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Knot placement in B-spline curve approximation

Knot placement in B-spline curve approximation. Reporter:Cao juan Date:2006.54.5. Outline:. Introduction Some relative paper discussion. Introduction:. Background: The problem is…. It is a multivarate and multimodal nonlinear optimization problem. The NURBS Book

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Knot placement in B-spline curve approximation

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  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

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

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

  7. Use chordlength parameterization and average knot:

  8. Disadvantage: • Time-consuming • Relate to initial knots

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

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

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

  12. Adaptive Knot Sequence Generation (AKSG)

  13. Based on curvature only Using origial tangents

  14. 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

  15. Chord length parameter:

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

  17. Characters: approximate the curve once

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

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

  20. Initial population: Fitness function: Bayesian information criterion

  21. Example of two-point crossover:

  22. 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

  23. 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

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

  25. a heuristic rule for knot placement Su BQ,Liu DY:<<Computational geometry—curve and surface modeling>> approximation interpolation best select points

  26. 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

  27. 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

  28. 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

  29. 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

  30. Example:

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

  32. summary: • torsion • arc length • multi-knots (discontinue,cusp)

  33. 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

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