B-spline curve approximation
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Transcript of B-spline curve approximation
B-spline curve approximation
zhu ping
08.02.20
Outline
1. Application
2. Some works
3. Discussion
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
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
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.
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
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
Associate Professor in the Division of Computing Studies, CAD,CAGD&CGFarin’s student
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.
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
Estimate the number of points required to interpolate (ENP)
Approximated by a finite number of circular arc segments
Finding the interpolating points(independent of parametrization):
1. arc length
2. curvature
(1) curvature extrema
(2)inflection point
Only baesd on arc length :
Based on curvature distribution:
Adaptive Knot Sequence Generation(AKSG)
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
Adaptive knot placement in B-spline curve approximation Weishi Li, Shuhong Xu, Gang Zhao, Li Ping Goh
Computer-Aided Design 2005
a heuristic rule for knot placement
Su BQ,Liu DY:<<Computational geometry—curve and surface modeling>>
approximation interpolation
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
smoothing of discrete curvature:
Lowpass fliter
divide into several subsets:inflection points
iteratively bisect each segment untill satisfy the heuristic rule:
curvature integration
Newton-Cotes formulae
check the adjacent intervals that joint at a feature point
Example:
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
Advantage:
1. compare with KTP and NKTP:when |m-n| is small, it is sensitive to parameter values.
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
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
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
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
Base curve with 16 control points
10 initial dominant points
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
10 dominant points 13 dominant points
Experimental results
Comparing:
Future:
1. parameterization
2. optimal selection of dominant points as genetic
algorithm
3. B-spline surface and spatial curve
Thanks !