Exponential Quasi-interpolato Exponential Quasi-interpolato ry Subdivision Scheme ry Subdivision Scheme Yeon Ju Lee and Jungho Yoon Yeon Ju Lee and Jungho Yoon Department of Mathematics, Ewha W. University Seoul, Korea
Jan 04, 2016
Exponential Quasi-interpolatory SubdiviExponential Quasi-interpolatory Subdivision Schemesion Scheme
Yeon Ju Lee and Jungho Yoon Yeon Ju Lee and Jungho Yoon
Department of Mathematics, Ewha W. University Seoul, Korea
Exponential quasi-interpolatory s.s.
ContentsContents
Subdivision scheme – several type of s.s. Quasi-interpolatory subdivision scheme Construction Smoothness & accuracy Example Exponential quasi-interpolatory subdivision scheme Construction Smoothness Example
Exponential quasi-interpolatory s.s.
Subdivision schemeSubdivision scheme
Useful method to construct smooth curves and surfaces in CAGD
The rule :
Exponential quasi-interpolatory s.s.
Subdivision schemeSubdivision scheme
Rule :
Interpolatory s.s. & Non-interpolatory s.s
Stationary s.s. & Non-stationary s.s
Exponential quasi-interpolatory s.s.
B-spline subdivision schemeB-spline subdivision scheme
It has maximal smoothness Cm-1 with minimal support. It has approximation order only 2 for all m. Cubic-spline :
Exponential quasi-interpolatory s.s.
Interpolatory subdivision schemeInterpolatory subdivision scheme
4-point interpolatory s.s. :
The Smoothness is C1 in some range of w. The Approximation order is 4 with w=1/16.
Exponential quasi-interpolatory s.s.
Goal
We want to construct a new scheme
which has good smoothness and approximation order.
Exponential quasi-interpolatory s.s.
Quasi-interpolatory subdivision schemeQuasi-interpolatory subdivision scheme Construction
Exponential quasi-interpolatory s.s.
Quasi-interpolatory subdivision schemeQuasi-interpolatory subdivision scheme Advantage L : odd (L+1,L+2)-scheme. So in even pts case, it has
tension. L : even (L+2,L+2)-scheme. It has tension in both
case.
This scheme has good smoothness.
It has approximation order L+1.
Exponential quasi-interpolatory s.s.
Quasi-interpolatory subdivision schemeQuasi-interpolatory subdivision scheme
The mask set of cubic case
In cubic case, the mask can reproduce polynomials up to degree 3.
odd case : use 4-pts
even case : use 5-pts with tension v
Exponential quasi-interpolatory s.s.
Quasi-interpolatory subdivision schemeQuasi-interpolatory subdivision scheme
Various basic limit function which start with
-6 -4 -2 0 2 4 6-0.2
0
0.2
0.4
0.6
0.8
1
1.2
v=0
Limit functions
v=0.02
v=0.04
v=0.06
Cubic Spline
Exponential quasi-interpolatory s.s.
Quasi-interpolatory subdivision schemeQuasi-interpolatory subdivision scheme
Exponential quasi-interpolatory s.s.
Quasi-interpolatory subdivision schemeQuasi-interpolatory subdivision scheme
Exponential quasi-interpolatory s.s.
Quasi-interpolatory subdivision schemeQuasi-interpolatory subdivision scheme
Comparison of schemes which use cubic
Cubic-spline 4-pts interp. s.s. Sa
Where L=3
Support of limit ftn [-2, 2] [-3, 3] [-4, 4]
MaximalSmoothnes
sC2 C1 C3
Approximation
Order2 4 4
Exponential quasi-interpolatory s.s.
ExampleExample
1 2 3 4 5 6 70.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
v=0
v=0.02
v=0.04
v=0.06
Cubic-spline
Exponential quasi-interpolatory s.s.
Comparison with some Comparison with some exampleexample
Example < cubic-spline > < Sa >
E=0.8169 E=0.1428
1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8-10
-5
0
5
10
1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.80
0.2
0.4
0.6
0.8
1
1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8-10
-5
0
5
10
1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.80
0.05
0.1
0.15
0.2
Exponential quasi-interpolatory s.s.
Quasi-interpolatory subdivision schemeQuasi-interpolatory subdivision scheme
General caseL Mask set Sm
.Range of tension
3 O=[-1/16,9/16,9/16,-1/16] E= [-v, 4v,1-6v,4v,-v]
C3 0.0208<v<0.0404
4 O=[-v,–77/2048+5v,385/512-10v, 385/1024+10v,-55/512-5v,35/2048+v] E(i)=O(7-i) for i=1:6
C3 -0.0106<v<-0.0012
5 O=[3,–25,150,150,–25,3]/256] E=[-v,6v,–15v,1+20v,-15v,6v,-v]
C4 -0.0084<v<-0.0046
6 O=[-v,385/65536+7v,–2079/32768-21v, 51975/65536+35v,5775/16384-35v, -7245/65536+21v,945/32768-7v,-231/65536+v] E(i)=O(9-i) for i=1:8
C4 0.0007<v<0.0017
7 O=[-5,49,–245,1225,1225,–245,49,–5]/2048 E=[-v, 8v,–28v,56v,1-70v,56v,–28v,8v,–v]
C5 0.0012<v<0.0015
Exponential quasi-interpolatory s.s.
Exponential quasi-interpolatory s.s.Exponential quasi-interpolatory s.s. Construction
Exponential quasi-interpolatory s.s.
Analysis of non-stationary Analysis of non-stationary s.s.s.s.
Exponential quasi-interpolatory s.s.
Exponential quasi-interpolatory s.s.Exponential quasi-interpolatory s.s.
Exponential quasi-interpolatory s.s.
Exponential quasi-interpolatory s.s.Exponential quasi-interpolatory s.s.
Example
E=7.7716e-016 E=0.1434
1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8-10
-5
0
5
10
1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.80
0.05
0.1
0.15
0.2
-1 -0.5 0 0.5 1-1
-0.5
0
0.5
1
Exponential quasi-interpolatory s.s.
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