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.

Next StudyNext Study