on total least squares for quadratic form...

366 Stud. Geophys. Geod., 59 (2015), 366379, DOI: 10.1007/s11200-014-0267-x © 2015 Inst. Geophys. CAS, Prague

On total least squares for quadratic form estimation

XING FANG1, JIN WANG2*, BOFENG LI3, WENXIAN ZENG1 AND YIBIN YAO1

1 School of Geodesy & Geomatics, Wuhan University, Wuhan, China 2 Beijing Key Laboratory of Traffic Engineering, Beijing University of Technology, Beijing,

China ([email protected]) 3 College of Surveying and Geo-Informatics, Tongji University, Shanghai, China * Corresponding author

Received: October 29, 2014; Revised: December 29, 2014; Accepted: February 6, 2015

ABSTRACT

The mathematical approximation of scanned data by continuous functions like quadratic forms is a common task for monitoring the deformations of artificial and natural objects in geodesy. We model the quadratic form by using a high power structured errors-in-variables homogeneous equation. In terms of Euler-Lagrange theorem, a total least squares algorithm is designed for iteratively adjusting the quadratic form model. This algorithm is proven as a universal formula for the quadratic form determination in 2D and 3D space, in contrast to the existing methods. Finally, we show the applicability of the algorithm in a deformation monitoring.

Ke y wo rd s : total least squares, quadratic forms, high power structured errors-in-

variables homogeneous equation, deformation monitoring

1. INTRODUCTION

Terrestrial laser scanning technique for collecting three-dimension (3D) points has become a more and more popular in geodesy and surveying engineering in the past a few years (Hesse, 2007; Eling, 2009; Paffenholz, 2012). Due to its high-precision, high-resolution and efficiency, it has been widely employed to the cultural heritage, as-built documentation and structural deformation monitoring. The resolution of scanned 3D points can be up to a few millimeters so far, depending on the distance of laser scanning equipment to the object’s surface. Such high-resolution cloud points allow to depict the object surface with a representative and ‘continuous’ model. Therefore, to process this huge amount of scanned cloud data without losing useful information is a real challenge. As in a typical deformation task the figure of objects always changes because of bending and flexing, a thorough parameter reduction is required when fitting or reconstructing the surface by characteristic parameters, which preserve the relevant information of the scanned surface. In this study, we focus on making use of the total least-squares (TLS) estimation to recover the representative parameters of the object’s surface that is often described by quadratic forms.


Stud. Geophys. Geod., 59 (2015) 367

The TLS adjustment initialized by Golub and van Loan (1980) is widely applied to adjust the errors-in-variables (EIV) models. In geodesy, it is well known that Teunissen (1988) formulated and solved an EIV or TLS problem for the first time. The solution form provided was exact, closed and non-iterative. In addition, the planar similarity transformation problem was illustrated as a TLS problem and then solved. Recently, a number of literatures published on this topic in community, for instance Schaffrin and Wieser (2008), Shen et al. (2011), Amiri-Simkooei and Jazaeri (2012), Grafarend and Awange (2012), Xu et al. (2012), Amiri-Simkooei (2013), Li et al. (2012, 2013), Fang (2013, 2014a,b, 2015), Xu and Liu (2014).

The algorithm to fit the quadratic form was introduced by Drixler (1993) in industrial surveying. The author transformed the nonlinear adjustment system into a special eigenvalue problem via an equivalent Gauss-Markov model. This approach was also applied in the automated form recognition of laser scanned deformable objects (Hesse and Kutterer, 2006). However, the solution to eigenvalue problem based on eigenvalue decomposition is not a rigorous solution due to the high power structure of the coefficient matrix. The high power structure means that random elements in the coefficient matrix appear as a high power in terms of other random elements.

Many algorithms to adjust a circle have been developed over the last several decades in mathematics and pattern recognition, including those by Bookstein (1979), Coope (1993), Gander et al. (1994), Späth (1997), Davis (1999) and Nievergelt (2001), to mention just a few. In geodesy, Kupferer (2005) presented the Gauss-Helmert model (GHM) to linearize the circle’s functional model. Unfortunately, some linearization pitfalls discussed by Pope (1972) are not avoided in his proposed approach. A rigorous solution for the circle fitting can be found in Schaffrin and Snow (2010), who also used the iterative GHM method but the linearization pitfalls are avoided by using the correct strategy of updating model matrices.

For other quadratic form, Späth (1996, 1997a,b) describe geometric fitting algorithms in parametric form for ellipse/hyperbola/parabola. By Späth’s ellipse fitting algorithm, it is also assumed that the two axis lengths of ellipse are different. Later on, fitting methods, specific for ellipsoids, as opposed to the more general conic sections are fifor proposed in Fitzgibbon (1999). A statistical point of view on the ellipsoid fitting problem is taken in Kanatani (1994), which proposed an unbiased estimation method, namely a renormalization procedure. Only recently Markovsky et al. (2004) proved that this TLS solution will be statistically inconsistent, and proposed a “correction term” for which they, however, need to know the true value of the noise variance.

However, these orthogonal fitting of a single quadratic form could not represent actual applications when scanned objects were figured as other quadratic forms. Furthermore, although all of these TLS algorithms may prove quite practical, errors of the measured points are assumed to be i.i.d in most papers.

In this contribution, a universal model of quadratic form is established by considering high power structured random errors in the coefficient matrix. The TLS algorithm is then developed to solve this model based on the Euler-Lagrange theorem. The solution is proven to be a correctly converged solution. Finally, a real deformation monitoring is implemented to show its applicability in geodesy.

X. Fang et al.

368 Stud. Geophys. Geod., 59 (2015)

2. MATHEMATICAL FORMULATION OF QUADRATIC FORMS

In most applications of laser scanning, reducing the complexity of point clouds or segmenting the point clouds is required by fitting them with some simple geometric shapes specified by the relevant mathematical forms. Such mathematical models are usually formulated in linear and quadratic, or even higher degree forms. In this study, we focus on the reconstruction of quadratic form, which can be, of course, easily extended to the higher degree forms. The mathematical formulation of a quadratic form for a single 3D point reads (Drixler, 1993):

T T0 0 x Mx m x , (1)

where Tx y zx is the vector of the true values of the 3D coordinates of the single point,

1 4 5

4 2 6

5 6 3

M and T7 8 9 m

are the symmetric matrix and the vector of unknown parameters, respectively. Obviously, the above equation has a trivial solution, such that one solution multiplying a constant can obtain another solution. In order to avoid situation, we fix 0 to be a non-zero constant as the constant). For the i-th point, Eq. (1) can be explicitly formulated as

2 2 2 21 2 3 4 5 6 7 8 9 02 2 2 0i i i i i i i i i i i ix y z x y y z x z x y z . (2)

By collecting the observations of all point cloud, the compact model formulation can be expressed by

2 2 2 1

,

i i i i i i i i i i i i i i i h

h h

x x y y z z x y x z y z x y z

ξ 0

A ξ 0

(3)

where hA and hξ are the random coefficient matrix and the unknown parameter vector in a homogenous equation, respectively. Because the elements in the coefficient matrix are from real error-affected observations and then they are random, the equation system is thus referred to as the EIV model. When the elements in the coefficient matrix appear repeatedly in identical and negative forms or fixed, the EIV model is called ‘structured’. Here, a high power structured EIV homogenous equation should be taken into consideration for the quadratic form model, where the elements in the coefficient matrix appear repeatedly in a high degree form of other elements.

For the different applications, some of unknown parameters may be non-available. For example, in a sphere fitting problem, the parameters, 4 , 5 and 6 , are vanished. In this



case, one can directly formulate the associated quadratic form without these parameters. Without loss of generality, in this paper, we present some model still with Eq. (3) but now introducing a set of constraints. Then the universal formulae of quadratic form is symbolized as

0 0or ,,

n A n

Aξ 1 A E ξ 1 yKξ κ

(4)

where n1 is the n-column vector of all elements of 1; A is the same as hA but excluding the elements of the last column; ξ contains the nine unknown parameters ( 1 to 9 ). AE is the random error matrix of matrix A (nota bene: AE contains the first and second order terms of random errors). It should be noted that the elements within the matrix AE are not zero-mean errors. For the random properties of the squared residuals, we refer to Amiri-Simkooei (2007) and Teunissen and Amiri-Simkooei (2008). The fixed vector y is defined as 0: ny 1 , thus it is known as a non-zero fixed vector. Kξ κ is a set of the linear constraints for eliminating some unnecessary parameters according to the known form. It must be noted that the known form can be statistically determined after all the unknown parameters have been estimated (no constraints), see Kutterer and Schön (1999). This statistical determination distinguishes the necessary and unnecessary parameters. For example, if the quadratic form is statistically determined as a sphere, the parameters 4 ,

5 and 6 are restricted to zeros. The determined form is then estimated by a constrained estimator, i.e., only the necessary parameters for the quadratic form are considered in the adjustment in the next section.

After formulating the functional model of the quadratic form, the stochastic model is established according to the independent random errors:

2 2 10 0 ,

,

,i i i

ll

i i i

x y z

D D

x y z

l ε Q P

l

ε

(5)

where ε is the vector of random errors corresponding to the x, y and z coordinates. The relationship between the random error vector ε and the vectorized vector : vecA Ae E

can be written as A e Jε , using the Jacobian matrix Tvec A J E ε .

3. TOTAL LEAST SQUARES SOLUTION OF THE QUADRATIC FORM

From a geodetic point of view, Teunissen (1988) as well as Teunissen (1985) actually reformulated the EIV model to the standard nonlinear Gauss-Markov model, and therefore they have the advantage that all available knowledge (numerical and statistical, see Teunissen, 1989a,b) of solving nonlinear observation equations by least-squares can be directly applied. However, for the quadratic form estimation, the right-hand side of the

X. Fang et al.


model given by Eq. (4) is non-random and the error matrix contains high power terms of errors, which may yield direct observation equations with nonlinear constraints. Therefore, we prefer to use a Lagrangian approach for solving our problem to keep the original meaning of the quadratic form. According to the traditional Lagrange approach, the target function is given as follows:

T T T

T T T T

, , , 2 2

2 vec E 2 ,

A

n

A

Φ ε ξ λ μ ε Pε λ y Aξ E ξ μ Kξ κ

ε Pε λ y Aξ ξ I μ Kξ κ (6)

where λ and μ are the Lagrange multiplier vectors. Setting the partial derivatives of the target function with respect to ξ , ε , λ , μ equal to 0 individually, gives the necessary conditions as

T T T

ˆ ˆ, ,

1 ˆ ˆ ˆ2 A

ε λ μ

ΦA λ E λ K μ 0

ξ

, (7)

T

ˆ ˆ, ,

1 ˆ ˆ2 n

ξ ε λ

ΦPε J ξ I λ 0

ε , (8)

T

ˆ ,

1 ˆ ˆ vec2 n A

ξ ε

Φy Aξ ξ I E 0

λ

, (9)

ˆ

1 ˆ2

ξ

ΦKξ κ 0

μ . (10)

From Eq. (8) the error vector is estimated as

T ˆ ˆll n ε Q J ξ I λ . (11)

Using Eqs (9) and (11), λ̂ can be solved as

1T ˆˆll

λ BQ B y Aξ , (12)

with

ˆTn B ξ I J .

By inserting Eq. (12) into Eq. (8), the functional independent error vector ε and the extended error vector Ae can be estimated as

1T T ˆll ll

ε Q B BQ B y Aξ (13)



and

1T T Â ll ll

e JQ B BQ B y Aξ , (14)

respectively. Therefore the error matrix is

1T T învecA ll ll

E JQ B BQ B y Aξ , (15)

where the function invec is inverse to function vec , transforming a vector back

to a matrix. It is rather easy to see that the estimated objective function Tε Pε is equal to

1TA ll A

e JQ J e , which is the weighted sum of squared errors Ae . Now, returning to Eq. (7), we derive the nonlinear constrained normal equation as

1T T Tˆ Â ll

A E BQ B y Aξ K μ 0 . (16)

Combining Eqs (16) and (10), the solution of constrained quadratic form can be obtained as follows

T

1T T

1T T

ˆ,

,

.

A ll

A ll

nξN KκμK 0

N A E BQ B A

n A E BQ B y

(17)

After given an initial value of the parameter vector, we compute the solution with Eq. (17) iteratively. Note that, in each iteration, the matrices and vectors containing the parameter vector and the error matrix should be updated. Since the nonlinear Lagrangian equations provide the necessary condition for the local solution, it should be further verified that the Hessian matrix with respect to the variables is positive definite to fulfill the sufficient condition (Teunissen, 1990).

Therefore, the TLS procedure for quadratic form estimation is summarized as follows:

Algorithm 1: TLS for quadratic form estimation

Begin 1. Fixing the constraints Kξ = κ , i.e. determination of the quadratic form according

to Kutterer and Schön (1999) and giving initial parameter values. 2. Repeat

solving T 1î

nN K ξκK 0 μ

iteratively

X. Fang et al.


with

T

1T T

T 1T

T 1T

ˆ ,

ˆinvec ,

,

.

in

i iA ll ll

iA ll

iA ll

B ξ I J

E JQ B BQ B y Aξ

N A E BQ B A

n A E BQ B y

3. If 1ˆ ˆ ˆi i i ξ ξ ξ ( is a sufficiently small positive value)

Stop

In the proposed iterative method, the matrix N to be inverted is not symmetric, which may show new algorithmic aspect in comparison to the standard Gauss Newton method. Since the proposed iterative method does not relate to any existing iterative techniques presented in Teunissen (1990), the convergence behavior should be further investigated.

4. NUMERICAL EXAMPLES

In order to demonstrate the performance of Algorithm 1, in this section the algorithm will be applied to a circle fitting problem representing the quadratic form problem, compared with the algorithms proposed by Drixler (1993) and Schaffrin and Snow (2010).

The data used are from Gander et al. (1994) who fitted a parametric form of the circle with six data points. The coordinates of these six points are presented in Table 1.

In following, we first outline the methods of Drixler (1993) and Schaffrin and Snow (2010) for fitting circle as the special form of the quadratic form in brief. Then, Algorithm 1 is demonstrated to compare the methods in the case of the known quadratic form.

Table 1. The data (dimensionless) for circle fitting from Gander et al. (1994).

Point Number X Coordinate Y Coordinate

1 1 7 2 2 6 3 5 8 4 7 7 5 9 5 6 3 7



4 . 1 . M e t h o d o f D r i x l e r ( 1 9 9 3 ) f o r c i r c l e f i t t i n g

The quadratic form for the 2D space is given in Drixler (1993) as

1 3 46

3 2 50i

i i i ii

a a x ax y x y a

a a y a

,

where i ix y represents the 2D coordinates of the i-th point. If the quadratic form specifies a circle, the estimation of the quadratic parameters is obtained in Eq. (5.2.17) in Drixler (1993) as

4 1T T5 2 2 2 1

6

ˆˆˆ

aaa

A A A A ,

where 2 21 i ix y

A

and 2 n A x y 1 , with ix

x

and iy

y

.

The first 3 parameters are set to 1, 1, 0, respectively for the circle fitting (see Drixler, 1993).

4 . 2 . M e t h o d o f S c h a f f r i n a n d S n o w ( 2 0 1 0 ) f o r c i r c l e f i t t i n g

For the circle fitting, Schaffrin and Snow (2010) formulated the original functional model of a circle as

2 2 2, , 0i i m i mf x x y y r β x y ,

where T, ,m mx y rβ . Since the model is typically nonlinear system, it can be linearized by neglecting higher than second-order as follows

01 2, ,f d β x y J β J ε = 0 ,

where

0 0 01 2 , 2 , 2m n m n m nx y r

J x 1 y 1 1 , 0 02 2 , 2m n m nx y

J x 1 y 1 .

The above linearized model refers to as a classical GHM. The original equation can be adjusted by the iterative GHM proposed by Pope (1972), where the linearization is repeated for each iteration.

X. Fang et al.


4 . 3 . M e t h o d p r o p o s e d i n t h i s p a p e r

For a circle, some parameters are fixed if one formulates it using the proposed quadratic form Eq. (4). Here, we formulate the constraints Kξ κ for the circle fitting problem as

0 0 1 0 0 0 0 0 00 0 0 1 0 0 0 0 00 0 0 0 1 0 0 0 00 0 0 0 0 1 0 0 00 0 0 0 0 0 0 0 11 1 0 0 0 0 0 0 0

ξ 0 .

The constraints are written in a matrix form by the unnecessary parameters equaling zero (3, 4, 5, 6, 9 = 0) and 1 = 2 due to the circle.

If the constraints are initialized, we can implement Algorithm 1 to derive the TLS solution in Table 2.

The approach of Drixler (1993) without any iteration cannot obtain the optimal solution, which can be seen by comparing the sum of the squared residuals. As this method is relatively simple, it is more suitable for the approximation of the exact solution.

Table 2 indicates that our results are exactly same to the TLS solutions proposed in Schaffrin and Snow (2010) for the circle fitting (we transform the parameters ξ to the parameters of origin and radius). However, the advantage of Algorithm 1 is that the solution form of the quadratic form is very simple (without linearization) and general (also for the 3D case). Schaffrin and Snow (2010) proposed the linearized method for fitting a circle. If one generalizes the method in another quadratic form, the linearization for the original function with respect to the parameters and observations may be very difficult. Furthermore, one needs to linearize the original function for any special case. For example, in 3D case except the sphere, three orientation parameters should be considered for the quadratic form. Here, the original function of the rotated ellipsoid is given as

Table 2. The results of the parameters and the weighted sum of squared residuals for the circle fitting example. xm, ym - coordinates of the circle center, r - radius of the circle (dimensionless).

Drixler (1993) Schaffrin and Snow (2010) Algorithm 1

mx 4.742331 4.739782 4.739782

my 3.835123 2.983533 2.983533 r 4.108762 4.714226 4.714226

Total sum of squared residuals 1.398289 1.227599 1.227599



2 2 2

0 0 02 2 2

1m m mx x y y z z

a b c

with

0

0 3 3

0

x xy yz z

R ,

where a, b and c are three semi-principal axes of the ellipsoid, Tix y z are the

original observations, T

0 0 0 ix y z are the observations transformed by the rotation

matrix 3 3R , which contains 3 parameters (3 rotation angles). In this case, the linearization explicitly with respect to the parameters (e.g.,

translations, rotations) and the observations would be very complicated. However, with the method proposed in this paper, the estimation is relatively simple, because the solution of the quadratic form problem does not need linearization of any special quadratic form.

5. APPLICATION

The Oker dam situated in the Harz mountains in Lower Saxony, Germany (Fig. 1a), was scanned four times by using the Trimble GX 3D scanner over two years, see e.g., Eling (2009). The dam is of an arch shape with height of 75 m and length of 260 m. One aim of the measurement is to detect possible deformation of the dam surface. Based on the (transformed) point clouds scanned by terrestrial laser scanner, the whole dam surface was fitted by the quadratic form estimation. A unique framework was established by these parameters. Then, this framework was compared with point clouds collected from other epochs to detect the possible deformation of the dam surface. Backsight targets and identical points were applied in the establishment of the geodetic network (Fig. 1b). In each epoch, four scanner stations (Station 1000, 2000, 3000 and 4000) were placed in

Fig. 1. a) Location map of the Oker dam in Germany. The filled square is the location of the dam. b) Geodetic monitoring network of the Oker dam (re-drawn after Eling, 2009, the four emphasized points are used for monitoring). © 2009 German Geodetic Commission, Munich, Germany).

X. Fang et al.


front of the dam surface for data collection. Because the scanner Station 3000 has only scanned part of the dam surface, point clouds obtained from this station are excluded in the following data processing.

A quasi indirect error adjustment (Fang, 2014c) was used to estimate the transformation parameters in order to transform all the point clouds collected in the last epoch into the local reference coordinate system. With the transformation parameters, all the point clouds were transformed into this local reference coordinate system. After that, in order to mathematically fit the shape of the dam surface, the quadratic form estimation presented in this manuscript was used based on these point clouds. If outliers are available, the outliers in the scanned points can be eliminated by applying the data snooping to the EIV model in Amiri-Simkooei and Jazaeri (2013). The data snooping method use the statistical test to detect the outliers in one observation equation instead of a single observation within the equation.

The fitted results of the dam surface as the elliptical cylinder are demonstrated in Fig. 2, where the left part shows the photo of the dam and the right one shows the modeled results with quadratic form estimation. In addition, these quadratic form estimates are shown in Table 3.

In Fig. 2b, the large area represents the registered point clouds, the smaller area inside represents the modeled elliptical cylinder by using the estimated parameters. ˆcx and ˆcy are the center coordinates of the elliptical cylinder in the directions of x and y, ˆcz is the minimum value of the point clouds in the direction of z.

Fig. 2. a) Photo of the Oker dam, b) result of the elliptical cylinder fitting. Xr, Yr and Zr denote the reference coordinate system, ˆ ˆ ˆ, ,c c cx y z represents the origin of the elliptical cylinder system

, ,e e eX Y Z .

Table 3. The estimated parameters of the elliptical cylinder (see Fig. 2b) to model the dam surface (values in m).

1̂ 2̂ 3̂ 4̂ 5̂ 6̂ 7̂ 8̂ 9̂ 0̂

Estimate 0.0008 0.0008 0 0 0 0 0.0476 0.0773 0 1.0000



Based on the quadratic form estimates, the whole dam surface was represented by the elliptical cylinder. This elliptical cylinder was used as a unique framework for further deformation analysis between epochs (for details see Wang, 2013b).

6. CONCLUSIONS

This work has proposed the total least-squares (TLS) solution with particular emphasis on the high power structured coefficient matrix within the errors-in-variables homogenous model. It is indeed interesting to see the progression from linearly dependent elements within the coefficient matrix to the high power structured elements that is exemplified by quadratically dependent elements. The TLS solution is obtained by the traditional Lagrange approach. The proposed iterative algorithm (Algorithm 1) differs from the classical method (e.g., Drixler, 1993) based on eigenvalue decomposition from the perspective of the solution accuracy. In addition, the proposed solution does not need the complicated linearization process of any specific quadratic form. Finally, the proposed solution is showcased in a real deformation monitoring task based on point clouds collected from the dam surface.

Acknowledgements: The work presented in this paper was mainly conducted during doctoral

studies of the first and second authors at Leibniz University Hanover (LUH), Germany. We are grateful to the President at the Federal Agency for Cartography and Geodesy (BKG) in Germany, Prof. Kutterer and all staff of the laser scanning team in LUH for providing source data and their helpful comments. This research was also supported by the National Natural Science Foundation of China (41404005；41474006）and the Fundamental Research Funds for the Central Universities(2042014kf053).

References Amiri-Simkooei A.R., 2007. Least-Squares Variance Component Estimation: Theory and GPS

Applications. Ph.D. Thesis, Delft University of Technology. Publication on Geodesy, 64, Netherlands Geodetic Commission, Delft, The Netherlands.

Amiri-Simkooei A.R. and Jazaeri S., 2012. Weighted total least squares formulated by standard least squares theory. J. Geod. Sci., 2(2), 113124.

Amiri-Simkooei A.R., 2013. Application of least squares variance component estimation to errors-in-variables models. J. Geodesy, 87, 935944.

Amiri-Simkooei A.R. and Jazaeri S., 2013. Data-snooping procedure applied to errors-in-variables models. Stud. Geophys. Geod., 57, 426441.

Bookstein F.L., 1979. Fitting conic sections to scattered data. Comput. Graph. Image Process., 9, 5971.

Coope D., 1993. Circle fitting by linear and nonlinear least-squares. J. Optim. Theory Appl., 76, 381388.

Davis T.G., 1999. Total least-squares spiral curve fitting. J. Surv. Eng.-ASCE, 125, 159176. Drixler E., 1993. Analyse der Lage und Form von Objekten im Raum. Dissertation. Deutsche

Geodätische Kommission, Reihe C, Heft Nr. 409, München, Germany (in German).

X. Fang et al.


Eling D., 2009. Terrestrisches Laserscanning für die Bauwerksüberwachung. Dissertation. Deutsche Geodätische Kommission, Reihe C, Nr. 641, München, Germany (in German, http://www.dgk.badw.de/fileadmin/docs /c-641.pdf).

Fang X., 2011. Weighted Total Least Squares Solution for Application in Geodesy. PhD Thesis, No. 294, Leibniz University, Hannover, Germany.

Fang X., 2013. Weighted total least squares: necessary and sufficient conditions, fixed and random parameters. J. Geodesy, 87, 733749. DOI: 10.1007/s00190-013-0643-2.

Fang X., 2014a. A structured and constrained total least-squares solution with cross-covariances. Stud. Geophys. Geod., 58, 116, DOI: 10.1007/s11200-012-0671-z.

Fang X., 2014b. On non-combinatorial weighted total least squares with inequality constraints. J. Geodesy, 88, 805816.

Fang X., 2014c. A total least squares solution for geodetic datum transformations. Acta Geod. Geophys., 49, 189207.

Fang X., 2015. Weighted total least-squares with constraints: a universal formula for geodetic symmetrical transformations. J. Geodesy, DOI: 10.1007/s00190-015-0790-8 (in print).

Fitzgibbon A., Pilu M. and Fisher R., 1999. Direct least-squares fitting of ellipses. IEEE Trans. Pattern Anal. Machine Intell., 21, 476480.

Gander W., Golub G.H. and Strebel R., 1994. Least-squares fitting of circles and ellipses. BIT, 34, 558578.

Golub G. and Van Loan C., 1980. An analysis of the total least-squares problem. SIAM J. Numer. Anal., 17, 883893, DOI: 10.1137/0717073.

Grafarend E. and Awange J.L., 2012. Applications of Linear and Nonlinear Models. Fixed Effects, Random Effects, and Total Least Squares. Springer-Verlag, Berlin, Germany.

Hesse C., 2007. Hochauflösende kinematische Objekterfassung mit terrestrischen Laserscannern. Dissertation. Deutsche Geodätische Kommission, Reihe C, Nr. 608, München, Germany (in German).

Hesse C. and Kutterer H., 2006. Automated form recognition of laser scanned deformable objects. In: Sansò F. and Gil A. (Eds), Geodetic Deformation Monitoring: From Geophysical to Engineering Roles. International Association of Geodesy Symposia, 131, Springer-Verlag, Heidelberg, Germany.

Kanatani K., 1994. Statistical bias of conic fitting and renormalization. IEEE Trans. Pattern Anal. Machine Intell., 16, 320326.

Kupferer S., 2005. Application of the TLS Technique to Geodetic Problems. PhD Thesis. Geodetic Institute, University of Karlsruhe, Karlsruhe, Germany.

Kutterer H. and Schön S., 1999. Statistische Analyse quadratischer Formen - der Determinantenansatz. AVN, 10/1999, 322330 (in German, http://www.wichmann-verlag.de/images/stories/avn/artikelarchiv/1999/10/5f134c46d35.pdf).

Li B., Shen Y. and Lou L., 2013. Seamless multivariate affine error-in-variables transformation and its application to map rectification. Int. J. GIS, 27, 15721592.

Markovsky I., Kukush A. and Van Huffel S., 2004. Consistent least-squares fitting of ellipsoids. Numer. Math., 98, 177194.

Nievergelt Y., 2001. Hyperspheres and hyperplanes fitting seamlessly by algebraic constrained total least-squares. Linear Alg. Appl., 331, 145155.

Paffenholz J.A., 2012. Direct Geo-Referencing of 3D Point Clouds with 3D Positioning Sensors. Dissertation. Deutsche Geodätische Kommission, Reihe C, Nr. 689, München, Germany (in German)



Pope A.J., 1972. Some pitfalls to be avoided in the iterative adjustment of nonlinear problems. In: Proceedings of the 38th Annual Meeting, American Society of Photogrammetry, Washington, D.C., Mar. 12-17, 1972. American Society of Photogrammetry, Bethesda, MD, 449477.

Schaffrin B. and Wieser A., 2008. On weighted total least-squares adjustment for linear regression. J. Geodesy, 82, 415421, DOI: 10.1007/s00190-007-0190-9.

Schaffrin B. and Snow K., 2010. Total Least-Squares regularization of Tykhonov type and an ancient racetrack in Corinth. Linear Alg. Appl., 432, 20612076.

Shen Y., Li B. and Chen Y., 2011. An iterative solution of weighted total least-squares adjustment. J. Geodesy, 85, 229238, DOI: 10.1007/s00190-010-0431-1.

Späth H., 1996. Orthogonal squared distance fitting with parabolas. In: Alefeld G. and Herzberger J. (Eds), Proceedings of the IMACS-GAMM International Symposium on Numerical Methods and Error-Bounds. Akademie-Verlag, Berlin, Germany, 261269.

Späth H., 1997a. Least-squares fitting of ellipses and hyperbolas. Comput. Stat., 12, 329341. Späth H., 1997b. Orthogonal distance fitting by circles and ellipses with given area. Comput. Stat.,

12, 343354. Teunissen P.J.G., 1985. The Geometry of Geodetic Inverse Linear Mapping and Nonlinear

Adjustment. Netherlands Geodetic Commission, Publications on Geodesy, New Series, 8(1), 186 pp.

Teunissen PJG (1988) The non-linear 2D symmetric Helmert transformation: an exact non-linear least-squares solution. Journal of Geodesy, 62 (1):1-15

Teunissen P.J.G., 1989a. A note on the bias in the symmetric Helmert transformation, Festschrift Krarup, 18.

Teunissen P.J.G., 1989b. First and second moments of nonlinear least-squares estimators. J. Geodesy, 63, 253262.

Teunissen P.J.G., 1990. Nonlinear least squares. Manuscripta Geodaetica, 15, 137150. Teunissen P.J.G. and Amiri-Simkooei A.R., 2008. Least-squares variance component estimation.

J. Geodesy, 82, 6582. Wang J., Kutterer H. and Fang X., 2012. On the detection of systematic errors in terrestrial laser

scanning data. J. Appl. Geod., 6, 187192. Wang J., 2013a. Block-to-point fine registration in terrestrial laser scanning. Remote Sens., 5,

69216937, DOI: 10.3390/rs5126921. Wang J., 2013b. Towards Deformation Monitoring with Terrestrial Laser Scanning Based on

External Calibration and Feature Matching Methods. PhD Thesis, No. 308, Leibniz University, Hannover, Germany.

Xu P.L., Liu J.N. and Shi C., 2012. Total least squares adjustment in partial errors-in-variables models: algorithm and statistical analysis. J. Geodesy, 86, 661675, DOI: 10.1007/s00190-012-0552-9.

Xu P.L. and Liu J.N., 2014. Variance components in errors-in-variables models: estimability, stability and bias analysis. J. Geodesy, 88, 719734, DOI: 10.1007/s00190-014-0717-9.

on total least squares for quadratic form...

Documents