Interpolation and elevation

1. CAGD&CG Interpolation and elevation zhu ping 07.10.10

2. Problem • T0,T1,T2,…,Tn∈R2 , Ti≠Ti+1 • Find a curve P(t): P:[0,1]—> R2 , satisfying P(ti)=Tii=0,1,…,n

3. Paper • 1. <<Constructing Iterative Non-Uniform B-spline Curve and Surface to Fit Data Points>> Lin Hongwei Science in China 04 • 2.<<Totally Positive Bases and Progressive Iteration Approximation>> Lin Hongwei Computers and Mathematics with Application 05 • 3.<<Progressive iterative approximation and bases with the fastest convergence rates>> J.Delgado CAGD 07 • 4.<<Geometric interpolation by planar cubic polynomial curves>> Jernej Kozak CAGD 07 • 5.<<An alternative method of curve interpolation>> Les A.Piegl The Visual Computer 05 • 6.<<On the degree elevation of B-spline curve and corner cutting>> Wang Guozhao CAGD07 • 7.<<Interpolation by geometric algorithm>> Takashi Maekawa CAD 07

4. Constructing Iterative Non-Unifrom B-spline Curve and Surface to Fit Data Points • Problem:Progressive iterative approximation • Proposed by Prof.Qi Dongxu and de Boor • In 1991, cubic uniform B-spline and convergence • Shortage:NURBS,convexity-preserving,explict expression

5. Iterative algorithm (1) parameter (2)normal

6. Parameter

7. Normal

8. Deduce:

9. Iterative error:

10. Iterative convergence

11. Convexity-preserving algorithm Polyline L is convexity and if sgn(αi)=sgn(βi) polyline L’is also convexity

13. Iterative formulae

14. Surface iterative

15. Example: 30 times convexity-preserving property of non-uniform B-spline curve

16. Totally Positive Bases and Progressive Iteration Approximation • Totally Positive Bases Definition: Given a basis {Bi(t) ≥0|i=0,1,…,n} , an increasing sequence the collocation matrix is Basis {Bi} is totally positive basis if the matrix is a totally positive matrix

17. Theorem. If the basis is totally positive and its collocation matrix B at {t0,t1,…,tn} is non-singular,the curve has progressive iteration appromation. Proof.

18. Example: Bezier,B-spline and NUBRS curve and surface Bezier 20 times 60 times

19. Progressive iterative approximation and bases with the fastest convergence rates • Outline: The normalized B-basis has optimal shape preserving properties and we prove that it satisfies the progressive iterative approximation property with the fastest convergence rates. A similar result for tensor product surfaces is also derived.

20. Theorem: Given a space U with an TP basis,the normalized B-basis of U provided a progressive iterative approximation with the fastest convergence rates among all TP bases of U.

21. Proof. will be minimum when the smallest eigenvalue of B is maximum. U:unique normalized B-basis (b0,…,bn),TP basis (v0,…,vn) . (v0,…,vn)=(b0,…,bn)K . K is a stochastic TP matrix. prove that the smallest eigenvalue of B is greater than the smallest eigenvalue of V

22. Consider matrix since is similar to is similar to .We should prove

24. Tensor product surfaces: • Kronecker product:

25. Numerical test: Bernstein Basis VS Said-Ball Basis Definition:

26. Comparion:

27. Error compare:

28. Geometric Method for Hermite Iterpolation by a class of PH Quintics • Outline: Geometric relationship among Bezier control points of PH Quintics is developed.And choose the best curve in the resulant PH quintics.

31. Theorem 1: A Bezier curve is PH curve when two points A,B exist making

32. Hermite Method:

33. On the degree elevation of B-spline curves and corner cutting • Outline: Our idea is making bi-degree B-spline to elevate B-spline. Old Method:splite into pieces of Bezier curves

36. Theorem 1: , are defined and Theorem 2:

37. Property:positivity,partition of unity,linear independence globally • Theorem 3:

39. An alternative method of curve interpolation • Author: Les A.Piegl, South Florida University, research in CAD/CAM,geometric modeling,computer graphics Wayne Tiller,in GeomWare, The NURBS Book

40. Outline:given a point data set that contains several fairly unevently distributed random points,this paper presents a new paradigm of curve interpolation to fit a curve to the data with end tangent vector constraints.

41. Major components: 1. Base curve 2. Localization 3. Constrained shape manipulation 4. Parametrization adjustment

42. Definition:a B-spline curve of degree p

43. Constrained shaping:

44. Parametrization: 1.Obtain a polygonal aproximation of the base curve 2.Project all points onto the polygon and find the closest vertex or polygon leg 3.Obtain an approximation parameter from the parameters of the closest leg or vertex

45. Geometric decomposition: 1.Decompose the NURBS curve into piecewise Bezier segments 2.Bezier curve decomposition

46. Location: Greville abscissae: degree p degree p-1

47. Procedure: 1.Go through the brackets (ηi,ηi+1) i=1,…,n-1 to see how many parameters fall within the brackets 2.If there are two or more parameters,insert a knot and recomputer the brackets 3.Repeat the process until each bracket contains no more than one parameter

48. Let w∈ (ηi,ηi+1) be the new abscissa

49. The algorithm: 1.Computer the Grevile brackets 2.Check end conditions and insert knots if necessary 3.While there are brakets with more than one parameter 3.1 Find first offending span 3.2 Average parameters lying inside the span.The average is to become the new Greville abscissa 3.3 Get the knot to be inserted 3.4 If the knot falls on an existing knot,select another parameter representative 3.5 Insert the knot and recompute Greville brackets

50. Base curve:two methods (1)1.Obtain a local cubic C1 interpolation to the points. 2.Sample the interpolant 3.Approximate the smple points by least-squares.