Interpolation to Data Points

1
Interpolation to Data Points

• Lizheng Lu
• Oct. 24, 2007

2
Problem
3
Interpolation VS. Approximation
Interpolation
Approximation
4
Classification

• Curve
• Constraint

5
Outline

• Some classical methods
• Some recent methods on geometric interpolation
• Estimate the tangent

6
C2k-1 Hermite Interpolation
Cubic Interpolation
7
C2 Cubic B-spline Interpolation

• Given A set of points and
• a knot sequence
• Find A cubic B-spline curve, s.t.

8
Geometric Hermite Interpolation (GHI)
de Boor et al., 1987

• Given Planar points pi, with positions,
  tangents and curvatures
• Result Piecewise cubic Bezier curves, having
• G2 continuity
• 6th order accuracy
• Convexity preservation

9
Comments on GHI

• Independent of parameterization
• High accuracy
• But, it usually includes nonlinear problems
• Questions on the existence of solution and
  efficient implement
• Difficult to estimate approximation order, etc

10
References on GHI
11
High Order Approximationof Rational Curves
Floater, 2006

• Given A rational curve ,
  where
• f and g are of degree M and N, let k MN,
• with parameters values
• Find A polynomial p of degree at most nk-2,
• and scalar values satisfying
  the
• 2n interpolation conditions

12
Geometric Interpolation by Planar Cubic
Polynomial Curves

• Comp. Aided Geom. Des. 2007, 24(2) 67-78
• Jernej Kozak Marjeta Krajnc
• FMFIMFM IMFM
• Jadranska 19, Ljubljana, Slovenia

13
Problem

• Given six points
• Find a cubic polynomial parameter curve
• which satisfies

14
An Alternative SolutionQuintic Interpolating
Curves

• Find a quintic curve
• s.t.,
• where ti are chosen to be the uniform
• and chord length parameterization.

15
Essential of Problem
Know t0, t5, p0, p3 Unknown t1, t2, t3,
t4 , p1, p2 Equations P3(ti) Ti, i 2, 3, 4
16
Solution of Problem
Know t0, t5, p0, p3 Unknown t1, t2, t3,
t4 , p1, p2 Equations P3(ti) Ti, i 2, 3, 4
Solved by Newton Iteration with initial values
17
Existence of Solution

• Provide two sufficient conditions guaranteeing
  the existence
• Summarize cases in a table which does not allow a
  solution

18
Comparison
19
On Geometric Interpolation by Planar Parametric
Polynomial Curves

• Mathematics of Computation 76(260) 1981-1993

20
Problem

• Given 2n points
• Find a cubic polynomial parameter curve
• which satisfies

21
Main Results

• If the data, sampled from a convex smooth
• curve, are close enough, then
• equations that determine the interpolating
  polynomial curve are derived for general n
  (Theorem 4.5)
• if the interpolating polynomial curve exists, the
  approximation order is 2n for general n (Theorem
  4.6)
• the interpolating polynomial curve exists for n
  5 (Theorem 4.7)

22
On Geometric Interpolation of Circle-like Curves

• Comp. Aided Geom. Des. 2007, 24(4) 241-251

23
What is Circle-like Curve?

• A circular arc of an arclength is
  defined by
• Suppose that a convex curve is parameterized by
  the
• same parameter as . The curve will be
  called
• circle-like, if it satisfies
• (1)
• (2)

24
The Result
25
Outline

• Some classical methods
• Some methods on geometric interpolation
• Estimate the tangent

26
Tangent Estimation Methods

• FMill , 1974
• Circle Method
• Bessel
• Ackland, 1915
• Akima, 1970
• G. Albrecht, J.-P. Bécar, G. Farin, D. Hansford,
  2005, 2007

27
Problem
?
28
FMILL
29
Circle Method
30
Bessel
Parabola f (t)
31
Bessel
32
Akimas Method
33
Albrechts Method

• Albrecht G., Bécar J.P.
• Univ. de Valenciennes et du HainautCambrésis,
  France
• Farin G., Hansford D.
• Dep. Comp. Sci., Arizona State Univ.
• Détermination de tangentes par lemploi de
  coniques dapproximation.
• On the approximation order of tangent estimators.
  CAGD, in press

34
Main Idea

• Method Estimate the tangent by using the
  interpolating conic of the given five points
• Solution solved by Pascals theorem in
  projective geometry
• Advantages
• Conic precision
• Less computations without computing the implicit
  conic

35
Idea Derivation
Farin, 2001

• Any conic section is uniquely determined by five
  distinct points in the plane, pi(xi, yi).

36
Idea Derivation
Pascal, 1640
37
Projective Geometry in CAGD

• Express rational forms
• Implicit representation of rational forms

38
Projective Geometry in CAGD

• Express rational forms
• Implicit representation of rational forms
• Chen, Sederberg

Line conics
Conic section
39
Projective Geometry
40
Projective Geometry

• A line in is represented by
• The line joining the two points is
• The intersection of two lines is

41
Estimate the Tangent
42
Estimate the Tangent
43
Degenerate Cases
(b)
(a)
(c)
44
Examples
45
Experimental results
46
Non-convex Case
Conic method
Akima
Bessel
Circle method
47
Approximation order
48
Theoretical Analysis

• Consider a planar curve

49
Theoretical Analysis

• Consider a planar curve

Take five points
50
Theoretical Analysis

• Consider a planar curve

Take five points
Let
51
Theoretical Analysis

• Taylor expansion

Exact tangent
Exact norm
52
Theoretical Analysis

• For a point , with the tangent

Its corresponding tangent in the projective
space is
53
Compute the Approximation Order
To solve the k in

• Taylor expansion
• Symbolic computation MAPLE

54
Numerical Result (1)
55
Numerical Result (2)
56
Summary

• Obtain order four approximation for the convex
  case, two for the inflection point
• Estimate the approximation order with theoretical
  justification
• Estimate the direction of the tangent only, not
  the vector!

57

• Thank You!