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Journal ArticleDOI

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

01 Jun 2016-Acta Geodaetica Et Geophysica Hungarica (Springer Netherlands)-Vol. 51, Iss: 2, pp 245-256

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.

AbstractThe 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.

Topics: Helmert transformation (65%), Coordinate system (53%), Computer technology (53%), Rotation matrix (53%), Geodetic datum (52%)

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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A comparison of different solutions of the Bursa–Wolf
model and of the 3D, 7-parameter datum transformation
Jo
´
zsef Za
´
voti
1
Ja
´
nos Kalma
´
r
1
Received: 18 March 2015 / Accepted: 12 June 2015 / Published online: 11 July 2015
Ó Akade
´
miai Kiado
´
2015
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 pho-
togrammetry 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.
Keywords Quaternion-algebra Bursa–Wolf model Rotation matrix Scale factor
3D or 7-parameter datum (Helmert) transformation
1 Introduction
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). Subsequently Awange et al. (2004) has added extensions to the solutions.
& Jo
´
zsef Za
´
voti
zavoti@ggki.hu;
https://www.ggki.hu
Ja
´
nos Kalma
´
r
kalmar.janos@csfk.mta.hu
1
Research Centre for Astronomy and Earth Sciences, Hungarian Academy of Sciences,
Csatkai u. 6-8, Sopron 9400, Hungary
123
Acta Geod Geophys (2016) 51:245–256
DOI 10.1007/s40328-015-0124-6

Za
´
voti (1999) has given a solution with limited conditions in L1 norm. In the application of
computer algebraic systems to datum transformations, Awange and Grafarend (2002,
2003a, b, c) have taken new directions. In Hungary, Za
´
voti (2005) gives the first algebraic
solution to the problem, and his solution also proposes correction to the mathematical
model. Za
´
voti and Jancso
´
(2006) give the basic idea for bringing the problem to linear
form, and this concept is described in more detail in Za
´
voti (2012). Battha and Za
´
voti
(2009a, b) have applied computerized algebra to the problem of the intersection problem.
Za
´
voti 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 Za
´
voti (2012). Za
´
voti
and Kalma
´
r (2014) give a good summary of the differences between the two solutions.
2 The model of the new solution for the 3D, 7-parameter similarity
transformation
Suppose that we have two distinct coordinate systems with n common points given by their
coordinates.
The 3D, 7-parameter (Helmert) overdetermined similarity transformation is given by
the matrix equation:
s
i
¼ t þ kRp
i
; i ¼ 1; 2; ...; n; ð1Þ
where s
i
¼ X
i
; Y
i
; Z
i
½
T
are the target coordinates, t ¼ X
0
; Y
0
; Z
0
½
T
is the unknown shift, k is
the unknown scale-factor, R(a, b, c) is the rotation matrix, p
i
¼ x
i
; y
i
; z
i
½
T
are the coor-
dinates of the object points.
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.
R ¼ R
1
aðÞR
2
bðÞR
3
cðÞ: ð2Þ
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
r
23
r
33

; b ¼ arcsinðr
13
Þ; c ¼arctan
r
12
r
11

; ð3Þ
where r
ij
is the j-th entry in the i-th row of the matrix R. Thus our goal is the determination
of the rotation matrix.
Awange and Grafarend (2002) have introduced the (5) skew-symmetric matrix C
0
that
has the property:
R ¼ I
3
C
0

1
I
3
þ C
0

; ð4Þ
where I
3
is the 3-dimesional identity matrix, and C
0
is given by
C
0
¼
0 cb
c 0 a
ba 0
2
4
3
5
; ð5Þ
with parameters a, b and c.
246 Acta Geod Geophys (2016) 51:245–256
123

Multiplying from the left Eq. (1) with I
3
C
0

, and applying Eq. (4), we obtain:
1 c b
c 1 a
b a 1
2
6
4
3
7
5
X
i
Y
i
Z
i
2
6
4
3
7
5
¼
1 c b
c 1 a
b a 1
2
6
4
3
7
5
X
0
Y
0
Z
0
2
6
4
3
7
5
þ k
1 cb
c 1 a
ba 1
2
6
4
3
7
5
x
i
y
i
z
i
2
6
4
3
7
5
;
i ¼ 1; 2; ...; n:
ð6Þ
These equations are the basis of the algebraic solution of the 3D, 7-parameter (Helmert)
transformation.
3 Determination of the scale-factor of the 3D, 7-parameter similarity
transformation
Za
´
voti (2012) has eliminated the shift parameters by reducing the coordinates to the center
of gravity. He also has shown that during the solution of the overdetermined system of
equations parameters a, b and c are eliminated, and the following overdetermined system
of equations, quadratic in the unknown k parameter, are obtained:
k
2
x
2
is
þ y
2
is
þ z
2
is

¼ X
2
is
þ Y
2
is
þ Z
2
is
; i ¼ 1; 2; ...; n; ð7Þ
where X
is
¼ X
i
X
s
; Y
is
¼ Y
i
Y
s
; Z
is
¼ Z
i
Z
s
i ¼ 1; 2; ...; n; x
is
¼ x
i
x
s
; y
is
¼
y
i
y
s
; z
is
¼ z
i
z
s
; i ¼ 1; 2; ...; n; X
s
; Y
s
; Z
s
ðÞ; x
s
; y
s
; z
s
ðÞare coordinates the centre of
gravity.
The system of Eq. (7) is overdetermined, with several solutions according to a chosen
error function.
3.1 Solution I
The (7) system of equations is written as a product:
k
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x
2
is
þ y
2
is
þ z
2
is
q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
2
is
þ Y
2
is
þ Z
2
is
q

k
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x
2
is
þ y
2
is
þ z
2
is
q
þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
2
is
þ Y
2
is
þ Z
2
is
q

¼ 0;
i ¼ 1; 2; ...; n: ð8Þ
Let us consider the first factor in the above product. The system of equations to be
solved is:
k
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x
2
is
þ y
2
is
þ z
2
is
q
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
2
is
þ Y
2
is
þ Z
2
is
q
; i ¼ 1; 2; ...; n: ð9Þ
Add up all these equations! The solution for k of the overdetermined system of equa-
tions—taking into account that only the positive root has a physical meaning for us—is
given, according to Za
´
voti (2012), by the following well known equation:
k
1
¼
P
n
i¼1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
2
is
þ Y
2
is
þ Z
2
is
p
P
n
i¼1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x
2
is
þ y
2
is
þ z
2
is
p
: ð10Þ
Acta Geod Geophys (2016) 51:245–256 247
123

Albertz and Kreiling (1975) have shown that the k scale-factor can be obtained as the
quotient of the sums of the coordinates of the points, taken in the coordinate system with
origin in the center of gravity. Thus we 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
Adding up all equations in (7) gives
k
2
X
n
i¼1
x
2
is
þ y
2
is
þ z
2
is

¼
X
n
i¼1
X
2
is
þ Y
2
is
þ Z
2
is

: ð11Þ
There is a simple solution of this equation in non-negative real numbers without trans-
forming it to a product. 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:
k
2
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P
n
i¼1
X
2
is
þ Y
2
is
þ Z
2
is
ðÞ
P
n
i¼1
x
2
is
þ y
2
is
þ z
2
is
ðÞ
v
u
u
u
u
u
t
: 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).
3.3 Solution III
Our starting point is again the system of Eq. (7). We want to obtain a least squares solution
for k using intermediary equations. Elementary steps, given in full detail in (23)–(26),
yield:
k
3
¼
P
n
i¼1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x
2
is
þ y
2
is
þ z
2
is
ðÞX
2
is
þ Y
2
is
þ Z
2
is
ðÞ
p
P
n
i¼1
x
2
is
þ y
2
is
þ z
2
is
ðÞ
: ð13Þ
Thus there are three different solution procedures (estimations) for the k scale factor of the
3D, 7-parameter (Helmert) 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.
Za
´
voti (2013) has determined the normal matrix of the problem:
248 Acta Geod Geophys (2016) 51:245–256
123

P
n
i¼1
ky
is
þ Y
is
ðÞ
2
þ kz
is
þ Z
is
ðÞ
2
hi
P
n
i¼1
kx
is
þ X
is
ðÞky
is
þY
is
ðÞ
P
n
i¼1
kx
is
þX
is
ðÞkz
is
þ Z
is
ðÞ
P
n
i¼1
kx
is
þ X
is
ðÞ
2
þ kz
is
þ Z
is
ðÞ
2
hi
P
n
i¼1
ky
is
þ Y
is
ðÞkz
is
þ Z
is
ðÞ
P
n
i¼1
kx
is
þ X
is
ðÞ
2
þ ky
is
þ Y
is
ðÞ
2
hi
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
:
ð14Þ
(The symmetric entries of the normal matrix are not listed).
The normal vector is obtained in a similar way:
2k
P
n
i¼1
ðy
is
Z
is
z
is
Y
is
Þ
P
n
i¼1
ðz
is
X
is
x
is
Z
is
Þ
P
n
i¼1
x
is
Y
is
y
is
X
is
ðÞ
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
: ð15Þ
There are several procedures to obtain the parameters a, b and c from the 3 9 3 normal
system of equations, we have chosen the singular value decomposition because of its
stability. Utilizing special properties of the normal matrix the rotation parameters in (3) can
also be determined.
The shift parameters X
0
, Y
0
and Z
0
can be determined after Eq. (1) is rewritten to a
system with the center of gravity as origin
X
0
Y
0
Z
0
2
4
3
5
¼
X
s
Y
s
Z
s
2
4
3
5
kR
x
s
y
s
z
s
2
4
3
5
: ð16Þ
Parameters for the precision, variance and covariance are computed in the conventional
way.
5 The Bursa–Wolf model of the datum-transformation
Equation (13) can also be obtained in the following way (
s and
p are the centers of gravity
in the two systems):
Ds
i
¼ s
i
s ) s
i
¼ Ds
i
þ
s ;
Dp
i
¼ p
i
p ) p
i
¼ Dp
i
þ
p :
ð17Þ
Substituting this into the transformation Eq. (1) gives:
Ds
i
þ
s ¼ t þ k R Dp
i
þ
p
ðÞ
; i ¼ 1; 2; ...; n: ð18Þ
Reordering terms gives:
Ds
i
þ
s ¼ t þk R
p þ k R Dp
i
; i ¼ 1; 2; ...; n: ð19Þ
Since (1) holds for the centers of gravity
s and
p too, so the term in the middle of (19)
can be omitted and thus we are left with:
Ds
i
¼ k R Dp
i
; i ¼ 1; 2; ...; n: ð20Þ
Acta Geod Geophys (2016) 51:245–256 249
123

Citations
More filters

Journal ArticleDOI
Abstract: Rigid transformation including rotation and translation can be elegantly represented by a unit dual quaternion. Thus, a non-differential model of the Helmert transformation (3D seven-parameter similarity transformation) is established based on unit dual quaternion. This paper presents a rigid iterative algorithm of the Helmert transformation using dual quaternion. One small rotation angle Helmert transformation (actual case) and one big rotation angle Helmert transformation (simulative case) are studied. The investigation indicates the presented dual quaternion algorithm (QDA) has an excellent or fast convergence property. If an accurate initial value of scale is provided, e.g., by the solutions no. 2 and 3 of Zavoti and Kalmar (Acta Geod Geophys 51:245–256, 2016) in the case that the weights are identical, QDA needs one iteration to obtain the correct result of transformation parameters; in other words, it can be regarded as an analytical algorithm. For other situations, QDA requires two iterations to recover the transformation parameters no matter how big the rotation angles are and how biased the initial value of scale is. Additionally, QDA is capable to deal with point-wise weight transformation which is more rational than those algorithms which simply take identical weights into account or do not consider the weight difference among control points. From the perspective of transformation accuracy, QDA is comparable to the classic Procrustes algorithm (Grafarend and Awange in J Geod 77:66–76, 2003) and orthonormal matrix algorithm from Zeng (Earth Planets Space 67:105, 2015. https://doi.org/10.1186/s40623-015-0263-6 ).

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)....

    [...]

  • ...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 ....

    [...]

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

    [...]

  • ...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,...

    [...]

  • ...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…...

    [...]


Journal ArticleDOI
TL;DR: The results showed an accurate transforming datum using ANN technique for both common and check points, and the novel model improved the transformation coordinates by 37 to 72% in space directions.
Abstract: The geodetic datum transformation in-between local and global systems seen in the world are inspiring for the engineering applications. In this context, the Egyptian geodetic network has a limited observation for the terrestrial and satellite of the geodetic networks. Transforming the coordinates of the Egyptian datum, here we demonstrate the datum transformation in three directions from global to local coordinates that utilized the artificial neural network (ANN) technique as a new tool of datum transformation in Egypt. A conventional, which are the Helmert and Molodensky, and numerical, which are the regression, minimum curvature surface, and ANN, datum transformation techniques are investigated and compared over the available data in Egypt. The results showed an accurate transforming datum using ANN technique for both common and check points, and the novel model improved the transformation coordinates by 37 to 72% in space directions. A comparison between the conventional and numerical techniques shows that the accuracy of the developed ANN model is 20.16 cm in terms of standard deviation based on the residuals of the projected coordinates.

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):...

    [...]


Journal ArticleDOI
Orhan Kurt1
Abstract: To find 3D similarity transformation parameters (a scale, three rotational angles, and three translation elements) between two orthogonal coordinate systems in 3D is an ill-posed non-linear inverse problem by means of common points (their Cartesian components are known in the both systems). The problem can be solved via Linearized Least Squares (LLS) or Direct (non-iterative) Least Squares (DLS). Since the parameters in LLS take different quantities (and units) from each other, the condition problems can arise during the solution of normal equations. In this paper, we propose a combined solution to reducing ill-conditioning and to perform precision analysis and global outlier test in LLS accordingly. The way is based on column norming and uses the normalized unknowns instead of the original ones at the solution stage of the normal equations. While the global outlier test is fulfilled on the normalized unknowns, the original unknowns and their precisions obtained using the normalized matrix with li...

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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Journal ArticleDOI
Abstract: The rigid motion involving both rotation and translation in the 3D space can be simultaneously described by a unit dual quaternion. Considering this excellent property, the paper constructs the Helmert transformation (seven-parameter similarity transformation) model based on a unit dual quaternion and then presents a rigid iterative algorithm of Helmert transformation using a unit dual quaternion. Because of the singularity of the coefficient matrix of the normal equation, the nine parameter (including one scale factor and eight parameters of a dual quaternion) Helmert transformation model is reduced into five parameter (including one scale factor and four parameters of a unit quaternion which can represent the rotation matrix) Helmert transformation one. Besides, a good start estimate of parameter is required for the iterative algorithm, hence another algorithm employed to compute the initial value of parameter is put forward. The numerical experiments involving a case of small rotation angles i.e. geodetic coordinate transformation and a case of big rotation angles i.e. the registration of LIDAR points are studied. The results show the presented algorithms in this paper are correct and valid for the two cases, disregarding the rotation angles are big or small. And the accuracy of computed parameter is comparable to the classic Procrustes algorithm from Grafarend and Awange (J Geod 77:66–76, 2003), the orthonormal matrix algorithm from Zeng (Earth Planets Space 67:105, 2015), and the algorithm from Wang et al. (J Photogramm Remote Sens 94:63–69, 2014).

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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Journal ArticleDOI
TL;DR: The experimental results show that the calibration approach for Camera-IMU pose parameters with adaptive constraints of multiple error equations improves the measurement accuracy by 84.0% and can effectively suppress IMU drift with good robustness.
Abstract: In the calibration of the pose parameters of a camera and inertial measurement unit (Camera-IMU), the camera depth information is unreliable due to the uneven spatial distribution of calibration points, because the calibration points have random errors due to the IMU drift and the inadequate robustness of stereovision and because the Camera-IMU pose parameters lack self-adaptation. This paper proposes a high-precision calibration approach for Camera-IMU pose parameters with adaptive constraints of multiple error equations (adaptive constraint calibration approach, ACCA). The approach calibrates pose parameters of the Camera-IMU jointly via error equations, such as lens distortion correction, camera parallax correction and error compensation of the inertial sensor. The experimental results show that the calibration approach for Camera-IMU pose parameters with adaptive constraints of multiple error equations improves the measurement accuracy by 84.0% and can effectively suppress IMU drift with good robustness.

3 citations


References
More filters

Journal ArticleDOI
TL;DR: A closed-form solution to the least-squares problem for three or more paints is presented, simplified by use of unit quaternions to represent rotation.
Abstract: Finding the relationship between two coordinate systems using pairs of measurements of the coordinates of a number of points in both systems is a classic photogrammetric task . It finds applications i n stereoph and in robotics . I present here a closed-form solution to the least-squares problem for three or more paints . Currently various empirical, graphical, and numerical iterative methods are in use . Derivation of the solution i s simplified by use of unit quaternions to represent rotation . I emphasize a symmetry property that a solution to thi s problem ought to possess . The best translational offset is the difference between the centroid of the coordinates i n one system and the rotated and scaled centroid of the coordinates in the other system . The best scale is equal to th e ratio of the root-mean-square deviations of the coordinates in the two systems from their respective centroids . These exact results are to be preferred to approximate methods based on measurements of a few selected points . The unit quaternion representing the best rotation is the eigenvector associated with the most positive eigenvalue o f a symmetric 4 X 4 matrix . The elements of this matrix are combinations of sums of products of correspondin g coordinates of the points .

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....

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

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  • ...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)....

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  • ...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)....

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Journal ArticleDOI
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.

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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Journal Article

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)....

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Journal ArticleDOI
Abstract: Several procedures for solving in a closed form the three-dimensional resection problem have already been presented. In the present contribution, the overdetermined three-dimensional resection problem is solved in a closed form in two steps. In step one a combinatorial minimal subset of observations is constructed which is rigorously converted into station coordinates by means of the Groebner basis algorithm or the multipolynomial resultant algorithm. The combinatorial solution points in a polyhedron are then reduced to their barycentric in step two by means of their weighted mean. Such a weighted mean of the polyhedron points in ℝ3 is generated via the Error Propagation law/variance–covariance propagation. The Fast Nonlinear Adjustment Algorithm was proposed by C.F. Gauss, whose work was published posthumously, and C.G.I. Jacobi. The algorithm, here referred to as the Gauss–Jacobi Combinatorial algorithm, solves the overdetermined three-dimensional resection problem in a closed form without reverting to iterative or linearization procedures. Compared to the actual values, the obtained results are more accurate than those obtained from the closed-form solution of a minimano of three known stations.

27 citations


DissertationDOI
01 Jan 2002
Abstract: Die Methode der Grobner-Basen und Multipolynomialen Resultante wird als wirksames algebraische Hilfsmittel zur expliziten Losung nichtlinearer geodatischer Problem vorgestellt. Wir nutzen dir Grobner-Basen und Multipolynomialen Resultante als Rechenhilfsmittel bei der Losung des nichtlinearen Gauss-Markov Modells mit Hilfe des kombinatorischen Gauss-Jacobi-Algorithmus. The algebraic techniques of Grobner bases and Multipolynomial resultants are presented as efficient algebraic tools for solving the nonlinear geodetic problems. The capability of the Grobner bases and the Multipolynomial resultants to solve explicitly nonlinear geodetic problems enables us to use them as the computational engine in the Gauss-Jacobi combinatorial algorithm to solve the nonlinear Gauss-Markov model.

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)....

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  • ...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....

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  • ...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....

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

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  • ...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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