Parameterization for Curve Interpolation

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Parameterization for Curve Interpolation
Topics in Multivariate Approximation and
Interpolation

• Michael S. Floater and Tatiana Surazhsky

Speaker CAI Hong-jie Date Oct. 11, 2007
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The First Author

• Michael S. Floater
• Main Posts
• Professor of the University of Oslo
• Editor of the journal Computer Aided
  Geometric Design
• Research
• Geometric modeling
• Numerical analysis
• Approximation theory

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The Second Author

• Tatiana Surazhsky
• Post
• 3D Researcher of Samsung Electronics,
• Samsung Telecom Research Israel
• Research
• Geometric modeling
• Computer graphics

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Outline

• Background
• Metric for approximation error
• Approximation order
• Cubic polynomial
• Cubic spline
• higher degree polynomial
• Hermite interpolation

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Background

• Concept Parameterization for interpolation
• Given
• points P0,P1,,Pn in Rk, k 2 or 3
• To find
• t0ltt1ltlttn and parametric curve P(t)
• such that P(ti)Pi, i0,,n.

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Background

• Selection of parametric curve
• Polynomial curve
• Spline curve
• Selection of knot vector
• To determine diti1-ti, i0,1,,n-1.

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Choices for di

• Uniform di 1
• Chordal di Pi1-Pi
• J. H. Ahlberg, E. N. Nilson, and J. L. Walsh
• The theory of splines and their
  applications, 1967
• M. P. Epstein
• On the influence of parametrization in
  parametric interpolation, 1976
• Centripetal di Pi1-Pi1/2
• E. T. Y. Lee
• Choosing nodes in parametric curve
  interpolation, 1989
• Affine invariant
• T. A. Foley and G. M. Nielson
• Knot selection for parametric spline
  interpolation, 1989

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Comparison of Four Choices
Original Curve thin black Spline Curves thick
gray
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Comparison of Three Choices
Original curve blue uniform green
Chordal black centripetal magenta
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Comparison of Three Choices
Original curve blue uniform green
Chordal black centripetal magenta
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Metric for Approximation Error

• Hausdorff distance
• Let A,B be point sets in Rk (k2,3), define
• where E is Euclidean distance, then
  Hausdorff distance between A and B is

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Metric for Approximation Error

• Illustration for Hausdorff distance
• d(A,B)1
• d(B,A)3
• dH(A,B)3
• Application of Hausdorff distance
• Image matching

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Hausdorff distance for curves

• Definition
• P0,P1,,Pn sampled from parametric curve
  fa,b? Rk, Pi f(si), as0lts1ltlt snb.
  Interpolate Pi by P(t)t0,tn? Rk, then the
  distance between them is

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Metric for Approximation Error

• Parametric distance
• where ? t0,tn ?s0,sn is strictly
  increasing, C1 functions such that ?(t0)s0,
  ?(tn)sn.
• T. Lyche and K. MØrken, A metric for
  parametric approximation, Curves and Surfaces,
  1994

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Approximation Order

• Why not distances
• Hard to calculate
• Even bounds are difficult to achieve
• Approximation order instead
• where h Length(f s0,sn ) sn-s0.
• Larger approximation order m, better
  interpolation

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Cubic Polynomial Interpolation

• Theorem
• Given f?C4a,b, samples as0lts1lts2lts3 b, let
  t00, ti1- ti f(si1) - f(si)(i0,1,2), and
  P(t)t0,t3 ? Rk be cubic polynomial such that
• P(ti)f(si),i0,1,2,3.
• Then dP(fs0,s3, P) O(h4), h ?0, where
  hs3-s0.

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Cubic Polynomial Interpolation

• Lemma 1
• If f?C2a,b, then
• Tip for proof let u(sisi1)/2, then

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Cubic Polynomial Interpolation

• Lemma 2
• If ?t0, t3 ?R cubic polynomial such that
  ?(ti)si, i 0,1,2,3, then
• Tip for proof Newton interpolation formula

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Extension to Cubic Spline

• Theorem
• Given f?C4a,b, samples as0ltltsn b, let
  t00, ti1- ti f(si1) - f(si), 0 iltn, and
  s(t)t0,tn ? Rk be the cubic spline curve such
  that
• Then dP(fs0,sn, s) O(h4), h ?0, where

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Parameterization Improvement for higher degree

• Case polynomial degree n2,3
• Uniform O(h2)
• Chordal O(hn1)
• Case polynomial degree n 4,5
• Uniform O(h2)
• Chordal O(h4)
• Improvement O(hn1)
• diLength(chordal cubic polynomial between
  Pi,Pi1)

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Hermite Interpolation

• Cubic two-point
• Given f?C4a,b, t1- t0 f(s1) - f(s0), and let
  P(t)t0,t1 ? Rk be cubic polynomial such that
• Then dP(fs0,s1, P) O(h4), as h ?0.

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Hermite Interpolation

• Quintic two-point
• Given f?C6a,b, let u0, u1 be chordal
  parametric knot vector, and t0, t1 be improved
  knot vector, P(t)t0,t1 ? Rk be quintic
  polynomial such that
• Then dP(fs0,s1, P) O(h6), as h ?0.

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Numerical Examples
Original curve
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Numerical Examples
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Comparison with Cubic Spline

1. Samples from a glass cup
2. Chordal C2 cubic spline curve
3. Improved C2 quintic Hermite spline curve

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Reference

• M.S. Floater ,T. Surazhsky. Parameterization for
  curve interpolation. Topics in Multivariate
  Approximation and Interpolation, 2007.
• M.S. Floater. Arc Length Estimation and The
  Convergence of Polynomial Curve Interpolation.
  Numerical Mathematics, to appear.
• T. Surazhsky, V. Surazhsky. Sampling Planar
  Curves Using Curvature-Based Shape Analysis.
  Mathematical Methods for Curves and Surfaces,
  Tromsø 2004.
• ???,???,???. ????,?4?,2003.

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Thanks!QA