq. (15b), the solu-

tion of dn can be derived as below:

AT1Q

�1l ðL1 � u1ðe01; n0Þ � A1dnþ B1e

01Þ ¼ 0 ð21Þ

dn ¼ ðAT1Q

�1l A1Þ�1

AT1Q

�1l ðL1 � u1ðe01; n0Þ þ B1e

01Þ ð22Þ

Thereby, the parameter vector is updated as:

n ¼ n0 þ dn.Inserting Eq. (22) into Eq. (19) then into Eqs. (17a)–(17c),

we obtain the specific expression for each error vector:

~eL1¼ ðQL1L1

þQL1a1BT

1 ÞQ�1l ðL1 � u1ðe01; n0Þ � A1dnþ B1e

01Þ

ð23aÞ~e1 ¼ ðQa1L1

þQa1a1BT


01Þ

ð23bÞ~e2 ¼ ðQa2L1

þQa2a1BT


01Þ

ð23cÞFinally, from Eq. (15d), we can do the prediction step to

obtain ~L2 as:

~L2 ¼ u2ðe02; n0Þ þ A2dnþ B2ð~e2 � e02Þ ð24ÞThe above procedure can be implemented iteratively, and asmall positive threshold e0 should be primarily presented to

terminate the iteration until kdnk < e0, where k � k denotesthe l2–norm of a vector. It’s important to note that A1, B1,A2, and B2 should be updated during the iterative process.

From Eqs. (17a)–(22), we find that the prediction equa-tion has no effect on parameter estimation. Therefore, Eqs.(23c) and (24) can be carried out after the iteration. If thethreshold e0 is sufficiently small to stop the iteration proce-

dure, i.e., kdnk < e0 ! 0, as a result, ~e2 � e02 ! 0, Eqs.(23c) and (24) will become:

~e2 ¼ ðQa2L1þQa2a1

BT1 ÞQ�1

l ðL1 � u1ð~e1; nÞ þ B1~e1Þ ð25Þ~L2 ¼ u2ð~e2; nÞ ð26ÞIn addition, Eq. (19) can be simplified as:

k ¼ Q�1l ðL1 � u1ð~e1; nÞ þ B1~e1Þ ð27Þ

From Eq. (16), we find that the sum of weighted squaredresiduals ~eTP~e may be expressed as below:

~eTP~e ¼ ½ kT kTB1 0 �QePQe

k

BT1 k

0

264

375

¼ ½ kT kTB1 0 �Qe

k

BT1 k

0

264

375

¼ kTðQL1L1þQL1a1

BT1 þ B1Qa1L1

þ B1Qa1a1BT

1 Þk¼ kTQlk ð28Þ

As a consequence, the variance factor of the unit weightis estimated as following:

r20 ¼

~eTP~e

r¼ kTðL1 � u1ð~e1; nÞ þ B1~e1Þ

rð29Þ


where the degree of freedom r is:

r ¼ ðmþ pÞ � ðt þ pÞ ¼ m� t ð30ÞIn Eq. (30), the total number of equations includes m

equations for common points and p equations for newpoints. The total number of necessary unknowns includest parameters and p target values to be predicted.

The covariance matrix of the estimated parameters canbe obtained via linearly approximate variance propagationlaw to Eq. (22) as:

Dn ¼ r20ðAT

1Q�1l A1Þ�1 ð31Þ

Since GTLSP algorithm is derived based on the theoryof nonlinear least squares, the solution along with its pre-cision assessment is biased, though the bias may be so smallin practice. This bias is related to the precision of the obser-vations and the geometrical properties of the nonlinearmanifold (see Teunissen, 1984, 1985, 1990). To work outthe bias formulae, one can refer to the theory and methodsproposed by Box (1971), which has also been applied in thepartial EIV model (Xu et al., 2012) and constrained TLSproblem (Fang, 2015).

4. Application of the GTLSP algorithm for 3D similarity

transformation

In the application of this new algorithm for 3D similar-ity transformation, related matrices (A1, B1, A2, B2) shouldbe specifically determined and updated during the iteration.

Assuming that there are k common points and l newpoints in total. It is certain that the corresponding dimen-sions mentioned in Section 3 are: m = n = 3k, p = q = 3land t = 7, respectively. For the ith common point,

u1i ¼ lM

x1iy1iz1i

264

375�

ex1iey1iez1i

264

375

0B@

1CAþ

Dx

Dy

Dz

264

375 ð32Þ

A1i ¼ @u1i

@nT

� �0

¼ @u1i

@DX

� �0@u1i

@l

� �0@u1i

@h

� �0� �ð33Þ

In Eq. (33),

@u1i

@DX

� �0

¼ @u1i

@Dx

� �0@u1i

@Dy

� �0@u1i

@Dz

� �0� �¼ I3 ð34aÞ

@u1i

@l

� �0

¼ M0 � C i ð34bÞ

@u1i

@h

� �0

¼ @u1i

@a

� �0@u1i

@b

� �0@u1i

@c

� �0� �

¼ l0 � @M

@a

� �0

C i@M

@b

� �0

C i@M

@c

� �0

C i

� �ð34cÞ

B1i ¼ @u1i

@eT1i

� �0

¼ @u1i

@ex1i

� �0@u1i

@ey1i

� �0@u1i

@ez1i

� �0� �¼�l0 �M0

ð35Þ

where I3 is a 3–order identity matrix. The subscripts ‘‘i” inabove equations indicate the ith point. The superscripts

‘‘0” represent substituting approximate values ðe01; n0Þ intorelated expressions, and all of these values should be con-tinuously updated during the iteration. In addition,

C i ¼x1iy1iz1i

264

375�

ex01iey01iez01i

264

375 ð36Þ

@M

@a¼ M3 �M2 � @M1

@a;

@M

@b¼ M3 � @M2

@b�M1;

@M

@c¼ @M3

@c�M2 �M1 ð37Þ

@M1

@a¼

0 0 00 � sin a cos a0 � cos a � sin a

24

35;

@M2

@b¼

� sin b 0 � cos b0 0 0

cos b 0 � sin b

24

35;

@M3

@c¼

� sin c cos c 0� cos c � sin c 0

0 0 0

24

35

ð38Þ

Combining related expressions of each point together, wecan obtain:

A1 ¼ @u1

@nT

� �0

¼

A11

A12

..

.

A1k

266664

377775;

B1 ¼ @u1

@eT1

� �0

¼

B11

B12

. ..

B1k

266664

377775 ð39Þ

The initial values of e01 and n0 are given as: e01 ¼ 0,

n0 ¼ ½ 0 0 0 1 0 0 0 �T, respectively. In case theapproximate value of the scale ratio becomes non-positive (i.e., l0 < 0) in the iterative procedure, it shouldbe reset to 1 so as to avoid convergence to the incorrectsolution.

The necessary matrices for new points (A2 and B2) can bederived and updated in a similar way as described above.

In practice, the coordinates in the original system andtarget system are usually statistically uncorrelated. In otherwords, QL1a1

¼ 0, Qa2L1¼ 0, thus the related formulae

become:

Ql ¼ QL1L1þ B1Qa1a1

BT1 ð40Þ

~eL1¼ QL1L1

�Q�1l ðL1 � u1ðe01; n0Þ � A1dnþ B1e

01Þ ð41aÞ

~e1 ¼ Qa1a1BT

1 �Q�1l ðL1 � u1ðe01; n0Þ � A1dnþ B1e

01Þ ð41bÞ

~e2 ¼ Qa2a1BT

1 �Q�1l ðL1 � u1ðe01; n0Þ � A1dnþ B1e

01Þ ð41cÞ


If the coordinates of the common and new points in theoriginal system are independent (Qa2a1

¼ 0), the error vec-

tor ~e2 ¼ 0, which means that the coordinates of the newpoints in the original system will not be corrected and theresults of GTLSP algorithm will be equivalent to those ofexisting WTLS algorithms for the universal 3D similaritytransformation (see e.g., Felus and Burtch, 2009; Fang,2014b; Fang, 2015).

5. Experiment analysis

Assuming that there are 18 points distributing in a spacedomain. The true coordinates of each point in the originalcoordinate system (system I) and target coordinate system(system II) are known, and the corresponding transforma-tion parameters from systems I to II are as below:Dx ¼ 1000, Dy ¼ 1000, Dz ¼ 1000, l ¼ 2:0, a ¼ 1:0 rad,b ¼ 1:5 rad, c ¼ 2:5 rad.

Eight of these points are designated as common points,and the remaining ten points are specified as new points(check points).

We assume that the coordinates of the common andcheck points in system I are statistically correlated(Qa2a1

– 0). Random errors with the given covariance

matrices are generated and added to the coordinates ofall points in system I and the common points in systemII. This simulation is repeated 10,000 times.

Assuming that the square root of the apriori variancecomponent r0 is 0.01. The following two computationalschemes are employed to implement the transformationof check points from system I to system II.

(1) WTLS algorithm with universal constraints for the3D similarity transformation (see Fang, 2015);

(2) Generalized Total least squares prediction (GTLSP)algorithm proposed in this paper.

The coordinates of the check points in system II:

½ xi;II yi;II zi;II �T can be obtained by these two schemes.

By utilizing the known true coordinates ½ xi;II yi;II zi;II �T ,we can calculate the root mean square errors (RMSEs)for the x, y and z components of the transformed coordi-nates as follows:

Table 1Statistical results of coordinate transformation for Cases 1–4.

Case Scheme Ave (rx) Ave (ry) Ave (rz) A

1 (1) 0.0357 0.0371 0.0347 0(2) 0.0340 0.0316 0.0314 0

2 (1) 0.0340 0.0371 0.0336 0(2) 0.0314 0.0293 0.0291 0

3 (1) 0.0324 0.0377 0.0330 0(2) 0.0286 0.0269 0.0265 0

4 (1) 0.0305 0.0377 0.0323 0(2) 0.0254 0.0238 0.0236 0

rx ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX10

i¼1ðxi;II � xi;IIÞ2

.10

rð42aÞ

ry ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX10

i¼1ðyi;II � yi;IIÞ2

.10

rð42bÞ

rz ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX10

i¼1ðzi;II � zi;IIÞ2

.10

rð42cÞ

Therefore, the positional RMSE is obtained as:

rp ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2x þ r2

y þ r2z

qð42dÞ

The experiment is carried out in four cases. In Case 1,the correlation coefficients between the coordinate compo-nents of common and check points are all set as 0.35. InCases 2–4, we increase them to 0.45, 0.55, and 0.65 respec-tively while maintaining other conditions unchanged (Theprecision for each coordinate component is the same asthat in Case 1).

The statistics on the calculation results of these fourcases, including the mean and maximum values of the rx,ry , rz, and rp are derived and tabulated in Table 1. In addi-tion, Figs. 1–4 present the experiment sequences for Cases1–4, respectively.

According to the results in Table 1 and Figs. 1–4, wefind the followings:

(i) The corresponding mean values of the rx, ry, rz, andrp from Scheme (2) are 95.2%, 85.4%, 90.6%, and90.1% of those from Scheme (1) in Case 1, 92.5%,78.9%, 86.6%, and 85.5% of those from Scheme (1)in Case 2, 88.5%, 71.3%, 80.4%, and 79.2% of thosefrom Scheme (1) in Case 3, and 83.5%, 63.2%,73.3%, and 72.1% of those from Scheme (1) in Case4, respectively.

(ii) Moreover, there exist some big RMSEs fromScheme (1) during the simulation, and the maximalRMSEs of all directions and positions fromScheme (2) are obviously smaller than those fromScheme (1) in all cases (Cases 1–4).

(iii) Additionally, we also count the number of timeswhen each RMSE obtained by Scheme (2) is smallerthan that obtained by Scheme (1). The results for x

ve (rp) Max (rx) Max (ry) Max (rz) Max (rp)

.0634 0.0843 0.1093 0.0882 0.1439

.0571 0.0744 0.0705 0.0653 0.0960

.0618 0.0914 0.1113 0.0948 0.1431

.0529 0.0656 0.0649 0.0656 0.0934

.0610 0.0848 0.1312 0.1174 0.1912

.0483 0.0596 0.0704 0.0570 0.0833

.0596 0.0930 0.1576 0.1171 0.2071

.0430 0.0583 0.0555 0.0523 0.0764

0 2000 4000 6000 8000 10000

0.02

0.04

0.06

0.08

0 2000 4000 6000 8000 10000

0.02

0.04

0.06

0.08

0.10

0 2000 4000 6000 8000 10000

0.02

0.04

0.06

0.08

0.10

0 2000 4000 6000 8000 10000

0.040.060.080.100.120.14

σ x

Experiment index

σ y

Experiment index

σ z

Experiment index

σ p

Experiment index

Fig. 1. Experiment sequences for Case 1. The green dash line represents the results obtained by Scheme (1); the red solid line represents the resultsobtained by Scheme (2). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

0 2000 4000 6000 8000 10000

0.02

0.04

0.06

0.08

0 2000 4000 6000 8000 10000

0.02

0.04

0.06

0.08

0.10

0 2000 4000 6000 8000 10000

0.02

0.04

0.06

0.08

0.10

0 2000 4000 6000 8000 10000

0.040.060.080.100.120.14

σ x

Experiment index

σ y

Experiment index

σ z

Experiment index

σ p

Experiment index


0 2000 4000 6000 8000 10000

0.02

0.04

0.06

0.08

0 2000 4000 6000 8000 10000

0.020.040.060.080.100.120.14

0 2000 4000 6000 8000 10000

0.02

0.04

0.06

0.08

0.10

0.12

0 2000 4000 6000 8000 10000

0.04

0.08

0.12

0.16

0.20

σ x

Experiment index

σ y

Experiment index

σ z

Experiment index

σ p

Experiment index



0 2000 4000 6000 8000 10000

0.02

0.04

0.06

0.08

0.10

0 2000 4000 6000 8000 10000

0.04

0.08

0.12

0.16

0 2000 4000 6000 8000 10000

0.02

0.04

0.06

0.08

0.10

0.12

0 2000 4000 6000 8000 10000

0.04

0.08

0.12

0.16

0.20

σ x

Experiment index

σ y

Experiment index

σ z

Experiment index

σ p

Experiment index



direction, y direction, z direction, and position are6160, 6912, 6584, and 7327 times in Case 1, 6522,7269, 6878, and 7746 times in Case 2, 6736, 7531,7213, and 8044 times in Case 3, and 7099, 7911,7539, and 8427 times in Case 4, respectively. Thereis no doubt that Scheme (2) has more probabilitiesto derive better transformation results.

In general, Scheme (2) shows improvement in statisticalaccuracy of the transformed coordinates than Scheme (1)to some extent. This improvement is more remarkableespecially when there is stronger stochastic correlationbetween the coordinates of the common and check pointsin coordinate system I.

In this example, WTLS algorithm with universal con-straints [Scheme (1)] ignores the statistical correlationbetween the common and check points in coordinate sys-tem I. However, GTLSP algorithm [Scheme (2)] takes thiscorrelation into account, and the coordinates of the newpoints in system I will be corrected when implementingthe coordinate transformation. Thus it is more reasonablein theory. Additionally, from Eq. (41c) we know that oncethe correlation becomes stronger, these coordinates ofcheck points will be corrected more adequately. The supe-riority of Scheme (2) to Scheme (1) is exactly attributed tothese reasons.

6. Conclusions

In this study, we proposed a GTLSP algorithm for theuniversal 3D similarity transformation which combinesthe parameter calculation and new point transformationrigorously. The theoretical and experimental conclusionsare summarized as followings:

(1) GTLSP algorithm can take the random errors of newpoints and the statistical correlation between com-

mon and new points in the original coordinate systeminto account, therefore it is more rigorous than otherTLS algorithms for the 3D similarity transformationin theory.

(2) The experiment shows that after the original coordi-nates of the check points be corrected throughGTLSP algorithm, the statistical accuracy of thetransformed coordinates can be improved to someextent compared with that obtained by WTLS algo-rithm with universal constraints. The extent ofimprovement depends on the stochastic correlationbetween the common and the check points in theoriginal coordinate system. The stronger the correla-tion is, the greater improvement the GTLSP algo-rithm will make.

(3) GTLSP algorithm is derived in the form of abstractfunctions, thus it can be applied to solve many otherproblems as well, such as the 2D datum transforma-tion, the GNSS height fitting, the LiDAR point reg-istration and the image processing.

(4) The algorithm proposed in this paper is based on theassumption that the errors of coordinates observed intwo systems have the same variance component.However they may not be exactly same in practice.In further work, the variance component estimation(VCE) method for GTLSP algorithm will be investi-gated to solve this problem.

Acknowledgments

This research was supported by the DAAD ThematicNetwork Project (No. 57173947), the National NaturalScience Foundation of China (No. 41404004), the ChinaPostdoctoral Science Foundation (No. 2014M551790),and the National Basic Research Program of China (No.2013CB733300).


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