parameterization for curve interpolation michael s. floater and tatiana surazhsky topics in...
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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 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
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.