Evolution-based least-squares fitting using Pythagorean hodograph spline curves Speaker: Ying.Liu November 29. 2007

Evolution-based least-squares fitEvolution-based least-squares fitting using Pythagorean hodograting using Pythagorean hodogra

ph spline curvesph spline curves

Speaker: Ying .LiuSpeaker: Ying .Liu

November 29. 2007November 29. 2007

Institute of Applied Geometry, JphaInstitute of Applied Geometry, Jphannes Kepler University ,Linz, Austirnnes Kepler University ,Linz, Austir

aawww.ag.jkwww.ag.jku.atu.at

Martin Aigner

Bert Juttler

Author:Author:

Martin Aigner:Martin Aigner:– Dr. Mag., research assistant Dr. Mag., research assistant – Email: martin. aigner @ jku .at Email: martin. aigner @ jku .at Zbynek Sir:Zbynek Sir:– Dr.; research assistant at Dr.; research assistant at FWFFWF-Projekt P1738-Projekt P1738

7-N12 7-N12 – Alumni Alumni

Author: Bert JuttlerAuthor: Bert JuttlerSelected scientific activities:Selected scientific activities:– Since 2003:associated editor Since 2003:associated editor

of CAGDof CAGD– Organizer of various Mini symposia Organizer of various Mini symposia – Member of program committeesMember of program committees

of numerous conferences of numerous conferences

Research interests:Research interests:– CAGD, Applied Geometry, CAGD, Applied Geometry,

Kinematics, Robotics, Kinematics, Robotics, Differential GeometryDifferential Geometry

IntroductionIntroduction

Using PH spline curves to evoluted fittinUsing PH spline curves to evoluted fitting a given set of data points or a curve g a given set of data points or a curve For example:For example:

Steps:Steps:

Introduce a general framework for Introduce a general framework for abstract curve fittingabstract curve fitting

Apply this framework to PH curvesApply this framework to PH curves

Discuss the relationship between this Discuss the relationship between this method and Gauss-Newton iterationmethod and Gauss-Newton iteration

An abstract framework for curve An abstract framework for curve fitting via evolutionfitting via evolution

Parameterized family of curves:Parameterized family of curves:（（ s, us, u ）） -> ->

– u is the curve parameteru is the curve parameter– s is the vector of shape parameterss is the vector of shape parameters

Let s depend smoothly on an evolution Let s depend smoothly on an evolution parameter t, s( t)=( )parameter t, s( t)=( )Approximately compute the limit Approximately compute the limit

)(uCs

)(),......,(),( 21 tststs n

)(lim tst

],[ bau nRs


Each point travels with the velocity:Each point travels with the velocity:

Normal velocity of the inner points:Normal velocity of the inner points:

)(|)(

)()( )(1

)()( tss

uCuCuv itss

n

i i

ststs

n

itsitss

i

stststs unts

s

uCunuvuv

1)()()()()( )())(|

)(()()()(


Assume a set of data points is Assume a set of data points is given.given.

Let and Let and

Expected to toward their associated Expected to toward their associated data points if data points if then then

NjjP ......1}{

)()( jtsj uCf ||)(||minarg )(],[

uCPu tsjbau

j

)()()( )( jtsjjjj unfPduv },{ bau j },{ bau j jjjj fPduv

)(



Time derivatives of the shape Time derivatives of the shape parameters satisfied the following parameters satisfied the following equation in least-squares senseequation in least-squares sense

N

bauj

N

baujRjjjtsvjjts

s j j

RwfPuvwduvws},{,1 },{,1

2)(

2)( ||)()(||))((minarg

Necessary condition for a minimum

))(()())(( tsrtstsM

))(())(())(()( 1 tsrtsMtsFts


Definition:Definition: – A given curve:A given curve:– a set of parameters U is said to be a set of parameters U is said to be

regular:regular:– A set parameters: that A set parameters: that

andand– Unit normal vectorsUnit normal vectors

That the matrix has That the matrix has a maximal ranka maximal rank

)(uCs

NjjuU .....1}{ Uba },{

0)()(

uC ts

)()( jtsj unn

)(, |)(

tssk

jsjkj s

uCnA

nN


LemmaLemma: in a regular case and if all : in a regular case and if all closet points are neither singular nor closet points are neither singular nor boundary points, then any solution of boundary points, then any solution of the usual least-squares fitting the usual least-squares fitting

is a stationary point of the is a stationary point of the differential equation derived from the differential equation derived from the evolution processevolution process

N

jjtsj

bausuCP

j1

2)(

],[||)(||minminarg

Evolution of PH splinesEvolution of PH splines

Ordinary PH curves c (u)=[x ( u) ,y (u)] satOrdinary PH curves c (u)=[x ( u) ,y (u)] satisfied the following conditions:isfied the following conditions:

Regular PH curves: let w=1.Regular PH curves: let w=1.The difference : gcd (x’ ( u ),y’ (u)) is a The difference : gcd (x’ ( u ),y’ (u)) is a square of a polynomialsquare of a polynomial called preimage curvecalled preimage curve

)(w'x 22 )2(w'y

)]u(),u([


Proposition: if a regular PH curve c Proposition: if a regular PH curve c (u) and then:(u) and then:– Smooth field of unit tangent vectors for Smooth field of unit tangent vectors for

all uall u– Parametric speed and arc-length are Parametric speed and arc-length are

polynomial functions polynomial functions – Its offsets are rational curvesIts offsets are rational curves

)0,0())u(),u((


Let an open integral B-spline curve,Let an open integral B-spline curve, and and

Let Let

)]u(),u([

km2m1mm1kk1k10u...uu,u,...,u,u,u...uu

m

0ik,ii

)u(N)u(

m

0ik,ii

)u(N)u(

)u(K2y

xud

)u()u(2

)u()u(

y

x)u(C

j,i

m

0i

m

0j

ji

jiji

0

0u

u

22

0

0

s1k

u

uk,jk,ij,i

1k

ud)u(N)u(N)u(K


In the evolution we fix the knot vector, sIn the evolution we fix the knot vector, so the shape parameters areo the shape parameters are

the velocitythe velocity

The unit normalsThe unit normals

],...,,...,y,x[sm0m000

m

0i

m

0jj,iij

ijij

ij0

0s

)u(K],[2]y,x[)u(v

m

0i

m

0j k,jk,ijiji

k,jk,i

m

0i

m

0j

jiji

ji

22

s

s )u(N)u(N)(

)u(N)u(N2

)u()u(

)u('C)u(n

Evolution of PH splinesEvolution of PH splinesThe length of PH spline:The length of PH spline:

The regularization term:The regularization term:

Which forces the length to converge tWhich forces the length to converge to some constant value o some constant value

1m

1k

u

u

m

0i

m

0j1mj,ijiji

22

s)u(K)(du))u()u((L

m

0jj,ij

i

s )u(K2L

m

0jj,ij

i

s )u(K2L

2

sse)LLL(:R

s

Le

L

Examples of PH splines evolutioExamples of PH splines evolutionn

Simple example: Simple example: – fitting two circular arcs with radius 1.fitting two circular arcs with radius 1.– Two cubic PH segments depending on 8 Two cubic PH segments depending on 8

shape parametersshape parameters– Initial position: straight lineInitial position: straight line

e

LRv

www

Examples of PH splines evolutioExamples of PH splines evolutionn

Examples of PH splinesExamples of PH splinesInitial: two Initial: two straight straight segments For the segments For the global shape global shape =8,=8,

Gradually raised Gradually raised length to 14length to 14Fix end pointsFix end points

Insert knotsInsert knots

eL

1 Rv www

1.0Rw 100vw

Examples of PH splinesExamples of PH splinesInitial: two straight Initial: two straight segments For the segments For the global shape =8,global shape =8,

Gradually raised Gradually raised length to 14length to 14

Fix end pointsFix end points


eL

1 Rv www

1.0Rw 100vw

Examples of PH splinesExamples of PH splinesInitial: two straight Initial: two straight segments For the glsegments For the global shape =8,obal shape =8,

Gradually raised lengGradually raised length to 14th to 14Fix end pointsFix end points


eL

1 Rv www

1.0Rw 100vw





eL

1 Rv www

1.0Rw 100vw





eL

1 Rv www

1.0Rw 100vw

Example of PH splinesExample of PH splinesInitial: two straight Initial: two straight segments For the segments For the global shape =8,global shape =8,




eL

1 Rv www

1.0Rw 100vw





eL

1 Rv www

1.0Rw 100vw





eL

1 Rv www

1.0Rw 100vw

Examples of PH splinesExamples of PH splines

Initial value by Hermite interpolation Initial value by Hermite interpolation – Split data points at estimated inflectionsSplit data points at estimated inflections

Speed of convergenceSpeed of convergence

LemmaLemma: the Euler update of the : the Euler update of the shape parameters for the evolution shape parameters for the evolution with step h is equivalent to a Gauss-with step h is equivalent to a Gauss-Newton step with the same h of the Newton step with the same h of the problemproblem

Provided that Provided that

m

js

jsj uCP1

2 min||)(|| ||)(||minarg],[

uCPu sjbau

j

}...1|{},{ Njuba j 0Rw



Quadratic convergence of the Quadratic convergence of the methodmethod

Concluding remarksConcluding remarks

Least-squares fitting by PH spline cuves iLeast-squares fitting by PH spline cuves is not necessarily more complicated than s not necessarily more complicated than othersothersFuture work is devoted to using the apprFuture work is devoted to using the approximation procedure in order to obtain oximation procedure in order to obtain more compact representation of NC tool more compact representation of NC tool pathspaths

Q&AQ&A

Thanks!Thanks!

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Evolution-based least-squares fitting using Pythagorean hodograph spline curves Speaker: Ying.Liu November 29. 2007