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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