Orthonormal matrices, Procrustes and quaternion analysis are closed form solutions of the configuration matching problem, common in geodesy as in the datum transformation problem. Literature reports more Procrustes based geodetic applications than Quaternions, which are more used in other application fields, such as aerospace navigation, robotics and computer vision. The large popularity of Procrustes in geodesy is mainly due to its capability to take into account a priori observation weighting in a simple wa

De Agostino, Mattia

Porporato, Chiara Maria

Roggero, Marco

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Politecnico di Torino
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[Proceeding] A review on non iterative closed form configuration matching
and rotations estimation
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De Agostino M.; Porporato C.; Roggero M. (2012). A review on non iterative closed
form configuration matching and rotations estimation. In: VII Hotine-Marussi Symposium on
Mathematical Geodesy, Roma, 6-10 giugno 2009. pp. 347-352
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A review on non iterative closed form configuration 
matching and rotations estimation 
M. De Agostino, C. Porporato, M. Roggero 
 
 
 
 
 
 
 
 
Abstract. Orthonormal matri ces, Procrustes  and qua-
ternion analysis are closed form solutions of the con-
figuration matching problem, common in geodesy as 
in the datum transformation problem. Literature re-
ports more Procrustes based geodetic applications 
than Quaternions, which are more used in other appli-
cation fields, such as aerospace navigation, robotics 
and computer vision. The large popularity of Pro-
crustes in geodesy is mainly due to its capability to 
take into account a priori observation weighting in a 
simple way. 
Keywords. Rotations, quaternions, ortonormal ma-
trices, Procrustes. 
1  Introduction  
A rotation is a transformation of the Euclidean 
space that rigidly moves objects leaving fixed at least 
one point (the origin of Euclidean space). In geodesy 
rotations are involved in  many problems, especi ally 
for solving the transformation between reference 
frames. More in detail, a rotation is an isometry of an 
Euclidean space that preserves the orientation, and it 
is described by an orthogonal matrix. In a Euclidean 
space of two or three dimensions each orthogonal ma-
trix expresses a rotation around a point or an axis, a 
reflection, or a combination of these two transforma-
tions. 
As mentioned above, the representation of a rot a-
tion expresses the orientation of an object with respect  
to a reference system, or the relative orientation of 
two or more reference systems. Euler's theorem shows 
that any space rotation can be decomposed into the 
product of the three rotations i j kR R R   , where 
i j k   and 3i, j, k  , and where iR  indicates a 
rotation of αradians counterclockwise around the i  
axis. According to Euler's theorem, the attitude of a 
rigid body can be described by a rotation around only 
one axis. Furthermore, this rotation can be defined un-
iquely by a minimum of three parameters, such as the 
directors cosines matrices, that represent the most 
widespread method for estimating rotations into geo-
detic networks. 
However, for various reasons, there are several  
ways to represent rotations, making use of a number 
of parameters even higher than three, although also 
those redundant representations have always only 
three degrees of freedom. Some of these methods, 
presented over the years, are hereinafter described and 
compared.  
2  The Quaternion-based approach 
What Sir William Rowan Hamilton wrote on the 
16th of October 1843 on a stone of Brougham Bridge 
in Dublin, is simply: 
 2 2 2i j k ijk 1      ( 1 ) 
 
M. De Agostino, C. Porporato 
Department of Land, Environment and Geoengineering 
Politecnico di Torino, Corso Duca degli Abruzzi 24.          
10129 Torino, Italy 
M. Roggero 
Department of Human Settlements Science and Technology Po-
litecnico di Torino, Viale Mattioli 39.                             10129 
Torino, Italy 
 
 
 
   
Fig. 1 The Brougham Bridge in Dublin and the Hamilton  
commemorative plaque 
The combination: 
 0 1 2 3q q iq jq kq     ( 2 ) 
where q0, q1, q2 and q3 are real numbers, defines the 
generic quaternion. Quaternions satisfy all the laws of 
algebra, except the multiplication commutative law. 
In fact:  
 ij ij, jk kj, ki ik       ( 3 ) 
which represents a serious violation of the commuta-
tive law ab ba . It is also apparent: 
 
ij k jk i ki j
ji k kj i ik j
  
     
 ( 4 ) 
The fundamental values of quat ernions, i, j and k, 
can be handled as three mutually perpendicular 
clockwise axes in a common three-dimensional Euc-
lidean space. Each unit quaternion, in particular, de-
fine a rot ation in 3  space. These rotations are given 
by the conjuge: 
 1 *r qrq qrq      ( 5 ) 
It can be veri fied that i f r is purely imaginary (the 
real part is equal to zero), also r  is purely imaginary; 
therefore it can be defined an action of the group of 
unit quaternions on 3 . Each action defined in this 
way is indeed a rotation, since it preserves the norm: 
 **r qrq q r q r           ( 6 ) 
It is possible to show the equivalence between the 
conjuge and the product of the 3x3 Rodriguez rotation 
matrix and a real vector. In fact, the operation: 
      * * T Tqrq Qr q Q Qr Q Q r        ( 7 ) 
produce a 4x4 rotation matrix TQ Q , whose lower 
right 3x3 sub-matrix is orthonormal, and it is the rota-
tion matrix that takes r  to r . 
 
Given the three-dimensional rotation between two 
frames r  and s  with a scale-change ρ:  
 *s qrq     ( 8 ) 
where *qq 1  , the measurement equation is: 
 *i i is qr q        ( 9 ) 
where i  are still purely imaginary quaternions. In 
the following, we will present two methods in the lite-
rature to solve the problem (9), respectively by mini-
mizing the residual vector   (Sansò, 1973) or alterna-
tively maximizing the scalar product s r   (Horn, 
1987). 
2.1 Residual vector minimization 
In accordance with the least squares  approach, we 
must compute the minimum of the function:  
   *i i
i
q,        ( 10 ) 
where 2 *q qq 1    . Differentiating  with respect to 
 and q  and introducing a real Lagrange multiplier , 
we obtain, after some mathematical steps: 
     2 22i i i i
i i i
s r q 2 s qr 0      
 
        ( 11 ) 
It is possible to show that: 
 2 22i i
i i
s r      ( 12 ) 
Using equation (12) in (11), through some mathe-
matical steps we reach the final equation: 
 
2
i i i
i i
rs q r q
Aq q
   
    
   

     
 
 ( 13 ) 
As it is possible to see, the unknowns  and q  are re-
spectively an eigenvalue and eigenvector of a symme-
tric matrix A, that can be built directly from the data ri 
and si. In particular, using the products expansion 
rules of quaternions, we find: 
   
   i i i i i i i i
i i i i
i i i
i i
r s q s qr S q r R S q
R S q A q
   
   
    
   
   
 
       
 
( 14 ) 
 
 
from which it is possible to derive the expressions of 
iA  and A. 
2.2 Scalar product maximization
 
In this second approach, we seek the quaternion q  
to maximize the scalar product: 
  
n n
*
i i i i
i 1 i 1
qrq s r s
 
          ( 15 ) 
Reminding the geometric meaning of the scal ar 
product of vectors, we have:  
 i i i ir s r s cos        ( 16 ) 
where θ is the angle subtended. Since i ir r    and is  
are constants, the maximization of the scalar product  
is equal to minimize the parameter θ (or maximize 
cos ). Using the above results, we can rewrite the 
scalar product as: 
    
n
i i
i 1
qr s q

      ( 17 ) 
The products  iqr   and  is q   can be expressed by 
means of the matrices R and S, therefore: 
 
   
n n
T T
i i i i
i 1 i 1
n
T T
i i
i 1
n
T T
i
i 1
qr s q q R S q
q R S q
q Å q q Åq
 


  
 
  
 
 
  
 
 


     
 
   
 ( 18 ) 
It is now simple to derive the values of the sub-
matrices iÅ , and consequently of the matrix Å , 
where Ti iÅ Å  and 
TÅ Å . Recalling that we were 
seeking: 
  
n
T
i iq q
i 1
Max r s Max q Åq

   
 
      ( 19 ) 
It is possible to note that Å A  . 
3  The Orthonormal matrices approach 
Among the existing ways to represent rotation we 
present one that is most often used in photogramme-
try: the orthonormal matrices. Again r is the position 
vector in  the original RS and s  the position vector in  
the final RS. The aim is to find the rotation that mi-
nimizes the residual errors. Therefore, we have to find 
the orthonormal matrix B, 3×3 matrix,  that maximize  
  
n n
T
i i i i
i 1 i 1
s Br s Br
 
   ( 20 ) 
Being  
  T T Ta Bb TR B ab  ( 21 ) 
it is possible to write the (20) as 
  
n
T T T
i i
i 1
Tr B s r Tr B M

 
 
 
  ( 22 ) 
where 
n
T
i i
i 1
M s r

  and 
 
 
1 1 1 2 1 3
i i i i i i
i i i
2 1 2 2 2 3
i i i i i i
i i i
3 1 3 2 3 3
i i i i i i
i i i
s r s r s r
M s r s r s r
s r s r s r
 
 
 
  
 
 
  
  
  
  
 ( 23 ) 
It follows that the rotation that minimize the resi-
dual errors corresponds to the orthonormal matrix B 
that maximizes  TTr B M . 
A square matrix M could always be decomposed 
into the product of an orthonormal matrix U and a 
positive semi-definite matrix S. When M is non singu-
lar, the matrices U and S are univocally determined 
and it allows to write  
 
 
1
T 2
1
T 2
MM M M US
M M
   ( 24 ) 
In this expression,  
1
T 2U M M M

  is an or-
thonormal matrix and  
1
T 2S M M  is the square root 
positive semi-definite of the symmetric matrix TM M . 
 
 
It is possible to write this matrix TM M  using its ei-
genvalues  i  and eigenvectors iuˆ  as following: 
T T T T
1 1 1 2 2 2 3 3 3ˆ ˆ ˆ ˆ ˆ ˆM M u u u u u u      ( 25 ) 
Since TM M  is positive semi-definite, its eigenva-
lues are positive and their square root is Real and it is 
possible to write the symmetrical matrix S 
T T T
1 2 31 1 2 2 3 3ˆ ˆ ˆ ˆ ˆ ˆS u u u u u u       ( 26 ) 
As the eigenvectors are orthogonal, it follows that  
2 TS M M . This expression of the S matrix is al-
lowed also when some eigenvectors are null. For this 
reason, the result is positive semi-definite instead of 
positive definite. If all the eigenvectors are positive, S 
becomes  
1 T T T
1 1 2 2 3 3
1 2 3
1 1 1ˆ ˆ ˆ ˆ ˆ ˆS u u u u u u   
  
          ( 27 ) 
It is useful to calculate the U matrix 
 
1
1 T 2U MS M M M

  . It is possible to note that 
the sign of the determinant of U is the same of the de-
terminant of M matrix. In fact  
       1 1det U det MS det M det S    ( 28 ) 
And the  1det S  is positive because its eigenva-
lues are positive. The U matrix is a rotation when 
 det M 0  and it represents a reflection if 
 det M 0 . It is necessary to minimize this expres-
sion    T TTr B M det B US  that, substituting the ex-
pression (26) becomes  
   
   
T T T
1 1
1
T T T T
2 2 3 3
2 3
1 ˆ ˆTr B US Tr B Uu u
1 1ˆ ˆ ˆ ˆTr B Uu u Tr B Uu u
 

 
 
         ( 29 ) 
For each X and Y matrices such that the XY and 
YX products are square, it follows that 
   Tr XY Tr YX and 
   
 
T T T T
i i i i
i i i i
ˆ ˆ ˆ ˆTr B Uu u Tr u B Uu
ˆ ˆ ˆ ˆTr Bu Uu Bu Uu
 
 
 ( 30 ) 
Since  iuˆ  is a unit vector and both U and B are 
ortogonal transformations, it is veri fied that  
1 1ˆ ˆBu Uu 1 . It follows that  
   T 1 2 3Tr B US Tr S                ( 31 ) 
And there is the maximum value of 
 TTr B US when TB U I or B U . The sought or-
tonormal matrix is the one that arises from the de-
composition of M decomposed into the product of an 
orthonormal matrix and symmetric one. When M is 
non-singular matrix, then 
  
1
T 2B M M M

           ( 32 ) 
The presented method is a closed-form solution us-
ing orthonormal matrices and their eigenvalues-
eigenvector decomposition. 
4  Procrustes approach 
The minimization problem known as “Procrustes” is 
the technique of matching one configuration R  into 
another configuration S in order to produce a measure 
of match, by an orthogonal transformation matrix T 
such that the sum of squares of the residual matrix  
E RT S  is minimum: 
 TTr E E min      ( 33 ) 
Expanding the product TE E , the  can be ex-
pressed as function of T: 
   T T T T T2Tr T R S Tr T R RT S S   
 
( 34 )
 
which partial derivative with respect to T is 
   
 
T T T
T T
2Tr M 2Tr T R R
T
Tr MT R R

  

 
 
( 35 )
 
The condition 2RT S min   it is equivalent to 
   T TTr S RT Tr M T max  . Be TUDV  the singu-
lar value decomposition of M where 
1 2 3D diag( , , )    , then 
   
 
T T T T
ii i ii i
Tr T UDV Tr V T UD
Tr ZD z
 
       
( 36 )
 
where T TZ V T U , that has a maximum in Z = I.  
Finally TT UV  is the optimal rotation matrix. 
 
 
 
5  Different Approaches comparisons 
The methods that are presented in this paper are 
widely used in geodesy, photogrammetry, robotics 
and computer graphic. Here we can underline some 
common or different aspects for each approach.  
The presented approaches di ffer in term of rotation 
representation and optimization method, while the op-
timization criteria is always least squares. 
The two quaternion approach investigated are for-
mally equivalent. In fact, it was shown that the two 
matrix A  and Å  defined respectively for the residual  
vector minimization approach and for the scalar prod-
uct maximization one, are related by Å A  . More-
over, it is find that these matrices depend only from 
the data sets: r  and s .  
The orthonormal matrix approach has been studied 
in a closed-form solution performed using orthonor-
mal matrices. This method requires the computation 
of the square root of a symmetric matrix to  solve the 
rotation problem.  
Finally the Procrustes approach is often used in  
photogrammetry, in order to solve the orient ation 
problem. When the weight (both in the data set and 
for each 3D component) in the transformation is in-
troduce we are dealing with the Generalized Pro-
crustes Analysis. It is possible to shown that the used 
orthonormal matrix T is equivalent to the Rodriguez 
matrix. 
The described algorithm have been implemented in  
a FORTRAN90 software and numeri cally veri fied on 
real data examples. 
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A review on non iterative closed form configuration matching and rotations estimation

Orthonormal matrices, Procrustes and quaternion analysis are closed form solutions of the configuration matching problem, common in geodesy as in the datum transformation problem. Literature reports more Procrustes based geodetic applications than Quaternions, which are more used in other application fields, such as aerospace navigation, robotics and computer vision. The large popularity of Procrustes in geodesy is mainly due to its capability to take into account a priori observation weighting in a simple way

DE AGOSTINO, MATTIA

PORPORATO, CHIARA MARIA

ROGGERO, MARCO

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

05 August 2020POLITECNICO DI TORINORepository ISTITUZIONALEA review on non iterative closed form configuration matching and rotations estimation / De Agostino M.; Porporato C.;Roggero M.. - STAMPA. - 137(2012), pp. 347-352. ((Intervento presentato al convegno VII Hotine-Marussi Symposiumon Mathematical Geodesy tenutosi a Roma nel 6-10 giugno 2009.OriginalA review on non iterative closed form configuration matching and  rotations estimationPublisher:PublishedDOI:10.1007/978-3-642-22078-4_52Terms of use:openAccessPublisher copyright(Article begins on next page)This article is made available under terms and conditions as specified in the  corresponding bibliographic description inthe repositoryAvailability:This version is available at: 11583/2372830 since:Springer  A review on non iterative closed form configuration matching and rotations estimation M. De Agostino, C. Porporato, M. Roggero         Abstract. Orthonormal matri ces, Procrustes  and qua-ternion analysis are closed form solutions of the con-figuration matching problem, common in geodesy as in the datum transformation problem. Literature re-ports more Procrustes based geodetic applications than Quaternions, which are more used in other appli-cation fields, such as aerospace navigation, robotics and computer vision. The large popularity of Pro-crustes in geodesy is mainly due to its capability to take into account a priori observation weighting in a simple way. Keywords. Rotations, quaternions, ortonormal ma-trices, Procrustes. 1  Introduction  A rotation is a transformation of the Euclidean space that rigidly moves objects leaving fixed at least one point (the origin of Euclidean space). In geodesy rotations are involved in  many problems, especi ally for solving the transformation between reference frames. More in detail, a rotation is an isometry of an Euclidean space that preserves the orientation, and it is described by an orthogonal matrix. In a Euclidean space of two or three dimensions each orthogonal ma-trix expresses a rotation around a point or an axis, a reflection, or a combination of these two transforma-tions. As mentioned above, the representation of a rot a-tion expresses the orientation of an object with respect  to a reference system, or the relative orientation of two or more reference systems. Euler's theorem shows that any space rotation can be decomposed into the product of the three rotations i j kR R R   , where i j k   and 3i, j, k  , and where iR  indicates a rotation of αradians counterclockwise around the i  axis. According to Euler's theorem, the attitude of a rigid body can be described by a rotation around only one axis. Furthermore, this rotation can be defined un-iquely by a minimum of three parameters, such as the directors cosines matrices, that represent the most widespread method for estimating rotations into geo-detic networks. However, for various reasons, there are several  ways to represent rotations, making use of a number of parameters even higher than three, although also those redundant representations have always only three degrees of freedom. Some of these methods, presented over the years, are hereinafter described and compared.  2  The Quaternion-based approach What Sir William Rowan Hamilton wrote on the 16th of October 1843 on a stone of Brougham Bridge in Dublin, is simply:  2 2 2i j k ijk 1      ( 1 )  M. De Agostino, C. Porporato Department of Land, Environment and Geoengineering Politecnico di Torino, Corso Duca degli Abruzzi 24.          10129 Torino, Italy M. Roggero Department of Human Settlements Science and Technology Po-litecnico di Torino, Viale Mattioli 39.                             10129 Torino, Italy       Fig. 1 The Brougham Bridge in Dublin and the Hamilton  commemorative plaque The combination:  0 1 2 3q q iq jq kq     ( 2 ) where q0, q1, q2 and q3 are real numbers, defines the generic quaternion. Quaternions satisfy all the laws of algebra, except the multiplication commutative law. In fact:   ij ij, jk kj, ki ik       ( 3 ) which represents a serious violation of the commuta-tive law ab ba . It is also apparent:  ij k jk i ki jji k kj i ik j        ( 4 ) The fundamental values of quat ernions, i, j and k, can be handled as three mutually perpendicular clockwise axes in a common three-dimensional Euc-lidean space. Each unit quaternion, in particular, de-fine a rot ation in 3  space. These rotations are given by the conjuge:  1 *r qrq qrq      ( 5 ) It can be veri fied that i f r is purely imaginary (the real part is equal to zero), also r  is purely imaginary; therefore it can be defined an action of the group of unit quaternions on 3 . Each action defined in this way is indeed a rotation, since it preserves the norm:  **r qrq q r q r           ( 6 ) It is possible to show the equivalence between the conjuge and the product of the 3x3 Rodriguez rotation matrix and a real vector. In fact, the operation:       * * T Tqrq Qr q Q Qr Q Q r        ( 7 ) produce a 4x4 rotation matrix TQ Q , whose lower right 3x3 sub-matrix is orthonormal, and it is the rota-tion matrix that takes r  to r .  Given the three-dimensional rotation between two frames r  and s  with a scale-change ρ:   *s qrq     ( 8 ) where *qq 1  , the measurement equation is:  *i i is qr q        ( 9 ) where i  are still purely imaginary quaternions. In the following, we will present two methods in the lite-rature to solve the problem (9), respectively by mini-mizing the residual vector   (Sansò, 1973) or alterna-tively maximizing the scalar product s r   (Horn, 1987). 2.1 Residual vector minimization In accordance with the least squares  approach, we must compute the minimum of the function:     *i iiq,        ( 10 ) where 2 *q qq 1    . Differentiating  with respect to  and q  and introducing a real Lagrange multiplier , we obtain, after some mathematical steps:      2 22i i i ii i is r q 2 s qr 0               ( 11 ) It is possible to show that:  2 22i ii is r      ( 12 ) Using equation (12) in (11), through some mathe-matical steps we reach the final equation:  2i i ii irs q r qAq q                 ( 13 ) As it is possible to see, the unknowns  and q  are re-spectively an eigenvalue and eigenvector of a symme-tric matrix A, that can be built directly from the data ri and si. In particular, using the products expansion rules of quaternions, we find:       i i i i i i i ii i i ii i ii ir s q s qr S q r R S qR S q A q                         ( 14 )   from which it is possible to derive the expressions of iA  and A. 2.2 Scalar product maximization In this second approach, we seek the quaternion q  to maximize the scalar product:   n n*i i i ii 1 i 1qrq s r s           ( 15 ) Reminding the geometric meaning of the scal ar product of vectors, we have:   i i i ir s r s cos        ( 16 ) where θ is the angle subtended. Since i ir r    and is  are constants, the maximization of the scalar product  is equal to minimize the parameter θ (or maximize cos ). Using the above results, we can rewrite the scalar product as:     ni ii 1qr s q      ( 17 ) The products  iqr   and  is q   can be expressed by means of the matrices R and S, therefore:     n nT Ti i i ii 1 i 1nT Ti ii 1nT Tii 1qr s q q R S qq R S qq Å q q Åq                      ( 18 ) It is now simple to derive the values of the sub-matrices iÅ , and consequently of the matrix Å , where Ti iÅ Å  and TÅ Å . Recalling that we were seeking:   nTi iq qi 1Max r s Max q Åq          ( 19 ) It is possible to note that Å A  . 3  The Orthonormal matrices approach Among the existing ways to represent rotation we present one that is most often used in photogramme-try: the orthonormal matrices. Again r is the position vector in  the original RS and s  the position vector in  the final RS. The aim is to find the rotation that mi-nimizes the residual errors. Therefore, we have to find the orthonormal matrix B, 3×3 matrix,  that maximize    n nTi i i ii 1 i 1s Br s Br    ( 20 ) Being    T T Ta Bb TR B ab  ( 21 ) it is possible to write the (20) as   nT T Ti ii 1Tr B s r Tr B M     ( 22 ) where nTi ii 1M s r  and   1 1 1 2 1 3i i i i i ii i i2 1 2 2 2 3i i i i i ii i i3 1 3 2 3 3i i i i i ii i is r s r s rM s r s r s rs r s r s r                ( 23 ) It follows that the rotation that minimize the resi-dual errors corresponds to the orthonormal matrix B that maximizes  TTr B M . A square matrix M could always be decomposed into the product of an orthonormal matrix U and a positive semi-definite matrix S. When M is non singu-lar, the matrices U and S are univocally determined and it allows to write    1T 21T 2MM M M USM M   ( 24 ) In this expression,  1T 2U M M M  is an or-thonormal matrix and  1T 2S M M  is the square root positive semi-definite of the symmetric matrix TM M .   It is possible to write this matrix TM M  using its ei-genvalues  i  and eigenvectors iuˆ  as following: T T T T1 1 1 2 2 2 3 3 3ˆ ˆ ˆ ˆ ˆ ˆM M u u u u u u      ( 25 ) Since TM M  is positive semi-definite, its eigenva-lues are positive and their square root is Real and it is possible to write the symmetrical matrix S T T T1 2 31 1 2 2 3 3ˆ ˆ ˆ ˆ ˆ ˆS u u u u u u       ( 26 ) As the eigenvectors are orthogonal, it follows that  2 TS M M . This expression of the S matrix is al-lowed also when some eigenvectors are null. For this reason, the result is positive semi-definite instead of positive definite. If all the eigenvectors are positive, S becomes  1 T T T1 1 2 2 3 31 2 31 1 1ˆ ˆ ˆ ˆ ˆ ˆS u u u u u u               ( 27 ) It is useful to calculate the U matrix  11 T 2U MS M M M  . It is possible to note that the sign of the determinant of U is the same of the de-terminant of M matrix. In fact         1 1det U det MS det M det S    ( 28 ) And the  1det S  is positive because its eigenva-lues are positive. The U matrix is a rotation when  det M 0  and it represents a reflection if  det M 0 . It is necessary to minimize this expres-sion    T TTr B M det B US  that, substituting the ex-pression (26) becomes        T T T1 11T T T T2 2 3 32 31 ˆ ˆTr B US Tr B Uu u1 1ˆ ˆ ˆ ˆTr B Uu u Tr B Uu u            ( 29 ) For each X and Y matrices such that the XY and YX products are square, it follows that    Tr XY Tr YX and     T T T Ti i i ii i i iˆ ˆ ˆ ˆTr B Uu u Tr u B Uuˆ ˆ ˆ ˆTr Bu Uu Bu Uu   ( 30 ) Since  iuˆ  is a unit vector and both U and B are ortogonal transformations, it is veri fied that  1 1ˆ ˆBu Uu 1 . It follows that     T 1 2 3Tr B US Tr S                ( 31 ) And there is the maximum value of  TTr B US when TB U I or B U . The sought or-tonormal matrix is the one that arises from the de-composition of M decomposed into the product of an orthonormal matrix and symmetric one. When M is non-singular matrix, then   1T 2B M M M           ( 32 ) The presented method is a closed-form solution us-ing orthonormal matrices and their eigenvalues-eigenvector decomposition. 4  Procrustes approach The minimization problem known as “Procrustes” is the technique of matching one configuration R  into another configuration S in order to produce a measure of match, by an orthogonal transformation matrix T such that the sum of squares of the residual matrix  E RT S  is minimum:  TTr E E min      ( 33 ) Expanding the product TE E , the  can be ex-pressed as function of T:    T T T T T2Tr T R S Tr T R RT S S    ( 34 ) which partial derivative with respect to T is     T T TT T2Tr M 2Tr T R RTTr MT R R    ( 35 ) The condition 2RT S min   it is equivalent to    T TTr S RT Tr M T max  . Be TUDV  the singu-lar value decomposition of M where 1 2 3D diag( , , )    , then     T T T Tii i ii iTr T UDV Tr V T UDTr ZD z        ( 36 ) where T TZ V T U , that has a maximum in Z = I.  Finally TT UV  is the optimal rotation matrix.    5  Different Approaches comparisons The methods that are presented in this paper are widely used in geodesy, photogrammetry, robotics and computer graphic. Here we can underline some common or different aspects for each approach.  The presented approaches di ffer in term of rotation representation and optimization method, while the op-timization criteria is always least squares. The two quaternion approach investigated are for-mally equivalent. In fact, it was shown that the two matrix A  and Å  defined respectively for the residual  vector minimization approach and for the scalar prod-uct maximization one, are related by Å A  . More-over, it is find that these matrices depend only from the data sets: r  and s .  The orthonormal matrix approach has been studied in a closed-form solution performed using orthonor-mal matrices. This method requires the computation of the square root of a symmetric matrix to  solve the rotation problem.  Finally the Procrustes approach is often used in  photogrammetry, in order to solve the orient ation problem. When the weight (both in the data set and for each 3D component) in the transformation is in-troduce we are dealing with the Generalized Pro-crustes Analysis. It is possible to shown that the used orthonormal matrix T is equivalent to the Rodriguez matrix. The described algorithm have been implemented in  a FORTRAN90 software and numeri cally veri fied on real data examples. References Akca D. (2003). Full automatic registration of laser scanner point clouds. ETH, Swiss Federal Institute of Technology Zurich , Institute of Geodesy and Photogrammetry. Akca D. (2003). Generalized Procrustes Analysis and its appli-cations in Photogrammetry. 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A review on non iterative closed form configuration matching and  rotations estimation

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A review on non iterative closed form configuration matching and rotations estimation

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