Degree reduction of Bézier curve/surface
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Transcript of Degree reduction of Bézier curve/surface
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Outline
Introduction of degree reduction in CAGD Related work Degree reduction of curves Degree reduction of tensor product Bézier surfaces Degree reduction of triangular Bézier surfaces Our work and future work
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Problem Statement
Degree from to
Input: control points of
Output: control points of
Objective function:
di p R P
di q R Q
mind P,Q
n m
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Applications
Data transfer and exchange Data compression Data comparison Surface intersection Curve smoothness Boolean operations and rendering
Michael S. Floater, High order approximation of rational curves by polynomial curves, Computer Aided Geometric Design 23 (2006) 621–628
CONSURF BUILD UNISURF CATIA COMPAC Geomod PADL GEMS
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Early work
Based on the control points approaching Inverse of elevation
Forrest, A.R., Interactive interpolation and approximation by Bézier curve,
The Computer Journal, 15(1972), 71-79. G. Farin, Algorithms for rational Bezier curves, Computer Aided Design 15
(1983) 73–77.
Approximate conversion Danneberg, L., and Nowacki, H., Approximate conversion of surface represe
ntations with polynomial bases, Computer Aided Geometric Design, 2(198
5), 123-132. Hoschek, J., Approximation of spline curves, Computer Aided Geometric D
esign, 4(1987), 59-66.
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Early work
Constrained optimization Moore, D. and Warren, J., Least-square approximation to Bezier curves and
surfaces in James Arvo eds. Computer Gemes (II), Academic Press, New Yo
rk, 1991. Lodha, S. and Warren, J., Degree reduction of Bezier simplexes, Computer
Aided Design, 26(1994), 735-746. Perturbing control points
胡事民, CAD 系统数据通讯中若干问题的研究 : [ 博士学位论文 ], 杭州 , 浙江大学数学系 , 1996.
Hu, S.M., Sun, J.G., Jin T.G., et al., Approximate degree reduction of Bezier curves, Tsinghua Science and Technology, 3(1998), 997-1000.
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Early work
Based on the basis transformation Watkins, M. and Worsey, A., Degree reduction for Bézier curves, Computer Ai
ded Design, 20(1988), 398-405. Lachance, M.A., Chebyshev economization for parametric surfaces. Computer
Aided Geometric Design, 5(1988), 195-208. Eck, M., Degree reduction of Bézier curves, Computer Aided Geometric Desig
n, 10(1993), 237-257.69 Bogacki, P., Weinstein, S. and Xu, Y., Degree reduction of Bézier curves by un
iform approximation with endpoint interpolation, Computer Aided Design, 27(1995), 651-661.
Eck, M., Least squares degree reduction of Bézier curves, Computer Aided Design, 27(1995), 845-851.48
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Recent work
Optimal multi-degree reduction Chen Guodong, Wang Guojin, Optimal multi-degree reduction of Bézier curves
with constraints of endpoints continuity. Computer Aided Geometric Design, 2002,19: 365-377
Zheng, J., Wang, G., Perturbing Bézier coefficients for best constrained degree reduction in the -norm. Graphical Models 2003, 65, 351–368.
Zhang Renjiang and Wang Guojin, Constrained Bézier curves’ best multi-degree reduction in the -norm, Progress in Natural Science, 2005, 15(9): 843-850
Others
2L
2L
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Key progress
( , ) ( , ) ( , )00,0 0,1 0,
( , ) ( , ) ( , )1,1 1, 1, 1 I I I1
0 1 1 2 1
( , )( , ) ( , ),, , 1
1
2
1
, , , , , ,
nm n m n m nn m
m n m n m n nn m n m
r r r n s
m nm n m n nr n m rr r r r n r m
nr
nr
nn s
Bb b b
b b b B
bb b B
B
B
B
Q Q Q P P P
( , ) ( , ), ,
1 ( , )( , )1, 11, 1 1
( , ) ( , ) ( , ), , 1 ,
0 1 1 1
, , ,
, , , , , , , , ,
nm n m nm sm s m s m s n s
m s m m m nm n nm nm m n
m n m n m n nm m m n m n n
r r n s n s
Bb b
bb B
b b b B
Q Q Q
P P P P P P
0
1
,
n
n nnnn
B
B
B
P
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Key progress
I 1 1 II 1 1 II( ) (1 ) ( ) (1 )s r s rn N N Nt t t t t t P P P B
II II 2 2 2 2 (2 2,2 2) III (2 2,2 2)( r , s ) r s r sN N N N N N N N
P B P E J P J
Jacobi
1, 1 III (2 2,2 2)1 r ssr s rMm Mt t t
P JJ
1
III (2 2,21 ) V
1
1 1 121 1s sr
m sr s m
M M M M ii
ri
r
t t t tt
P J P B Q B
B--J
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Strength
Optimal Multi-degree reduction Explicit expression Precise error Less time consuming
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Idea
Jacobi polynomial
Basis transformation
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Key progress
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Jacobi polynomial
.
.
.
mn
mndxxJxJxx rs
n
rsm
rsn
rs
,
,011 ,
,1
1
,
, , , , , ,1 1 0 0
s r s r s r s r s r s rn n n n nx b J x b J x b J x Q
n
k
krs
n
x
k
krsn
kn
snxJ
0
),(
2
1)(
1,1 , 1, 1 .x s r
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Others
Lutterkort, D., Peters, J., Reif, U., 1999. Polynomial degree reduction in the -norm equals best Euclidean approximation of Bézier coefficients. Computer Aided Geometric Design 16, 607–612.
Ahn, Y.J., Lee, B.G., Park, Y., Yoo, J., 2004. Constrained polynomial degree reduction in the -norm equals best weighted Euclidean approximation of Bézier coefficients. Computer Aided Geometric Design 21, 181–191.
2L
2L
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Optimal multi-degree reduction of Bézier
curves with -continuity
Lizheng Lu , Guozhao Wang
Computer Aided Geometric Design 23 (2006) 673–683
2G
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Problem statement
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Motivation
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condition1G
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Algorithm for -constrained degree reduction
1G
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Least square method
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Influence of the parameters , 0 1
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A weakness
The approximation curve will be singular at the endpoint when or is nearly equal to 0.0v v
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Regularization
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Conjugate gradient method
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Algorithm 1
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Improvement of the singularities
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Remark
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-constrained degree reduction2G
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Example 1
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Example 2
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Example 3
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Degree reduction of tensor product Bézier surfaces
1 1
0 0 0 0
0 0 , 0 0
, , 0 0
, ,
0 ,0 ; 0,1; 0,1.
nm n m
u v u u v v
u v u vu v u v
o o u v
P Q
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Related work 陈发来 , 丁友东 , 矩形域上参数曲面的插值降阶逼近 , 高等学校计算数学学
报 ( 计算几何专辑 ),1993,7,22-32 Hu Shimin, Zheng Guoqin, Sun Jiaguang. Approximate degree reduction of recta
ngular Bézier surfaces, Journal of Software, 1997, 4(4): 353-361 周登文 , 刘芳 , 居涛 , 孙家广 , 张量积 Bézier 曲面降阶逼近的新方法 , 计
算机辅助设计与图形学学报 , 2002 14(6), 553-556 Chen Guodong and Wang Guojin, Multi-degree reduction of tensor product Bézi
er surfaces with conditions of corners interpolations, SCIENCE IN CHINA, Series F,2002, 45(1): 51~58
郭清伟 , 朱功勤 , 张量积 Bézier 曲面降多阶逼近的方法 , 计算机辅助设计与图形学学报 , 2004,16(6)
章仁江 , CAGD 中曲线曲面的降阶与离散技术的理论研究 : [ 博士学位论文 ], 杭州 , 浙江大学数学系, 2004.
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1 1
1 1 1 1
1 1
1 1
1 1 1 1 1 1
1 1
1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1
0 0 ,1 1
1 1
0 0 , 00 0 01 1
0, 0 ,0 0 ,
( , ) ( ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
m mm n m n
nm j j n j j nj j
n nn m n m n m
i i i m j mi i
n m n m n mm m n n n m n m
u v B v B u B v B u
B u B v B u B v B u B v
B u B v B u B v B u B
P Q Q
Q Q Q
Q Q Q
1
1
1 1
1 1 1 1
III III III00 01 0,
III IIITT 10 1,
III III III,0 ,1 ,
T M
( ))
( ) ( ) ( ) ( )
( ) ( )
0
m
mn n m m n m
n n n m
n n m m
v
u v u v
u v
P P P
P PB B B B
P P P
B B
P
P
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mm
mnnn
m
m
nnmnmnijmn
1
1111
1
1
1
III,
III1,
III0,
III,1
III10
III,0
III01
III00
T
)1()1(
MM 0 E EPP
PPP
PP
PPP
P
-
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1
1
2
0
2
0
22M1
1
M )()(~
)1()1()()(n
i
n
i
m
j
mj
niij
m
j
mj
niij vBuBvvuuvBuB PP
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Our work
Multi-degree reduction of tensor
product Bézier surfaces in norm
Without constraintsWith interpolation
at endpoints
With boundary
constraints
Best Best locally Better
2L
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Fruit 1
Control points
Approximate error
1 1 1 1 1 1
1 1
1
1
, , , , , , , , , ,
, , , , .
T T Tn n U m m U n n m m U n n U m mUR DL
Tn n
n
TmU m m UDR
Q B P B G B H H B G
H B H
F
F
1 1 1 1 1 1
1 1 1 1 1 1
, , , ,1 1 1 1 1 1 1 1
1
2
T T Tn m m n n m n n m mUR DL DRn m m n n m n n m mI Z B Z Z B Z Z B Z
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Fruit 2
Control points are
Error bound is
1 1
1 11 1
1 1 1 11 1
,00 0, 0,
, ,,0 ,11 1
,,0 1, ,
1 1
TI u c Im m
v c C v cn nn m
TI u c In m n m
n m
p P p
Q P Q P
p P p
1 1 1 1
2, , 0 1 0 1
1 1 1( , )= .
1 1 16u u v v
nm n m nm n m Ld E
n m
P Q P Q
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Example 1
Guo 0.4121 Chen 0.3183 Zhou 0.0421
Given a degree 6 6 surface,we will present the 1 1 reduced surface with
corners 0 continuous.
Bézier
Original surface
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Error surface
Guo 0.4121 Chen 0.3183
Zhou 0.0421
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Example 2
Given a degree 6 5 surface,we will present the 2 1 reduced surface with
corners 1 continuous.
Bézier
0.Gu 54o 18 0C .he 67n 04 0Z .ho 26u 04
Original surface
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Error surface
0.Gu 54o 18 0C .he 67n 04
0Z .ho 26u 04
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Example 3
Given a degree 7 7 surface,we will present the 2 2 reduced surface with
corners 1 continuous.
Bézier
0.Gu 35o 46 0Z .ho 32u 02 0C .he 07n 22
Original surface
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Error surface
0.Gu 35o 46 0C .he 07n 22
0Z .ho 32u 02
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Key progress1 1 2 2
M M 2 2
1 1 0 0
( ) ( ) (1 ) (1 ) ( ) ( )n m n m
n m n mij i j ij i j
i j i j
B u B v u u v v B u B v
P P
2 2 2 2
1,1 1,1M 2 2 M
0 0 0 0
( ) ( ) (2 1) (2 1)n m n m
n mij i j ij i j
i j i j
B u B v J u J v
P P
1 1
1 1
1 1
1 1
1 1
2 21,1 1,1M
0 0
2 22 2M
0 0
1 1
1 1
(1 ) (1 ) (2 1) (2 1)
ˆ(1 ) (1 ) ( ) ( )
( ) ( )
n m
ij i ji j
n mn m
ij i ji j
n mn m
ij i ji j
u u v v J u J v
u u v v B u B v
B u B v
P
P
Q Jacobi
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Degree reduction of triangular Bézier surfaces
Refer to the report of Lizheng Lu in the Ph.D student seminar on Sep. 13
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Related work Hu SM, Zuo Z, Sun JG. Approximate degree reduction of triangular Bézier surface.
Tsinghua Science and Technology 1998;3(2):1001–4 Rababah A. degree reduction of triangular Bézier surfaces with common tangent
planes at vertices. International Journal of Computational Geometry & Applications 2005;15(5):477–90.
郭清伟 , 陶长虹 , 三角 Bézier 曲面的降多阶逼近 . 复旦学报 ( 自然科学版 ) 2006 Vol.45 No.2 P.270-276
Lizheng Lu, Guozhao Wang, Multi-degree reduction of triangular Bézier surfaces with boundary constraints. Computer-Aided Design 38 (2006) 1215–1223
2L
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Future work
Optimal approximation in various norm
Geometry continuous
Reduce the degree of a Bézier surface composed of some small Bézier surface holistically
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Thanks!Thanks!
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A lemma
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A lemma
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Gerald Farin
Degree: Ph.DUniversity of Braunschweig, 1979
Biography:
Gerald Farin joined ASU in 1987. He has also worked at the University of Utah and spent four years in CAD/CAM development at Mercedes-Benz, Stuttgart, Germany. He has taught CAGD tutorials worldwide and has given more than 100 invited lectures worldwide.
Research:- Computer Aided Geometric Design
- NURBS
- Modeling 3D