Knot placement in B-spline curve approximation

Reporter:Cao juanDate: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:

1

1

1

int( )

(1 ) 1,....,p j i i

md

n p

i jd jd i

u u u j 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 requiredto interpolate (ENP)

Adaptive Knot Sequence Generation (AKSG)

1

1

1 1

i i

i ior

i i

i x x

Based on curvatur

e 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:

1 2,

1 1

1

1

{0,0,...,0, , ....., ,1,1,....,1}

1( 1,2,...., )

n

p p

j p

j ii j

U V V V

V u j n pp

P M Q

1 2 1 2( , ,...., ) ( , ,..., )n nQ Q Q u u u

Problem: data are noise & unequal distribution

Aim: reconstruction (B-spline curve with a “good shape”)

' i -1 i i+1i

x +x +xx =

3

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

Fitness function:Bayesian information criterion

Initial population:

Example of two-point crossover:

Mutation method: for each individual

for counter = 1 to individual length

Generate a random number

Generate a random number

add a gene randomly Delete a gene randomly

>PmYN

Counter + 1

>0.5

N Y

Character: insert or delete knots adaptivelyQuasi-multiple knots Don’t need error tolerance Independent with initial estimation of

the knot locationsOnly one –dimensional case

Adaptive knot placement in B-spline 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 interpolationbest 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 adjacentintervals that joint at a feature point

Interpolate

inflectionpoints

smooth the discrete curvature

divide into several subsets

iteratively bisect each segment till satisfy the heuristic rule

check the adjacentintervals that joint at a feature point

Interpolate

curvature integration

smooth the discrete curvature

divide into several subsets

iteratively bisect each segment till satisfy the heuristic rule

check the adjacentintervals that joint at a feature point

Interpolate

curvature integration

Example:

character:

smooth discrete curvatureautomaticallysensitive 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