Parameterization for Curve Interpolation

Michael S. Floater and Tatiana Surazhsky

Topics in Multivariate Approximation and Interpolation

Speaker: CAI Hong-jie

Date: Oct. 11, 2007

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

The Second Author

Tatiana Surazhsky

• Post 3D Researcher of Samsung Electronics,

Samsung Telecom Research Israel

• Research Geometric modeling

Computer graphics

Outline

• Background

• Metric for approximation error

• Approximation order Cubic polynomial Cubic spline higher degree polynomial

• Hermite interpolation

Background

• Concept: Parameterization for interpolation Given

points P0,P1,…,Pn in Rk, k= 2 or 3 To find

t0<t1<…<tn and parametric curve P(t)

such that P(ti)=Pi, i=0,…,n.

P0

Pn-1

PnP1

Background

• Selection of parametric curve Polynomial curve Spline curve

• Selection of knot vector

To determine di:=ti+1-ti, i=0,1,…,n-1.

Choices for di

• Uniform di = 1

• Chordal di = |Pi+1-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 = |Pi+1-Pi|1/2 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

Comparison of Four Choices

Original Curve: thin black

Spline Curves: thick gray

Comparison of Three Choices

-1 0 1 2 3 4 5 6 7

-1

-0.5

0

0.5

1

1.5

Original curve: blue uniform: green

Chordal: black centripetal: magenta

Comparison of Three Choices

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1.5

-1

-0.5

0

0.5

1

1.5

Original curve: blue uniform: green

Chordal: black centripetal: magenta

Metric for Approximation Error

• Hausdorff distance Let A,B be point sets in Rk (k=2,3), define

where ||·||E is Euclidean distance, then Hausdorff distance between A and B is

( , ) max{min }E

d

b Ba A

A B a b

( , ) max{ ( , ), ( , )}.Hd d dA B A B B A

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

Hausdorff distance for curves• Definition P0,P1,…,Pn sampled from parametric curve f:[a,

b]→ Rk, Pi= f(si), a≤s0<s1<…< sn≤b. Interpolate Pi by P(t):[t0,tn]→ Rk, then the distance between them is

000

00

[ , ] [ ]( | , | ) max max min | ( ) ( ) |,

max min | ( ) ( ) |

n 0 nnn

nn

H s s t ,t s s st t t

t t ts s s

d s t

s t

f P f P

f P

Metric for Approximation Error

• Parametric distance

where Ф: [t0,tn] →[s0,sn] is strictly increasing, C1 functions such that Ф(t0)=s0, Ф(tn)=sn.

T. Lyche and K. MØrken, A metric for parametric approximation, Curves and Surfaces, 1994

0[ , ] [ ]( | , | ) infn 0 nP s s t ,td

f P f P

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

0[ , ] [ ]( | , | ) ( ), 0n 0 n

mP s s t ,td O h h f P

Cubic Polynomial Interpolation

• Theorem

Given f∈C4[a,b], samples a≤s0<s1<s2<s3 ≤b, let t0=0, ti+1- ti= |f(si+1) - f(si)|(i=0,1,2), and P(t):[t0,t3] → Rk be cubic polynomial such that

P(ti)=f(si)， i=0,1,2,3.

Then dP(f|[s0,s3], P)= O(h4), h →0, where h=s3-s0.

Cubic Polynomial Interpolation

• Lemma 1 If f∈C2[a,b], then

Tip for proof: let u=(si+si+1)/2, then

1

3 ''1 1 1

10 ( ) | ( ) ( ) | ( ) max | ( ) | .

12 i ii i i i i i

s s ss s s s s s s

f f f

1

1 1 1

1

( ) ( ) '( )( ) / 2 ( ) ''( ) ,

( ) ( ) '( )( ) / 2 ( ) ''( ) .

i

i

s

i i i iu

u

i i i is

s u u s s s t t dt

s u u s s t s t dt

f f f f

f f f f

Cubic Polynomial Interpolation

• Lemma 2

If Ф:[t0, t3] →R cubic polynomial such that Ф(ti)=si, i = 0,1,2,3, then

Tip for proof: Newton interpolation formula

4( ).O h

f P

3

0 1 2 30( )[ , , , , ]( )iit t t t t t t

f P f

Extension to Cubic Spline• Theorem

Given f∈C4[a,b], samples a≤s0<…<sn ≤b, let t0=0, ti

+1- ti= |f(si+1) - f(si)|, 0 ≤ i<n, and σ(t):[t0,tn] → Rk be the cubic spline curve such that

　　 Then dP(f|[s0,sn], σ)= O(h4), h →0, where

( ) ( ), 0,1,..., ,

'( ) '( ), 0, .i i

i i

t s i n

t s i n

f

f

10max( ).i i

i nh s s

Parameterization Improvement for higher degree

• Case: polynomial degree n=2,3 Uniform O(h2) Chordal O(hn+1)

• Case: polynomial degree n= 4,5 Uniform O(h2) Chordal O(h4) Improvement O(hn+1)

di=Length(chordal cubic polynomial between Pi,Pi+1)

Hermite Interpolation

• Cubic two-pointGiven f∈C4[a,b], t1- t0= |f(s1) - f(s0)|, and let P(t):[t0,

t1] → Rk be cubic polynomial such that

Then dP(f|[s0,s1], P)= O(h4), as h →0.

( ) ( )( ) ( ), 0,1, 0,1.l li it s i l P f

Hermite Interpolation

• Quintic two-point

Given f∈C6[a,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(f|[s0,s1], P)= O(h6), as h →0.

( ) ( )( ) ( ), 0,1, 0,1,2.l li it s i l P f

Numerical Examples

Original curve

Numerical Examples

Comparison with Cubic Spline

(a) Samples from a glass cup

(b) Chordal C2 cubic spline curve

(c) Improved C2 quintic Hermite spline curve

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. Mathematica

l Methods for Curves and Surfaces, Tromsø 2004. • 李庆杨，王能超，易大义 . 数值分析，第 4版， 2003.

Thanks!

Q&A