Download - B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion
![Page 1: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/1.jpg)
B-spline curve approximation
zhu ping
08.02.20
![Page 2: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/2.jpg)
Outline
1. Application
2. Some works
3. Discussion
![Page 3: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/3.jpg)
Arctile:
1. The NURBS Book. Les Pigel&Wayne Tiller, 2nd 1996
2. 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
![Page 4: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/4.jpg)
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
Wayne Tiller,in GeomWare,
The NURBS Book
![Page 5: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/5.jpg)
A normal method(least square)(The Nurbs Book) Les Piegl&Wayne Tiller
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.
![Page 6: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/6.jpg)
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.
1
1
1
int( )
(1 ) 1,....,p j i i
md
n p
i jd jd i
u u u j n p
0max | ( ) |k mk k kQ C u
0max(min | ( ) | (0 1))k mk kQ C u u
![Page 7: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/7.jpg)
Curve appromation is iterative process.
Disvantage:
1. Time-consuming;
2. Relate to initial knots
![Page 8: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/8.jpg)
Knot Placement for B-Spline Curve Approximation
Anshuman Razdan, Arizona State University
Technical Report, 1999
Associate Professor in the Division of Computing Studies, CAD,CAGD&CGFarin’s student
![Page 9: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/9.jpg)
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.
![Page 10: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/10.jpg)
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
![Page 11: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/11.jpg)
Estimate the number of points required to interpolate (ENP)
Approximated by a finite number of circular arc segments
![Page 12: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/12.jpg)
Finding the interpolating points(independent of parametrization):
1. arc length
2. curvature
(1) curvature extrema
(2)inflection point
![Page 13: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/13.jpg)
Only baesd on arc length :
![Page 14: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/14.jpg)
Based on curvature distribution:
![Page 15: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/15.jpg)
![Page 16: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/16.jpg)
![Page 17: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/17.jpg)
Adaptive Knot Sequence Generation(AKSG)
![Page 18: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/18.jpg)
![Page 19: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/19.jpg)
AKSG:
if , insert a auxiliary knot in the middle of the
segment
1
1
1 1
i i
i ior
i i
i x x
![Page 20: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/20.jpg)
Adaptive knot placement in B-spline curve approximation Weishi Li, Shuhong Xu, Gang Zhao, Li Ping Goh
Computer-Aided Design 2005
![Page 21: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/21.jpg)
a heuristic rule for knot placement
Su BQ,Liu DY:<<Computational geometry—curve and surface modeling>>
approximation interpolation
![Page 22: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/22.jpg)
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
![Page 23: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/23.jpg)
smoothing of discrete curvature:
Lowpass fliter
![Page 24: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/24.jpg)
divide into several subsets:inflection points
![Page 25: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/25.jpg)
iteratively bisect each segment untill satisfy the heuristic rule:
curvature integration
Newton-Cotes formulae
![Page 26: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/26.jpg)
check the adjacent intervals that joint at a feature point
![Page 27: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/27.jpg)
Example:
![Page 28: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/28.jpg)
![Page 29: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/29.jpg)
![Page 30: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/30.jpg)
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
![Page 31: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/31.jpg)
![Page 32: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/32.jpg)
Advantage:
1. compare with KTP and NKTP:when |m-n| is small, it is sensitive to parameter values.
![Page 33: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/33.jpg)
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
![Page 34: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/34.jpg)
Determination of konts:
2
1 ( )
1 ( 1,..., 1)
1
i p
p i f jj i
t t i n pp
are the parameter values of pointskt
kpkp
jd ( ) :f j
![Page 35: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/35.jpg)
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
{ }kp
![Page 36: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/36.jpg)
Selection of seed points:
local curvature maximum(LCM) points, inflection points
LCM: and1i ik k 1i ik k
exclude / 4i low avgk k k
base curve
251 input points
![Page 37: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/37.jpg)
Base curve with 16 control points
10 initial dominant points
![Page 38: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/38.jpg)
Choice of a new dominant points:
max deviation:
The segment is to be refined.
,s eS j sd p 1j ed p
|| ( ) ||i iC t p
, ,min | |w s w w e where s w e
choosing wp
0 1r shape index
![Page 39: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/39.jpg)
10 dominant points 13 dominant points
![Page 40: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/40.jpg)
Experimental results
Comparing:
![Page 41: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/41.jpg)
![Page 42: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/42.jpg)
![Page 43: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/43.jpg)
Future:
1. parameterization
2. optimal selection of dominant points as genetic
algorithm
3. B-spline surface and spatial curve
![Page 44: B-spline curve approximation zhu ping 08.02.20. Outline 1. Application 2. Some works 3. Discussion](https://reader030.vdocuments.mx/reader030/viewer/2022033022/56649e055503460f94af1acc/html5/thumbnails/44.jpg)
Thanks !