theoretical foundations of itrf determination the algebraic and the kinematic approach the vii...
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Theoretical foundations of ITRF determination The algebraic and the kinematic approach
The VII Hotine-Marussi SymposiumRome, July 6–10, 2009
Zuheir Altamimi1 & Athanasios Dermanis2
(1) IGN-LAREG - (2) Aristotle University of Thessaloniki
THE ITRF FORMULATION PROBLEM
given a time sequence of sub-network coordinates
(one from each technique T = VLBI, SLR, GPS, DORIS)combine them into coordinates for the whole network
obeying a time-evolution model
Essentially: Determine the model parametersfor each network point i
t
( ) ( , )i it tx F a
, ( )i T ktx
( )i tx
ia
THE ITRF FORMULATION PROBLEM
t
( ) ( , )i it tx F a
, ( )i T ktx
( )i tx
ia
0 0( )i it t x v
0( , )i i x v
given a time sequence of sub-network coordinates
(one from each technique T = VLBI, SLR, GPS, DORIS)combine them into coordinates for the whole network
obeying a time-evolution model
Essentially: Determine the model parametersfor each network point i
THE COORDINATE-FREE APPROACH (ALMOST) TO THEITRF FORMULATION PROBLEM
Given a time sequence of sub-network shapes(one from each technique: VLBI, SLR, GPS, DORIS)
t
THE COORDINATE-FREE APPROACH (ALMOST) TO THEITRF FORMULATION PROBLEM
Replace them with a smooth sequence of shapes
t
THE COORDINATE-FREE APPROACH (ALMOST) TO THEITRF FORMULATION PROBLEM
Replace them with a smooth sequence of shapes
t
THE COORDINATE-FREE APPROACH (ALMOST) TO THEITRF FORMULATION PROBLEM
Note that although shape variation is insignificantcoordinates may vary significantly due to temporal instabilityin reference system maintenance
t
THE COORDINATE-FREE APPROACH (ALMOST) TO THEITRF FORMULATION PROBLEM
To remove coordinate variation assign a different reference system at each epoch
t
THE COORDINATE-FREE APPROACH (ALMOST) TO THEITRF FORMULATION PROBLEM
To remove coordinate variation assign a different reference system at each epoch
t
THE COORDINATE-FREE APPROACH (ALMOST) TO THEITRF FORMULATION PROBLEM
To remove coordinate variation assign a different reference system at each epochsuch that when networks are viewed in the “same” system
t
THE COORDINATE-FREE APPROACH (ALMOST) TO THEITRF FORMULATION PROBLEM
To remove coordinate variation assign a different reference system at each epochsuch that when networks are viewed in the “same” systemcoordinates vary in a smooth way
t
( ) ( , )i it tx F a
THE COORDINATE-FREE APPROACH (ALMOST) TO THEITRF FORMULATION PROBLEM
To remove coordinate variation assign a different reference system at each epochsuch that when networks are viewed in the “same” systemcoordinates vary in a smooth wayin conformance with a coordinate time-variation model
t
THE COORDINATE-FREE APPROACH (ALMOST) TO THEITRF FORMULATION PROBLEM
00( ) ( , ) ( )i i ii t t t t xx a vFcurrently:
To remove coordinate variation assign a different reference system at each epochsuch that when networks are viewed in the “same” systemcoordinates vary in a smooth wayin conformance with a coordinate time-variation model
t
STACKING FOR EACH PARTICULAR TECHNIQUE
t
t
t
data:
coordinatetransformationparameters:
modelparameters:
0 0( ) ( )i i it t t x x v
, ( )i T ktx
0 ,i ix v
( )T ktθ
( )T ktd
( )T ks t
coordinate variationmodel:
THE ITRF FORMULATION PROBLEM= SIMULTANEOUS STACKING FOR ALL TECHNIQUES
t
SLRVLBI
DORISGPS
ITRF
THE ITRF FORMULATION PROBLEMIN AN OPERATIONALLY CONVENIENT COMPROMISE
Separation into 2 steps: (1) Separate stackings one for each technique:
Provides initial coordinates and velocitiesfor the subnetwork of each technique
0 , ,, ,
VLBI,SLR,GPS,DORISi T i T
T
x v
THE ITRF FORMULATION PROBLEMIN AN OPERATIONALLY CONVENIENT COMPROMISE
Separation into 2 steps: (1) Separate stackings one for each technique:
Provides initial coordinates and velocitiesfor the subnetwork of each technique
(2) Combination of initial coordinates and velocities:
Provides initial coordinates and velocitiesfor the whole ITRF network
0 , ,, ,
VLBI,SLR,GPS,DORISi T i T
T
x v
0 ,i ix v
The (general) model:
( ) ( , )i it tx F aPoint Pi coordinates:
ai = point Pi parameters
The current model:
0 0 0( ) ( , , ) ( )i i i i it t t t x F x v x v
0ii
i
xa
v
THE MODEL FOR TIME EVOLUTION OF COORDINATES
SMOOTHINGReplaces observes time sequences of sub-network shapeswith a single smooth time sequencefor the whole ITRF network
INTERPOLATIONProvides shapes expressed by coordinates for epochs other than observation ones
IMPOSES THE USE OF A REFERENCE SYSTEMso that network shapes are represented by coordinates
WHAT THE MODEL DOES
WHAT THE MODEL DOES NOT DO
SMOOTHINGReplaces observes time sequences of sub-network shapeswith a single smooth time sequencefor the whole ITRF network
MAIN ITRF FORMULATIONPROBLEM
Assign a reference systemfor each epoch
INTERPOLATIONProvides shapes expressed by coordinates for epochs other than observation ones
IMPOSES THE USE OF A REFERENCE SYSTEMso that network shapes are represented by coordinates
It does not resolve theproblem of the choiceof the reference system
WHAT THE MODEL DOES NOT DO
MAIN ITRF FORMULATIONPROBLEM
Assign a reference systemfor each epoch
It does not resolve theproblem of the choiceof the reference system
WHAT THE MODEL DOES NOT DO
MAIN ITRF FORMULATIONPROBLEM
Assign a reference systemfor each epoch
It does not resolve theproblem of the choiceof the reference system
PROBLEM SOLUTION: Introduce additional minimal constraints in theLeast-Square data analysis problem
Minimal constraints: At any epoch t they determine the reference systemwithout affecting the optimal network shapeuniquely determined by the least-squares principlefor the determination of ITRF parameters
WHAT THE MODEL DOES NOT DO
MAIN ITRF FORMULATIONPROBLEM
Assign a reference systemfor each epoch
It does not resolve theproblem of the choiceof the reference system
PROBLEM SOLUTION: Introduce additional minimal constraints in theLeast-Square data analysis problem
Minimal constraints: At any epoch t they determine the reference systemwithout affecting the optimal network shapeuniquely determined by the least-squares principlefor the determination of ITRF parameters
How to choose the minimal inner constraints?
2 approaches: (1) The algebraic approach (classical Meissl inner constraints)(2) The kinematic approach (new!)
THE ALGEBRAIC APPROACH
Formulation of Least Squares problem
b Ax v minT v Pv
with infinite solutions for different choices of reference system
THE KINEMATIC APPROACH
THE ALGEBRAIC APPROACH
Formulation of Least Squares problem
b Ax v minT v Pv
with infinite solutions for different choices of reference system
minT T x x E x 0
or partial inner constraints:
1 1 1minT T x x E 0 x 0
Choice of unique solutionby inner constraints:
THE KINEMATIC APPROACH
THE ALGEBRAIC APPROACH THE KINEMATIC APPROACH
Formulation of Least Squares problem
b Ax v minT v Pv
with infinite solutions for different choices of reference system
minT T x x E x 0
or partial inner constraints:
1 1 1minT T x x E 0 x 0
Choice of unique solutionby inner constraints:
Choice of reference system by minimization of apparent variationof coordinate for network points
THE ALGEBRAIC APPROACH
Discrete Tisserand Reference System
THE KINEMATIC APPROACH
Formulation of Least Squares problem
b Ax v minT v Pv
with infinite solutions for different choices of reference system
minT T x x E x 0
or partial inner constraints:
1 1 1minT T x x E 0 x 0
Choice of unique solutionby inner constraints:
Choice of reference system by minimization of apparent variationof coordinate for network points
(3) constant mean quadratic scale
Measures of coordinate variation:
(1) Minimum relative kinetic energy == vanishing relative angular momentum
(2) constant network barycenter
( )
( ) ( )
( )
t
t t
t
ψ
p g
Inner constraints determined from the linear variation of unknown parameters xwhen coordinate system changes with small transformation parameters p
THE ALGEBRAIC APPROACH – INNER CONSTRAINTS
x xrotation angles
translation vector
scale parameter
( )
( ) ( )
( )
t
t t
t
ψ
p g
Inner constraints determined from the linear variation of unknown parameters xwhen coordinate system changes with small transformation parameters p
x x Ep
THE ALGEBRAIC APPROACH – INNER CONSTRAINTS
x x
Determine the parameter variation equations
rotation angles
translation vector
scale parameter
( )
( ) ( )
( )
t
t t
t
ψ
p g
Inner constraints determined from the linear variation of unknown parameters xwhen coordinate system changes with small transformation parameters p
T E x 0
Then the (total) inner constraints are
x x Ep
THE ALGEBRAIC APPROACH – INNER CONSTRAINTS
x x
Determine the parameter variation equations
rotation angles
translation vector
scale parameter
MODEL PRESERVING APPROXIMATE TRANSFORMATIONS OF INITIAL COORDINATES AND VELOCITIES
( ) ( ) ( ) ( ) [ ( ) ] ( ) ( )t t t t t t t x x x x ψ g0 0( ) ( ) [ ] ( ) ( )t t t t v v x x ψ g
Transformation of coordinates in first order approximation
MODEL PRESERVING APPROXIMATE TRANSFORMATIONS OF INITIAL COORDINATES AND VELOCITIES
( ) ( ) ( ) ( ) [ ( ) ] ( ) ( )t t t t t t t x x x x ψ g0 0( ) ( ) [ ] ( ) ( )t t t t v v x x ψ g
0 0( ) ( )t t t ψ ψ ψ
0 0( ) ( )t t t
0 0( ) ( )t t t g g g
Model preserving transformations
Transformation of coordinates in first order approximation
0 0( ) ( )t t t x x v 0 0( ) ( )t t t x x v
MODEL PRESERVING APPROXIMATE TRANSFORMATIONS OF INITIAL COORDINATES AND VELOCITIES
Transformation of model parameters
( ) ( ) ( ) ( ) [ ( ) ] ( ) ( )t t t t t t t x x x x ψ g0 0( ) ( ) [ ] ( ) ( )t t t t v v x x ψ g
0 0( ) ( )t t t ψ ψ ψ
0 0( ) ( )t t t
0 0( ) ( )t t t g g g
Model preserving transformations
Transformation of coordinates in first order approximation
0 0( ) ( )t t t x x v 0 0( ) ( )t t t x x v
0 0 0 0 0 0 0[ ] x x x x ψ g0 0 0[ ] v v x v x ψ g
THE ALGEBRAIC APPROACH – INNER CONSTRAINTS PER STATION
T
i ii
T E aE x 0The (total) inner constraints are
i i i a a E p
For each station Pi
determine the parameter variation equations
The inner constraints per station are
Ti i E a 0
MODEL PRESERVING APPROXIMATE TRANSFORMATIONS OF INITIAL COORDINATES AND VELOCITIES
0
0 0
0
( ) ( )t t t
ψ ψ
p g g
The use of model preserving transformations
( )
( ) ( )
( )
t
t t
t
ψ
p ginstead of arbitrary transformations
leads to a sub-optimal solution:
No matter what the optimality criterion, there exist an arbitrary transformationleading to a better solution
which does not conform with the chosen model
( )i tx
0 0( ) ( )i i it t t x x v
( )tp
Strict optimality leads the solution OUTSIDE the adopted model !
ap ap0, 0, 0 0, 0, 0 0[ ]i i i i x x x x ψ g
ap ap0, 0,[ ]i i i i v v x x ψ g
MODEL PRESERVING APPROXIMATE TRANSFORMATIONS OF INITIAL COORDINATES AND VELOCITIES
Transformation of model parameters
0, 0, 0 0, 0, 0 0[ ]i i i i x x x x ψ g0, 0 0,[ ]i i i i i v v x v x ψ g
in terms of corrections to approximate values
ap0, 0, 0,i i i x x x ap
i i i v v v
Transformation of corrections to model parameters
Transformation of model parameters
in terms of corrections to approximate values
ap0, 0, 0,i i i x x x ap
i i i v v v
Transformation of corrections to model parameters
MODEL PRESERVING APPROXIMATE TRANSFORMATIONS OF INITIAL COORDINATES AND VELOCITIES
0, 0, 0 0, 0, 0 0[ ]i i i i x x x x ψ g0, 0 0,[ ]i i i i i v v x v x ψ g
ap ap0, 0,
ap ap0,
0,
0
0,
, 00
0
[ ]
[ ] i
i ii ii i
i ii i
a
x I x 0 0 0E
0 0 0 x I x
ψ
g
pψ
g
x xa a
v v
Transformation of model parameters
in terms of corrections to approximate values
ap0, 0, 0,i i i x x x ap
i i i v v v
Transformation of corrections to model parameters
MODEL PRESERVING APPROXIMATE TRANSFORMATIONS OF INITIAL COORDINATES AND VELOCITIES
0, 0, 0 0, 0, 0 0[ ]i i i i x x x x ψ g0, 0 0,[ ]i i i i i v v x v x ψ g
ap ap0, 0,
ap ap0,
0,
0
0,
, 00
0
[ ]
[ ] i
i ii ii i
i ii i
a
x I x 0 0 0E
0 0 0 x I x
ψ
g
pψ
g
x xa a
v v
inner constraints sub-matrix
THE STACKING PROBLEM
0 0( ) (1 ) ( )[ ( ) ]k ki i k k k i k i k it s t t x x R θ x v d e
Transformation parametersfrom ITRF system to
technique-system at epoch tk
Observed coordinatesin particular technique
at epoch tk
ITRF model coordinates at epoch tk
GIVEN SOUGHT
NUISANCE
Original observation model
THE STACKING PROBLEM
0 0( ) (1 ) ( )[ ( ) ]k ki i k k k i k i k it s t t x x R θ x v d e
0 0 0 0( ) [ ]k ki i k i k i i k k it t s x x v x x θ d e
ap ap0 0 0 0( ) [ ]k k
i i k i k i i k k it t s x x v x x θ d e
In first order approximation
In terms of corrections to approximate values
Transformation parametersfrom ITRF system to
technique-system at epoch tk
Observed coordinatesin particular technique
at epoch tk
ITRF model coordinates at epoch tk
GIVEN SOUGHT
Original observation model
NUISANCE
INNER CONSTRAINTS FOR THE STACKING PROBLEM
0 ap ap0 0 0( ) [ ]
kik k
i k i i k ii
k
t t
s
θx
x I I x I x d ev
0ii
i
xa
v
k
k k
ks
θ
z d
0
i
ii i
i
a
xa a E p
v
i
k
k k k
ks
z
θ
d z z E p
ap ap0 0
ap ap0 0
[ ]
[ ]i
i i
i i
a
x I x 0 0 0E
0 0 0 x I x
0
0
0
( )
( )
0 0 1 0 0 ( )k
k
k
k
t t
t t
t t
z
I 0 0 I 0 0
E 0 I 0 0 I 0
0 0 0~ ( , , , , , ) p ψ g ψ g
Change of ITRF reference system
INNER CONSTRAINTS FOR THE STACKING PROBLEM
ap0 0
1 1
01 1
ap0 0
1 1
1 1 ap0 0
1 1
01 1
ap0 0
1 1
[ ]
( )
[ ] ( )
( )
( ) ( )
i k
N M
i i ki k
N M
i ki k
N MT
i i kN Mi kT T
i k N Mi k
i i k ki k
N M
i k ki k
N MT
i i k ki k
s
t t
t t
t t s
a z
x x θ
x d
x x
E a E z
x v θ
v d
x v
0
(Total) inner constraints
initial orientation
initial translation
initial scale
orientation rate
translation rate
scale rate
INNER CONSTRAINTS FOR THE STACKING PROBLEM
ap0 0
1
01
ap0 0
1
1 ap0
1
1
ap0
1
[ ]
( )
[ ]
( )
i
N
i ii
N
ii
NT
i iNiT
i Ni
i ii
N
ii
NT
i ii
a
x x
x
x x
E a 0
x v
v
x v
Partial inner constraints – Coordinates & velocities
initial orientation
initial translation
initial scale
orientation rate
translation rate
scale rate
INNER CONSTRAINTS FOR THE STACKING PROBLEM
1
1
1
10
1
01
01
( )
( )
( )
k
M
kk
M
kk
M
kMkT
k Mk
k kk
M
k kk
M
k kk
s
t t
t t
t t s
z
θ
d
E z 0
θ
d
Partial inner constraints – Transformation parameters
initial orientation
initial translation
initial scale
orientation rate
translation rate
scale rate
THE COMBINATION PROBLEM
ap ap0 0[ ]
TiTTi Ti Ti is vx xv dv θ e
00ap ap0 0 00 0 0[ ]
T iT T Tiii iT s xx dx x eθx
Transformation parametersfrom ITRF system to
technique (stacking) system
Initial coordinatesand velocities fromeach technique T
UnknownITRF initial coordinates
and velocities
GIVEN SOUGHT NUISANCE
Observation model
INNER CONSTRAINTS FOR THE COMBINATION PROBLEM
0ii
i
xa
v0
i
ii i
i
a
xa a E p
v
0
0
0
T
T
T
T
T T
T
T
T
s
s
z
θ
d
z z E pθ
d
0 0 0~ ( , , , , , ) p ψ g ψ g
Change of ITRF reference system
ap ap0 0[ ]
TiTi i i T T i Ts vv v x θ x d e
0
ap ap0 0 0 0 0 0 0[ ]
T iT i i i T T i Ts xx x x θ x d e
0
0
0
T
T
TT
T
T
T
s
s
θ
d
zθ
d
ap ap0 0
ap ap0 0
[ ]
[ ]i
i i
i i
a
x I x 0 0 0E
0 0 0 x I x TzE I
INNER CONSTRAINTS FOR THE COMBINATION PROBLEM
ap0 0 0
1 1
0 01 1
ap0 0 0
1 1
1 1 ap0
1 1
1 1
ap0
1 1
[ ]
( )
[ ]
( )
i T
N K
i i Ti T
N K
i Ti T
N KT
i i TN Ki TT T
i T N Ki T
i i Ti T
N K
i Ti T
N KT
i i Ti T
s
s
a z
x x θ
x d
x x
E a E z
x v θ
v d
x v
0
(Total) inner constraints
initial orientation
initial translation
initial scale
orientation rate
translation rate
scale rate
INNER CONSTRAINTS FOR THE COMBINATION PROBLEM
ap0 0
1
01
ap0 0
1
1 ap0
1
1
ap0
1
[ ]
( )
[ ]
( )
i
N
i ii
N
ii
NT
i iNiT
i Ni
i ii
N
ii
NT
i ii
a
x x
x
x x
E a 0
x v
v
x v
initial orientation
initial translation
initial scale
orientation rate
translation rate
scale rate
Partial inner constraints – Coordinates & velocities
INNER CONSTRAINTS FOR THE COMBINATION PROBLEM
ap0 0
1
01
ap0 0
1
1 ap0
1
1
ap0
1
[ ]
( )
[ ]
( )
i
N
i ii
N
ii
NT
i iNiT
i Ni
i ii
N
ii
NT
i ii
a
x x
x
x x
E a 0
x v
v
x v
initial orientation
initial translation
initial scale
orientation rate
translation rate
scale rate
Partial inner constraints – Coordinates & velocities
Same as for the stacking problem !
INNER CONSTRAINTS FOR THE COMBINATION PROBLEM
01
01
01
1
1
1
1
T
K
TT
K
TT
K
TKTT
T KT
TT
K
TT
K
TT
s
s
z
θ
d
E z 0
θ
d
initial orientation
initial translation
initial scale
orientation rate
translation rate
scale rate
Partial inner constraints – Transformation parameters
THE KINEMATIC APPROACH
Translation
Orientation
Scale
Establish a reference system in such a way that the apparent motion of network points(variation of their coordinates) is minimized with respect to:
THE KINEMATIC APPROACH
Establish a reference system in such a way that the apparent motion of network points(variation of their coordinates) is minimized with respect to:
Translation: The network barycenter does not move
1
1( ) ( ) const,
n
B ii
t t tn
x x
Scale
Orientation
THE KINEMATIC APPROACH
Establish a reference system in such a way that the apparent motion of network points(variation of their coordinates) is minimized with respect to:
Translation: The network barycenter does not move
Orientation: The relative kinematic energy is minimized = = the relative angular momentum vanishes
1
1( ) ( ) const,
n
B ii
t t tn
x x
1 1
( ) ( ) ( ) min, ( ) [ ]( ) ( ) ,Tn n
i i iR R i
i i
d d dT t t t t t t t t
dt dt dt
x x x
h x 0
Scale
THE KINEMATIC APPROACH
Establish a reference system in such a way that the apparent motion of network points(variation of their coordinates) is minimized with respect to:
Translation: The network barycenter does not move
Orientation: The relative kinematic energy is minimized = = the relative angular momentum vanishes
Scale: The network mean quadratic scale remains constant
1
1( ) ( ) const,
n
B ii
t t tn
x x
2
1 1
( ) [ ( ) ( )] [ ( ) ( )] const,n n
TiB i B i B
i i
S t S t t t t t
x x x x
1 1
( ) ( ) ( ) min, ( ) [ ]( ) ( ) ,Tn n
i i iR R i
i i
d d dT t t t t t t t t
dt dt dt
x x x
h x 0
MINIMAL CONSTRAINTS IN THE KINEMATIC APPROACH
Initial translation:
Initial orientation:
Initial scale:
ap ap0 0 0
1 1
1 1n n
i ii in n
x x x
Translation rate:
Orientation rate:
Scale rate:
ap ap
1 1
1 1n n
i ii in n
v v v
ap ap ap ap ap0 0 0
1 1 1
[ ] [ ] [ ]n n n
i i i i R i ii i i
v x x v h x v
ap ap0 0 0 0
1 1
( ) ( )n n
T Ti i i
i i
x x x x 0 ap ap ap ap ap ap0 0 0 0
1 1
( ) ( ) ( )n n
T T Ti i i i
i i
n
x x v x v x v
NOT available (to be borrowed from the algebraic approach)
MINIMAL CONSTRAINTS IN THE KINEMATIC APPROACH
ap ap0 0 0
1 1
1 1n n
i ii in n
x x x
Translation rate:
Orientation rate:
Scale rate:
ap ap
1 1
1 1n n
i ii in n
v v v
ap ap ap ap ap0 0 0
1 1 1
[ ] [ ] [ ]n n n
i i i i R i ii i i
v x x v h x v
ap ap0 0 0 0
1 1
( ) ( )n n
T Ti i i
i i
x x x x 0 ap ap ap ap ap ap0 0 0 0
1 1
( ) ( ) ( )n n
T T Ti i i i
i i
n
x x v x v x v
Under the choiceap ap0 , ,i i x 0 v 0
Initial translation:
Initial orientation:
Initial scale:
NOT available (to be borrowed from the algebraic approach)
MINIMAL CONSTRAINTS IN THE KINEMATIC APPROACH
01
n
ii
x 0
Translation rate:
Orientation rate:
Scale rate:
ap ap
1 1
1 1n n
i ii in n
v v v
ap ap ap ap ap0 0 0
1 1 1
[ ] [ ] [ ]n n n
i i i i R i ii i i
v x x v h x v
ap ap0 0 0 0
1 1
( ) ( )n n
T Ti i i
i i
x x x x 0 ap ap ap ap ap ap0 0 0 0
1 1
( ) ( ) ( )n n
T T Ti i i i
i i
n
x x v x v x v
Under the choiceap ap0 , ,i i x 0 v 0
Initial translation:
Initial orientation:
Initial scale:
NOT available (to be borrowed from the algebraic approach)
MINIMAL CONSTRAINTS IN THE KINEMATIC APPROACH
01
n
ii
x 0
Translation rate:
Orientation rate:
Scale rate:
1
n
ii
v 0
ap ap ap ap ap0 0 0
1 1 1
[ ] [ ] [ ]n n n
i i i i R i ii i i
v x x v h x v
ap ap0 0 0 0
1 1
( ) ( )n n
T Ti i i
i i
x x x x 0 ap ap ap ap ap ap0 0 0 0
1 1
( ) ( ) ( )n n
T T Ti i i i
i i
n
x x v x v x v
Under the choiceap ap0 , ,i i x 0 v 0
Initial translation:
Initial orientation:
Initial scale:
NOT available (to be borrowed from the algebraic approach)
MINIMAL CONSTRAINTS IN THE KINEMATIC APPROACH
01
n
ii
x 0
Translation rate:
Orientation rate:
Scale rate:
1
n
ii
v 0
ap0
1
[ ]n
i ii
x v 0
ap ap0 0 0 0
1 1
( ) ( )n n
T Ti i i
i i
x x x x 0 ap ap ap ap ap ap0 0 0 0
1 1
( ) ( ) ( )n n
T T Ti i i i
i i
n
x x v x v x v
Under the choiceap ap0 , ,i i x 0 v 0
Initial translation:
Initial orientation:
Initial scale:
NOT available (to be borrowed from the algebraic approach)
MINIMAL CONSTRAINTS IN THE KINEMATIC APPROACH
01
n
ii
x 0
Translation rate:
Orientation rate:
Scale rate:
1
n
ii
v 0
NOT available (to be borrowed from the algebraic approach)
ap0
1
[ ]n
i ii
x v 0
ap0 0
1
( )n
Ti i
i
x x 0 ap ap ap ap ap ap0 0 0 0
1 1
( ) ( ) ( )n n
T T Ti i i i
i i
n
x x v x v x v
Under the choiceap ap0 , ,i i x 0 v 0
Initial translation:
Initial orientation:
Initial scale:
MINIMAL CONSTRAINTS IN THE KINEMATIC APPROACH
01
n
ii
x 0
Translation rate:
Orientation rate:
Scale rate:
1
n
ii
v 0
ap0
1
[ ]n
i ii
x v 0
ap0 0
1
( )n
Ti i
i
x x 0 ap0
1
( )n
Ti i
i
x v 0
Under the choiceap ap0 , ,i i x 0 v 0
Initial translation:
Initial orientation:
Initial scale:
NOT available (to be borrowed from the algebraic approach)
MINIMAL CONSTRAINTS IN THE KINEMATIC APPROACH
01
n
ii
x 0
Translation rate:
Orientation rate:
Scale rate:
1
n
ii
v 0
ap0
1
[ ]n
i ii
x v 0
ap0 0
1
( )n
Ti i
i
x x 0 ap0
1
( )n
Ti i
i
x v 0
Under the choiceap ap0 , ,i i x 0 v 0
Same as the partial inner constraints of the algebraic approach !
Initial translation:
Initial orientation:
Initial scale:
NOT available (to be borrowed from the algebraic approach)
SUMMARY AND CONCLUSIONS
00( ) ( , ) ( )i i ii t t t t xx a vF
MODEL for smooth shape variation(removal of data noise)
OPTIMALITY CRITERION Best reference system amongall equivalent ones connected by arbitrary transformations
( ) 1 ( ) ( ) ( ) ( )t t t t t x R θ x d
INCONCISTENT
SUMMARY AND CONCLUSIONS
00( ) ( , ) ( )i i ii t t t t xx a vF
MODEL for smooth shape variation(removal of data noise)
OPTIMALITY CRITERION Best reference system amongall equivalent ones connected by arbitrary transformations
( ) 1 ( ) ( ) ( ) ( )t t t t t x R θ x d
INCONCISTENT
SUMMARY AND CONCLUSIONS
00( ) ( , ) ( )i i ii t t t t xx a vF
MODEL for smooth shape variation(removal of data noise)
OPTIMALITY CRITERION Best reference system amongall equivalent ones connected by approximate transformations
( ) ( ) [ ( ) ] ( ) ( ) ( ) ( )t t t t t t t x x x θ x d
INCONCISTENT
SUMMARY AND CONCLUSIONS
00( ) ( , ) ( )i i ii t t t t xx a vF
MODEL for smooth shape variation(removal of data noise)
OPTIMALITY CRITERION Best reference system amongall equivalent ones connected by approximate transformations
( ) ( ) [ ( ) ] ( ) ( ) ( ) ( )t t t t t t t x x x θ x d
CONCISTENT
which preserve the model
0 0( ) ( )t t t θ θ θ
0 0( ) ( )t t t d d d
0 0( ) ( )t t t
SUMMARY AND CONCLUSIONS
00( ) ( , ) ( )i i ii t t t t xx a vF
MODEL for smooth shape variation(removal of data noise)
OPTIMALITY CRITERION Best reference system amongall equivalent ones connected by approximate transformations
( ) ( ) [ ( ) ] ( ) ( ) ( ) ( )t t t t t t t x x x θ x d
CONCISTENT
which preserve the model
0 0( ) ( )t t t θ θ θ
0 0( ) ( )t t t d d d
0 0( ) ( )t t t
SUB-OPTIMALITY
SUMMARY AND CONCLUSIONS
SUB-OPTIMAL REFERENCE SYSTEM (close to the identity, model preserving transformations)
BY USING MINIMAL CONSTRAINTS
SUMMARY AND CONCLUSIONS
SUB-OPTIMAL REFERENCE SYSTEM (close to the identity, model preserving transformations)
BY USING MINIMAL CONSTRAINTS
ALGEBRAIC APPROACH Minimization of parameter
sum of squares
PARTIAL INNER CONSTRAINTS
KINEMATIC APPROACH Minimization of apparent
coordinate variation
MINIMAL CONSTRAINTS
SUMMARY AND CONCLUSIONS
SUB-OPTIMAL REFERENCE SYSTEM (close to the identity, model preserving transformations)
BY USING MINIMAL CONSTRAINTS
ALGEBRAIC APPROACH Minimization of parameter
sum of squares
PARTIAL INNER CONSTRAINTS
IDENTICAL RESULTSunder proper choice of approximate values
ap ap ap0 0
1, ,i i i i
N x x 0 v 0
KINEMATIC APPROACH Minimization of apparent
coordinate variation
MINIMAL CONSTRAINTS
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