# A comparison of different solutions of the Bursa–Wolf model and of the 3D, 7-parameter datum transformation

TL;DR: Various methods for calculating the scale factor are discussed and solutions based on quaternion with those that are based on rotation matrix making use of skew-symmetric matrix are compared.

Abstract: The present work deals with an important theoretical problem of geodesy: we are looking for a mathematical dependency between two spatial coordinate systems utilizing common pairs of points whose coordinates are given in both systems. In geodesy and photogrammetry the most often used procedure to move from one coordinate system to the other is the 3D, 7 parameter (Helmert) transformation. Up to recent times this task was solved either by iteration, or by applying the Bursa–Wolf model. Producers of GPS/GNSS receivers install these algorithms in their systems to achieve a quick processing of data. But nowadays algebraic methods of mathematics give closed form solutions of this problem, which require high level computer technology background. In everyday usage, the closed form solutions are much more simple and have a higher precision than earlier procedures and thus it can be predicted that these new solutions will find their place in the practice. The paper discusses various methods for calculating the scale factor and it also compares solutions based on quaternion with those that are based on rotation matrix making use of skew-symmetric matrix.

## Summary (2 min read)

### 2 The model of the new solution for the 3D, 7-parameter similarity transformation

- Suppose that the authors have two distinct coordinate systems with n common points given by their coordinates.
- The R rotation matrix is parametrized using the three independent and unknown a, b and c Cardan-angles (Awange 2002), which belong to rotations about axes z-y-x by the angle c, b and a consecutively.
- Obviously, changing the order of the three rotations or the direction of the angles leads to rotation matrix.
- The rotation angles can be obtained from the elements of the rotation matrix: a ¼ arctan r23 r33 ; b ¼ arcsinðr13Þ; c ¼ arctan r12 r11 ; ð3Þ where rij is the j-th entry in the i-th row of the matrix R. Thus their goal is the determination of the rotation matrix.
- These equations are the basis of the algebraic solution of the 3D, 7-parameter transformation.

### 3 Determination of the scale-factor of the 3D, 7-parameter similarity transformation

- Závoti (2012) has eliminated the shift parameters by reducing the coordinates to the center of gravity.
- The system of Eq. (7) is overdetermined, with several solutions according to a chosen error function.

### 3.1 Solution I

- Let us consider the first factor in the above product.
- Thus the authors have transformed the quadratic Eq. (7) to linear equations, as opposed to the procedure given in Awange and Grafarend (2002), which requires the cumbersome separation of the roots of a polynomial of degree 4.

### 3.2 Solution II

- After taking into account that a root with physical meaning must be positive, the authors obtain for the lambda scale-factor the following equation, given in Horn (1987) using quaternions, which is also the solution of the Bursa–Wolf model: k2 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPn i¼1 X2is þ Y2is þ Z2isð Þ Pn i¼1 x2is þ y2is þ z2isð Þ vuuuuut : ð12Þ.
- Thus the authors can obtain the unique k scale-factor from the quadratic equations as opposed to the complicated separation procedure of the roots of a polynomial of degree 4 given by Awange and Grafarend (2002).

### 3.3 Solution III

- The authors starting point is again the system of Eq. (7).
- Thus there are three different solution procedures for the k scale factor of the 3D, 7-parameter transformation.

### 4 Determination of the linear and shift parameters

- After having determined the scale-factor, the problem can be written in linear form, and the adjustment model for the linear problem can be given.
- This procedure makes it possible to include arbitrarily many equations (common points), and give a solution for the parameters a, b, and c. Závoti (2013) has determined the normal matrix of the problem: Utilizing special properties of the normal matrix the rotation parameters in (3) can also be determined.

### 5 The Bursa–Wolf model of the datum-transformation

- The authors now want to deal with the scale-factor.
- In order to make the comparison more simple, the authors rewrite Eq. (10) using the notations of: k1 ¼ Xn i¼1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DsTi Dsi q ,Xn i¼1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DpTi Dpi q : ð22Þ.
- Now the authors can verify that substituting (24) into (10) and into (12) leads to equality k = k1 = k2, and thus the two statistical estimates (the theoretical scale) is identical.
- Xn i¼1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DsTi Dsið Þ DpTi Dpið Þ q ,Xn i¼1 DpTi Dpi: ð27Þ.

### 7 The solution of the minimax problem using quaternion-algebra

- Using the quaternion q, the authors obtain the rotation matrix R = (rij) from (37), and the rotation angles from (3).
- The shift vector t is then obtained from (21) by taking averages.

### 9 Summary

- In the present work the authors have given a general model for the 3D, 7-parameter similarity transformation, which can be used to obtain several different solutions, and which includes as a special case the Bursa–Wolf model.
- In their method the authors introduce an overdetermined system of equations for the scale-factor, which is then solved using different principles.
- The advantage of their model is that the determination of the scale-factor makes it possible to obtain the solution for the original nonlinear problem from the solution of a linear problem.
- The authors have given a new least-squares solution for the scale-factor, which is numerically identical to the relevant parameter of the Bursa–Wolf model.
- The authors have shown that there is a functional connection between the quaternions of the Bursa–Wolf model and the entries of the skew-symmetric matrix given by Awange and Grafarend.

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##### Citations

9 citations

### Cites background or methods from "A comparison of different solutions..."

...DQA sets the threshold τ to 1.0 × 10−10 and adopts six initial values of , i.e., the solution of Han (2010), the three solutions of Závoti and Kalmár (2016), a biased one (solution 5 with initial value as 10) and a seriously biased one (solution 6 with initial value as 100)....

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...For this situation, because the Han (2010) and Závoti and Kalmár (2016) do not consider the weight of control point when computing the solution of scale, all the four solutions (solution 1 to solution 4) just offer slightly biased initial values of ....

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...2014; Závoti and Kalmár 2016; Zeng 2014; Zeng 2015; Zeng et al....

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...The case study shows the presented algorithm requires one iteration to recover the transformation parameter if accurate initial value of scale is provided like the solutions no. 2 and 3 of Závoti and Kalmár (2016) for the situation that the weights are identical; otherwise,...

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...Závoti and Kalmár (2016) presented three solutions of scale as follows. where xis, yis, zis are the centrobaric coordinates in the original coordinate system, i.e., (68) H = mean ���dx′ij ��� ��dxij �� i �= j, (69) Z1 = ∑n i=1 √ X2is + Y 2 is + Z 2 is ∑n i=1 √ x2is + y 2 is + z 2 is , (70) Z2…...

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7 citations

### Cites methods from "A comparison of different solutions..."

...In the well-known Bursa–Wolf similarity 3D, seven parameter transformation (Helmert) model can be presented as follows (Deakin 2006; Závoti and Kalmár 2016):...

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5 citations

### Cites background or methods from "A comparison of different solutions..."

...Indeed, we run all non-iterative methods (PGH, PSH, DLS = ALS, QBA) with produced data according to all rotation angle combination in the one by one....

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...Equalizing an eigenvector (corresponding to largest one of eigenvalues) to ̂, QBA solution is completed with ̂ = (q̂20 − ̂ T ̂) + 2 (̂ ̂T + q̂0 ̂ ) which is reflected a relation between the quaternion vector and the rotation matrix (see [6] in detail), where q̂0 and ̂ = [ â b̂ ĉ ] T are real and imaginary parts of the estimated quaternion ̂ = [ q̂0 ̂ T ]T and the elements of ̂ are directly related to the skew-symmetric matrix ̂ in Equation (4.3) [6–18]....

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...Some direct LS methods are developed under the rotational invariant situation (DLS, QBA, PGH, and PSH) and some under small rotational angles (DBW, DMB)....

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...Some techniques take advantage of a skew-symmetric matrix related with the rotation matrix, these are QBA [16,18], GJC [19,20] and DLS [12–16] = ALS [17]....

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...• In QBA, after solving ̂ = and ŝ with hierarchically, LS minimization on the rotation matrix is provided by max ( ∑ j ̄Tj ̄j) for ensuring T → min [6]....

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5 citations

### Cites background from "A comparison of different solutions..."

...2006; Zeng and Yi 2011; Závoti and Kalmár 2016), Rodrigues matrix and Gibbs vector see (e.g. Zeng and Huang 2008; Zeng and Yi 2010; Závoti and Kalmár 2016; Zeng et al....

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...Some papers have presented formulas of computing scale factor which is independent to the rotation angles and translation, e.g. Han (2010), Závoti and Kalmár (2016)....

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...2006; Zeng 2015; Závoti and Kalmár 2016; Zeng et al....

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3 citations

##### References

4,263 citations

### "A comparison of different solutions..." refers methods in this paper

...(31): k ¼ 1 k X i DsTi Dsi ,X i DpTi Dpi ; ð33Þ which yields the well known Horn-Eq....

[...]

...Zeitschrift für Vermessungswesen 122:323–333 Horn BKP (1987) Closed form solution of absolute orientation using unit quaternions....

[...]

...Horn (1987) is one of the earliest works to give a solution to the absolute orientation problem, but his solution is different from Závoti (2012)....

[...]

...After taking into account that a root with physical meaning must be positive, we obtain for the lambda scale-factor the following equation, given in Horn (1987) using quaternions, which is also the solution of the Bursa–Wolf model: k2 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPn i¼1 X2is þ Y2is þ Z2isð Þ Pn i¼1 x2is þ y2is þ z2isð Þ vuuuuut : ð12Þ Thus we can obtain the unique k scale-factor from the quadratic equations as opposed to the complicated separation procedure of the roots of a polynomial of degree 4 given by Awange and Grafarend (2002)....

[...]

62 citations

### "A comparison of different solutions..." refers background in this paper

...Shen et al. (2006) give the following equation for the rotation matrix R and the q quaternion: R ¼ q20 qT q I3 þ 2 q qT þ q0 C qð Þ : ð37Þ Now we can rewrite (35): max R X i DsTi R Dpi ¼ max q X i sTi Qþ Pþi q ¼ max q qT N q; ð38Þ where the 4x4 matrix N is given by N ¼ X i DsTi Dpi DsTi C Dpið Þ C…...

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...…Þ C Dsið Þ Dpi Dsi DpTi þ C Dsið Þ C Dpið Þ : ð39Þ The maximum of the quadratic form (38) is obtained when q is an eigenvector of N, and then its value is equal to the eigenvalue of N (Shen et al. 2006), and thus we have to determine the maximal eigenvalue of N and the eigenvector q (the…...

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46 citations

### "A comparison of different solutions..." refers methods in this paper

...The conventional treatment of the 3D, 7-parameter datum transformation is given in Grafarend and Krumm (1995), in Grafarend and Kampmann (1996), and in Grafarend and Shan (1997)....

[...]

27 citations

21 citations

### "A comparison of different solutions..." refers background or methods in this paper

...In the application of computer algebraic systems to datum transformations, Awange and Grafarend (2002, 2003a, b, c) have taken new directions. In Hungary, Závoti (2005) gives the first algebraic solution to the problem, and his solution also proposes correction to the mathematical model. Závoti and Jancsó (2006) give the basic idea for bringing the problem to linear form, and this concept is described in more detail in Závoti (2012). Battha and Závoti (2009a, b) have applied computerized algebra to the problem of the intersection problem. Závoti and Fritsch (2011) have given a totally new solution for the outer orientation problem of photogrammetry. Horn (1987) is one of the earliest works to give a solution to the absolute orientation problem, but his solution is different from Závoti (2012)....

[...]

...In the application of computer algebraic systems to datum transformations, Awange and Grafarend (2002, 2003a, b, c) have taken new directions. In Hungary, Závoti (2005) gives the first algebraic solution to the problem, and his solution also proposes correction to the mathematical model. Závoti and Jancsó (2006) give the basic idea for bringing the problem to linear form, and this concept is described in more detail in Závoti (2012). Battha and Závoti (2009a, b) have applied computerized algebra to the problem of the intersection problem. Závoti and Fritsch (2011) have given a totally new solution for the outer orientation problem of photogrammetry. Horn (1987) is one of the earliest works to give a solution to the absolute orientation problem, but his solution is different from Závoti (2012). Závoti and Kalmár (2014) give a good summary of the differences between the two solutions....

[...]

...In the application of computer algebraic systems to datum transformations, Awange and Grafarend (2002, 2003a, b, c) have taken new directions. In Hungary, Závoti (2005) gives the first algebraic solution to the problem, and his solution also proposes correction to the mathematical model....

[...]

...Subsequently Awange et al. (2004) has added extensions to the solutions....

[...]

...In the application of computer algebraic systems to datum transformations, Awange and Grafarend (2002, 2003a, b, c) have taken new directions. In Hungary, Závoti (2005) gives the first algebraic solution to the problem, and his solution also proposes correction to the mathematical model. Závoti and Jancsó (2006) give the basic idea for bringing the problem to linear form, and this concept is described in more detail in Závoti (2012). Battha and Závoti (2009a, b) have applied computerized algebra to the problem of the intersection problem. Závoti and Fritsch (2011) have given a totally new solution for the outer orientation problem of photogrammetry....

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