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

Outline:  Introduction  Some relative paper  discussion

Introduction: BBackground: T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:

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)

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:

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

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 randomlyDelete a gene randomly >Pm Y N Counter + 1 >0.5 NY

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 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: > 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 iteratively bisect each segment till satisfy the heuristic rule check the adjacent intervals 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 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;  Razdan A. Knot Placement for B-spline curve approximation. Tempe,AZ: Arizona State University;  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: