1. Knot placement in B-spline curve approximation Reporter:Cao juan Date:2006.54.5

2. Outline:  Introduction  Some relative paper  discussion

3. Introduction: BBackground: TThe problem is…

4. It is a multivarate and multimodal nonlinear optimization problem

6. The NURBS Book Author:Les Piegl & Wayne Tiller

7. They are iterative processes : 1.Start with the minimum or a small number of knots 2.Start with the maximum or many knots

8. Use chordlength parameterization and average knot:

9. Disadvantage:  Time-consuming  Relate to initial knots

10. Knot Placement for B-spline Curve Approximation Author: Anshuman Razdan (Arizona State University, Technical Director, PRISM)

11. Assumptions:  A parametric curve  evaluated at arbitrary discrete values Goals:  closely approximate with B-spline

12. Estimate the number of points required to interpolate (ENP)

16. Adaptive Knot Sequence Generation (AKSG)

17. Based on curvature only Using origial tangents

19. The Pre-Processing of Data Points for Curve Fitting in Reverse Engineering Author: Ming-Chih Huang & Ching-Chih Tai Department of Mechanical Engineering, Tatung University, Taipei, Taiwan Advanced Manufacturing Technology 2000

20. Chord length parameter:

21. Problem: data are noise & unequal distribution Aim: reconstruction (B-spline curve with a “good shape”)

25. Characters: approximate the curve once

26. Data fitting with a spline using a real-coded genetic algorithm Author:Fujiichi Yoshimoto, Toshinobu Harada, Yoshihide Yoshimoto Wakayama University CAD(2003)

27. About GA: 60’s by J.H,Holland some attractive points: Global optimum Robust... fitness

28. Fitness function: Bayesian information criterion Initial population:

29. Example of two-point crossover:

30. Mutation method: for each individual for counter = 1 to individual length Generate a random number Generate a random number add a gene randomlyDelete a gene randomly >Pm Y N Counter + 1 >0.5 NY

36. Character:  insert or delete knots adaptively  Quasi-multiple knots  Don’t need error tolerance  Independent with initial estimation of the knot locations  Only one –dimensional case

37. Adaptive knot placement in B- spline curve approximation author: Weishi Li, Shuhong Xu, Gang Zhao, Li Ping Goh CAD(2005)

38. a heuristic rule for knot placement Su BQ,Liu DY: > approximation interpolation best select points

39. Algorithm: smooth the discrete curvature divide into several subsets iteratively bisect each segment till satisfy the heuristic rule check the adjacent intervals that joint at a feature point Interpolate

40. smooth the discrete curvature divide into several subsets iteratively bisect each segment till satisfy the heuristic rule check the adjacent intervals that joint at a feature point Interpolate inflection points

41. smooth the discrete curvature divide into several subsets iteratively bisect each segment till satisfy the heuristic rule check the adjacent intervals that joint at a feature point Interpolate curvature integration

42. smooth the discrete curvature divide into several subsets iteratively bisect each segment till satisfy the heuristic rule check the adjacent intervals that joint at a feature point Interpolate curvature integration

43. Example:

46. character:  smooth discrete curvature  automatically  sensitive to the variation of curvature  torsion?  arc length?

47. summary:  torsion  arc length  multi-knots (discontinue,cusp)

48. reference:  Piegl LA, Tiller W. The NURBS book. New York: Springer; 1997.  Razdan A. Knot Placement for B-spline curve approximation. Tempe,AZ: Arizona State University; 1999 http://3dk.asu.edu/archives/publication/publication.html  Huang MC, Tai CC. The pre-processing of data points for curve fittingin reverse engineering. Int J Adv Manuf Technol 2000;16:635–42  Yoshimoto F, Harada T, Yoshimoto Y. Data fitting with a spline using a real-coded genetic algorithm. Comput Aided Des 2003;35:751–60.  Weishi Li,Shuhong Xu,Gang Zhao,Li Ping Goh.Adaptive knot placement in B-spline curve approximation.Computr-Aided Design.2005;37:791-797