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B-spline curve approximation. zhu ping. 08.02.20. 1. Application 2. Some works 3. Discussion. Outline. Arctile:. The NURBS Book. Les Pigel&Wayne Tiller, 2nd 1996 Knot Placement for B-Spline Curve Approximation. Anshuman Razdan, Technical Report 1999

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Presentation Transcript
outline
1. Application

2. Some works

3. Discussion

Outline
slide3

Arctile:

  • The NURBS Book. Les Pigel&Wayne Tiller, 2nd 1996
  • Knot Placement for B-Spline Curve Approximation.
  • Anshuman Razdan, Technical Report 1999
  • 3. Surface approximation to scanned data. Les Piegl&Wayne Tiller, The Visual Computer 2000
  • 4. Adaptive knot placement in B-spline curve approximation. Weishi Li, Shuhong Xu, Gang Zhao, Li Ping Goh, Computer-Aided Design 2005
  • 5. B-spline curve fitting based on adaptive curve refinement using domiant points. Hyungjun Park, Joo-Haeng Lee, Computer-Aided Design 2007
a normal method least square the nurbs book les piegl wayne tiller
Les A.Piegl, South Florida University,

research in CAD/CAM,geometric

modeling,computer graphics

A normal method(least square)(The Nurbs Book) Les Piegl&Wayne Tiller

Wayne Tiller,in GeomWare,

The NURBS Book

a normal method least square the nurbs book les piegl wayne tiller1
Given:

1. Data points;

2. End Interpolation;

Goals:

error bound:

Process

1. Parametrization(chord parameteration)

2. Knot placement

3. Select end conditions

4. Solve the tri-diagonal linear systems of equations.

A normal method(least square)(The Nurbs Book) Les Piegl&Wayne Tiller
slide6
Knot placement:

1. Start with the minimum or a small number of knots

2. Start with the maximum or many knots

Error bounds:

1.

2.

slide7
Curve appromation is iterative process.

Disvantage:

1. Time-consuming;

2. Relate to initial knots

knot placement for b spline curve approximation
Anshuman Razdan, Arizona State University

Technical Report, 1999

Knot Placement for B-Spline Curve Approximation

Associate Professor in the Division of Computing Studies, CAD,CAGD&CG

Farin’s student

slide9
Assumption:

1. Given a parametric curve.

2. Evaluated at arbitrary discrete values within the parameter range.

Goals:

1. Closely approximate with a C2 cubic B-spline curve.

slide10
Process:

1. Pick appropriate points on the given curve

2. Parametrization

3. Select end conditions

4. Solve the tri-diagonal linear systems of equations

How to obtain sampling points

1. Estimate the number of sampling points;

2. Find samping points on the given curve

slide11
Estimate the number of points required to interpolate (ENP)

Approximated by a finite number of circular arc segments

slide12
Finding the interpolating points(independent of parametrization):

1. arc length

2. curvature

(1) curvature extrema

(2)inflection point

slide19
AKSG:

if , insert a auxiliary knot in the middle of the

segment

slide21
a heuristic rule for knot placement

Su BQ,Liu DY:<<Computational geometry—curve and surface modeling>>

approximation

interpolation

slide22
Algorithm:

1. smoothing of discrete curvature

2. divide the initial parameter-curvature set into several subsets

3. iteratively bisect each segment untill satisfy the heuristic rule

4. check the adjacent intervals that joint at a feature point

5. interpolate

slide25
iteratively bisect each segment untill satisfy the heuristic rule:

curvature integration

Newton-Cotes formulae

b spline curve fitting based on adaptive curve refinement using domain points
B-spline curve fitting based on adaptive curve refinement using domain points

Hyungjun Park, since 2001,a faculty member of Industrial Engineering at Chosun University,

geometric modeling, CAD/CAM/CG application

Joo-Haeng Lee, a senior researcher in ETRI

CAD&CG, robotics application

slide32
Advantage:

1. compare with KTP and NKTP:when |m-n| is small, it is sensitive to parameter values.

slide33
2. compare with KRM and Razdon’s method:

stability, robustness to noise and error-boundedness

Proposed approach:

1. parameterization;

2. dominant point selection

3. knot placement(adaptive using the parameter values of the selected dominant points)

4. least-squares minimization

slide34
Determination of konts:

are the parameter values of points

slide35
Selection of dominant points:

1. Selection of seed points from

2. Choice of a new dominant point

Based on the adaptive refinement paradigm

fewer dominant points at flat regions and more at

complex regions

slide36
Selection of seed points:

local curvature maximum(LCM) points, inflection points

LCM:and

exclude

base curve

251 input points

slide37

Base curve with 16 control points

10 initial dominant points

slide38
Choice of a new dominant points:

max deviation:

The segment is to be refined.

choosing

shape index

where

slide39

10 dominant points

13 dominant points

slide43
Future:

1. parameterization

2. optimal selection of dominant points as genetic

algorithm

3. B-spline surface and spatial curve