Evolutionbased leastsquares fitting using Pythagorean hodograph spline curves

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Evolution-based least-squares fitting using
Pythagorean hodograph spline curves

• Speaker Ying .Liu
• November 29. 2007

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Institute of Applied Geometry, Jphannes Kepler
University ,Linz, Austira
Bert Juttler

• www.ag.jku.at

Martin Aigner
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Author

• Martin Aigner
• Dr. Mag., research assistant
• Email martin. aigner _at_ jku .at
• Zbynek Sir
• Dr. research assistant at FWF-Projekt P17387-N12
  
• Alumni

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Author Bert Juttler

• Selected scientific activities
• Since 2003associated editor
• of CAGD
• Organizer of various Mini symposia
• Member of program committees
• of numerous conferences
• Research interests
• CAGD, Applied Geometry, Kinematics,
  Robotics, Differential Geometry

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Introduction

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

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Steps

• Introduce a general framework for abstract curve
  fitting
• Apply this framework to PH curves
• Discuss the relationship between this method and
  Gauss-Newton iteration

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An abstract framework for curve fitting via
evolution

• Parameterized family of curves
• (s, u)-gt
• u is the curve parameter
• s is the vector of shape parameters
• Let s depend smoothly on an evolution parameter
  t, s( t)( )
• Approximately compute the limit

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An abstract framework for curve fitting via
evolution

• Each point travels with the velocity
• Normal velocity of the inner points

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An abstract framework for curve fitting via
evolution

• Assume a set of data points is given.
• Let and
  
• Expected to toward their associated data points
  if
  then

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

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

Necessary condition for a minimum
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An abstract framework for curve fitting via
evolution

• Definition
• A given curve
• a set of parameters U is said to be regular
• A set parameters that
  and
• Unit normal vectors
• That the matrix has a
  maximal rank

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An abstract framework for curve fitting via
evolution

• Lemma in a regular case and if all closet points
  are neither singular nor boundary points, then
  any solution of the usual least-squares fitting
  
• is a stationary point of the differential
  equation derived from the evolution process

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Evolution of PH splines

• Ordinary PH curves c (u)x ( u) ,y (u)
  satisfied the following conditions
• Regular PH curves let w1.
• The difference gcd (x ( u ),y (u)) is a
  square of a polynomial
• called preimage curve

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Evolution of PH splines

• Proposition if a regular PH curve c (u) and
  then
• Smooth field of unit tangent vectors for all u
• Parametric speed and arc-length are
  polynomial functions
• Its offsets are rational curves

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Evolution of PH splines

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

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Evolution of PH splines

• In the evolution we fix the knot vector, so the
  shape parameters are
• the velocity
• The unit normals

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Evolution of PH splines

• The length of PH spline
• The regularization term
• Which forces the length to converge to some
  constant value

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Examples of PH splines evolution

• Simple example
• fitting two circular arcs with radius 1.
• Two cubic PH segments depending on 8 shape
  parameters
• Initial position straight line

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Examples of PH splines evolution
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Examples of PH splines

• Initial two straight segments For the global
  shape 8,
• Gradually raised length to 14
• Fix end points
• Insert knots

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Examples of PH splines

• Initial two straight segments For the global
  shape 8,
• Gradually raised length to 14
• Fix end points
• Insert knots

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Examples of PH splines

• Initial two straight segments For the global
  shape 8,
• Gradually raised length to 14
• Fix end points
• Insert knots

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Examples of PH splines

• Initial two straight segments For the global
  shape 8,
• Gradually raised length to 14
• Fix end points
• Insert knots

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Examples of PH splines

• Initial two straight segments For the global
  shape 8,
• Gradually raised length to 14
• Fix end points
• Insert knots

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Example of PH splines

• Initial two straight segments For the global
  shape 8,
• Gradually raised length to 14
• Fix end points
• Insert knots

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Examples of PH splines

• Initial two straight segments For the global
  shape 8,
• Gradually raised length to 14
• Fix end points
• Insert knots

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Examples of PH splines

• Initial two straight segments For the global
  shape 8,
• Gradually raised length to 14
• Fix end points
• Insert knots

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Examples of PH splines

• Initial value by Hermite interpolation
• Split data points at estimated inflections

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Speed of convergence

• Lemma the Euler update of the shape parameters
  for the evolution with step h is equivalent to a
  Gauss-Newton step with the same h of the problem
• Provided that

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Speed of convergence
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Speed of convergence

• Quadratic convergence of the method

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Concluding remarks

• Least-squares fitting by PH spline cuves is not
  necessarily more complicated than others
• Future work is devoted to using the approximation
  procedure in order to obtain more compact
  representation of NC tool paths

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• QA

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• Thanks!