Knot placement in B-spline curve approximation

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

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

3. Introduction: • Background: • 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. Initial population: Fitness function: Bayesian information criterion

29. Example of two-point crossover:

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

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:<<Computational geometry—curve and surface modeling>> approximation interpolation best select points

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

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

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

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

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