Knot placement in Bspline curve approximation

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Knot placement in B-spline curve approximation

• ReporterCao juan
• Date2006.54.5

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Outline

• Introduction
• Some relative paper
• discussion

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Introduction

• Background
• The problem is

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It is a multivarate and multimodal nonlinear
optimization problem
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• The NURBS Book
• AuthorLes Piegl Wayne Tiller

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They are iterative processes

1.Start with the minimum or a small number of
knots
2.Start with the maximum or many knots
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• Use chordlength parameterization
• and average knot

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• Disadvantage
• Time-consuming
• Relate to initial knots

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

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

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• Estimate the number of points required
• to interpolate (ENP)

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Adaptive Knot Sequence Generation (AKSG)
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Based on curvature only
Using origial tangents
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• 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

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Chord length parameter
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Problem data are noise unequal distribution
Aim reconstruction (B-spline curve with a
good shape)
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• Characters
• approximate the curve once

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• Data fitting with a spline using
• a real-coded genetic algorithm
• AuthorFujiichi Yoshimoto, Toshinobu Harada,
  Yoshihide Yoshimoto
• Wakayama University
• CAD(2003)

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About GA

• 60s by J.H,Holland
• some attractive points
• Global optimum
• Robust
• ...

fitness
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Initial population
Fitness function
Bayesian information criterion
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Example of two-point crossover
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Mutation method

• for each individual
• for counter 1 to individual length

Generate a random number
Counter 1
gtPm
N
Y
Generate a random number
gt0.5
N
Y
add a gene randomly
Delete a gene randomly
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• Character
• insert or delete knots adaptively
• Quasi-multiple knots
• Dont need error tolerance
• Independent with initial estimation of the knot
  locations
• Only one dimensional case

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

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a heuristic rule for knot placement
Su BQ,Liu DYltltComputational geometrycurve and
surface modelinggtgt
approximation
interpolation
best select points
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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
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smooth the discrete curvature
inflection points
divide into several subsets
iteratively bisect each segment till satisfy
the heuristic rule
check the adjacent intervals that joint at a
feature point
Interpolate
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smooth the discrete curvature
divide into several subsets
curvature integration
iteratively bisect each segment till satisfy
the heuristic rule
check the adjacent intervals that joint at a
feature point
Interpolate
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smooth the discrete curvature
divide into several subsets
iteratively bisect each segment till satisfy
the heuristic rule
curvature integration
check the adjacent intervals that joint at a
feature point
Interpolate
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Example
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character

• smooth discrete curvature
• automatically
• sensitive to the variation of curvature
• torsion?
• arc length?

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summary

• torsion
• arc length
• multi-knots (discontinue,cusp)

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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/publi
  cation/publication.html
• Huang MC, Tai CC. The pre-processing of data
  points for curve fittingin reverse engineering.
  Int J Adv Manuf Technol 20001663542
• Yoshimoto F, Harada T, Yoshimoto Y. Data fitting
  with a spline using a real-coded genetic
  algorithm. Comput Aided Des 20033575160.
• Weishi Li,Shuhong Xu,Gang Zhao,Li Ping
  Goh.Adaptive knot placement in B-spline curve
  approximation.Computr-Aided Design.200537791-797