Modifying Cadzow's algorithm to generate the optimal TLS-solution for the structured EIV-Model of a similarity transformation

Schaffrin, Burkhard; Neitzel, Frank; Uzun, Sibel; Mahboub, Vahid

doi:10.2478/v10156-011-0028-5

Cited by 26 publications

(16 citation statements)

References 15 publications

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“…However, Neitzel [14] showed that their solution needed modifications and provided the correct TLS results by evaluating the solution obtained by an iteratively linearised GHM. Additionaly, an iterative TLS solution for the same problem was presented in [15].…”

Section: Introductionmentioning

confidence: 99%

Götterdämmerung over total least squares

Malissiovas

Neitzel

2016

Journal of Geodetic Science

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Abstract:The traditional way of solving non-linear least squares (LS) problems in Geodesy includes a linearization of the functional model and iterative solution of a nonlinear equation system. Direct solutions for a class of nonlinear adjustment problems have been presented by the mathematical community since the 1980s, based on total least squares (TLS) algorithms and involving the use of singular value decomposition (SVD). However, direct LS solutions for this class of problems have been developed in the past also by geodesists. In this contribution we attempt to establish a systematic approach for direct solutions of non-linear LS problems from a "geodetic" point of view. Therefore, four non-linear adjustment problems are investigated: the fit of a straight line to given points in 2D and in 3D, the fit of a plane in 3D and the 2D symmetric similarity transformation of coordinates. For all these problems a direct LS solution is derived using the same methodology by transforming the problem to the solution of a quadratic or cubic algebraic equation. Furthermore, by applying TLS all these four problems can be transformed to solving the respective characteristic eigenvalue equations. It is demonstrated that the algebraic equations obtained in this way are identical with those resulting from the LS approach. As a by-product of this research two novel approaches are presented for the TLS solutions of fitting a straight line to 3D and the 2D similarity transformation of coordinates. The derived direct solutions of the four considered problems are illustrated on examples from the literature and also numerically compared to published iterative solutions.

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Section: Introductionmentioning

confidence: 99%

Götterdämmerung over total least squares

Malissiovas

Neitzel

2016

Journal of Geodetic Science

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“…The structure of the last two columns ofẼ A is highlighted by drawing a box around the rst two rows. This replication of structure in the residual matrix had already been pointed out by Fang (2011), Mahboub (2012), and Scha rin et al (2012, for EIV-models with dispersion matrices having full rank. The property holds here as well in the new algorithms that handle rank-de cient dispersion matrices.…”

Section: Example: 2-d Similarity Transformationmentioning

confidence: 88%

“…Scha rin et al, 2012). The data for the 2-D similarity transformation are taken from Neitzel and Scha rin (2013).…”

Section: Example: 2-d Similarity Transformationmentioning

confidence: 99%

On The Errors-In-Variables Model With Singular Dispersion Matrices

Schaffrin

Snow

Neitzel³

2014

Journal of Geodetic Science

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While the Errors-In-Variables (EIV) Model has been treated as a special case of the nonlinear GaussHelmert Model (GHM) for more than a century, it was only in 1980 that Golub and Van Loan showed how the Total Least-Squares (TLS) solution can be obtained from a certain minimum eigenvalue problem, assuming a particular relationship between the diagonal dispersion matrices for the observations involved in both the data vector and the data matrix. More general, but always nonsingular, dispersion matrices to generate the "properly weighted" TLS solution were only recently introduced by Scha rin and Wieser, Fang, and Mahboub, among others. Here, the case of singular dispersion matrices is investigated, and algorithms are presented under a rank condition that indicates the existence of a unique TLS solution, thereby adding a new method to the existing literature on TLS adjustment. In contrast to more general "measurement error models," the restriction to the EIV-Model still allows the derivation of (nonlinear) closed formulas for the weighted TLS solution. The practicality will be evidenced by an example from geodetic science, namely the over-determined similarity transformation between di erent coordinate estimates for a set of identical points.

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“…A typical EIV model is similar to a GaussMarkov (GM) model but all the variables are subject to random errors. For further reading, see e.g., Van Huffel and Vandewalle (1991), Schaffrin and Wieser (2008), Felus (2004), Schaffrin et al (2012a, b), Mahboub et al (2012), Fang (2013Fang ( , 2014, Snow and Schaffrin (2012), Snow (2012) and Mahboub (2014), etc. Meanwhile, some other researchers investigated this problem traditionally; see e.g., Neitzel (2010) and Shen et al (2011).…”

Section: Introductionmentioning

confidence: 99%

Adjustment of non-typical errors-in-variables models

2015

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In this contribution, non-typical errors-in-variables (EIV) model is introduced as a more general form of the so-called EIV model which is highly relevant for geodetic data processing. Total least-squares (TLS) algorithms within a typical EIV model cannot deal with the non-typical EIV models since the first order moment/mean of the random design matrix in a non-typical EIV model is not linear. In this paper, we propose a new algorithm besides a standard solution to deal with this model. To achieve this goal, first we review a classic algorithm in order to solve this problem using traditional non-linear leastsquares method within a non-linear mixed model. Then by comparison, weighted TLS algorithm within the typical EIV model can be replaced by the proposed approach after some slight modifications which results in an excellent approximate solution. This foundation is important because there is no need for linearization in the TLS algorithms. The proposed way is not sensitive to the approximate initial values of the unknown parameters and it is applicable to curve fitting, surface reconstruction or other non-typical EIV models. Here we employ it to the curve fitting. Also two examples convincingly demonstrate that the standard TLS solution of the non-typical EIV model is not admissible when it is incorrectly considered as a typical EIV model; i.e., the non-linear relationships of the elements of the random design matrix are neglected.

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Modifying Cadzow's algorithm to generate the optimal TLS-solution for the structured EIV-Model of a similarity transformation

Cited by 26 publications

References 15 publications

Götterdämmerung over total least squares

Götterdämmerung over total least squares

On The Errors-In-Variables Model With Singular Dispersion Matrices

Adjustment of non-typical errors-in-variables models

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