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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
Author:Les Piegl & Wayne Tiller
They are iterative processes:
1.Start with the minimum or a
small number of knots
2.Start with the maximum or
many knots
Use chordlength parameterization
and average knot:
m 1
d
n  p 1
i  int( jd )
  jd  i
u p  j  (1   )ui 1   ui j  1,...., n  p
Disadvantage:

Time-consuming

Relate to initial knots
Knot Placement for B-spline
Curve Approximation
Author: Anshuman Razdan
(Arizona State University , Technical
Director, PRISM)
Assumptions:


A parametric curve
evaluated at arbitrary discrete values
Goals:
 closely approximate with B-spline
Estimate the number of points required
to interpolate (ENP)
Adaptive Knot Sequence Generation
(AKSG)
i
i
1
  or

i 1
i 1 
i  xi  xi 1
Based on
curvature
only
Using
origial
tangents
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
Chord length parameter:
(Q1, Q2 ,...., Qn )  (u1, u2 ,..., un )
U  {0, 0,..., 0,V1 ,V2, .....,Vn ,1,1,....,1}
p 1
1
Vj 
p
p 1
j  p 1
u
i j
i
( j  1, 2,...., n  p)
1
PM Q
Problem: data are noise & unequal distribution
Aim: reconstruction (B-spline curve with a “good
shape”)
xi-1 + xi + xi+1
x =
3
'
i
Characters:
approximate the curve once
Data fitting with a spline using
a real-coded genetic
algorithm
Author:Fujiichi Yoshimoto, Toshinobu Harada,
Yoshihide Yoshimoto
Wakayama University
CAD(2003)
About GA:
60’s by J.H,Holland
some attractive
points:
•Global optimum
•Robust
•...
fitness
Initial population:
Fitness
function:
Bayesian
information
criterion
Example of two-point crossover:
Mutation method:
for each individual
for counter = 1 to individual length
Counter + 1
N
Generate a random
number
>Pm
Y
Generate a random
number
>0.5
N
add a gene randomly
Y
Delete a gene randomly
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
Adaptive knot placement in Bspline curve approximation
author: Weishi Li, Shuhong Xu, Gang Zhao, Li
Ping Goh
CAD(2005)
a heuristic rule for knot placement
Su BQ,Liu DY:<<Computational geometry—curve
and surface modeling>>
approximation
interpolation
best select points
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
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
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
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
Example:
character:
smooth discrete curvature
 automatically
 sensitive to the variation of curvature

torsion?
 arc length?

summary:

torsion

arc length

multi-knots (discontinue,cusp)
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