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Interpolation to Data Points

Interpolation to Data Points. Lizheng Lu Oct. 24, 2007. Problem. Interpolation VS. Approximation. Interpolation. Approximation. (piecewise) Bezier curves B-spline curves Rational Bezier/B-spline curves. Classification. Curve Constraint . Outline. Some classical methods

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Interpolation to Data Points

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  1. Interpolation to Data Points Lizheng Lu Oct. 24, 2007

  2. Problem

  3. Interpolation VS. Approximation Interpolation Approximation

  4. (piecewise) Bezier curves • B-spline curves • Rational Bezier/B-spline curves 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 = M+N, with parameters values Find:A polynomial p of degree at most n+k-2, and scalar values satisfying the 2ninterpolation conditions:

  12. Geometric Interpolation by Planar Cubic Polynomial Curves Comp. Aided Geom. Des. 2007, 24(2): 67-78 Jernej Kozak Marjeta Krajnc FMF&IMFM IMFM Jadranska 19, Ljubljana, Slovenia

  13. Problem Given: six points Find: a cubic polynomial parameter curve which satisfies

  14. An Alternative Solution:Quintic Interpolating Curves Find a quintic curve s.t., where tiare 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. cubic uniform chord length 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 2nfor 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. Akima’s Method

  33. Albrecht’s Method • Albrecht G., Bécar J.P. • Univ. de Valenciennes et du Hainaut–Cambrésis, France • Farin G., Hansford D. • Dep. Comp. Sci., Arizona State Univ. • Détermination de tangentes par l’emploi de coniques d’approximation. • 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 Pascal’s 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:

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