Nonlinear analysis of the three-dimensional datum transformation[conformal group C7(3)]

doi:10.1007/S00190-002-0299-9

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Nonlinear analysis of the three-dimensional datum transformation[conformal group C7(3)]

Journal Article•DOI•

Nonlinear analysis of the three-dimensional datum transformation[conformal group C7(3)]

Erik W. Grafarend¹, Joseph L. Awange•Institutions (1)

University of Stuttgart¹

01 May 2003-Journal of Geodesy (Springer-Verlag)-Vol. 77, Iss: 1, pp 66-76

TL;DR: The problem of incorporating the stochasticity measures of both systems of coordinates involved in the seven parameter datum transformation problem [conformal group ℂ7(3)] which is free of linearization and any iteration procedure can be considered to be solved.

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Abstract: The weighted Procrustes algorithm is presented as a very effective tool for solving the three-dimensional datum transformation problem In particular, the weighted Procrustes algorithm does not require any initial datum parameters for linearization or any iteration procedure As a closed-form algorithm it only requires the values of Cartesian coordinates in both systems of reference Where there is some prior information about the variance–covariance matrix of the two sets of Cartesian coordinates, also called pseudo-observations, the weighted Procrustes algorithm is able to incorporate such a quality property of the input data by means of a proper choice of weight matrix Such a choice is based on a properly designed criterion matrix which is discussed in detail Thanks to the weighted Procrustes algorithm, the problem of incorporating the stochasticity measures of both systems of coordinates involved in the seven parameter datum transformation problem [conformal group ℂ7(3)] which is free of linearization and any iterative procedure can be considered to be solved Illustrative examples are given

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Journal Article•DOI•

Computing Helmert transformations

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G. A. Watson¹•Institutions (1)

University of Dundee¹

15 Dec 2006-Journal of Computational and Applied Mathematics

TL;DR: In this article, it is shown how a Gauss-Newton method in the rotation parameters alone can easily be implemented to determine the parameters of the nine-parameter transformation (when different scale factors for the variables are needed).

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101 citations

Cites background from "Nonlinear analysis of the three-dim..."

...The basic requirement is just the solution of an orthogonal Procrustes problem, see for example [1], [4], [5]....
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Journal Article•DOI•

A Quaternion-based Geodetic Datum Transformation Algorithm

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Yunzhong Shen¹, Yunzhong Shen², Y. Chen², Y. Chen¹, D. H. Zheng¹ - Show less +1 more•Institutions (2)

Tongji University¹, State Bureau of Surveying and Mapping²

07 Jun 2006-Journal of Geodesy

TL;DR: In this paper, the authors introduce quaternions to represent rotation parameters and derive the formulae to compute quaternion, translation and scale parameters in the Bursa-Wolf geodetic datum transformation model from two sets of co-located 3D coordinates.

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Abstract: This paper briefly introduces quaternions to represent rotation parameters and then derives the formulae to compute quaternion, translation and scale parameters in the Bursa–Wolf geodetic datum transformation model from two sets of co-located 3D coordinates. The main advantage of this representation is that linearization and iteration are not needed for the computation of the datum transformation parameters. We further extend the formulae to compute quaternion-based datum transformation parameters under constraints such as the distance between two fixed stations, and develop the corresponding iteration algorithm. Finally, two numerical case studies are presented to demonstrate the applications of the derived formulae.

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72 citations

Cites background or methods or result from "Nonlinear analysis of the three-dim..."

...these coordinates, we compute the transformation parameters from a local geodetic system (system A) to WGS84 (system B) and list the results in Table 2 and give the corresponding quaternion and rotation matrix in Table 3. The transformation parameters and transformed coordinates are exactly equal to those reported by Grafarend and Awange (2003) ....
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...It should be pointed out that the elements of the rotation matrix in the 1st row and 2nd column should be exchanged with that in the 1st row and 3rd column in Table 8 of Grafarend and Awange (2003) ; otherwise the rotation matrix is different from ours and is also inconsistent with the rotation matrix of their Table 9. The inconsistency among various implementations of the seven-parameter model are explained in, e.g., Soler (1988)....
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...The Cartesian coordinates of seven stations, as shown in Table 1, are taken from Grafarend and Awange (2003) ....
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... Grafarend and Awange 2003 ), some simplifications, such as isotropy and independence of the errors at each station, have to be made; otherwise, no analytical solution may be possible....
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Journal Article•DOI•

On symmetrical three-dimensional datum conversion

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Yaron A. Felus¹, Robert C. Burtch¹•Institutions (1)

Ferris State University¹

01 Jan 2009-Gps Solutions

TL;DR: In this article, the similarity transformation problem is analyzed with respect to the EIV model, and a novel algorithm is described to obtain the transformation parameters, which can be used to convert GPS-WGS84-based coordinates to those in a local datum using a set of control points with coordinate values in both systems.

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Abstract: A 3-D similarity transformation is frequently used to convert GPS-WGS84-based coordinates to those in a local datum using a set of control points with coordinate values in both systems. In this application, the Gauss-Markov (GM) model is often employed to represent the problem, and a least-squares approach is used to compute the parameters within the mathematical model. However, the Gauss–Markov model considers the source coordinates in the data matrix (A) as fixed or error-free; this is an imprecise assumption since these coordinates are also measured quantities and include random errors. The errors-in-variables (EIV) model assumes that all the variables in the mathematical model are contaminated by random errors. This model may be solved using the relatively new total least-squares (TLS) estimation technique, introduced in 1980 by Golub and Van Loan. In this paper, the similarity transformation problem is analyzed with respect to the EIV model, and a novel algorithm is described to obtain the transformation parameters. It is proved that even with the EIV model, a closed form Procrustes approach can be employed to obtain the rotation matrix and translation parameters. The transformation scale may be calculated by solving the proper quadratic equation. A numerical example and a practical case study are presented to test this new algorithm and compare the EIV and the GM models.

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69 citations

Journal Article•DOI•

On least-squares solution to 3D similarity transformation problem under Gauss–Helmert model

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Guobin Chang

11 Mar 2015-Journal of Geodesy

TL;DR: In this paper, the 3D similarity datum transformation problem with Gauss-Helmert model was studied and the closed-form least-squares solution to this problem was derived.

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Abstract: In this note, the 3D similarity datum transformation problem with Gauss–Helmert model, also known as the 3D symmetric Helmert transformation, is studied. The closed-form least-squares solution, i.e., without iteration, to this problem is derived. It is found that the rotation parameters in this solution are the same to that for the transformation with Gauss–Markov model, while the scale and translation parameters differ from each other.

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46 citations

Cites background or result from "Nonlinear analysis of the three-dim..."

...Note that the rotation parameters can be represented in different forms, e.g., Euler angles (Fang 2014b; Yang 1999), direction cosine matrix (Grafarend and Awange 2003; Teunissen 1984a), normalized quaternion (Sanso 1973; Shen et al. 2006), etc....
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...From Grafarend and Awange (2003); Teunissen (1984a), we find that it is also c that is to be maximized through estimating R in the LS solution to the GMM, and the estimated R is thus the same as (21). Note that, if R is parameterized using unit quaternion which is equivalent to the direction cosinematrix, the quaternion can be calculated through eigen decomposition of a 4×4 matrix elaborately constructed from xi and yi . And it will be found that this quaternion is the same to that forGMMbySanso (1973) andShen et al. (2006). Sowe establish the equivalence between the rotation parameter estimated for the GHM and that for the GMM....
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...From Grafarend and Awange (2003); Teunissen (1984a), we find that it is also c that is to be maximized through estimating R in the LS solution to the GMM, and the estimated R is thus the same as (21)....
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..., Euler angles (Fang 2014b; Yang 1999), direction cosine matrix (Grafarend and Awange 2003; Teunissen 1984a), normalized quaternion (Sanso 1973; Shen et al....
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...This process follows (Grafarend and Awange 2003; Teunissen 1984a). c = N∑ i=1 yTi R xi = trace [ N∑ i=1 R xi yTi ] = trace [ R N∑ i=1 xi yTi ] = trace [RH] (16) where H = N∑ i=1 xi yTi (17) is implicitly defined in (16)....
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Journal Article•DOI•

A total least squares solution for geodetic datum transformations

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Xing Fang¹•Institutions (1)

Wuhan University¹

23 Apr 2014-Acta Geodaetica Et Geophysica Hungarica

TL;DR: In this article, the authors classified the symmetrical total least squares adjustment for 3D datum transformations as quasi indirect errors adjustment (QIEA), which is a traditional geodetic adjustment category invented by Wolf (Ausgleichungsrechnung nach der Methode der kleinsten Quadrate, 1968).

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Abstract: In this contribution, the symmetrical total least squares adjustment for 3D datum transformations is classified as quasi indirect errors adjustment (QIEA). QIEA is a traditional geodetic adjustment category invented by Wolf (Ausgleichungsrechnung nach der Methode der kleinsten Quadrate, 1968), which is specifically used for quasi nonlinear models. The form of the QIEA objective function contains the information of the functional model, and presents an unconstrained minimization problem referring simply to the transformation parameters. Based on QIEA, a solution is presented through a quasi-Newton approach, specially, the Broyden–Fletcher–Goldfarb–Shanno method. In order to justify the solutions of the QIEA, three validation conditions are proposed to check the correctness of the symmetrical treatment by comparison between the transformation and its reverse transformation. Finally, the applicability of the proposed algorithm was tested in a deformation monitoring task.

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45 citations

Cites methods from "Nonlinear analysis of the three-dim..."

...Two fundamental algorithms for adjusting the GMM are the Procrustes analysis (e.g., Grafarend and Awange 2003; Grafarend 2006; Awange et al. 2010; Grafarend and Awange 2012) and the unit quaternion-based approach (e.g., Shen et al. 2006)....
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References

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Computational geometry. an introduction

[...]

Franco P. Preparata¹, Michael Ian Shamos²•Institutions (2)

University of Illinois at Urbana–Champaign¹, Carnegie Mellon University²

01 Jan 1985

TL;DR: This book offers a coherent treatment, at the graduate textbook level, of the field that has come to be known in the last decade or so as computational geometry.

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Abstract: From the reviews: "This book offers a coherent treatment, at the graduate textbook level, of the field that has come to be known in the last decade or so as computational geometry...The book is well organized and lucidly written; a timely contribution by two founders of the field. It clearly demonstrates that computational geometry in the plane is now a fairly well-understood branch of computer science and mathematics. It also points the way to the solution of the more challenging problems in dimensions higher than two."

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6,525 citations

"Nonlinear analysis of the three-dim..." refers background in this paper

...…Procrustes algorithm include: Green (1952), Schönemann (1966), Gower (1975), Ten Berge (1977), Mardia (1978), Brokken (1983), Crosilla (1983a,b), Preparata and Shamos (1985), Golub (1987), Chu and Driessel (1990), Goodall (1991), Bingham et al. (1992), Mathias (1993), Cox and Cox (1994), Chang…...
[...]
...References to the Procrustes algorithm include: Green (1952), Scho¨ nemann (1966), Gower (1975), Ten Berge (1977), Mardia (1978), Brokken (1983), Crosilla (1983a,b), Preparata and Shamos (1985) , Golub (1987), Chu and Driessel (1990), Goodall (1991), Bingham et al. (1992), Mathias (1993), Cox and Cox (1994), Chang and Ko (1995), Gullikson (1995a,b), Borg and Groenen (1997), Mathar (1997), Chu and Trendafilov (1998), Voigt (1998), and ......
[...]

Book•

Conformal Field Theory

[...]

Philippe Di Francesco, Pierre Mathieu, David Sénéchal

13 Dec 1996

TL;DR: This paper developed conformal field theory from first principles and provided a self-contained, pedagogical, and exhaustive treatment, including a great deal of background material on quantum field theory, statistical mechanics, Lie algebras and affine Lie algesas.

...read moreread less

Abstract: Filling an important gap in the literature, this comprehensive text develops conformal field theory from first principles. The treatment is self-contained, pedagogical, and exhaustive, and includes a great deal of background material on quantum field theory, statistical mechanics, Lie algebras and affine Lie algebras. The many exercises, with a wide spectrum of difficulty and subjects, complement and in many cases extend the text. The text is thus not only an excellent tool for classroom teaching but also for individual study. Intended primarily for graduate students and researchers in theoretical high-energy physics, mathematical physics, condensed matter theory, statistical physics, the book will also be of interest in other areas of theoretical physics and mathematics. It will prepare the reader for original research in this very active field of theoretical and mathematical physics.

...read moreread less

3,440 citations

Book•

Computational Geometry: An Introduction

[...]

Franco P. Preparata¹, Michael Ian Shamos²•Institutions (2)

University of Illinois at Urbana–Champaign¹, Carnegie Mellon University²

01 Jan 1978

TL;DR: In this article, the authors present a coherent treatment of computational geometry in the plane, at the graduate textbook level, and point out the way to the solution of the more challenging problems in dimensions higher than two.

...read moreread less

3,419 citations

Book•

Numerical Linear Algebra

[...]

Lloyd N. Trefethen, David Bau

01 Jan 1997

3,140 citations

Journal Article•DOI•

Generalized procrustes analysis

[...]

John C. Gower

01 Mar 1975-Psychometrika

TL;DR: In this article, the authors investigated the problem of translating, rotating, reflecting and scaling configurations to minimize the goodness-of-fit criterion, where Gi is the centroid of the points in p-dimensional space.

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Abstract: SupposePi(i) (i = 1, 2, ...,m, j = 1, 2, ...,n) give the locations ofmn points inp-dimensional space. Collectively these may be regarded asm configurations, or scalings, each ofn points inp-dimensions. The problem is investigated of translating, rotating, reflecting and scaling them configurations to minimize the goodness-of-fit criterion Σi=1m Σi=1n Δ2(Pj(i)Gi), whereGi is the centroid of them pointsPi(i) (i = 1, 2, ...,m). The rotated positions of each configuration may be regarded as individual analyses with the centroid configuration representing a consensus, and this relationship with individual scaling analysis is discussed. A computational technique is given, the results of which can be summarized in analysis of variance form. The special casem = 2 corresponds to Classical Procrustes analysis but the choice of criterion that fits each configuration to the common centroid configuration avoids difficulties that arise when one set is fitted to the other, regarded as fixed.

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2,852 citations

"Nonlinear analysis of the three-dim..." refers background or methods in this paper

...References to the Procrustes algorithm include: Green (1952), Schönemann (1966), Gower (1975), Ten Berge (1977), Mardia (1978), Brokken (1983), Crosilla (1983a,b), Preparata and Shamos (1985), Golub (1987), Chu and Driessel (1990), Goodall (1991), Bingham et al. (1992), Mathias (1993), Cox and Cox…...
[...]
...References to the Procrustes algorithm include: Green (1952), Scho¨ nemann (1966), Gower (1975) , Ten Berge (1977), Mardia (1978), Brokken (1983), Crosilla (1983a,b), Preparata and Shamos (1985), Golub (1987), Chu and Driessel (1990), Goodall (1991), Bingham et al. (1992), Mathias (1993), Cox and Cox (1994), Chang and Ko (1995), Gullikson (1995a,b), Borg and Groenen (1997), Mathar (1997), Chu and Trendafilov (1998), Voigt (1998), and ......
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