Study of Subdivision Schemes and their impact on Geometric ...prr.hec.gov.pk/.../9098/...HSR_2017_IUB_28.03.2018.pdf · Study of Subdivision Schemes and their impact on Geometric

Study of Subdivision Schemes andtheir impact on Geometric Modeling

and Computer Graphics

By

Robina Bashir

A dissertation submitted in partial fulfillment

of the requirements for the degree of

Doctor of Philosophyin

Mathematics

Department of Mathematics

The Islamia University of Bahawalpur

Bahawalpur 63100, PAKISTAN

2017

Study of Subdivision Schemes andtheir impact on Geometric Modeling

and Computer Graphics

By

Robina Bashir

A dissertation submitted in partial fulfillment

of the requirements for the degree of

Doctor of Philosophyin

Mathematics

Supervised By

Prof. Dr. Ghulam Mustafa

Department of Mathematics

The Islamia University of Bahawalpur

Bahawalpur 63100, PAKISTAN

2017

Declaration

I, Robina Bashir, solemnly declares that the research work presented in this dis-

sertation entitled "Study of Subdivision Schemes and their impact on Geometric

Modeling and Computer Graphics" is my own otherwise acknowledged. This

work has not been submitted as a whole or in part for any other degree to any

other university in Pakistan or abroad.

ROBINA BASHIR

Email: [email protected]

Approval

It is certified that Robina Bashir has completed this dissertation/research work

entitled "Study of Subdivision Schemes and their impact on Geometric Model-

ing and Computer Graphics" for the degree of Doctor of Philosophy in Mathe-

matics under my supervision.

(Supervisor/Chairman)

PROF. DR. GHULAM MUSTAFA

The Islamia University of Bahawalpur, Pakistan

Email: [email protected]

[email protected]

Certificate

It is hereby certified that work presented by Ms. Robina Bashir D/O Muham-

mad Bashir in the thesis titled "Study of Subdivision Schemes and their impact

on Geometric Modeling and Computer Graphics" has been successfully pre-

sented/defended and is accepted in its present form as satisfying the require-

ments for the degree of Doctor of Philosophy in the Department of Mathematics

and Faculty of Sciences The Islamia University of Bahawalpur.

Candidate’s Name

Robina Bashir

Supervisor/Chairman

Prof. Dr. Ghulam Mustafa

External Examiner

External Examiner

Department Name Mathematics

Dean

Faculty Name Sciences

Date: ——–

Dedication

I WOULD LIKE TO DEDICATE MY THESIS TO MY

Beloved Fatherand

Sweet Mother

WHO ALWAYS PICKED ME UP ON TIME AND

ENCOURAGED ME TO GO ON EVERY ADVENTURE

ESPECIALLY THIS ONE

Acknowledgments

I offer all the praises and deepest gratitude to Almighty ALLAH, the most

gracious, the most merciful and to His Holy Prophet Muhammad (Peace be

upon him), a teacher of the whole humanity and a source of inspiration and

guidance throughout my life.

I owe a scholarly debt of gratitude to Prof. Dr. Ghulam Mustafa, my super-

visor & Chairman Department of Mathematics, whose charisma, skill and con-

cern surpassed all understanding. This task would not have been accomplished

without his brilliant and devoted supervision. I extend my deepest thanks and

felicitation for his monumentally scholarly enterprise and giving me the chance

to make an enchanting voyage into the conglomerates of the present study.

Special thanks are for my husband, parents and family for their continu-

ous support and encouragement throughout my whole educational period and

Ph.D. study. Without their support, and prayers for me, I cannot finish my Ph.D.

studies.

I acknowledge that this research work is supported by Indigenous Ph. D 5000

Fellowship Program and National Research Program for Universities (NRPU)

Project No. 3183 of Higher Education Commission (HEC) of Pakistan.

Robina Bashir

Abstract

Subdivision is an efficient tool to explain curves and surfaces in geometric mod-

eling and computer aided geometric design. Subdivision schemes are very help-

ful techniques to produced smooth curves and surfaces from finite set of con-

trol points. The aim of this dissertation is to introduce variety of subdivision

schemes for curve and surface designing based on complexity, arity and param-

eter. Several simple and well-organized formulae are presented which gener-

ate the different kind of parametric and non-parametric subdivision schemes.

Many well known existing schemes are generated by proposed formulae. Con-

vergence and smoothness of curves and surfaces subdivision schemes are p-

resented by using Laurent polynomial method. Shape preserving properties

such as monotonicity, convexity and concavity preservation of data fitting are

derived. Some of significant properties of proposed subdivision schemes such

as Hölder regularity, polynomial generation, polynomial reproduction, approx-

imation order and support of basic limit function are also discussed. Visual

performances of the schemes have also been demonstrated through different

examples.

Contents

Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Approval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Certificate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 Introduction 1

1.1 Literature survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Convergence and smoothness analysis . . . . . . . . . . . . . . . 9

1.4 Our contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5 Outline of dissertation . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Four-point n-ary interpolating subdivision schemes 14

2.1 Multi-step Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1.2 Analysis of subdivision schemes . . . . . . . . . . . . . . . 19

2.2 Properties of subdivision schemes . . . . . . . . . . . . . . . . . . 21

2.2.1 Hölder regularity . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.2 Polynomial generation . . . . . . . . . . . . . . . . . . . . . 24

2.2.3 Polynomial reproduction and approximation order . . . . 25

2.3 Numerical examples and conclusion . . . . . . . . . . . . . . . . . 29

3 A class of shape preserving 5-point n-ary approximating schemes 30

3.1 Algorithm for construction of schemes . . . . . . . . . . . . . . . . 30

3.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.1.2 Smoothness analysis of proposed schemes . . . . . . . . . 33

3.2 Shape preserving properties . . . . . . . . . . . . . . . . . . . . . 34

3.2.1 Monotonicity preservation . . . . . . . . . . . . . . . . . . 34

3.2.2 Convexity preservation . . . . . . . . . . . . . . . . . . . . 43

3.2.3 Concavity preservation . . . . . . . . . . . . . . . . . . . . 54

3.2.4 Demonstration . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.3 Traditional properties of schemes . . . . . . . . . . . . . . . . . . 67

3.3.1 Hölder exponent . . . . . . . . . . . . . . . . . . . . . . . . 67

3.3.2 Polynomial generation . . . . . . . . . . . . . . . . . . . . . 73

3.3.3 Polynomial reproduction and approximation order . . . . 74

3.3.4 Basic limit function . . . . . . . . . . . . . . . . . . . . . . . 76

3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4 A family of 6-point n-ary interpolating subdivision schemes 83

4.1 Three-step Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.1.2 Smoothness Analysis of Proposed schemes . . . . . . . . . 86

4.2 Properties of subdivision schemes . . . . . . . . . . . . . . . . . . 87


4.2.2 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . 93

4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5 3n-point quaternary shape preserving subdivision schemes 95

5.1 Shape preserving subdivision schemes of higher order . . . . . . 95

5.1.1 Convexity preservation . . . . . . . . . . . . . . . . . . . . 98

5.1.2 Concavity preservation . . . . . . . . . . . . . . . . . . . . 101

5.2 Numerical examples and comparison . . . . . . . . . . . . . . . . 104

5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6 Univariate approximating schemes and their non-tensor product gen-

eralization 108

6.1 Algorithm for univariate schemes . . . . . . . . . . . . . . . . . . . 109

6.1.1 Smoothness analysis of univariate schemes . . . . . . . . . 110

6.1.2 Response of univariate schemes to polynomial and mono-

tone data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113


6.1.4 Numerical experiments of univariate schemes . . . . . . . 118

6.2 Algorithm for non-tensor product schemes . . . . . . . . . . . . . 122

6.2.1 Smoothness analysis of bivariate proposed schemes . . . . 126

6.2.2 Response of non-tensor product schemes to polynomial

and monotone data . . . . . . . . . . . . . . . . . . . . . . . 128

6.2.3 Numerical experiments of non-tensor product schemes . . 135

6.2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

7 Generalization of binary tensor product schemes depending upon four

parameters 139

7.1 Algorithm for tensor product schemes . . . . . . . . . . . . . . . . 140

7.1.1 Univariate schemes . . . . . . . . . . . . . . . . . . . . . . . 140

7.1.2 Bivariate schemes . . . . . . . . . . . . . . . . . . . . . . . . 141

7.2 Polynomial generation and reproduction of bivariate schemes . . 145

7.3 Numerical examples and comparison . . . . . . . . . . . . . . . . 147

7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Bibliography 152

Publications of Robina Bashir 164

List of Tables

3.1 Monotone data set . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.2 Convex data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.3 Concave data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.1 Monotone data set . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.1 Convex data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.2 Concave data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.3 Smoothness of proposed schemes with existing schemes. . . . . . 106

6.1 The order of continuityO(C) of proposed binary approximating schemes

for certain ranges of parameter. . . . . . . . . . . . . . . . . . . . . . 111

6.2 Continuity of some members of the family of schemes . . . . . . . . . . 112

6.3 Monotone data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.4 The order of continuity O(C) of proposed non-tensor product schemes

with some existing non-tensor product schemes. . . . . . . . . . . . . 128

6.5 Monotone data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

7.1 Show the Continuity (C), polynomial generation (P. G) and poly-

nomial reproduction (P. R) of bivariate schemes . . . . . . . . . . . 151

List of Figures

2.1 Labeling of a sample control polygon. The newly inserted point between

old vertices b and c are referred to as p1, p2, . . . , pn−1 respectively. . . . 15


old vertices b and c are referred to as p1 and p2, respectively. . . . . . . 16


old vertices b and c are referred to as p1, p2 and p3 respectively. . . . . . 19

2.4 Comparison of the limit curves generated by proposed 4-point 2-ary, 3-

ary, 4-ary, 5-ary, 6-ary and 7-ary interpolating subdivision schemes at

1st subdivision level. . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1 Labeling of a control polygon. . . . . . . . . . . . . . . . . . . . . . . 32

3.2 The curves (a), (b), (c), (d) and (e) are generated by cubic Hermite

spline, Hussan and Bashir (2011), Tan et al. (2014), scheme (3.2) and

(3.3) by using monotone data set. . . . . . . . . . . . . . . . . . . . . 66

3.3 The curves (a), (b) and (c) are generated by rational cubic function

Hussan and Bashir (2011), scheme (3.2) and (3.3) respectively by using

monotone data set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.4 The convex curves (a), (b), (c), (d) and (e) are generated by Hao et al.

(2011), Tan et al. (2014), Cai (2009), Dyn et al. (1999), schemes (3.2)

and (3.3) respectively by using convex data set. . . . . . . . . . . . . . 69

3.5 The concave curves (a) and (b) are generated by scheme (3.2) and (3.3)

respectively by using concave data set. . . . . . . . . . . . . . . . . . 70

3.6 (a) Graph of the Hölder exponent against µ for the scheme (3.2). (b)

Graph of the Hölder exponent against µ for the scheme (3.3). . . . . . . 78

3.7 (a) and (b) show the effect of parameter on the shape of the basic limit

function of the scheme (3.2) and (3.3) respectively. . . . . . . . . . . . 81

3.8 (a) and (b) show the effect of parameter on the shape of limit curves of

the scheme (3.2) and (3.3) respectively. . . . . . . . . . . . . . . . . . 82


old vertices b and c are referred to as p1, p2, . . . , pn−1 respectively. . . . 84

4.2 The curves (a)and (b) are produced by schemes (4.5) and (4.6) respec-

tively by using monotone data set. . . . . . . . . . . . . . . . . . . . . 93

4.3 Both (a) and (b) show limit curves of the schemes (4.5) and (4.6) respec-

tively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.1 (a) and (b) are the convex curves generated by schemes Saβ,3and Saβ,7

respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.2 (a) and (b) are the concave curves generated by schemes Saβ,3and Saβ,7

respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.3 (a) and (b) Shows the increase in tightness of the curve with decreasing β.106

6.1 The curves (a), (b), (c) and (d) are generated by the schemes fa1,0,µ ,

fa1,1,µ , fa1,2,µ and fa1,3,µ by using monotone data set. . . . . . . . . . . . 119

6.2 Most expanded and most shrinked curves: The curves (a), (b), (c) and

(d) are generated by the schemes fa2,0,µ , fa1,2,µ , fa2,2,µ and Romani (2015)

respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.3 Interpolating behavior: The curves (a) , (b) and (c) are generated by the

schemes fa2,0,µ , fa1,2,µ and Romani (2015) respectively. . . . . . . . . . 121

6.4 Most expanded and most shrinked curves: The curves (a), (b) and (c)

are generated by the schemes fa1,0,µ , fa1,1,µ and Romani (2015) respectively.121

6.5 Interpolating behavior: The curves (a), (b) and (c) are generated by the

schemes fa1,0,µ , fa1,1,µ and Romani (2015) respectively. . . . . . . . . . 122

6.6 (a) Initial monotone data. (b) A monotonicity preserving surface ob-

tained by the proposed scheme fa1,0,µ. . . . . . . . . . . . . . . . . . . . 136

6.7 (a) Control mesh. (b)-(d) Limit surfaces obtained by the proposed schemes

fa1,0,µafter 5 steps of refinement. . . . . . . . . . . . . . . . . . . . . . 137

6.8 (a) Control mesh. (b)-(d) Limit surfaces obtained by the proposed schemes

fa1,1,µafter 5 steps of refinement. . . . . . . . . . . . . . . . . . . . . . 138

7.1 (a) Show the initial mesh. (b)-(d) Show the different refinement steps . 148

7.2 (a) Show the initial mesh. (b)-(d) Show the different refinement steps . 149

Chapter 1

Introduction

Geometric modeling plays a pivotal role to fulfil the gap between computer sci-

ence study and industry. It’s crucial in various areas particularly in mechanical

industry such as manufacturing air-crafts, digital devices, automobiles indus-

try, scientific and medical instruments, household product both for functions

and designing. In a routine life matters and issues, there is a wide range of ge-

ometric techniques. The multifarious branches of geometric designing include

Computer Aided Geometric Design, Multi-resolution and Diffusion, Comput-

er Graphics, Solid Geometry, Shape Abstraction and Modeling, Computational

Geometry and Computer Vision etc. are prominent.

We keep our main attention and concentration on Computer Aided Geometric

Design (CAGD), which is derived from the broader areas of Geometry, Com-

puter Algebra, Numerical Analysis, Computer Graphics, Data Structure and

Approximation Theory. CAGD is a branch of computational mathematics that

is mainly dealing with construction and explanation of curves and surfaces.

CAGD bears broad applications in manufacturing, surface modeling arising

the structure of cars, ship and airplanes, analysis and computational graphics,

planning and controlling surgery, visualizing products, automatically produc-

ing sectional drawing, representation of large data sets.

1

Subdivision, is the most crucial, significant and widely applied methods of

CAGD. Subdivision is well flourished field. During subdivision, rough and

unrefined shapes could be polished to generate more versatile, aesthetic and

visually attractive shapes. Subdivision is based on the idea of refining the ini-

tial grid or control polygon. Subdivision defines a smooth curve and surface

as the limit of a sequence of successive refinements. Subdivision curve can be

generated by repeatedly applying a subdivision technique to the control poly-

gon and it is continually used in refining the shapes to produce smooth curves

and surfaces. Subdivision schemes are widely uses in application of computer

graphics, 3-D geometrical measurements, image reconstruction, animation and

geometric designs, the design of curves or surfaces, the approximation of arbi-

trary functions, shape preservation in data and geometric objects.

1.1 Literature survey

The basic idea of subdivision was used by a French mathematician Rham (1947),

he introduced a scheme on cutting the corners of a polygon to obtain a smoother

curve. Soon after, a famous graphics designer Chaikin (1974), gave a new tech-

nique to generate uniform and smooth curves. His scheme was corner cutting

approximating scheme which generate C1 smooth B-spline curve after succes-

sive refinements. Doo and Sabin (1978) extended the Chaikin’s corner cutting

method for surface. Subdivision scheme introduced by Doo and Sabin gener-

ate C2 limit curve. Bézier (1985) gave the idea that every polynomial curve

can be represented by its Bézier polygon and his idea is highly used in design

and modeling. A mathematical way which has many applications as importan-

t theoretical tool for curve formulation was introduced by a famous European

engineer Casteljau (1986).

Boor (1987) determined that generalization of corner cutting Chaikin’s method

2

to generate smooth curves. Chaikin’s technique becomes a special case of algo-

rithms interoduced by Rham (1974). In the similar year, very familiar 4-point

interpolatory scheme for curves also known as "Classical four point scheme" in-

troduced by Dyn et al. (1987). Following that, Deslauriers and Dubuc (1989)

worked in more details for generalization of 4-point binary scheme to b-ary 2N

point schemes using mimicking construction. Weissman (1990) improved this

method, and proposed a 6-point binary interpolation scheme which is C2 con-

tinuous.

After that Dyn 4-point binary and Weissman 6-point binary interpolating schemes

can be generated by taking a convex combination of two DD schemes in Dyn

(2002b). Hassan et al. (2002) also presented a 4-point ternary interpolating sub-

division scheme with tension parameter, which is C2 for certain range of pa-

rameter. Hassan and Dodgson (2003) introduced three point binary and ternary

approximating schemes that produce a C3 and C2 curve respectively. Tang et

al. (2005) used Laurent polynomial method to fined the convergence and s-

moothness of the 4-point DD scheme which is C1. Khan and Mustafa (2008)

constructed a ternary six-point interpolating scheme that is C2 continues. Hor-

mann and Sabin (2008) proposed a family of subdivision schemes with symbol

ak(z) by convolution of uniform B-spline with kernel. Mustafa et al. (2009) pre-

sented m-point binary approximating subdivision scheme. Zheng et al. (2009a,

2009b) developed even symmetric 2n-point ternary approximating and (2n−1)-

point ternary interpolatory subdivision scheme. Mustafa and Khan (2009) con-

structed a new 4-point quaternary approximating subdivision scheme with one

shape parameter. Aslam et al. (2011) offered an explicit formula for the mask

of (2n−1)-point ternary interpolating and approximating subdivision schemes.

Mustafa et al. (2011), Ghaffar and Mustafa (2012) introduced generalization of

the families of odd-point and even-point ternary approximating schemes. A

family of (2n − 1)-point binary approximating schemes with free parameter

3

for curve designing was offered by Mustafa et al. (2013). Conti and Romani

(2013) proposed a strategy for constructing dual m-ary approximating subdi-

vision schemes of de Rham-type, starting from two primal schemes of arity 2

and m respectively. Khan and Mustafa (2013) introduced a new approach to

construct a non-tensor product C1 subdivision scheme for quadrilateral mesh-

es. Zheng et al. (2014a) introduced a general formula to generate a family

of integer-point binary approximating subdivision schemes with a parameter.

Ashraf et al. (2014) applied six point varient on Lane-Riesenfeld algorithm to

generate a family of subdivision schemes. Mustafa et al. (2014) presented a

family of binary univariate dual and primal subdivision schemes. Zheng et al.

(2014b) devised a multi-parameter method which generate a class of existing bi-

nary subdivision schemes. By using their method continuity of existing schemes

can be increased up to Ck+n by multiplying the factor(1+z2

)k with the symbol

of existing scheme. Romani (2015) introduced an algorithm which generate the

univariate and bivariate non-tensor product subdivision schemes with tension

parameter. Mustafa et al. (2016) introduced the 6-point interpolating subdivi-

sion scheme and also discussed the fractal properties of the scheme. An efficient

algorithm to design a family of binary approximating schemes was offered by

Mustafa et al. (2016).

Higher arity subdivision schemes give better results and less computational cost

as compared to the lower arity schemes. It is also observed that higher ari-

ty schemes have higher smoothness and approximation order than lower arity

schemes. Thatswhy, higher arity schemes are more atrective than lower arity

schemes. Lian (2008a, 2008b) offered 3, 4, 5 and 6-point a-ary interpolating sub-

division schemes by using Wavelet theory. Lian (2009) also introduced (2m) and

(2m+ 1)-point non-parametric a-ary interpolating subdivision schemes. Zheng

et al. (2009c) constructed p-ary subdivision generalizing B-splines. The general

formulae for the mask of (2b + 4)-point n-ary interpolating and approximating

4

schemes for any integer b ≥ 0 and n ≥ 2 were offered by Mustafa and Rehman

(2010). Mustafa et al. (2012) presented an explicit method for the mask of odd

points n-ary, for any odd n ≥ 3, interpolating subdivision schemes. Ghaffar et

al. (2012) constructed unification of 3-point approximating subdivision schemes

of varying arity. Mustafa and Bashir (2013) discussed 4-point n-ary interpolat-

ing subdivision schemes. Ghaffar et al. (2013a) developed 4-point α-ary ap-

proximating subdivision schemes. Hameed and Mustafa (2017) interoduced a

generalized algorithm which generate a family of a-point b-ary approximating

subdivision schemes with bell-shaped mask.

Muti-stage approach is very helpful to construct subdivision schemes. This idea

is firstly used by Catmull and Clark (1978). Catmull and Clark (1978) used three-

stages technique to present the original description of subdivision in which

each refinement is expressed in three stages. Later on, Lane and Riesenfeld

(1980) presented a unified framework to represent the uniform B-spline curves

and their tensor product extensions by a subdivision process. This framework

consist of two stages, the first stage doubles the control point by taking each

point twice and the second stage is the midpoint averaging of these points.

Zorin and Schröder (2001) introduced an increasing sequence of alternating pri-

mal/dual quadrilateral subdivision schemes by using multi-step approach. Os-

wald and Schröder (2003) used the same method to produced families of subdi-

vision schemes. Augsdörfer et al. (2010) first derived and analyzed families of

variations on the four-point binary scheme, he also used three-step technique.

The generalization of Lane-Riesenfeld algorithm was offered by Cashman et

al.(2013), they used same operator to define the refine and smoothing stage.

Shape preserving properties have the key roll in subdivision schemes, which

are regarded as geometrical properties of subdivision schemes. Shalmon (1993)

offered a family subdivision scheme for curve design which preserved mono-

tonicity. Cai (1995) introduced a four point interpolatory subdivision scheme

5

which generates C1 continuous curves in nonuniform control points and dis-

cussed the monotonicity preservation of the limit curve. Dyn et al. (1999) de-

scribed the convexity in the useful sense and is realized for data fulfilling cer-

tain conditions in addition to the convexity conditions. Hussain and Hussain

(2007) developed schemes for the visualization of monotone data. They, in their

work, also attained the degree of smoothness as C1. The convexity preserving

properties of the subdivision scheme (Hassan et al. 2002) has been discussed

in Cai (2009). Hao et al. (2011) introduced a linear 6-point binary approximat-

ing subdivision scheme which preserves convexity while its support is large.

Tan et al. (2014) presented only a binary four point subdivision scheme which

preserve monotonicity and convexity of the limit curve. Hussain et al. (2012)

presented a piecewise rational cubic function to preserve the shape of monoton-

ic data. Pitolli (2013) introduced ternary shape-preserving subdivision schemes

generated by bell-shaped masks. Han (2015) presented a convexity-preserving

approximation method which is similar to the cubic spline interpolation.

Dyn (1990) introduced a parametric butterfly interpolating scheme for surface

modeling that provides flexibility of modeling. Kobbelt (1996) and Zorin et al.

(1996) described generalized form of surface modeling of univariate schemes

presented by Dyn et al. (1987). Ghaffar et al. (2013b) developed a unified tech-

nique to design tensor product scheme. Mustafa and Randhawa (2014) con-

structed a univariate and bivariate parametric 3-point approximating scheme.

Mustafa et al. (2014) offered generalized and unified families of p-ary, (2n)

and (2n − 1)-point interpolating subdivision schemes and also presented ten-

sor product version of these families of schemes. Mustafa and Hameed (2017)

introduced families of parameter dependent univariate and bivariate subdivi-

sion schemes originated from quartic B-spline.

6

1.2 Basic definitions

Definition 1.2.1. Subdivision scheme describes a smooth curve and surface as

a limit of sequence of consecutive refinements. By this technique at each re-

finement level, the new inserted points on a better grid are calculated by affine

combination of previously existing points. In the limit of the recursive proce-

dure, data are defined on a dense set of points.

Definition 1.2.2. Arity of subdivision scheme The number of points inserted

at level k + 1 between two consecutive points from level k is called arity of the

scheme. In the case when number of points inserted are 2, 3, . . . , n, the subdivi-

sion schemes are called binary, ternary, . . . , n-ary, respectively.

Definition 1.2.3. Even-ary and odd-ary subdivision scheme If the even num-

ber of points are inserted between two consecutive points then the scheme is

called even-ary scheme and if the odd number of points are inserted between

two consecutive points then the scheme is called odd-ary scheme.

Definition 1.2.4. Complexity of subdivision scheme The number of points in-

volved in the affine combination to insert a new point at next subdivision level

is called complexity of the scheme. If the number of points involved is even

then scheme is called to be even-point scheme otherwise odd-point scheme.

Definition 1.2.5. Interpolating subdivision scheme If the points of the limit

curve or surface pass through initial control polygon/mesh, then the scheme

is calles as interpolating subdivision scheme.

Definition 1.2.6. Approximating subdivision scheme If the points of the limit

curve or surface may or may not pass through initial control polygon/mesh,

then the scheme is called approximating subdivision scheme.

7

Definition 1.2.7. Support of the scheme Support is equal to the number of s-

pans of the curve influenced when one control point is moved, or to the amount

of control points affecting a given point or a given span of the limit curve. The

area, over which a control point effects the shape of the limiting curve, should

be finite and small.

Definition 1.2.8. Continuity of the scheme denotes to the differentiability of

the limit curve or surface generated by subdivision process. Subdivision schemes

should be continuous of a certain order preceding to construction i.e. Cm conti-

nuity means that the first through mth derivatives are equal and continuous at

the shared points.

Definition 1.2.9. Dyn and Levin (2002) and Rioul (1992). "Hölder continuity

is an extension of the notion of continuity which gives more information about

any scheme. A function ϕ : R → R is define to be regular of order m + ψ

(for m ∈ N0 and 0 < ψ ≤ 1) if it is m times continuously differentiable and ϕm

is Lipschitz of order ψ

∣∣ϕ(m)(x+ h)− ϕ(m)(x)∣∣ ≤ c |h|ψ

for all x and h in R and some constant c.

Continuity of a subdivision curve is defined by just saying that if mth deriva-

tive of a curve exists everywhere in an interval and is continuous, then curve is

said to be Cm continuous in that interval. But the Hölder continuity of a subdi-

vision curve is a measure of how many derivatives are continuous, and of how

continuous the highest derivative is. Therefore we also need to find Hölder

continuity of the schemes to further explore their smoothness."

Definition 1.2.10. Basic limit function "The basic limit function of a subdivision

8

scheme is defined as the limit function of the scheme for the data f 0i = δi,0, where

δi,0 is Kronecker delta."

Definition 1.2.11. Polynomial generation Conti and Hormann (2011). "A con-

vergent subdivision scheme generates polynomials up to degree d ( that is, πd is

contained in the space of all limit functions), if and only if

a(k)(αjn) = 0, j = 1, 2, . . . , n− 1 for k = 0, . . . , d, ” (1.1)

Definition 1.2.12. Polynomial reproduction Conti and Hormann (2011). "A

subdivision scheme Sa reproduces polynomials of degree d if it is convergent

and if S∞a f

0 = p for any polynomial p ∈ πd and initial data f 0 = p(t0i ), i ∈ Z."

Definition 1.2.13. Parameterization of the scheme Conti and Hormann (2011).

"For a convergent subdivision scheme Sa we denote by τ = a′(1)n

the correspond-

ing parametric shift and attach the data f li for i ∈ Z, l ∈ N to the parameter

values

tli = tl0 +i

nlwith tl0 = tl−1

0 − τ

nl.” (1.2)

1.3 Convergence and smoothness analysis

Dyn et al. (1991). "A general compact form of univariate n-ary subdivision

scheme S which maps polygon fk = {fki }i∈Z to a refined polygon fk+1 = {fk+1i }i∈Z

is defined by

fk+1i =

∑j∈Z

anj−ifkj , i ∈ Z, (1.3)

where the set a = {ai : i ∈ Z} of coefficients is called the mask at k-th level of

refinement. A necessary condition for the uniform convergence of subdivision

9

scheme (1.3) is that∑j∈Z

anj =∑j∈Z

anj+1 = . . . =∑j∈Z

anj+n−1 = 1. (1.4)

A subdivision scheme is uniformly convergent if for any initial data f 0 = {f 0i :

i ∈ Z}, there exists a continuous function f such that for any closed interval

I ⊂ R, it satisfies

limk→∞

supi∈nkI

|fki − f(n−ki)| = 0.

Obviously, f = S∞f 0

A symbol called Laurent polynomial

a(z) =∑i∈Z

aizi, (1.5)

of the mask a = {ai : i ∈ Z} plays an efficient role to analyze the convergence

and smoothness of the subdivision scheme. From (1.4) and (1.5) the Laurent

polynomial of convergent subdivision scheme satisfies

a(ςjn) = 0, j = 1, 2, . . . , n− 1 and a(1) = n. (1.6)

where ςjn = exp(2πijn) are the nth root of unity. This condition guarantees the

existence of a related subdivision scheme for the divided differences of the orig-

inal control points and the existence of an associated Laurent polynomial

a(1)(z) = nzn−1

(1− z

1− zn

)a(z).

The subdivision scheme S1 with Laurent polynomial a(1) (z) , is related to the

scheme S with Laurent polynomial a(z) by the following theorem."

Theorem 1.3.1. Aspert (2003). "Let S denote a subdivision scheme with Laurent poly-

nomial a(z) satisfying (1.6). Then there exists a subdivision scheme S1 with the prop-

erty

△fk = S1△fk−1,

10

where fk = Skf 0 and △fk ={(△fk)i = nk(fki+1 − fki ); i ∈ Z

}. Furthermore, S is

a uniformly convergent if and only if 1nS1 converges uniformly to zero function for all

initial data f 0, in the sense that

limk→∞

(1

nS1

)kf 0 = 0.

The above theorem indicates that for any given scheme S, with the mask a sat-

isfying (1.4), we can prove the uniform convergence of S by deriving the mask

of 1nS1 and computing

∥∥( 1nS1)

i∥∥∞ for i = 1, 2, 3..., L, where L is the first integer

for which∥∥( 1

nS1)

L∥∥∞ < 1. If such an L exists, then S converges uniformly. Since

there are “n” rules for computing the values at the next refinement level, so we

define the norm

∥S∥∞ = max

{∑j∈Z

|anj|,∑j∈Z

|anj+1|,∑j∈Z

|anj+2|, . . . ,∑j∈Z

|anj+n−1|

}, (1.7)

and ∥∥∥∥∥(1

nSβ

)L∥∥∥∥∥∞

= max

{∑j∈Z

∣∣∣b[β,L]i+nLj

∣∣∣ ; i = 0, 1, 2, . . . , nL − 1

}, (1.8)

where

b[β,L](z) =1

nL

L−1∏j=0

aβ(znj

), (1.9)

and

aβ(z) =

(nzn−1

(1− z

1− zn

))aβ−1(z) =

(nzn−1

(1− z

1− zn

))βa(z), β > 1.”

Theorem 1.3.2. Aspert (2003). "Let S be the subdivision scheme with a characteristic

f-polynomial a(z) =(

zn−1nzn−1(z−1)

)mq(z), q ∈ f. If the subdivision scheme Sm, corre-

sponding to the f-polynomial q(z), converges uniformly, then S∞f 0 ∈ Cm(R) for any

initial control polygon f 0."

Corollary 1.3.3. Aspert (2003). "If S is a subdivision scheme of the form above and

1nSm+1 converges uniformly to the zero function for all initial data f 0, then S∞f 0 ∈

Cm(R) for any initial control polygon f 0.

11

The above Corollary 1.3.3 indicates that for any given n-ary subdivision scheme

S, we can prove S∞f 0 ∈ Cm by first deriving the mask of 1nSm+1 and then com-

puting∥∥∥( 1nSm+1

)i∥∥∥∞

for i = 1, 2, 3, ..., L (where L is the first integer for which∥∥∥( 1nSm+1

)L∥∥∥∞< 1). If such an L exists, then S∞f 0 ∈ Cm."

Theorem 1.3.4. Conti and Hormann (2011). "A convergent subdivision scheme Sa

reproduces polynomials of degree d with respect to the parameterizations (1.2) if and

only if

a(k)(1) = n

k−1∏l=0

(τ − l) and a(k)(αjn) = 0, j = 1, 2, . . . , n− 1 for k = 0, . . . , d,

where αjn = exp

(2πi

nj

), j = 1, 2, . . . , n− 1.”

Theorem 1.3.5. Dyn (2002a). "A convergent subdivision scheme Sa that reproduces

polynomial πn (set of polynomials at most degree n) has an approximation order of

n+ 1."

1.4 Our contribution

In this dissertation, we construct a family of 4-point n-ary interpolating subdi-

vision schemes by using multi-step algorithm based on divided difference. An

efficient algorithm is presented which generate a new class of shape preserving

relaxed 5-point n-ary approximating subdivision schemes. We discuss about

shape preserving properties like monotonicity, convexity and concavity preser-

vation of interpolating, approximating and relaxed subdivision schemes. We al-

so construct the general formulae which generate the univariate approximating

subdivision schemes and their generalization of non-tensor product bivariate

subdivision schemes. By using four parameters we introduce a family of bivari-

ate interpolating, approximating and relaxed subdivision schemes. The behav-

12

ior, influence and comparison of proposed schemes and other existing schemes

are shown by numerical examples, graphs and tables.

1.5 Outline of dissertation

Chapter 2 presents a family of 4-point n-ary interpolating schemes by using a

simple and efficient multi-step algorithm instead of using Lagrange polynomial

and wavelets theory.

Chapter 3 provides a new class of shape preserving relaxed 5-point n-ary ap-

proximating subdivision schemes. The shape preserving properties that is mono-

tonicity, convexity and concavity preservation of the limit functions are derived.

Chapter 4 gives a general algorithm based on divided difference to generate

a family of 6-point n-ary interpolating subdivision schemes rather than using

polynomials.

Chapter 5 presents an algorithm to construct 3n-point quaternary approximat-

ing subdivision schemes. It is to be observed that the proposed schemes have

bell-shaped mask with high continuity as compere to the existing schemes.

Chapter 6 deals with univariate binary approximating subdivision schemes and

their generalization to non-tensor product bivariate subdivision schemes. The

graphical comparison of proposed schemes with some existing schemes is also

given.

Chapter 7 gives two general formulas of parametric bivariate subdivision schemes.

The generalization of bivariate schemes depends upon four parameters.

13

Chapter 2

Four-point n-ary interpolating

subdivision schemes

In this chapter, we present an efficient and simple algorithm to generate 4-point

n-ary interpolating schemes. Our algorithm is based on three simple steps: Sec-

ond divided differences, determination of position of vertices by using second

divided differences and computation of new vertices. It is observed that 4-point

n-ary interpolating schemes are generated by completely different frameworks

(i.e Lagrange interpolant and wavelet theory). Furthermore, we have discussed

continuity, Hölder regularly, degree of polynomial generation, polynomial re-

production and approximation order of the schemes.

2.1 Multi-step Algorithm

We construct 4-point n-ary interpolating subdivision schemes by using three-

step algorithm instead of using Lagrange polynomial and wavelets theory etc.

These three steps are as follows:

• Calculate second divided differences

14

Figure 2.1: Labeling of a sample control polygon. The newly inserted point between old

vertices b and c are referred to as p1, p2, . . . , pn−1 respectively.

At each old vertex compute the second divided difference D, i.e Db is the

second divided difference at point b and Dc is the second divided differ-

ence at point c (See Figure 2.1).

Db =c− 2b+ a

n2, (2.1)

Dc =d− 2c+ b

n2,

where n = 3, 4, . . .

• Determine the position of vertices by using divided differences

In n-ary subdivision scheme each segment is divided into n sub-segments

at each refinement level. First point is inserted at the position 1n

, second

point at the position 2n

and proceeding in the same way the (n−1)-th point

at the position n−1n

. By using divided differences Db and Dc, we calculate

the position of (n− 1)-th newly inserted points between two old vertices b

and c by

Dpj =

(n− j

n

)Db +

(j

n

)Dc, j = 1, 2, 3, . . . , n− 1. (2.2)

• Computation of new vertices

Finally, we calculate positions of new vertices p1, p2,. . . , pn−1 by using Dp1 ,

15


vertices b and c are referred to as p1 and p2, respectively.

Dp2 ,. . . , Dpn−1 respectively by

Dp1 = p2 − 2p1 + b,

Dpi = pi+1 − 2pi + pi−1, (2.3)

Dpn−1 = c− 2pn−1 + pn−2,

where i = 2, 3, . . . , n − 2. By solving above set of equations, we get the

position of new vertices p1, p2, . . . , pn−1.

2.1.1 Examples

A 4-point ternary interpolating scheme:

In ternary subdivision scheme each segment is divided into three sub-segments

at each refinement level. One point is inserted at the position 13

and another

point at the position 23

(See Figure 2.2). For n = 3 in (2.1), we get second divided

differences Db and Dc at point b and c

Db =c− 2b+ a

9, (2.4)

Dc =d− 2c+ b

9.

16

For n = 3 in (2.2), we get

Dpj =3− j

3Db +

j

3Dc, j = 1, 2.

By using (2.4), we get

Dp1 =2a− 3b+ d

27, (2.5)

Dp2 =a− 2b+ 2d

27.

For n = 3 in (2.3), we have

Dp1 = p2 − 2p1 + b,

Dp2 = c− 2p2 + p1.

This implies

p1 =2b+ c− 2Dp1 −Dp2

3,

p2 =b+ 2c−Dp1 − 2Dp2

3.


p1 =−5a+ 60b+ 30c− 4d

81,

p2 =−4a+ 30b+ 60c− 5d

81.

Now 4-point ternary scheme can be written asfk+13i = fki ,

fk+13i+1 = − 5

81fki−1 +

6081fki + 30

81fki+1 − 4

81fki+2,

fk+13i+2 = − 4

81fki−1 +

3081fki + 60

81fki+1 − 5

81fki+2.

(2.6)

A 4-point quaternary interpolating scheme:

In quaternary subdivision scheme each segment is divided into four sub-segments

17

at each refinement level. First, second and third points are inserted at the posi-

tions 14, 24

and 34

respectively (See Figure 2.3). For n = 4 in (2.1), we get second

divided differences Db and Dc at point b and c

Db =c− 2b+ a

16, (2.7)

Dc =d− 2c+ b

16.

For n = 4 in (2.2), we get

Dpj =4− j

4Db +

j

4Dc, j = 1, 2, 3.


Dp1 =3a− 5b+ c+ d

64,

Dp2 =2a− 2b− 2c+ 2d

64, (2.8)

Dp3 =a+ b− 5c+ 3d

64.

For n = 4 in (2.3), we have

Dp1 = p2 − 2p1 + b,

Dp2 = p3 − 2p2 + p1,

Dp3 = c− 2p3 + p2.

This implies

p1 =3b+ c− 3Dp1 − 2Dp2 −Dp3

4,

p2 =b+ c−Dp1 − 2Dp2 −Dp3

2,

p3 =b+ 3c−Dp1 − 2Dp2 − 3Dp3

4.


p1 =−7a+ 105b+ 35c− 5d

128.

p2 =−1a+ 9b+ 9c− 1d

16.

p3 =−5a+ 35b+ 105c− 7d

64.

18


vertices b and c are referred to as p1, p2 and p3 respectively.

Now 4-point quaternary scheme can be written as

fk+14i = fki ,

fk+14i+1 = − 7

128fki−1 +

105128fki + 35

128fki+1 − 5

128fki+2,

fk+14i+2 = − 1

16fki−1 +

916fki + 9

16fki+1 − 1

16fki+2,

fk+14i+3 = − 5

128fki−1 +

35128fki + 105

128fki+1 − 7

128fki+2.

(2.9)

The above schemes (2.6) and (2.9) were introduced by Deslauriers and Dubuc

(1989) by using Lagrange interpolant. Later on, this scheme was also re-constructed

by Lian (2009) by using wavelet theory.

Remark 2.1.1. By substituting n ≥ 3 in (2.1)-(2.3), we get the mask of 4-point

n-ary interpolating scheme of [Deslauriers and Dubuc (1989), Lian (2009)].

2.1.2 Analysis of subdivision schemes

Here we present the analysis of 4-point ternary and quaternary interpolating

subdivision schemes. Analysis of other schemes can be done in the similar way.

19

Analysis of 4-point ternary subdivision scheme

The Laurent polynomial a(z) for the scheme (2.6) is

a(z) =1

81

{−4z5 − 5z4 + 30z2 + 60z1 + 81 + 60z−1 + 30z−2

−5z−4 − 4z−5}. (2.10)

Using (1.9) for n = 3, β = 1, 2 and L = 1, we get

b[1,1](z) =1

3a1(z) = − 4

81z5 − 1

81z4 +

5

81z3 +

26

81z2 +

29

81z1 +

26

81+

5

81z−1

− 1

81z−2 − 4

81z−3, (2.11)

and

b[2,1](z) =1

3a2(z) = − 4

27z5 +

1

9z4 +

2

9z3 +

17

27z2 +

2

9z1 +

1

9− 4

27z−1.(2.12)

If Sβ is the scheme corresponding to aβ(z) then by (1.8)∥∥∥∥13Sβ∥∥∥∥∞

= max

{∑j∈Z

|b[β,1]i+3j| : i = 0, 1, 2

}, β = 1, 2.

Using (1.7), (2.11) and (2.12), we get∥∥∥∥13S1

∥∥∥∥∞

= max

{∣∣∣∣−4

81

∣∣∣∣+ ∣∣∣∣2681∣∣∣∣+ ∣∣∣∣ 581

∣∣∣∣ , ∣∣∣∣−1

81

∣∣∣∣+ ∣∣∣∣2981∣∣∣∣+ ∣∣∣∣−1

81

∣∣∣∣} ,and ∥∥∥∥13S2

∥∥∥∥∞

= max

{∣∣∣∣−4

27

∣∣∣∣+ ∣∣∣∣1727∣∣∣∣+ ∣∣∣∣−4

27

∣∣∣∣ , ∣∣∣∣19∣∣∣∣+ ∣∣∣∣29

∣∣∣∣} .As we see ∥ 1

3S1∥∞ < 1 then by Theorem 1.3.1 the scheme is C0. Similarly

∥ 13S2∥∞ < 1 then by Corollary 1.3.3 the scheme is C1.

Analysis of 4-point quaternary subdivision scheme

The Laurent polynomial a(z) for the scheme (2.9) is

a(z) =1

128{−5z7 − 8z6 − 7z5 + 35z3 + 72z2 + 105z1 + 128 + 105z−1 + 72z−2

+35z−3 − 7z−5 − 8z−6 − 5z−7}. (2.13)

20


b[1,1](z) =1

4a1(z) = − 5

128z7 − 3

128z6 +

1

128z5 +

7

128z4 +

30

128z3 +

34

128z2 +

34

128z

+30

128+

7

128z−1 +

1

128z−2 − −3

128z−3 − −5

128z−4. (2.14)

and

b[2,1](z) =1

4a2(z) = − 5

32z7 +

2

32z6 +

4

32z5 +

6

32z4 +

18

32z3 +

6

32z2 +

4

32z

+2

32− 5

32z−1. (2.15)

If Sβ is the scheme corresponding to aβ(z) then by (1.8)∥∥∥∥14S1

∥∥∥∥∞

= max

{∑j∈Z

|b[β,1]i+4j| : i = 0, 1, 2, 3

}, β = 1, 2.

Using (1.7), (2.14) and (2.15), we get∥∥∥∥14S1

∥∥∥∥∞

= max

{∣∣∣∣−5

128

∣∣∣∣+ ∣∣∣∣ 30128∣∣∣∣+ ∣∣∣∣ 7

128

∣∣∣∣ , ∣∣∣∣−3

128

∣∣∣∣+ ∣∣∣∣ 34128∣∣∣∣+ ∣∣∣∣ 1

128

∣∣∣∣} ,and ∥∥∥∥14S2

∥∥∥∥∞

= max

{∣∣∣∣−5

32

∣∣∣∣+ ∣∣∣∣1832∣∣∣∣+ ∣∣∣∣−5

32

∣∣∣∣ , ∣∣∣∣ 232∣∣∣∣+ ∣∣∣∣ 632

∣∣∣∣ , ∣∣∣∣ 432∣∣∣∣+ ∣∣∣∣ 432

∣∣∣∣} .As we see ∥ 1

4S1∥∞ < 1 then by Theorem 1.3.1 the scheme is C0. Similarly

∥ 14S2∥∞ < 1 then by Corollary 1.3.3 the scheme is C1

2.2 Properties of subdivision schemes

In this section, we show that how limit curve of 4-point ternary and 4-point

quaternary subdivision schemes give response to initial polynomial data. For

this we discuss Hölder regularity, degree of polynomial generation, polynomial

reproduction and approximation order of the schemes (2.6) and (2.9).

21

2.2.1 Hölder regularity

According to Dyn and Levin (2002) and Rioul (1992), "Hölder regularity is an

extension of the notion of continuity which gives more information about any

scheme. A function ϕ : R → R is define to be regular of order y + α (for y ∈ N0

and 0 < ψ ≤ 1) if it is y time continuously differentiable and ϕy is Lipschitz of

order α

∣∣ϕ(y)(x+ h)− ϕ(y)(x)∣∣ ≤ c |h|ψ (2.16)

for all x and h in R and some constant c.

The Hölder regularity of subdivision scheme with symbol a(z) can be computed

in the following way. Let a(z) =(

1+z+...+zn−1

n

)kb(z), without loss of generality

we can assume b0, . . . , bm to be the non-zero coefficients of b(z) and letB0,B1,. . . ,

Bm be the m×m matrices with elements

(Bq)ij = bm+i−nj+q, i, j = 1, . . . ,m and q = 0, 1, . . . ,m. (2.17)

Then the Hölder regularity is given by r = k − logn(µ), where µ is the joint

spectral radius of the matrices B0, B1,. . . , Bm i.e.

µ = ρ (B0, B1, . . . , Bm) = lim supl→∞

(max

{∥ Bil . . . Bi2Bi1∥1/l∞ : il ∈ {0, 1}

}).

and

max {ρ(B0), . . . , ρ(Bm)} ≤ ρ (B0, . . . , Bm) ≤ max {∥ B0∥∞, . . . , ∥ Bm∥∞} .

Since µ is bounded from below by the spectral radii and from above by the norm

of the metrics B0, B1,. . . , Bm then

max {ρ(B0), . . . , ρ(Bm)} ≤ µ ≤ max {∥ B0∥∞, . . . , ∥ Bm∥∞} .” (2.18)

Theorem 2.2.1. The Hölder regularity of scheme (2.6) is r = 4− log3(11) = 1.8173.

22

Proof. The Laurent polynomial (2.10) of the scheme (2.6) can be written as

a(z) =

(1 + z + z2

3

)4

b(z), (2.19)

where

b(z) =1

z5(−4 + 11z − 4z2).

From (2.17) and (2.19), b0 = −4, b1 = 11, b2 = −4, k = 4, m = 2 and n = 3, thus

q = 0, 1, 2 and then B0, B1, and B2 are the matrices with elements(B0)ij = b2+i−3j,

(B1)ij = b2+i−3j+1,

(B2)ij = b2+i−3j+2,

where i, j = 1, 2. This implies

B0 =

−4 0

11 0

, B1 =

11 0

−4 0

and B2 =

−4 0

0 −4

. (2.20)

From (2.18) and (2.20) we have

max {4, 11, 4} ≤ µ ≤ max {11, 11, 4} .

Since the largest eigenvalue and the max-norm of the metrics is 11, so

r = 4− log3(11) = 1.8173.

Theorem 2.2.2. The Hölder regularity of scheme (2.9) is r = 4− log4(24).

Proof. The Laurent polynomial (2.13) of scheme (2.9) can be written as

a(z) =

(1 + z + z2 + z3

4

)4

b(z), (2.21)

23

where

b(z) =1

z7(−10 + 24z − 10z2).

From (2.17) and (2.21), b0 = −10, b1 = 24, b2 = −10, k = 4, m = 2 and n = 4, thus

q = 0, 1, 2 and then B0, B1, and B2 are the matrices with elements(B0)ij = b2+i−4j,

(B1)ij = b2+i−4j+1,

(B2)ij = b2+i−4j+2,

where i, j = 1, 2. This implies

B0 =

−10 0

24 0

, B1 =

24 0

−10 0

and B2 =

−10 0

0 −10

. (2.22)

From (2.18) and (2.22) we have

max {10, 24, 10} ≤ µ ≤ max {24, 24, 10} .

Thus the largest eigenvalue and the max-norm of the metrics is 24, so

r = 4− log4(24) = 1.7077.

2.2.2 Polynomial generation

The generation degree of a subdivision scheme is the maximum degree of poly-

nomials that can potentially be generated by the scheme, provided that the ini-

tial data is chosen correctly. Suppose p0 is polynomial of degree d of initial data

f 0i and symbol of the scheme is

a(z) = (1 + z + . . .+ zn−1)d+1b(z),

24

then the limit curve of the refined data fki at any level k is polynomial of de-

gree d. So the condition is necessary and sufficient for the scheme being able to

generate polynomial of degree d.

Theorem 2.2.3. The degree of polynomial generation of scheme (2.6) is 3.

Proof. Since the Laurent polynomial a(z) of the scheme (2.6) is

a(z) = (1 + z + z2)(3+1)b(z),

where

b(z) =1

(3)4z5(−4 + 11z − 4z2),

then degree of polynomial generation is 3.


Proof. Since the Laurent polynomial of (2.9) can be written as

a(z) = (1 + z + z2 + z3)(3+1)b(z),

where

b(z) =1

(4)4z7(−10 + 24z − 10z2),

then degree of polynomial generation of scheme is 3.

2.2.3 Polynomial reproduction and approximation order

The polynomial reproduction property has its own importance, as the repro-

duction property of the polynomials up to a certain degree d implies that the

scheme has d + 1 approximation order. Polynomial reproduction of degree d

requires polynomial generation of degree d. For this, polynomial reproduction

25

can be made from initial data which has been sampled from some polynomial

function. In the view of Conti and Hormann (2011) the polynomial reproduction

property of the proposed scheme, can be obtain after having the parameteriza-

tions τ given in (1.2).

Theorem 2.2.5. A convergent subdivision scheme (2.6) reproduces polynomials of de-

gree 3 with respect to the parameterizations (1.2) if and only if

a(k)(1) = 3k−1∏l=0

(τ − l) and a(k)(αj3) = 0, j = 1, 2,

for k = 0,. . . ,3, αj3 = exp(2πi3j) and τ = a′(1)

3.

Proof. By taking first derivative of (2.10) and substituting z = 1 in it, we get

a(1)(1) = 0.

This implies that

τ =a(1)(1)

3= 0.

So from (1.2), the scheme (2.6) has primal parametrization. For k = 0, j = 1 and

from (2.10), we get

a(0)(α13) = a(e

2πi3 ) = 0.

Similarly, for j = 1, 2 and k = 0, 1, 2, 3 (k denotes the order of derivative)

a(k)(αj3) = 0.

By (2.10), we get a(1) = 3. Also 3∏−1

l=0(0 − l) = 3, which implies that a(1) =

3∏0−1

l=0 (τ − l). Similarly for k = 1, 2, 3, we can easily show that

a(k)(1) = 3k−1∏l=0

(τ − l).

Which completes the proof.

26

(a) 4-point 2-ary (b) 4-point 3-ary

(c) 4-point 4-ary (d) 4-point 5-ary

(e) 4-point 6-ary (f) 4-point 7-ary

Figure 2.4: Comparison of the limit curves generated by proposed 4-point 2-ary, 3-ary,

4-ary, 5-ary, 6-ary and 7-ary interpolating subdivision schemes at 1st subdivision level.

27

Since scheme (2.6) reproduces polynomial of degree 3, so by using Theorem

1.3.5, we get following theorem.

Theorem 2.2.6. A 4-point ternary interpolating scheme (2.6) has an approximation

order of 4.



a(k)(1) = 4k−1∏l=0

(τ − l) and a(k)(αj4) = 0, j = 1, 2, 3


4.


a(1)(1) = 0.

This implies that

τ =a(1)(1)

4= 0.

So from (1.2), the scheme (2.9) has primal parametrization. For k = 0, j = 1 and

from (2.13), we get

a(0)(α14) = a(e

2πi4 ) = 0.

Similarly, for j = 1, 2, 3 and k = 0, 1, 2, 3 (k denotes the order of derivative)

a(k)(αj4) = 0.

By (2.13), we get a(1) = 4. Also 4∏−1


4∏0−1


a(k)(1) = 4k−1∏l=0

(τ − l).


28

Again by Theorem 1.3.5, we get following theorem.

Theorem 2.2.8. A 4-point quaternary interpolating scheme (2.9) has an approximation

order of 4.

2.3 Numerical examples and conclusion

Six examples are depicted to show the usefulness of 4-point 2-ary, 3-ary, 4-ary,

5-ary, 6-ary and 7-ary interpolating subdivision schemes at 1st subdivision level

in Figure 2.4. In this figure the control polygons are drawn by dotted lines while

the subdivision curves are drawn by solid lines. From Figure 2.4, it is clear that

the initial polygon converges rapidly to limit curve as we increase the arity of

the subdivision scheme.

In this chapter, we have presented a multi-step algorithm which generate 4-

point n-ary interpolating subdivision schemes. We have also observed that the

4-point n-ary schemes generated by Lagrange polynomials and wavelet theo-

ry can also be generated by proposed multi-step algorithm. Some significant

properties like Hölder regularity, degree of polynomial generation, degree of

polynomial reproduction and approximation order have been also discussed.

29

Chapter 3

A class of shape preserving 5-point

n-ary approximating schemes

In this chapter, a new class of shape preserving relaxed 5-point n-ary approx-

imating subdivision schemes is presented. Furthermore, the conditions on the

initial data assuring monotonicity, convexity and concavity preservation of the

limit functions are derived. Moreover, some significant properties of schemes

have been elaborated such as continuity, Hölder exponent, polynomial genera-

tion, polynomial reproduction, approximation order and support of basic limit

function. Visual performance of schemes has also been demonstrated through

several examples.

3.1 Algorithm for construction of schemes

In this section, we present an algorithm for the construction of 5-point n-ary ap-

proximating subdivision schemes. This algorithm has two main steps. One step

has been borrowed by 4-point n-ary DD interpolating schemes of Deslauriers

and Dubuc (1989). That is during first step each segment of control polygon

30

is divided into n-subsegments by inserting n number of new points at position

1/n, 2/n,..., (n − 1)/n by 4-point DD-scheme. While the other step is to change

the interpolating rule of DD-scheme by 5-point approximating rule.

Consider the open polygon shown in Figure 3.1. Where z, a, b, c, d, e are coarse

points of control polygon. Let {p1, p2, . . . , pn−1}, {p′1, p′2, . . . , p′n−1} and

{p′′1, p′′2, . . . , p′′n−1} be the new inserted points (say DD-points) by DD-scheme

corresponding to the edges ab, bc and cd respectively. Then second step is to

modify all coarse points by using divided differences of coarse points and DD-

points. Here we only discuss the rule to modify one point say c. The point c can

be updated by following rule:

c′ =p′n−1 + p′′1 −Wc

2, (3.1)

whereWc is the affine combination of second divided difference of coarse points

and DD-points at point c defined below:

Wc = µ

{d− 2c+ b

n2

}+ (1− µ)

{p′′1 − 2c+ p′n−1

},

where µ ∈ [0, 1] while

p′′1 = A1b+ A2c+ A3d+ A4e,

p′n−1 = A4a+ A3b+ A2c+ A1d,

where

A1 =−(n− 1)(2n− 1)

6n3,

A2 =(n2 − 1)(2n− 1)

2n3,

A3 =(n+ 1)(2n− 1)

2n3,

A4 =−(n2 − 1)

6n3,

and n = 2, 3, 4, . . . .

31

Figure 3.1: Labeling of a control polygon.

3.1.1 Examples

Here we see that 5-point n-ary approximating schemes can be easily generated

by above algorithm.

• By substituting n = 2 in (3.1), we get the mask of 5-point binary approxi-

mating scheme of Augsdöefer (2010).

• If we substitute n = 3 in (3.1), we get following 5-point ternary schemefk+13i = − 10

162fki−1 +

120162fki + 60

162fki+1 − 8

162fki+2,

fk+13i+1 = − 8

162fki−1 +

60162fki + 120

162fki+1 − 10

162fki+2,

fk+13i+2 = − 4µ

162fki−1 +

16µ162fki + 162−24µ

162fki+1 +

16µ162fki+2 −

4µ162fki+3.

(3.2)

• For n = 4 in (3.1), we get following 5-point quaternary scheme.

fk+14i = − 14

256fki−1 +

210256fki + 70

256fki+1 − 10

256fki+2,

fk+14i+1 = − 16

256fki−1 +

144256fki + 144

256fki+1 − 16

256fki+2,

fk+14i+2 = − 10

256fki−1 +

70256fki + 210

256fki+1 − 14

256fki+2,

fk+14i+3 = − 5µ

256fki−1 +

20µ256fki + 256−30µ

256fki+1 +

20µ256fki+2 −

5µ256fki+3.

(3.3)

• By substituting µ = 0 in (3.2) and (3.3), we get the mask of 4 -point ternary

and quaternary interpolating scheme of Deslauriers and Dubuc (1989).

32

3.1.2 Smoothness analysis of proposed schemes

We discuss the analysis of relaxed 5-point ternary and quaternary approximat-

ing subdivision schemes. We use the theory of generating function Dyn and

Levin (2002) to examine the convergence and smoothness of the scheme (3.2)

and (3.3).

Theorem 3.1.1. The 5-point ternary approximating subdivision scheme (3.2) is C3 for

any µ ∈ (0.666, 0.700).

Proof. The Laurent polynomial a(z) for the scheme (3.2) is

a(z) =1

162{−4µz0 − 8z1 − 10z2 + 16µz3 ++60z4 + 120z5 + 162− 24µz6 (3.4)

+120z7 + 60z8 + 16µz9 − 10z10 − 8z11 − 4µz12}.

Now we consider

c(z) =

(3

1 + z + z2

)4

a(z)

=1

2(−4µ+ (16µ− 8)z + (22− 24µ)z2 + (16µ− 8)z3 − 4µz4).

Note that ∥∥∥∥13Sc∥∥∥∥∞

=1

3max

{∑j∈Z

|c3j|,∑j∈Z

|c3j+1|,∑j∈Z

|c3j+2|

}.

For µ ∈ (0.666, 0.700), we have∥∥∥∥13Sc∥∥∥∥∞

=1

3max

{∣∣∣∣−4µ

2

∣∣∣∣+ ∣∣∣∣16µ− 8

2

∣∣∣∣ , ∣∣∣∣−24µ+ 22

2

∣∣∣∣} < 1.

Hence Sc is contractive. Therefore, by Corollary 4.17 of Dyn and Levin (2002),

the scheme (3.2) is C3 for µ ∈ (0.666, 0.700).

Theorem 3.1.2. The 5-point quaternary approximating subdivision scheme (3.3) is C2

for any µ in (0.266, 1).

Proof of the above theorem is similar to the proof of Theorem 3.1.1.

33

3.2 Shape preserving properties

In this section, we will discuss that what condition should be imposed on the

initial points so that the limit curves generated by the subdivision schemes are

monotonicity, convexity and concavity preserving.

3.2.1 Monotonicity preservation

Definition 3.2.1. Hussain et al. (2012) "A univariate data (xi, fi), i = 0, 1, 2, . . . , n

is monotonically increasing if fi < fi+1 ∀ i = 0, 1, 2, . . . , n and the derivative at

the data points obey the condition di > 0 ∀ i = 0, 1, 2, . . . , n."

Here, we examine monotonicity preservation of 5-point ternary approximat-

ing subdivision scheme (3.2) and 5-point quaternary approximating scheme

(3.3).

Theorem 3.2.1. Let {f 0i }i∈Z be the sequence of initial points such that f 0

i < f 0i+1,

i ∈ Z. Let

Lki = fki+1 − fki , gki =Lki+1

Lki, Gk = max

i{gki ,

1

gki}, k ≥ 0, k ∈ Z, i ∈ Z.

Furthermore, let 0.3 ≤ µ ≤ 1 and ξ = − 1µ

, ξ ∈ R. If 1ξ≤ G0 ≤ ξ, {fki } is defined by

the subdivision scheme (3.2), then

Lki > 0,1

ξ≤ Gk ≤ ξ, k ≥ 0, k ∈ Z, i ∈ Z. (3.5)

Proof. (3.5) will be proved by mathematical induction. When k = 0,

L0i = f 0

i+1 − f 0i > 0, 1

ξ≤ G0 ≤ ξ, then (3.5) is true.

Suppose that (3.5) holds for k. i.e Lki = fki+1 − fki > 0, 1ξ≤ Gk ≤ ξ, since

Lk+13i = fk+1

3i+1 − fk+13i =

1

81{−(fki − fki−1) + 29(fki+1 − fki )− (fki+2 − fki+1)}.

34

This implies that

Lk+13i =

1

81{−Lki−1 + 29Lki − Lki+1}.

Similarly

Lk+13i+1 = fk+1

3i+2 − fk+13i+1 =

1

81{(2µ− 4)Lki−1 + (26− 6µ)Lki + (5 + 6µ)Lki+1

−2µLki+2},

Lk+13i+2 = fk+1

3i+3 − fk+13i+2 =

1

81{−2µLki−1 + (5 + 6µ)Lki + (26− 6µ)Lki+1

+(2µ− 4)Lki+2}.

Next we show that

Lk+13i > 0, Lk+1

3i+1 > 0 and Lk+13i+2 > 0.

Consider

Lk+13i =

1

81{−Lki−1 + 29Lki − Lki+1}.

This implies

Lk+13i =

Lki81

{− 1

gki−1

+ 29− gki }.

Again implies

Lk+13i ≥ Lki

81{−2ξ + 29}.

As we know that Lki > 0 and

1

81{−2ξ + 29} > 0, for 0.3 ≤ µ ≤ 1 and ξ = − 1

µ.

This further implies Lk+13i > 0. Again consider

Lk+13i+1 =

1

81{(2µ− 4)Lki−1 + (26− 6µ)Lki + (5 + 6µ)Lki+1 − 2µLki+2}.

35

This implies

Lk+13i+1 =

Lki81

{(2µ− 4)Lki−1

Lki+ (26− 6µ) + (5 + 6µ)

Lki+1

Lki− 2µ

Lki+2

Lki}.

Again implies

Lk+13i+1 =

Lki81

{(2µ− 4)1

gki−1

+ (26− 6µ) + (5 + 6µ)gki − 2µgki+1gki }.

This implies that

Lk+13i+1 =≥ Lki

81{(2µ− 4)

1

ξ+ (26− 6µ) + (5 + 6µ)

1

ξ− 2µ}.


1

81{(2µ− 4)

1

ξ+ (26− 6µ) + (5 + 6µ)

1

ξ− 2µ} > 0, for 0.3 ≤ µ ≤ 1 and ξ = − 1

µ.

This further implies that Lk+13i+1 > 0. Finally

Lk+13i+2 =

1

81{−2µLki−1 + (5 + 6µ)Lki + (26− 6µ)Lki+1(2µ− 4)Lki+2}.

This implies

Lk+13i+2 =

Lki+1

81{−2µ

Lki−1

Lki+1

+ (5 + 6µ)LkiLki+1

+ (26− 6µ) + (2µ− 4)Lki+2

Lki+1

}.

Furthermore

Lk+13i+2 =

Lki+1

81{−2µ

1

gki−1

1

gki+ (5 + 6µ)

1

gki+ (26− 6µ) + (2µ− 4)gki+1}.

This implies that

Lk+13i+2 =≥

Lki+1

81{−2µ+ (5 + 6µ)

1

ξ+ (26− 6µ) + (2µ− 4)

1

ξ}.

As we know that Lki+1 > 0 and

1

81{−2µ+ (5 + 6µ)

1

ξ+ (26− 6µ) + (2µ− 4)

1

ξ} > 0, for 0.3 ≤ µ ≤ 1 and ξ = − 1

µ.

36

This further implies that Lk+13i+2 > 0.

Now we prove that 1ξ≤ Gk+1 ≤ ξ, we first show that gk+1

3i − ξ ≤ 0.

gk+13i =

Lk+13i+1

Lk+13i

=181{(2µ− 4)Lki−1 + (26− 6µ)Lki + (5 + 6µ)Lki+1 − 2µLki+2}

181{−Lki−1 + 29Lki − Lki+1}

.

This implies that

gk+13i − ξ =

1

{−Lki−1 + 29Lki − Lki+1}{(2µ− 4)Lki−1 + (26− 6µ)Lki + (5 + 6µ)Lki+1

−2µLki+2 + ξLki−1 − 29ξLki + ξLki+1

}.

Again implies

gk+13i − ξ =

1

Lki−1{−1 + 29gki−1 − gki gki−1}

Lki

{(2µ− 4)

1

gki+ (26− 6µ) + (5 + 6µ)gki

−2µgki+1gki + ξ

1

gki− 29ξ + ξgki

}.

This further implies that

gk+13i − ξ ≤ Lki {2ξ2 + (8µ− 28)ξ − 8µ+ 26}

Lki−1{29ξ − 2}.

Since Lki {2ξ2 + (8µ − 28)ξ − 8µ + 26} is greater than zero and Lki−1{29ξ − 2} is

less than zero for 0.3 ≤ µ ≤ 1 and ξ = − 1µ

.

This implies that

gk+13i − ξ ≤ 0.

This further implies gk+13i ≤ ξ. Now we show that 1

gk+13i

− ξ ≤ 0.

1

gk+13i

=Lk+13i

Lk+13i+1

=181{−Lki−1 + 29Lki − Lki+1}

181{(2µ− 4)Lki−1 + (26− 6µ)Lki + (5 + 6µ)Lki+1 − 2µLki+2}

.

This implies that

gk+13i − ξ =

1

{(2µ− 4)Lki−1 + (26− 6µ)Lki + (5 + 6µ)Lki+1 − 2µLki+2}{−Lki−1 + 29Lki

−Lki+1 − (2µ− 4)ξLki−1 − (26− 6µ)ξLki − (5 + 6µ)ξLki+1 + 2µξLki+2

}.

37

Again implies

gk+13i − ξ =

1

Lki+1{(2µ− 4) 1gki−1

1gki

+ (26− 6µ) 1gki

+ (5 + 6µ)− 2µgki+1}Lki

{− 1

gki−1

+29− gki − (2µ− 4)ξ1

gki−1

− (26− 6µ)ξ − (5 + 6µ)ξgki + 2µξgki+1gki

}.


1

gk+13i

− ξ ≤Lki

81{2µξ3 + (6µ− 26)ξ − 21

ξ+ 28− 8µ}

Lki+1

81{(2µ− 4)ξ2 + (26− 6µ)ξ − 2µ1

ξ+ (6µ+ 5)}

.

Since Lki

81{2µξ3 + (6µ − 26)ξ − 21

ξ+ 28 − 8µ} is greater than zero and Lk

i+1

81{(2µ −

4)ξ2 + (26− 6µ)ξ − 2µ1ξ+ (6µ+ 5)} is less than zero for 0.3 ≤ µ ≤ 1 and ξ = − 1

µ.

This implies that

1

gk+13i

− ξ ≤ 0.

This further implies 1

gk+13i

≤ ξ. In the same way, we see that gk+13i+1 ≤ ξ, gk+1

3i+2 ≤ ξ,

1

gk+13i+1

≤ ξ and 1

gk+13i+2

≤ ξ. So Gk+1 ≤ ξ. Since Gk+1 = maxi{gk+1i , 1

gk+1i

}, it is obvious

that Gk+1 ≥ 1ξ.



i < f 0i+1,

i ∈ Z. Let

Lki = fki+1 − fki , gki =Lki+1

Lki, Gk = max

i{gki ,

1

gki}, k ≥ 0, k ∈ Z, i ∈ Z.

Furthermore, let 0.1 ≤ µ ≤ 1 and ξ = − 1µ

, ξ ∈ R. If 1ξ≤ G0 ≤ ξ, {fki } is defined by

the subdivision scheme (3.3), then

Lki > 0,1

ξ≤ Gk ≤ ξ, k ≥ 0, k ∈ Z, i ∈ Z. (3.6)

38

Proof. We use mathematical induction to prove (3.6). When k = 0,

L0i = f 0

i+1 − f 0i > 0, 1

ξ≤ G0 ≤ ξ, then (3.6) is true.

Suppose that (3.6) holds for k. i.e Lki = fki+1 − fki > 0, 1ξ≤ gk ≤ ξ, since

Lk+14i = fk+1

4i+1 − fk+14i =

1

128{Lki + 34Lki+1 − 3Lki+3},

Lk+14i+1 = fk+1

4i+2 − fk+14i+1 =

1

128{−3Lki + 34Lki+1 + Lki+2},

Lk+14i+2 = fk+1

4i+3 − fk+14i+2 =

(−5

128+

5µ

256

)Lki +

(15

64− 15µ

256

)Lki+1

+

(7

128+

15µ

256

)Lki+2 −

5µ

256Lki+3,

Lk+14i+3 = fk+1

4i+4 − fk+14i+3 = − 5µ

256Lki +

(7

128+

15µ

256

)Lki+1 +

(15

64− 15µ

256

)Lki+2

+

(−5

128+

5µ

256

)Lki+3.

Now we show that

Lk+14i > 0, Lk+1

4i+1 > 0, Lk+14i+2 > 0 and Lk+2

4i+3 > 0.

Now Consider

Lk+14i =

1

128{Lki + 34Lki+1 − 3Lki+2}.

This implies

Lk+14i =

Lki128

{1 + 34

Lki+1

Lki− 3

Lki+2

Lki+1

Lki+1

Lki

}.

Furthermore

Lk+14i =

Lki128

{1 + 34gki − 3gki+1g

ki

}.

39

This implies that

Lk+14i ≥ Lki

128

{1 + 34

1

ξ− 3ξ

}.


1

128

{1 + 34

1

ξ− 3ξ

}> 0, for 0.1 ≤ µ ≤ 0.9 and ξ =

1

µ.

This further implies that Lk+14i > 0. Further

Lk+14i+1 =

1

128{−3Lki + 34Lki+1 + Lki+2}.

Again implies

Lk+14i+1 =

Lki128

{−3 + 34

Lki+1

Lki+Lki+2

Lki+1

Lki+1

Lki

}.

Furthermore

Lk+14i+1 =

Lki128

{−3 + 34gki + gki+1g

ki

}.

This implies that

Lk+14i+1 ≥

Lki128

{−3 + 34

1

ξ+

1

ξ2

}.


1

128

{−3 + 34

1

ξ+ ξ

}> 0, for 0.1 ≤ µ ≤ 0.9 and ξ =

1

µ.

This further implies that Lk+14i+1 > 0. Furthermore

Lk+14i+2 =

(−5

128+

5µ

256

)Lki +

(15

64− 15µ

256

)Lki+1 +

(7

128+

15µ

256

)Lki+2

− 5µ

256Lki+3.

40

This implies that

Lk+14i+2 = Lki

{(−5

128+

5µ

256

)+

(15

64− 15µ

256

)Lki+1

Lki+

(7

128+

15µ

256

)Lki+2

Lki+1

Lki+1

Lki

− 5µ

256

Lki+3

Lki+2

Lki+2

Lki+1

Lki+1

Lki

}.

Furthermore

Lk+14i+2 = Lki

{(−5

128+

5µ

256

)+

(15

64− 15µ

256

)gki +

(7

128+

15µ

256

)gki+1g

ki

− 5µ

256gki+2g

ki+1g

ki

}.

This implies that

Lk+14i+2 ≥ Lki

{(−5

128+

5µ

256

)+

(15

64− 15µ

256

)1

ξ+

(7

128+

15µ

256

)1

ξ2− 5µ

256

1

ξ

}.

As we know that Lki > 0 and{(−5

128+

5µ

256

)+

(15

64− 15µ

256

)1

ξ+

(7

128+

15µ

256

)1

ξ2− 5µ

256

1

ξ

}> 0,

for 0.2 ≤ µ ≤ 0.9 and ξ = 1µ

.


Lk+14i+3 = − 5µ

256Lki +

(7

128+

15µ

256

)Lki+1 +

(15

64− 15µ

256

)Lki+2(

−5

128+

5µ

256

)Lki+3.

This implies

Lk+14i+3 = Lki

{− 5µ

256+

(7

128+

15µ

256

)Lki+1

Lki+

(15

64− 15µ

256

)Lki+2

Lki+1

Lki+1

Lki(−5

128+

5µ

256

)Lki+3

Lki+2

Lki+2

Lki+1

Lki+1

Lki

}.

Again implies

Lk+14i+3 = Lki

{− 5µ

256+

(7

128+

15µ

256

)gki +

(15

64− 15µ

256

)gki+1g

ki(

−5

128+

5µ

256

)gki+2g

ki+1g

ki

}.

41

Further implies that

Lk+14i+3 ≥ Lki

{− 5µ

256+

(7

128+

15µ

256

)1

ξ+

(15

64− 15µ

256

)1

ξ2

(−5

128+

5µ

256

)1

ξ3

}.

As we know that Lki > 0 and{− 5µ

256+

(7

128+

15µ

256

)1

ξ+

(15

64− 15µ

256

)1

ξ2

(−5

128+

5µ

256

)1

ξ3

}> 0,

for 0.2 ≤ µ ≤ 0.9 and ξ = 1µ

. This further implies that Lk+14i+3 > 0.

Now we prove that 1ξ≤ Gk+1 ≤ ξ, we first show that gk+1

4i − ξ ≤ 0.

gk+14i =

Lk4i+1

Lk4i=

1128

{−3Lki + 34Lki+1 + Lki+2}1

128{Lki + 34Lki+1 − 3Lki+3}

.

This implies that

gk+14i − ξ =

1128

{−3Lki + 34Lki+1 + Lki+2 − ξLki − 34ξLki+1 + 3ξLki+2}1

128{Lki + 34Lki+1 − 3Lki+3}

.

Again implies

gk+14i − ξ =

Lki+1

128{−3 1

gki+ 34 + gki+1 − ξ 1

gki− 34ξ + 3ξgki+1}

Lki

128{1 + 34gki − 3gki+1g

ki+1}

.


gk+14i − ξ ≤

Lki+1

128{3ξ2 − 36ξ + 33}

Lki

128{1 + 34ξ − 3}

.

Since Lki+1

128{3ξ2 − 36ξ + 33} is less than zero and Lk

i

128{1 + 34ξ − 3} is greater than

zero for 0.2 ≤ µ ≤ 0.9 and ξ = 1µ

.

This implies that

gk+14i − ξ ≤ 0.

This further implies that gk+14i ≤ ξ. Now we show that 1

gk+14i

− ξ < 0.

1

gk+14i

=Lk4iLk4i+1

=1

128{Lki + 34Lki+1 − 3Lki+3}

1128

{−3Lki + 34Lki+1 + Lki+2}.

42

This implies that

1

gk+14i

− ξ =1

128{Lki + 34Lki+1 − 3Lki+2 + 3ξLki − 34ξLki+1 − ξLki+2}

1128

{−3Lki + 34Lki+1Lki+3}

.

Again implies

1

gk+14i

− ξ =

Lki+1

128{ 1gki

+ 34− 3gki+1 + 3ξ 1gki

− 34ξ − ξgki+1}Lki

128{−3 + 34gki + gki+1g

ki+1}

.


1

gk+14i

− ξ ≤Lki+1

128{3ξ2 − 36ξ + 33}

Lki

128{−3 + 34ξ + ξ2}

.

Since Lki+1

128{3ξ2−36ξ+33} is less than zero and Lk

i

128{−3+34ξ+ ξ2} is greater than

zero for 0.2 ≤ µ ≤ 0.9 and ξ = 1µ

.

This implies that

1

gk+14i

− ξ ≤ 0.

In the same way, we can get gk+14i+1 ≤ ξ, gk+1

4i+2 ≤ ξ, gk+14i+3 ≤ ξ, 1

gk+14i+1

≤ ξ, 1

gk+14i+2

≤ ξ and

1

gk+14i+3

≤ ξ. So Gk+1 ≤ ξ. Since Gk+1 = maxi{gki , 1gki}, it is obvious that Gk+1 ≥ 1

ξ.

which completes the proof.

3.2.2 Convexity preservation

Definition 3.2.2. Mehaute and Uteras (1994). "Given a set of control points

pki ∈ Z, pki = (xki , fki ), fki is strictly convex at a point xki , if second order di-

vided difference dki = f [xki−1, xki , x

ki+1]

> 0."

We prove the convexity preservation of subdivision schemes (3.2) and (3.3)

with uniform initial control points. Tan et al. (2014) "Given a set of initial con-

trol points p0i ∈ Z, p0i = (x0i , f0i ) which are strictly convex, where x0i ∈ Z are

43

equidistant points. To simplify, we use △x0i = x0i+1 − x0i = 1. By the subdi-

vision scheme (3.2), we have △xk+1i = xk+1

i+1 − xk+1i = 1

3△ xki = 1

3k+1 . Denote

dki = f [xki−1, xki , x

ki+1] = 32k2−1(fki−1 − 2fki + fki+1) as the second order divided

differences. By the subdivision scheme (3.3), we have △xk+1i = xk+1

i+1 − xk+1i =

14△ xki = 1

4k+1 . Denote dki = f [xki−1, xki , x

ki+1] = 42k2−1(fki−1 − 2fki + fki+1) as the

second order divided differences. In the following, we will prove dki > 0, ∀k ≥

0, k ∈ Z, i ∈ Z."

Theorem 3.2.3. Suppose that the initial control points {p0i }i∈Z, {p0i } = (x0i , f0i ) are

strictly convex, i.e d0i > 0, for all i ∈ Z. Let

dki = 32k(2)−1(fki−1 − 2fki + fki+1), yki =dki+1

dki,

Y k = maxi

{yki ,1

yki}, ∀k ≥ 0, k ∈ Z, i ∈ Z.

Furthermore, let 0.3 ≤ µ ≤ 0.9 and δ = 1µ

, δ ∈ R. Then for 1δ≤ Y 0 ≤ δ,

dki > 0,1

δ≤ Y k ≤ δ, ∀k ≥ 0, k ∈ Z, i ∈ Z. (3.7)

That is, the limit function generated by the subdivision scheme (3.2) is strictly convex.

Proof. (3.7) will be proved by mathematical induction. When k = 0, d0i > 0,

1δ≤ Y 0 ≤ δ, then (3.7) is true.

Suppose that (3.7) holds for k. i.e dki > 0, 1δ≤ Y k ≤ δ, since

dk+13i = 32k(2)−1(fk+1

3i−1 − 2fk+13i + fk+1

3i+1).

This implies that

dk+13i = −2µ

9dki−1 +

(2

3+

4µ

9

)dki +

(1

3− 2µ

9

)dki+1.

Similarly

dk+13i+1 =

(−4

9+

4µ

9

)dki−1 +

(17

9− 8µ

9

)dki +

(−4

9+

4µ

9

)dki+1,

44

dk+13i+2 =

(1

3− 2µ

9

)dki−1 +

(2

3+

4µ

9

)dki −

2µ

9dki+1.

Next we show that

dk+13i > 0, dk+1

3i+1 > 0 and dk+13i+2 > 0.

Now consider

dk+13i = −2µ

9dki−1 +

(2

3+

4µ

9

)dki +

(1

3− 2µ

9

)dki+1.

This implies

dk+13i = dki

{−2µ

9

dki−1

dki+

(2

3+

4µ

9

)+

(1

3− 2µ

9

)dki+1

dki

}.

Further

dk+13i = dki

{−2µ

9

1

yki−1

+

(2

3+

4µ

9

)+

(1

3− 2µ

9

)yki

}.

This implies that

dk+13i ≥ dki

{−2µ

9δ +

(2

3+

4µ

9

)+

(1

3− 2µ

9

)1

δ

}.

As we know that dki > 0 and{−2µ

9δ +

(2

3+

4µ

9

)+

(1

3− 2µ

9

)1

δ

}> 0, for 0.3 ≤ µ ≤ 0.9 and δ =

1

µ.

This further implies that dk+13i > 0. Again consider

dk+13i+1 =

(−4

9+

4µ

9

)dki−1 +

(17

9− 8µ

9

)dki +

(−4

9+

4µ

9

)dki+1.

Again implies

dk+13i+1 = dki

{(−4

9+

4µ

9

)dki−1

dki+

(17

9− 8µ

9

)+

(−4

9+

4µ

9

)dki+1

dki

}.

45

Furthermore

dk+13i+1 = dki

{(−4

9+

4µ

9

)1

yki−1

+

(17

9− 8µ

9

)+

(−4

9+

4µ

9

)yki

}.

This implies that

dk+13i ≥ dki

{(−4

9+

4µ

9

)1

δ+

(17

9− 8µ

9

)+

(−4

9+

4µ

9

)1

δ

}.

As we know that dki > 0 and{(−4

9+

4µ

9

)1

yki−1

+

(17

9− 8µ

9

)+

(−4

9+

4µ

9

)yki

}> 0,

for 0.3 ≤ µ ≤ 0.9 and δ = 1µ

.

This further implies that dk+13i+1 > 0. Finally

dk+13i+2 =

(1

3− 2µ

9

)dki−1 +

(2

3+

4µ

9

)dki +−2µ

9dki+1.

This implies

dk+13i+2 = dki

{(1

3− 2µ

9

)dki−1

dki+

(2

3+

4µ

9

)− 2µ

9

dki+1

dki

}.

Again implies

dk+13i+2 = dki

{(1

3− 2µ

9

)1

yki−1

+

(2

3+

4µ

9

)− 2µ

9yki

}.

Furthermore

dk+13i+2 ≥ dki

{(1

3− 2µ

9

)1

δ+

(2

3+

4µ

9

)− 2µ

9δ

}.

As we know that dki > 0 and{(1

3− 2µ

9

)1

δ+

(2

3+

4µ

9

)− 2µ

9δ

}> 0, for 0.3 ≤ µ ≤ 0.9 and δ =

1

µ.

This further implies that dk+13i+2 > 0.

Now we prove that 1δ≤ Y k+1 ≤ δ, we first show that yk+1

3i − δ ≤ 0.

yk+13i =

dk+13i+1

dk+13i

=

(−4

9+ 4µ

9

)dki−1 +

(179− 8µ

9

)dki +

(−4

9+ 4µ

9

)dki+1

−2µ9dki−1 +

(23+ 4µ

9

)dki +

(13− 2µ

9

)dki+1

.

46

This implies that

yk+13i − δ =

1

dki−1

{−2µ

9+(23+ 4µ

9

) dkidki−1

+(13− 2µ

9

) dki+1

dki−1

}dki {(−4

9+

4µ

9

)dki−1

dki

+

(17

9− 8µ

9

)+

(−4

9+

4µ

9

)dki+1

dki−(1

3− 2µ

9

)δdki−1

dki

−(2

3+

4µ

9

)δ +

2µ

9δdki+1

dki

}.

Further implies

yk+13i − δ =

1

dki−1

{−2µ

9+(23+ 4µ

9

)yki−1

(13− 2µ

9

)yki y

ki−1

}dki {(−4

9+

4µ

9

)1

yki−1

+

(17

9− 8µ

9

)+

(−4

9+

4µ

9

)yki −

(1

3− 2µ

9

)δ

1

yki−1

−(2

3+

4µ

9

)δ +

2µ

9δyki

}.


yk+13i − δ ≤

dki{

2µ9δ2 +

(4µ9− 14

9

)δ − 6µ

9+ 14

9

}dki−1

{(13− 2µ

9

)δ2 +

(23+ 4µ

9

)δ − 2µ

9

} .Since dki

{2µ9δ2 +

(4µ9− 14

9

)δ − 6µ

9+ 14

9

}is less than zero and dki−1

{(13− 2µ

9

)δ2

+(23+ 4µ

9

)δ − 2µ

9

}is greater than zero for 0.3 ≤ µ ≤ 0.9 and δ = 1

µ.

This implies that

yk+13i − δ ≤ 0.

This implies that yk+13i ≤ δ. Now we show that 1

yk+13i

− δ ≤ 0.

1

yk+13i

=dk+13i

dk+13i+1

=−2µ

9dki−1 +

(23+ 4µ

9

)dki +

(13− 2µ

9

)dki+1(

−49+ 4µ

9

)dki−1 +

(179− 8µ

9

)dki +

(−4

9+ 4µ

9

)dki+1

.

47

This implies that

yk+13i − δ =

1

dki−1

{(−4

9+ 4µ

9

)+(179− 8µ

9

) dkidki−1

+(−4

9+ 4µ

9

) dki+1

dki−1

}dki {−2µ

9

dk−1i

dki

+

(2

3− 4µ

9

)+

(1

3− 2µ

9

)dk+1i

dki−(−4

9+

4µ

9

)δdk−1i

dki

−(17

9− 8µ

9

)δ −

(−4

9+

4µ

9

)δdk+1i

dki

}.

This further implies

yk+13i − δ =

1

dki−1

{(−4

9+ 4µ

9

)+(179− 8µ

9

)yki−1 +

(−4

9+ 4µ

9

)yki y

ki−1

}dki {−2µ

9

1

yki−1

+

(2

3− 4µ

9

)+

(1

3− 2µ

9

)yki −

(−4

9+

4µ

9

)δ

1

yki−1

−(17

9− 8µ

9

)δ −

(−4

9+

4µ

9

)δyki

}.

Again implies

yk+13i − δ ≤ 1

dki−1

{(−4

9+ 4µ

9

)+(179− 8µ

9

)δ +

(−4

9+ 4µ

9

)δ2}dki {−2µ

9

1

δ

+

(2

3− 4µ

9

)+

(1

3− 2µ

9

)δ −

(−4

9+

4µ

9

)−(17

9− 8µ

9

)δ

−(−4

9+

4µ

9

)}.


1

yk+13i

− δ ≤dki{(

−149+ 2µ

3

)δ − 2µ

91δ− 4µ

9+ 14

9

}dki−1

{(−4

9+ 4µ

9

)+(179− 8µ

9

)δ +

(−4

9+ 4µ

9

)δ2} .

48

Again since dki{(

−149+ 2µ

3

)δ − 2µ

91δ− 4µ

9+ 14

9

}is less than zero and dki−1

{(−4

9+ 4µ

9

)+(179− 8µ

9

)δ +

(−4

9+ 4µ

9

)δ2}

is greater than zero for 0.3 ≤ µ ≤ 0.9 and δ = 1µ

,

so

1

yk+13i

− δ ≤ 0.

This implies 1

yk+13i

≤ δ. In the same way, we see that yk+13i+1 ≤ δ, yk+1

3i+2 ≤ δ, 1

yk+13i+1

≤ δ

and 1

yk+13i+2

≤ δ. So Y k+1 ≤ δ. Since Y k+1 = maxi{yk+1i , 1

yk+1i

}, it is obvious that

Y k+1 ≥ 1δ.



strictly convex, i.e d0i > 0, for all i ∈ Z. Let


dki,

Y k = maxi

{yki ,1

yki}, ∀k ≥ 0, k ∈ Z, i ∈ Z.

Furthermore, let 0.1 ≤ µ ≤ 0.9 and δ = −1+µ2µ

, δ ∈ R. Then for 1δ≤ Y 0 ≤ δ,

dki > 0,1

δ≤ Y k ≤ δ, ∀k ≥ 0, k ∈ Z, i ∈ Z. (3.8)

That is, the limit function generated by the subdivision scheme (3.3) is strictly convex.




dk+14i = 42k+2(2)−1(fk+1

4i−1 − 2fk+14i + fk+1

4i+1).

This implies that

dk+14i =

1

2dki +

1

2dki+1.

49

Similarly

dk+14i+1 =

(1

4− 5µ

16

)dki +

(3

4+

5µ

8

)dki+1 −

5µ

16dki+2,

dk+14i+2 =

(−5

8+

5µ

8

)dki +

(9

4− 5µ

4

)dki+1 +

(−5

8+

5µ

8

)dki+2,

dk+14i+3 = −5µ

16dki +

(3

4+

5µ

8

)dki+1 +

(1

4− 5µ

16

)dki+2.

Now we show that

dk+14i > 0, dk+1

4i+1 > 0, dk+14i+1 > 0 and dk+2

4i+3 > 0.

Now consider

dk+14i =

1

2dki +

1

2dki+1.

This implies

dk+14i = dki

{1

2+

1

2

dki+1

dki

}.

Again implies

dk+14i = dki

{1

2+

1

2yki

}.

This implies that

dk+14i ≥ dki

{1

2+

1

2

1

δ

}.

As we know that dki > 0 and{1

2+

1

2

1

δ

}> 0, for 0.3 ≤ µ ≤ 0.9 and δ =

1

µ.

This further implies that dk+14i > 0. Further

dk+14i+1 =

(1

4− 5µ

16

)dki +

(3

4+

5µ

8

)dki+1 −

5µ

16dki+2.

50

Again implies

dk+14i+1 = dki

{(1

4− 5µ

16

)+

(3

4+

5µ

8

)dki+1

dki− 5µ

16

dki+2

dki+1

dki+1

dki

}.

Further

dk+14i+1 = dki

{(1

4− 5µ

16

)+

(3

4+

5µ

8

)yki −

5µ

16yki+1y

ki

}.

This implies that

dk+14i+1 ≥ dki

{(1

4− 5µ

16

)+

(3

4+

5µ

8

)1

δ− 5µ

16

}.


4− 5µ

16

)+

(3

4+

5µ

8

)1

δ− 5µ

16

}> 0,

for 0.3 ≤ µ ≤ 0.9 and δ = 1µ

.

This further implies that Lk+14i+1 > 0. Furthermore

dk+14i+2 =

(−5

8+

5µ

8

)dki +

(9

4− 5µ

4

)dki+1 +

(−5

8+

5µ

8

)dki+2.

Again implies

dk+14i+2 = dki+1

{(−5

8+

5µ

8

)dkidki+1

+

(9

4− 5µ

4

)+

(−5

8+

5µ

8

)dki+2

dki+1

}.

Further implies

dk+14i+2 = dki+1

{(−5

8+

5µ

8

)1

yki+

(9

4− 5µ

4

)+

(−5

8+

5µ

8

)1

yki+1

}.

This implies that

dk+14i+2 ≥ dki+1

{(−5

8+

5µ

8

)δ +

(9

4− 5µ

4

)+

(−5

8+

5µ

8

)δ2}.

As we know that dki+1 > 0 and{(−5

8+

5µ

8

)δ +

(9

4− 5µ

4

)+

(−5

8+

5µ

8

)δ2}> 0,

51

for 0.3 ≤ µ ≤ 0.9 and δ = 1µ

.


dk+14i+1 = −5µ

16dki +

(3

4+

5µ

8

)dki+1 +

(1

4− 5µ

16

)dki+2.

This implies

dk+14i+1 = dki

{−5µ

16+

(3

4+

5µ

8

)dki+1

dki

(1

4− 5µ

16

)dki+2

dki+1

dki+1

dki

}.

Furthermore

dk+14i+1 = dki

{−5µ

16+

(3

4+

5µ

8

)yki +

(1

4− 5µ

16

)yki+1y

ki

}.

This implies that

dk+14i+1 ≥ dki

{−5µ

16+

(3

4+

5µ

8

)1

δ+

(1

4− 5µ

16

)1

δ2

}.

As we know that dki > 0 and{−5µ

16+

(3

4+

5µ

8

)1

δ+

(1

4− 5µ

16

)1

δ2

}> 0,

for 0.3 ≤ µ ≤ 0.9 and δ = 1µ

. This further implies that Lk+14i+3 > 0.

Now we prove that 1δ≤ Y k+1 ≤ δ, first we show that yk+1

4i − δ ≤ 0.

yk+14i =

(14− 5µ

16

)dki +

(34+ 5µ

8

)dki+1 −

5µ16dki+2

12dki +

12dki+1

.

This implies that

yk+14i − δ =

{(14− 5µ

16

)dki +

(34+ 5µ

8

)dki+1 −

5µ16dki+2 − 1

2δdki − 1

2δdki+1}

{12dki +

12dki+1}

.

Again implies

yk+14i − δ =

dki+1{(14− 5µ

16

)1yki

+(34+ 5µ

8

)− 5µ

16yki+1 − 1

2δ 1yki

− 12δ}

dki {12+ 1

2yki }

.

52


yk+14i − δ ≤

dki+1{(14− 5µ

16

)δ +

(34+ 5µ

8

)− 5µ

161δ− 1

2− 1

2δ}

dki {12+ 1

2δ}

.

Since dki+1{(14− 5µ

16

)δ+

(34+ 5µ

8

)− 5µ

161δ− 1

2− 1

2δ} is less than zero and dki {1

2+ 1

2δ}

is greater than zero for 0.3 ≤ µ ≤ 0.9 and δ = 1µ

.

This implies that

yk+14i − δ ≤ 0.

Now we show that 1

yk+13i

− δ < 0.

1

yk+14i

=12dki +

12dki+1(

14− 5µ

16

)dki +

(34+ 5µ

8

)dki+1 −

5µ16dki+2

.

This implies that

yk+14i − δ =

{12dki +

12dki+1 −

(14− 5µ

16

)δdki −

(34+ 5µ

8

)δdki+1 +

5µ16δdki+2}

{(14− 5µ

16

)dki +

(34+ 5µ

8

)dki+1 −

5µ16dki+2.}

.

Again implies

yk+14i − δ =

dki+1{12

1yki

+ 12−(14− 5µ

16

)δ 1yki

−(34+ 5µ

8

)δ + 5µ

16δyki+1}

dki {(14− 5µ

16

)+(34+ 5µ

8

)yki −

5µ16yki+1y

ki }

.


yk+14i − δ ≤

dki+1{12δ + 1

2−(14− 5µ

16

)−(34+ 5µ

8

)δ + 5µ

16δ2}

dki {(14− 5µ

16

)+(34+ 5µ

8

)δ − 5µ

16}

.

Since dki+1{12δ + 1

2−(14− 5µ

16

)−(34+ 5µ

8

)δ + 5µ

16δ2} is less than zero and

dki{(

14− 5µ

16

)+(34+ 5µ

8

)δ − 5µ

16

}is greater than zero for 0.3 ≤ µ ≤ 0.9 and δ = 1

µ.

This implies that

1

yk+14i

− δ ≤ 0.

In the same way, we can get yk+14i+1 ≤ δ, yk+1

4i+2 ≤ δ, yk+14i+3 ≤ δ, 1

yk+14i+1

≤ δ, 1

yk+14i+2

≤ δ

and 1

yk+14i+3

≤ δ. So Y k+1 ≤ δ. Since Y k+1 = maxi{yk+1i , 1

yk+1i


Y k+1 ≥ 1δ.


53

3.2.3 Concavity preservation

Definition 3.2.3. Mehaute and Uteras (1994). "Given a set of control points pki ∈

Z, pki = (xki , fki ), fki is strictly concave at a point xki , if second order divided

difference Dki = f [xki−1, x

ki , x

ki+1] < 0."

we prove the concavity preservation of subdivision schemes (3.2) and (3.3)

with uniform initial control points. Tan et al. (2014) "Given a set of initial con-

trol points p0i ∈ Z, P 0i = (x0i , f

0i ) which are strictly concave, where x0i ∈ Z are

equidistant points. To simplify, we use △x0i = x0i+1 − x0i = 1. By the subdi-

vision scheme (3.2), we have △xk+1i = xk+1

i+1 − xk+1i = 1

3△ xki = 1

3k+1 . Denote

Dki = f [xki−1, x

ki , x

ki+1] = 32k2−1(fki−1 − 2fki + fki+1) as the second order divided

differences. By the subdivision scheme (3.3), we have △xk+1i = xk+1

i+1 − xk+1i =

14△ xki = 1

4k+1 . Denote Dki = f [xki−1, x

ki , x

ki+1] = 42k2−1(fki−1 − 2fki + fki+1) as the

second order divided differences. In the following, we will prove Dki < 0, ∀k ≥

0, k ∈ Z, i ∈ Z."


strictly concave, i.e D0i < 0, for all i ∈ Z. Let

Dki = 32k(2)−1(fki−1 − 2fki + fki+1), qki =

Dki+1

Dki

,

Qk = maxi

{qki ,1

qki}, ∀k ≥ 0, k ∈ Z, i ∈ Z.

Furthermore, let 0.3 ≤ µ ≤ 0.9 and β = 1µ

, β ∈ R. Then for 1β≤ Q0 ≤ β,

Dki < 0,

1

β≤ Qk ≤ β, ∀k ≥ 0, k ∈ Z, i ∈ Z. (3.9)

Namely, the limit function generated by the subdivision scheme (3.2) is strictly concave.



54


Dk+13i = 32k(2)−1(fk+1

3i−1 − 2fk+13i + fk+1

3i+1).

This implies that

Dk+13i = −2µ

9Dki−1 +

(2

3+

4µ

9

)Dki +

(1

3− 2µ

9

)Dki+1.

Similarly

Dk+13i+1 =

(−4

9+

4µ

9

)Dki−1 +

(17

9− 8µ

9

)Dki +

(−4

9+

4µ

9

)Dki+1,

Dk+13i+2 =

(1

3− 2µ

9

)Dki−1 +

(2

3+

4µ

9

)Dki −

2µ

9Dki+1.

Next we show that

Dk+13i < 0, Dk+1

3i+1 < 0 and Dk+13i+2 < 0.

Now consider

Dk+13i = −2µ

9Dki−1 +

(2

3+

4µ

9

)Dki +

(1

3− 2µ

9

)Dki+1.

This implies

Dk+13i = Dk

i

{−2µ

9

Dki−1

Dki

+

(2

3+

4µ

9

)+

(1

3− 2µ

9

)Dki+1

Dki

}.

Again implies

Dk+13i = Dk

i

{−2µ

9

1

qki−1

+

(2

3+

4µ

9

)+

(1

3− 2µ

9

)qki

}.

Furthermore

Dk+13i ≤ Dk

i

{−2µ

9

1

β+

(2

3+

4µ

9

)+

(1

3− 2µ

9

)β

}.

55

As we know that Dki < 0 and{

−2µ

9

1

β+

(2

3+

4µ

9

)+

(1

3− 2µ

9

)β

}> 0, for 0.3 ≤ µ ≤ 0.9 and β =

1

µ.

This further implies that Dk+13i < 0. Again consider

Dk+13i+1 =

(−4

9+

4µ

9

)Dki−1 +

(17

9− 8µ

9

)Dki +

(−4

9+

4µ

9

)Dki+1.

This implies

Dk+13i+1 = Dk

i−1

{(−4

9+

4µ

9

)+

(17

9− 8µ

9

)Dki

Dki−1

+

(−4

9+

4µ

9

)Dki+1

Dki−1

}.

Again implies

Dk+13i+1 = Dk

i

{(−4

9+

4µ

9

)+

(17

9− 8µ

9

)qki +

(−4

9+

4µ

9

)qki q

ki−1

}.

Furthermore

Dk+13i ≤ Dk

i

{(−4

9+

4µ

9

)+

(17

9− 8µ

9

)β +

(−4

9+

4µ

9

)β2

}.

As we know that Dki < 0 and{(

−4

9+

4µ

9

)+

(17

9− 8µ

9

)β +

(−4

9+

4µ

9

)β2

}> 0,

for 0.3 ≤ µ ≤ 0.9 and β = 1µ

.

This further implies that Dk+13i+1 < 0. Finally

Dk+13i+2 =

(1

3− 2µ

9

)Dki−1 +

(2

3+

4µ

9

)Dki +−2µ

9Dki+1.

Again implies

Dk+13i+2 = Dk

i

{(1

3− 2µ

9

)Dki−1

Dki

+

(2

3+

4µ

9

)− 2µ

9

Dki+1

Dki

}.

Furthermore

Dk+13i+2 = Dk

i

{(1

3− 2µ

9

)1

yki−1

+

(2

3+

4µ

9

)− 2µ

9yki

}.

56

This implies that

Dk+13i+2 ≤ Dk

i

{(1

3− 2µ

9

)β +

(2

3+

4µ

9

)− 2µ

9

1

β

}.


1

3− 2µ

9

)β +

(2

3+

4µ

9

)− 2µ

9

1

β

}> 0, for 0.3 ≤ µ ≤ 0.9 and β =

1

µ.

This further implies that Dk+13i+2 < 0.

Now we prove that 1β≤ Qk+1 ≤ β, we first show that qk+1

3i − β ≤ 0.

qk+13i =

Dk+13i+1

Dk+13i

=

(−4

9+ 4µ

9

)Dki−1 +

(179− 8µ

9

)Dki +

(−4

9+ 4µ

9

)Dki+1

−2µ9Dki−1 +

(23+ 4µ

9

)Dki +

(13− 2µ

9

)Dki+1

.

This implies that

qk+13i − β =

1

Dki−1

{−2µ

9+(23+ 4µ

9

) Dki

Dki−1

+(13− 2µ

9

) Dki+1

Dki−1

}Dki

{(−4

9+

4µ

9

)Dki−1

Dki

+

(17

9− 8µ

9

)+

(−4

9+

4µ

9

)Dki+1

Dki

−(1

3− 2µ

9

)βDki−1

Dki

−(2

3+

4µ

9

)β +

2µ

9βdki+1

Dki

}.

Again implies

qk+13i − β =

1

Dki−1

{−2µ

9+(23+ 4µ

9

)qki−1 +

(13− 2µ

9

)qki q

ki−1

}Dki

{(−4

9+

4µ

9

)1

qki−1

+

(17

9− 8µ

9

)+

(−4

9+

4µ

9

)qki −

(1

3− 2µ

9

)β

1

qki−1

−(2

3+

4µ

9

)β +

2µ

9βqki

}.

Further this implies that

qk+13i − β ≤

Dki

{2µ9β2 +

(4µ9− 14

9

)β − 6µ

9+ 14

9

}Dki−1

{(13− 2µ

9

)β2 +

(23+ 4µ

9

)β − 2µ

9

} .57

Since Dki

{2µ9β2 +

(4µ9− 14

9

)β − 6µ

9+ 14

9

}is less than zero and Dk

i−1

{(13− 2µ

9

)β2

+(23+ 4µ

9

)β − 2µ

9

}is greater than zero for 0.3 ≤ µ ≤ 0.9 and β = 1

µ, then

qk+13i − β ≤ 0.

This implies that qk+13i ≤ β. Now we show that 1

qk+13i

− β ≤ 0.

1

qk+13i

=Dk+1

3i

Dk+13i+1

=−2µ

9Dki−1 +

(23+ 4µ

9

)Dki +

(13− 2µ

9

)Dki+1(

−49+ 4µ

9

)Dki−1 +

(179− 8µ

9

)Dki +

(−4

9+ 4µ

9

)Dki+1

.

This implies that

qk+13i − β =

1

Dki−1{

(−4

9+ 4µ

9

)+(179− 8µ

9

) Dki

Dki−1

+(−4

9+ 4µ

9

) Dki+1

Dki−1

}Dki

{−2µ

9

Dk−1i

Dki

+

(2

3− 4µ

9

)+

(1

3− 2µ

9

)Dk+1i

Dki

−(−4

9+

4µ

9

)βDk−1i

Dki

−(17

9− 8µ

9

)β −

(−4

9+

4µ

9

)βDk+1i

Dki

}.

Again implies

qk+13i − β =

1

Dki−1{

(−4

9+ 4µ

9

)+(179− 8µ

9

)qki−1 +

(−4

9+ 4µ

9

)qki q

ki−1}

Dki

{−2µ

9

1

qki−1

+

(2

3− 4µ

9

)+

(1

3− 2µ

9

)qki −

(−4

9+

4µ

9

)β

1

qki−1

−(17

9− 8µ

9

)β −

(−4

9+

4µ

9

)βqki

}.

58

Furthermore

qk+13i − β ≤ 1

Dki−1{

(−4

9+ 4µ

9

)+(179− 8µ

9

)β +

(−4

9+ 4µ

9

)β2}

Dki

{−2µ

9

1

β

+

(2

3− 4µ

9

)+

(1

3− 2µ

9

)β −

(−4

9+

4µ

9

)−(17

9− 8µ

9

)β

−(−4

9+

4µ

9

)}.


1

qk+13i

− β ≤Dki {(−14

9+ 2µ

3

)β − 2µ

91β− 4µ

9+ 14

9}

Dki−1{

(−4

9+ 4µ

9

)+(179− 8µ

9

)β +

(−4

9+ 4µ

9

)β2}

.

Again sinceDki {(−14

9+ 2µ

3

)β−2µ

91β−4µ

9+14

9} is less than zero andDk

i−1{(−4

9+ 4µ

9

)+(

179− 8µ

9

)β +

(−4

9+ 4µ

9

)β2} is greater than zero for 0.3 ≤ µ ≤ 0.9 and β = 1

µ, so

1

qk+13i

− β ≤ 0.

This implies that 1

qk+13i

≤ β. In the same way, we see that qk+13i+1 ≤ β, qk+1

3i+2 ≤ β,

1

qk+13i+1

≤ β and 1

qk+13i+2

≤ β. So Qk+1 ≤ δ. Since Qk+1 = maxi{qk+1i , 1

qk+1i

}, it is obvious

that Qk+1 ≥ 1β

.



strictly concave, i.e D0i < 0, for all i ∈ Z. Let


Dki+1

Dki

,

Qk = maxi

{qki ,1

qki}, ∀k ≥ 0, k ∈ Z, i ∈ Z.

Furthermore, let 0.1 ≤ µ ≤ 0.9 and β = − 2µ1+µ

, β ∈ R. Then for 1β≤ Q0 ≤ β,

Dki < 0,

1

β≤ Qk ≤ β, ∀k ≥ 0, k ∈ Z, i ∈ Z. (3.10)

Namely, the limit function generated by the subdivision scheme (3.3) is strictly concave.

59

Proof. (3.10) will be proved by mathematical induction. When k = 0, D0i < 0,

1β≤ Q0 ≤ β, then (3.10) is true.

Suppose that (3.10) holds for k. i.e Dki < 0, 1

β≤ Qk ≤ β, since

Dk+14i = 42k+2(2)−1(fk+1

4i−1 − 2fk+14i + fk+1

4i+1).

This implies that

Dk+14i =

1

2Dki +

1

2Dki+1.

Similarly

Dk+14i+1 =

(1

4− 5µ

16

)Dki +

(3

4+

5µ

8

)Dki+1 −

5µ

16Dki+2,

Dk+14i+2 =

(−5

8+

5µ

8

)Dki +

(9

4− 5µ

4

)Dki+1 +

(−5

8+

5µ

8

)Dki+2,

Dk+14i+3 = −5µ

16Dki +

(3

4+

5µ

8

)Dki+1 +

(1

4− 5µ

16

)Dki+2.

Now we show that

Dk+14i < 0, Dk+1

4i+1 < 0 Dk+14i+1 < 0 and Dk+2

4i+3 < 0.

Now consider

Dk+14i =

1

2Dki +

1

2Dki+1.

This implies

Dk+14i = Dk

i

{1

2+

1

2

Dki+1

Dki

}.

Again implies

Dk+14i = Dk

i

{1

2+

1

2qki

}.

60

Furthermore

Dk+14i ≤ Dk

i

{1

2+

1

2β

}.


1

2+

1

2β

}> 0, for 0.3 ≤ µ ≤ 0.9 and β =

1

µ.

This further implies that Dk+14i < 0. Further

Dk+14i+1 =

(1

4− 5µ

16

)Dki +

(3

4+

5µ

8

)Dki+1 −

5µ

16Dki+2.

This implies

Dk+14i+1 = Dk

i

{(1

4− 5µ

16

)+

(3

4+

5µ

8

)Dki+1

Dki

− 5µ

16

Dki+2

Dki+1

Dki+1

Dki

}.

Again implies

Dk+14i+1 = Dk

i

{(1

4− 5µ

16

)+

(3

4+

5µ

8

)qki −

5µ

16qki+1q

ki

}.

Furthermore

Dk+14i+1 ≤ Dk

i

{(1

4− 5µ

16

)+

(3

4+

5µ

8

)β − 5µ

16

}.


1

4− 5µ

16

)+

(3

4+

5µ

8

)β − 5µ

16

}> 0, for 0.3 ≤ µ ≤ 0.9 and β =

1

µ.

This further implies that Dk+14i+1 < 0. Furthermore

Dk+14i+2 =

(−5

8+

5µ

8

)Dki +

(9

4− 5µ

4

)Dki+1 +

(−5

8+

5µ

8

)Dki+2.

This implies

Dk+14i+2 = Dk

i

{(−5

8+

5µ

8

)Dki

Dki+1

+

(9

4− 5µ

4

)+

(−5

8+

5µ

8

)Dki+2

Dki+1

}.

61

Again implies

Dk+14i+2 = Dk

i

{(−5

8+

5µ

8

)+

(9

4− 5µ

4

)qki +

(−5

8+

5µ

8

)qki+1q

ki

}.

Furthermore

Dk+14i+2 ≤ Dk

i

{(−5

8+

5µ

8

)+

(9

4− 5µ

4

)β +

(−5

8+

5µ

8

)β2

}.

As we know that Dki+1 < 0 and{(

−5

8+

5µ

8

)+

(9

4− 5µ

4

)β +

(−5

8+

5µ

8

)β2

}> 0,

for 0.3 ≤ µ ≤ 0.9 and β = 1µ

.

This further implies that Dk+14i+2 < 0. Finally

Dk+14i+1 = −5µ

16Dki +

(3

4+

5µ

8

)Dki+1 +

(1

4− 5µ

16

)Dki+2.

This implies

Dk+14i+1 = Dk

i

{−5µ

16+

(3

4+

5µ

8

)Dki+1

Dki

(1

4− 5µ

16

)Dki+2

Dki+1

Dki+1

Dki

}.

Further implies

Dk+14i+1 = Dk

i

{−5µ

16+

(3

4+

5µ

8

)qki +

(1

4− 5µ

16

)qki+1q

ki

}.

Furthermore

Dk+14i+1 ≤ Dk

i

{−5µ

16+

(3

4+

5µ

8

)β +

(1

4− 5µ

16

)β2

}.


−5µ

16+

(3

4+

5µ

8

)β +

(1

4− 5µ

16

)β2

}> 0, for 0.3 ≤ µ ≤ 0.9 and β =

1

µ.

This further implies that Dk+14i+3 < 0.

Now we prove that 1β≤ Qk+1 ≤ β, first we show that qk+1

4i − β ≤ 0.

qk+14i =

(14− 5µ

16

)Dki +

(34+ 5µ

8

)Dki+1 −

5µ16Dki+2

12Dki +

12Dki+1

.

62

This implies that

qk+14i − β =

{(14− 5µ

16

)Dki +

(34+ 5µ

8

)Dki+1 −

5µ16Dki+2 − 1

2βDk

i − 12βDk

i+1}{12Dki +

12Dki+1}

.

Again implies that

qk+14i − β =

Dki+1{

(14− 5µ

16

)1qki

+(34+ 5µ

8

)− 5µ

16qki+1 − 1

2β 1qki

− 12β}

Dki {1

2+ 1

2yki }

.


qk+14i − β ≤

Dki+1{

(14− 5µ

16

)β +

(34+ 5µ

8

)− 5µ

161β− 1

2− 1

2β}

Dki {1

2+ 1

2β}

.

SinceDki+1{

(14− 5µ

16

)β+(34+ 5µ

8

)− 5µ

161β− 1

2− 1

2β} is less than zero andDk

i {12+ 1

2β}

is greater than zero for 0.3 ≤ µ ≤ 0.9 and β = 1µ

.

This implies that

qk+14i − β ≤ 0.

Now we show that 1

qk+13i

− β < 0.

1

qk+14i

=12Dki +

12Dki+1(

14− 5µ

16

)Dki +

(34+ 5µ

8

)Dki+1 −

5µ16Dki+2

.

This implies that

qk+14i − β =

{12Dki +

12Dki+1 −

(14− 5µ

16

)βDk

i −(34+ 5µ

8

)βDk

i+1 +5µ16βDk

i+2}{(14− 5µ

16

)Dki +

(34+ 5µ

8

)Dki+1 −

5µ16Dki+2.}

.

Again implies that

qk+14i − β =

Dki+1{1

21qki

+ 12−(14− 5µ

16

)β 1qki

−(34+ 5µ

8

)β + 5µ

16βqki+1}

Dki {(14− 5µ

16

)+(34+ 5µ

8

)qki −

5µ16qki+1q

ki }

.


qk+14i − β ≤

Dki+1{1

2β + 1

2−(14− 5µ

16

)−(34+ 5µ

8

)β + 5µ

16β2}

Dki {(14− 5µ

16

)+(34+ 5µ

8

)β − 5µ

16}

.

63

SinceDki+1

{12β + 1

2−(14− 5µ

16

)−(34+ 5µ

8

)β + 5µ

16β2}

is less than zero andDki

{(14− 5µ

16

)+(34+ 5µ

8

)β − 5µ

16

}is greater than zero for 0.3 ≤ µ ≤ 0.9 and β = 1

µ.

This implies that

1

qk+14i

− β ≤ 0.

In the same way, we can get qk+14i+1 ≤ β, qk+1

4i+2 ≤ β, qk+14i+3 ≤ β, 1

qk+14i+1

≤ β, 1

qk+14i+2

≤ β

and 1

qk+14i+3

≤ β. So Qk+1 ≤ β. Since Qk+1 = maxi{qk+1i , 1

qk+1i


Qk+1 ≥ 1β

.


3.2.4 Demonstration

In this section, a numerical demonstration of monotonicity, convexity and con-

cavity preserving schemes given in previous section is presented.

Example 1.

Figures 3.2 and 3.3 are produced by using monotone data set presented in Table

3.1 borrowed by Hussain et al. (2012). In Figure 3.2(a), curve is generated by

using cubic Hermite spline which looses the monotone shape of the data, Figure

3.2(b)-3.2(e) are monotone curves obtained by rational cubic function of Hussan

and Bashir (2011), Tan et al. (2014), schemes (3.2) and (3.3) at µ = 0.5 respective-

ly. It is to be noted that rational cubic function tightly fit the data while scheme

(3.2) and (3.3) have relaxed data fitting. Scheme by Tan et al. (2014) also has too

relaxed data fitting.

Figures 3.3(a)-3.3(c) are generated by Hussan and Bashir (2011), schemes (3.2)

and (3.3) at different values of parameter. From these figures it is observed that

parameter in rational function of Hussan and Bashir (2011) effect the fitting over

some part of the domain while fitting remain tight over other part of the domain.

64

Schemes (3.2) and (3.3) give loose/tight fitting at different values of parameter.

These schemes also preserve the shape of data.

Example 2.

Convex data set is given in Table 3.2 borrowed by Samreen (2006) and Figure 3.4

is produced by using this data. Figure 3.4(a)-3.4(f) are convex curves obtained

by Hao et al. (2011), Tan et al. (2014), Cai (2009), Dyn et al. (1999), schemes

(3.2) and (3.3) for µ = 0.5 respectively. In Figure 3.4(a), Curve is far away from

initial and final points due to high shrinkage effect of the scheme. Curve is nice-

ly approximately fitted over the data in Figure 3.4(b). It is to be noted that in

Figure 3.4(c), curve exhibits slight fluctuation over the part of domain i.e. over

(2 3). Fluctuation area is shown by a circle. In Figure 3.4(d)-3.4(e), curves have

interpolatory effect and smoothly pass through the control points. The convexi-

ty preservation of proposed schemes (3.2) and (3.3) shown in Figure 3.4(e)-3.4(f)

give the better results as compare to the existing schemes of Hao et al. (2011),

Tan et al. (2014) and Cai (2009) shown in Figure 3.4(a)-3.4(c).

Example 3.

Figure 3.5 is produced by using concave data set given in Table 3.3. Concave

curves shown in Figure 3.5(a) and 3.5(b) is produced by schemes (3.2) and (3.3)

at µ = 0.5 respectively.

Table 3.1: Monotone data set

x 5 7 11 12 16

y 10 11 15 50 85

65

(a) (b) (c)

(d) (e)

Figure 3.2: The curves (a), (b), (c), (d) and (e) are generated by cubic Hermite spline,

Hussan and Bashir (2011), Tan et al. (2014), scheme (3.2) and (3.3) by using monotone

data set.

66

Table 3.2: Convex data set

x 0 1 2 3 4 5 6

y 9 5 3 2.40 2.20 2.15 2.10

Table 3.3: Concave data set

x 100 140 180 210 250 290 330

y 3.984 3.642 3 2.5 2.06 2.06 2.5

3.3 Traditional properties of schemes

In this section, we discuss some significant properties like Hölder exponent,

polynomial generation, polynomial reproduction, approximation order, sup-

port width of basic limit function of the schemes (3.2) and (3.3) by using tech-

niques developed in Conti and Hormann (2011), Dyn and Levin (2002), Mustafa

et al. (2015) and Rioul (1992):

3.3.1 Hölder exponent

In this section, we discuss the Hölder exponent of the schemes (3.2) and (3.3)

and we use Rioul’s (1992) method for Hölder exponent.

Theorem 3.3.1. The Hölder exponent of the scheme (3.2) is r = 3.052.

Proof. The Laurent polynomial (3.4) of the scheme (3.2) can be written as

a(z) =

(1 + z + z2

3

)4

b(z), (3.11)

67

(a) (b) (c)

Figure 3.3: The curves (a), (b) and (c) are generated by rational cubic function Hussan

and Bashir (2011), scheme (3.2) and (3.3) respectively by using monotone data set.

where

b(z) = {−2µz0 + (8µ− 4)z + (−12µ+ 11)z2 + (8µ− 4)z3 − 2µz4}. (3.12)

From (2.17) and (3.12) b0 = −2µ, b1 = 8µ − 4, b2 = −12µ + 11, b3 = 8µ − 4,

b4 = −2µ, k = 4, m = 4 and n = 3, thus q = 0, 1, . . . , 4 and then M0, M1, M2, M3

and M4 are the matrices with elements

(M0)ij = b4+i−3j,

(M1)ij = b4+i−3j+1,

(M2)ij = b4+i−3j+2,

(M3)ij = b4+i−3j+3,

(M4)ij = b4+i−3j+4,

(3.13)

where i, j = 1, . . . , 4. This implies

68

(a) (b) (c)

(d) (e) (f)

Figure 3.4: The convex curves (a), (b), (c), (d) and (e) are generated by Hao et al. (2011),

Tan et al. (2014), Cai (2009), Dyn et al. (1999), schemes (3.2) and (3.3) respectively by

using convex data set.

69

(a) (b)

Figure 3.5: The concave curves (a) and (b) are generated by scheme (3.2) and (3.3)

respectively by using concave data set.

M0 =

−12µ+ 11 0 0 0

8µ− 4 −2µ 0 0

−2µ 8µ− 4 0 0

0 −12µ+ 11 0 0

, M1 =

8µ− 4 −2µ 0 0

−2µ 8µ− 4 0 0

0 −12µ+ 11 0 0

0 8µ− 4 −2µ 0

,

M2 =

−2µ 8µ− 4 0 0

0 −12µ+ 11 0 0

0 8µ− 4 −2µ 0

0 −2µ 8µ− 4 0

, M3 =

0 −12µ+ 11 0 0

0 8µ− 4 −2µ 0

0 −2µ 8µ− 4 0

0 0 −12µ+ 11 0

,

M4 =

0 8µ− 4 −2µ 0

0 −2µ 8µ− 4 0

0 0 −12µ+ 11 0

0 0 8µ− 4 −2µ

. (3.14)

70

From (2.18)and (3.14) we have

max {2µ, 10µ− 4, 2µ, 10µ− 4, 2µ} ≤ λ ≤ max {10µ− 4, 10µ− 4, 10µ− 4,

10µ− 4, 10µ− 4} . (3.15)

Since the largest eigenvalue and the max-norm of the metrics is 2.833

for µ = 0.6833 where µ ∈ (0.666, 0.7), so we have

r = 4− log3(2.8333) = 3.052.

Theorem 3.3.2. The Hölder exponent of the scheme (3.3) is r = 2.888.

Proof. The Laurent polynomial of the scheme (3.3) can be written as

a(z) =

(1 + z + z2 + z3

4

)4

b(z) (3.16)

where

b(z) = {−5µz0 + (20µ− 10)z + (−30µ+ 24)z2 + (20µ− 10)z3 − 5µz4}. (3.17)

From (2.17) and (3.17) b0 = −5µ, b1 = 20µ − 10, b2 = −30µ + 24, b3 = 20µ − 10,

b4 = −5µ, k = 4, m = 4 and n = 4, thus q = 0, 1, . . . , 4 and then M0, M1, M2, M3

and M4 are the matrices with elements

(M0)ij = b4+i−4j,

(M1)ij = b4+i−4j+1,

(M2)ij = b4+i−4j+2,

(M3)ij = b4+i−4j+3,

(M4)ij = b4+i−4j+4,

(3.18)

71

where i, j = 1, . . . , 4. This impies

M0 =

20µ− 10 0 0 0

−30µ+ 24 0 0 0

20µ− 10 8µ− 4 0 0

−5µ −5µ 0 0

, M1 =

−30µ+ 24 0 0 0

20µ− 10 8µ− 4 0 0

−5µ −5µ 0 0

0 20µ− 10 0 0

,

M2 =

20µ− 10 8µ− 4 0 0

−5µ −5µ 0 0

0 20µ− 10 0 0

0 −30µ+ 24 0 0

, M3 =

−5µ −5µ 0 0

0 20µ− 10 0 0

0 −30µ+ 24 0 0

0 20µ− 10 0 0

,

M4 =

0 20µ− 10 0 0

0 −30µ+ 24 0 0

0 20µ− 10 0 0

0 −5µ −5µ 0

. (3.19)

From (2.18) and (3.19), we have

max{2µ− 10, 30µ− 24, 5µ, 5µ, 30µ− 24} ≤ λ ≤ max{10µ, 10µ, 10µ, (3.20)

10µ, 10µ}.

From (3.20), now we calculate the lower and upper bound of Hölder exponent

for µ = 0.733 where µ ∈ (0.266, 1.2).

Lower bound of Hölder exponent is

r ≥ 4− log4(∥b∥l)/l = 4− log4(4.666) = 2.888,

where b = 4.666, l = 1. Upper bound of Hölder exponent is

r ≤ 4− log4(λ) = 4− log4(4.666) = 2.888.

72

r = 2.888.

3.3.2 Polynomial generation

Polynomial generation of degree d is the ability of subdivision to generate the

full space of polynomials of up to d. This property is equivalent to sum rules of

order d + 1 on the subdivision mask or equivalently, to zero condition of order

d + 1 on the subdivision symbol. Obviously, generation degree is not less than

to reproduction degree.

Theorem 3.3.3. The degree of polynomial generation of the scheme (3.2) is 3.

Proof. By using (3.4), we have

a(0)(α13) = a(e

2πi3 ) = 0, for j = 1 and k = 0.

Similarly, we show that

a(k)(αj3) = 0, for j = 1, 2 and k = 0, 1, 2, 3 (3.21)

and

a(4)(αj3) = 0,

where k denotes the order of derivative.

Then by (1.1), degree of polynomial generation is 3.

Theorem 3.3.4. The degree of polynomial generation of the scheme (3.3) is 3.

73


a(0)(α14) = a(e

2πi4 ) = 0, for j = 1 and k = 0.


a(k)(αj4) = 0, for j = 1, 2, 3 and k = 0, 1, 2, 3 (3.22)

and

a(4)(αj4) = 0,



3.3.3 Polynomial reproduction and approximation order

The polynomial reproduction is a desirable property for a subdivision scheme,

because any convergent subdivision scheme that reproduce polynomial of de-

gree d has approximation order d + 1. A subdivision scheme reproduce poly-

nomials of degree d is that it must be able to generate polynomials of the same

degree as limit functions for some initial data. The degree of polynomial repro-

duction can never exceed the degree of polynomial generation.



a(k)(1) = 3k−1∏l=0

(τ − l) and a(k)(αj3) = 0, j = 1, 2,


3.


a(1)(1) = 0.

74

This implies that

τ =a(1)(1)

3= 0.

So from (1.2), the scheme (3.2) has primal parametrization. From (3.21), we have

a(k)(αj3) = 0.

By (3.4), we get a(1) = 3. Also 3∏−1


3∏0−1


a(k)(1) = 3k−1∏l=0

(τ − l),




Theorem 3.3.6. A 5-point ternary approximating scheme (3.2) has an approximation

order of 4.



a(k)(1) = 4k−1∏l=0

(τ − l) and a(k)(αj4) = 0, j = 1, 2, 3,


4.


a(1)(1) = 0.

This implies that

τ =a(1)(1)

4= 0.

75

So from (1.2), the scheme (3.3) has primal parametrization. From(3.22), we have

a(k)(αj3) = 0.

By (3.16), we get a(1) = 4. Also 4∏−1


4∏0−1


a(k)(1) = 4k−1∏l=0

(τ − l),



Theorem 3.3.8. A 5-point quaternary approximating scheme (3.3) has an approxima-

tion order of 4.

3.3.4 Basic limit function

The basic limit function F of a scheme is defined as the limit function of the

scheme for the data f 0i = δi,0, where δi,0 is Kronecker delta. By Theorems 3.1.1

and 3.1.2 it follows that the basic functions defined by the proposed schemes

(3.2) and (3.3) generate C3 and C2-continues limit curves respectively. These

functions are shown in Figure 3.7.

Now we derive a general relation to calcute support width of relaxed 5-point

n-ary scheme. We figure out that as we increase arity of relaxed 5-point n-ary

scheme the support width decreases, i.e for relaxed 5-point n-ary approximating

scheme arity and support width are reciprocal to each other.

Theorem 3.3.9. The basic function F defined by relaxed 5-point n-ary approximating

scheme has support width S = 4nn−1

, which implies that it vanishes outside the interval[− 4n

2(n−1), 4n2(n−1)

].

76

Proof. Since the basic function F is the limit function of the relaxed 5-point n-ary

scheme, its support width S can be determined by computing how far the effect

of the non zero vertex f 00 will propagate along by. As the mask of the scheme is

a (4n + 1)-long sequence by centering it on that vertex, the distances to the last

of its nonzero coefficients are equal to 2n on each side and after each subdivi-

sion level, its reduced by the factor 1n

. Therefore, at the first subdivision step,

the influence of the nonzero vertex extend a distance 2nn

on each side during the

second step, the last nonzero coefficient itself causes a further effect 2nn2 and suc-

cessive iterations give 2nn3 , 2n

n4 , . . . . Hence after k subdivision steps the furthest

nonzero vertex on the either side of center will be at

2n

(1

n+

1

n2+

1

n3+ . . .+

1

nk

)=

2n

n

(k−1∑j=0

1

nj

).

Since 1n< 1, the geometric sequence can be summed to give the extended dis-

tance on each side and we conclude that, in the limit, the total influence of the

original nonzero vertex will propagate along by

S = 2× 2n

n

(k−1∑j=0

1

nj

)=

4n

n

(1

1− 1n

)=

4n

n− 1.

Corollary 3.3.10. The basic function of the scheme (3.2) has support width S = 6,

which implies that it vanishes outside the interval [−3, 3].

Corollary 3.3.11. The basic function of the scheme (3.3) has support width S = 163

,

which implies that it vanishes outside the interval[−8

3, 83

].

In following theorem, we show that the basic limit function of 5-point n-ary

scheme is symmetric.

77

(a) (b)

Figure 3.6: (a) Graph of the Hölder exponent against µ for the scheme (3.2). (b) Graph

of the Hölder exponent against µ for the scheme (3.3).

Theorem 3.3.12. The basic limit function of 5-point n-ary scheme is symmetric about

Y-axis.

Proof. Let F denotes the basic limit function and define Rk := { ink | i ∈ Z} such

that the restriction of the basic limit function F to Rk satisfies F(ink

)= fki for all

i ∈ Z.

The symmetry of basic function is proved using mathematical induction on k.

First of all we note that F (i) = f 0i = f 0

−i = F (−i) for all i ∈ Z and thus F(ink

)=

F(− ink

)= f 0

−i for all i ∈ Z, k = 0.

Now we assume that F(ink

)= F

(− ink

)for all i ∈ Z, then it follows that fki =

F(ink

)= F

(− ink

)= fk−i for all i ∈ Z.

For α = 0, 1, . . . , n− 2, we have

fk+1ni+α = Anαf

ki−1 +Bn

αfki + Cn

αfki+1 +Dn

αfki+2,

for α = n− 1

fk+1ni+n−1 = En

1 fki−1 + En

2 fki + En

3 fki+1 + En

2 fki+2 + En

1 fki+3.

78

Let Anα, Bnα, Cn

α , Dnα, En

1 , En2 and En

3 are the coefficients of the 5-point n-ary ap-

proximating scheme. Therefore

F

(ni

nk+1

)= fk+1

ni = An0fki−1 +Bn

0 fki + Cn

0 fki+1 +Dn

0fki+2.

This implies that

F

(ni

nk+1

)= An0F

(i− 1

nk

)+Bn

0F

(i

nk

)+ Cn

0F

(i+ 1

nk

)+Dn

0F

(i+ 2

nk

).

So we have

F

(ni

nk+1

)= An0F

(−(i− 1)

nk

)+Bn

0F

(−ink

)+ Cn

0F

(−(i+ 1)

nk

)+Dn

0F

(−(i+ 2)

nk

).

Thus we have

F

(ni

nk+1

)= An0f

k−(i−1) +Bn

0 fk−i + Cn

0 fk−(i+1) +Dn

0fk−(i+2) = fk+1

−ni = F

(− ni

nk+1

).

Similarly, we can easily show that for α = 1, 2, . . . , n− 2.

F

(ni+ α

nk+1

)= F

(−ni+ α

nk+1

).

For α = n− 1, we have

F

(ni+ n− 1

nk+1

)= fk+1

ni+n−1 = En1 f

ki−1 + En

2 fki + En

3 fki+1 + En

2 fki+2 + En

1 fki+3.

This implies that

F

(ni+ n− 1

nk+1

)= En

1F

(i− 1

nk

)+ En

2F

(i

nk

)+ En

3F

(i+ 1

nk

)+ En

2F

(i+ 2

nk

)+En

1F

(i+ 3

nk

).


F

(ni+ n− 1

nk+1

)= En

1F

(−(i− 1)

nk

)+ En

2F

(−ink

)+ En

3F

(−(i+ 1)

nk

)+En

2F

(−(i+ 2)

nk

)+ En

1F

(−(i+ 3)

nk

).

79

Again implies

F

(ni+ n− 1

nk+1

)= En

1 fk−(i−1) + En

2 fk−i + En

3 fk−(i+1) + En

2 fk−(i+2) + En

1 fk−(i+3)

= fk+1−ni+n−1 = F

(−ni+ n− 1

nk+1

).


Figure 3.6(a) shows a graph of the Hölder exponent against µ ∈ (0.666, 0.7).

Notice that the highest smoothness of the 5-point ternary scheme (3.2) is achieved

at µ = 0.683 and its Hölder exponent is H = 3.052. Figure 3.6(b) shows a graph

of the Hölder exponent against µ ∈ (0.266, 1). It is also to be noted that the high-

est smoothness of the 5-point quaternary scheme (3.3) is achieved at µ = 0.633

and its Hölder exponent is H = 2.888.

In Figure 3.7, the basic limit functions of schemes (3.2) and (3.3) are shown. Here

doted lines show the control polygons where the solid lines indicate the basic

limit functions at different values of parameter. In 3.7(a), limit functions are

generated at µ = 0.1, 0.4 and 0.68 from top to bottom, while in 3.7(b), functions

are generated at µ = 0.15, 0.45 and 0.85 from top to bottom.

In Figure 3.8, we show performance of our schemes by setting different values

of shape parameter. The control polygons are drawn by doted lines and the

smooth curves by solid lines. Limit curves presented in 3.8(a) is obtained by

scheme (3.2) at µ = 0.15, 0.45 and 0.67 while limit curves shown in 3.8(b) is

obtained by scheme (3.3) at µ = 0.2, 0.45 and 0.85. These values are taken from

parameter range for C1-, C2- and C3- continuity of both schemes.

3.4 Conclusion

We give brief summary of work done so far. We present a simple and well-

organized two-step algorithm which generates a class of 5-point n-ary approx-

imating subdivision scheme. Shape preserving properties that is monotonicity,

80

(a) (b)

Figure 3.7: (a) and (b) show the effect of parameter on the shape of the basic limit

function of the scheme (3.2) and (3.3) respectively.

convexity and concavity preservation of data fitting have been discussed. It is

concluded that family of scheme has relaxed data fitting depending on the val-

ues of parameter.

We also present smoothness analysis of proposed scheme. Some importan-

t properties of proposed scheme like Hölder exponent, degree of polynomial

generation, polynomial reproduction, approximation order support and sym-

metry of basic limit function have been discussed. An explicit formula to calcu-

late support width of basic limit function is established. We deduced that arity

and support width of 5-point n-ary scheme are reciprocal to each other. Also as

we increase the arity of 5-point n-ary schemes, support of the schemes decreas-

es and it is generally observed that as we decrease the arity of the scheme the

Hölder exponent increases. For large arity of the scheme the Hölder exponent

approaches to 1. Visual performance of proposed schemes are shown by several

examples.

81

(a) (b)

Figure 3.8: (a) and (b) show the effect of parameter on the shape of limit curves of the

scheme (3.2) and (3.3) respectively.

82

Chapter 4

A family of 6-point n-ary

interpolating subdivision schemes

In this chapter, we derive three-step algorithm based on divided difference to

generate a family of 6-point n-ary interpolating subdivision schemes rather than

using polynomials. Further, some significant properties of ternary and qua-

ternary subdivision schemes have been elaborated such as continuity, degree

of polynomial generation, polynomial reproduction and approximation order.

Furthermore, a shape preserving property monotonicity is also derived. More-

over, the visual performance of proposed schemes has also been demonstrated

through several examples.

4.1 Three-step Algorithm

In this section, we define the method for the construction of 6-point n-ary in-

terpolating subdivision schemes by using three-step algorithm instead of using

Lagrange polynomial and wavelet theory. These three steps are as follows:

• Computation of the second divided differences

83


vertices b and c are referred to as p1, p2, . . . , pn−1 respectively.

At each old vertex we calculate second divided difference R. Ra is the

second divided difference at point a, i.e.

Ra =(b− a)− (a− z)

n2=b− 2a+ z

n2. (4.1)

Similarly we can compute second divided differences Rb, Rc and Rd at

point b, c and d respectively. See Fig. 4.1

• Computation of the new second divided differences

By using the second divided differences Ra, Rb, Rc, Rd and stencils of DD

schemes A = [AL,1, AL,2, AL,3, AL,4], B = [BS,1, BS,2, BS,3, BS,4], we calculate

the new divided differences. For odd a-ary, we construct new divided

differences as follows RpL = AL,1Ra + AL,2Rb + AL,3Rc + AL,4Rd,

RpV = An−V,1Ra + An−V,2Rb + An−V,3Rc + An−V,4Rd,(4.2)

for n ≥ 3, L = 1, 2, . . . , n− S, V = S . . . , n− 1 and S = n+12

.

In the case of even a-ary, we use the above new divided differences RpL ,

RpV and have to construct another new divided difference RpS , s.t

RpS = BS,1(Ra +Rd) +BS,2(Rb +Rc), (4.3)

for n ≥ 2, L = 1, 2, . . . , S − 1, V = S + 1 . . . , n− 1, u = 3S and S = n2,

84

where

AL,1 =−L(n− L)(2n− L)

6n3,

AL,2 =(n2 − L2)(2n− L)

2n3,

AL,3 =L(n+ L)(2n− L)

2n3,

AL,4 =−L(n2 − L2)

6n3,

and

BS,1 =−uS2

6n3,

BS,2 =u2S

2n3.

• Computation of modified vertices

By using (4.2) and (4.3), we compute positions of modified vertices i.e p1,

p2, . . . , pn−1 by using the following

Rp1 = p2 − 2p1 + b, Rpi = pi+1 − 2pi + pi−1, Rpn−1 = c− 2pn−1 + pn−2, (4.4)

where i = 2, 3, . . . , n− 2.

4.1.1 Examples

Herein, it can be seen that 6-point n-ary interpolating subdivision schemes can

be easily generated by above algorithm. In n-ary subdivision scheme each seg-

ment is divided into n sub-segments at each refinement level. One point is in-

serted at the position 1n

, second point is inserted at the position 2n

and so on

(n− 1)th point is inserted at the position n−1n

. By taking different values in (4.1)-

(4.4), we get different n-ary schemes.

85

• For n = 3 in (4.1), (4.2) and (4.4), the 6-point ternary interpolating scheme

isfk+13i = fki ,

fk+13i+1 =

142187

fki−2 − 1782187

fki−1 +16522187

fki + 8322187

fki+1 − 1462187

fki+2 +13

2187fki+2,

fk+13i+2 =

132187

fki−2 − 1462187

fki−1 +8322187

fki + 16522187

fki+1 − 1782187

fki+2 +14

2187fki+2.

(4.5)

• For n = 4 in (4.1)-(4.4), we have following 6-point quaternary interpolating

scheme

fk+14i = fki ,

fk+14i+1 =

428192

fki−2 − 5788192

fki−1 +68208192

fki + 23008192

fki+1 − 4308192

fki+2 +38

8192fki+2,

fk+14i+2 =

568192

fki−2 − 6808192

fki−1 +47208192

fki + 47208192

fki+1 − 6808192

fki+2 +56

8192fki+2,

fk+14i+3 =

388192

fki−2 − 4308192

fki−1 +23008192

fki + 68208192

fki+1 − 5788192

fki+2 +42

8192fki+2.

(4.6)

Remark 4.1.1. • By substituting n = 2 in (4.1)-(4.4), we have the mask of 6-

point binary interpolating scheme of Augsdöefer et al. (2010).

4.1.2 Smoothness Analysis of Proposed schemes

We discuss the analysis of a 6-point ternary and quaternary iterpolating sub-

division schemes. By using idea of Dyn and Levin (2002) which help to fined

convergence and smoothness of the schemes (4.5) and (4.6).

Theorem 4.1.1. The 6-point ternary interpolating subdivision scheme (4.5) is C2.

Proof. The Laurent polynomial a(z) for the scheme (4.5) is

a(z) =1

2187{13z0 + 14z1 − 146z3 − 178z4 + 832z6 + 1652z7 + 2187z8 (4.7)

+1625z9 + 832z10 − 178z12 − 146z13 + 14z15 + 13z16}.


b[1,1](z) =1

3a1(z) =

1

2187{13 + z − 14z2 − 133z3 − 31z4 + 164z5 + 699z6 + 789z7

+699z8 + 164z9 − 31z10 − 133z11 − 14z12 + z13 + 13z14}, (4.8)

86

and

b[2,1](z) =1

3a2(z) =

1

729{13− 12z − 15z2 − 106z3 + 90z4 + 180z5 + 429z6 + 180z7

+90z8 − 106z9 − 15z10 − 12z11 + 13z12}. (4.9)

If Sβ is the scheme corresponding to aβ(z) then by (1.8)∥∥∥∥13Sβ∥∥∥∥∞

= max

{∑j∈Z

|b[β,1]i+3j| : i = 0, 1, 2

}, β = 1, 2.

Using (1.7), (4.8) and (4.9), we get∥∥∥∥13S1

∥∥∥∥∞

= max

{∣∣∣∣ 13

2187

∣∣∣∣+ ∣∣∣∣−133

2187

∣∣∣∣+ ∣∣∣∣ 6992187

∣∣∣∣+ ∣∣∣∣ 1642187

∣∣∣∣+ ∣∣∣∣−14

2187

∣∣∣∣ ,∣∣∣∣ 1

2187

∣∣∣∣+ ∣∣∣∣−31

2187

∣∣∣∣+ ∣∣∣∣ 7892187

∣∣∣∣+ ∣∣∣∣−31

2187

∣∣∣∣+ ∣∣∣∣ 1

2187

∣∣∣∣} ,and∥∥∥∥13S2

∥∥∥∥∞

= max

{∣∣∣∣ 13729∣∣∣∣+ ∣∣∣∣−106

729

∣∣∣∣+ ∣∣∣∣429729

∣∣∣∣+ ∣∣∣∣−106

729

∣∣∣∣+ ∣∣∣∣ 13729∣∣∣∣ ,∣∣∣∣−12

729

∣∣∣∣+ ∣∣∣∣ 90729∣∣∣∣+ ∣∣∣∣180729

∣∣∣∣+ ∣∣∣∣−15

729

∣∣∣∣} ,As we see ∥ 1

3S1∥∞ < 1 then by Theorem 1.3.1 the scheme (4.5) is C0. Similarly

∥ 13S2∥∞ < 1 and ∥ (1

3S3)

6∥∞ < 1 then by Corollary 1.3.3 the scheme (4.5) is C1

and C2 respectively.

Theorem 4.1.2. The 6-point quaternary interpolating subdivision scheme (4.6) is C2.


4.2 Properties of subdivision schemes

In this section, we discuss some significant properties degree of polynomial gen-

eration, polynomial reproduction, approximation order of the schemes (4.5) and

(4.6).

87



a(0)(α13) = a(e

2πi3 ) = 0, for j = 1 and k = 0.


a(k)(αj3) = 0, for j = 1, 2 and k = 0, 1, 2, 3 (4.10)

and

a(4)(αj3) = 0,



Theorem 4.2.2. The degree of polynomial reproduction of the subdivision scheme (4.5)

is 3 with respect to the parameterizations (1.2) if and only if

a(k)(1) = 3k−1∏l=0

(τ − l) and a(k)(αj3) = 0, j = 1, 2,


3.


a(1)(1) = 0.

This implies that

τ =a(1)(1)

3= 0.

So from (1.2), the scheme (4.5) has primal parametrization. From (4.10), we have

a(k)(αj3) = 0.

88

By (4.7), we get a(1) = 3. Also 3∏−1


3∏0−1


a(k)(1) = 3k−1∏l=0

(τ − l),




Theorem 4.2.3. A 6-point ternary interpolating subdivision scheme (4.5) has an ap-

proximation order of 4.





a(k)(1) = 4k−1∏l=0

(τ − l) and a(k)(αj4) = 0, j = 1, 2, 3


4.



Theorem 4.2.6. A 6-point quaternary interpolating subdivision scheme (4.6) has an

approximation order of 4.

89


x 1760 2650 2760

y 500 1360 2940


Here, we examine monotonicity preservation of 6-point ternary interpolating

subdivision scheme (4.5) and 6-point quaternary interpolating scheme (4.6).


i < f 0i+1,

i ∈ Z. Let

dki = fki+1 − fki , gki =dki+1

dki, Gk = max

i{gki ,

1

gki}, k ≥ 0, k ∈ Z, i ∈ Z.

Furthermore, let 0.2 ≤ λ ≤ 0.9, λ ∈ R. If 1λ

≤ G0 ≤ λ, {fki } is defined by the

subdivision scheme (4.5), then

dki > 0,1

λ≤ Gk ≤ λ, k ≥ 0, k ∈ Z, i ∈ Z. (4.11)

Proof. (4.11) will be proved by mathematical induction. When k = 0,

d0i = f 0i+1 − f 0

i > 0, 1λ≤ G0 ≤ λ, then (4.11) is true.

Suppose that (4.11) holds for k. i.e dki = fki+1 − fki > 0, 1λ≤ Gk ≤ λ, next we will

prove that (4.11) holds for k + 1. Since

dk+13i =

1

2187{−14dki−2 + 164dki−1 + 699dki − 133dki+1 + 13dki+2}.

Similarly

dk+13i+1 =

1

2187{dki−2 − 31dki−1 + 789dki − 31dki+1 + dki+2},

dk+13i+2 =

1

2187{13dki−2 − 133dki−1 + 699dki + 164dki+1 − 14dki+2}.

90

Next we show that

dk+13i > 0, dk+1

3i+1 > 0 and dk+13i+2 > 0.

Now

dk+13i =

dki2187

{−141

gki−2

1

gki−1

+ 641

gki−1

+ 699− 133gki + 13gki+1gki }.

This implies

dk+13i ≥ dki

2187{685− 133λ+ 164

1

λ+ 13

1

λ2}.

As we know that dki > 0 and

1

2187{685− 133λ+ 164

1

λ+ 13

1

λ2} > 0, for 0.2 ≤ λ ≤ 0.9.

This further implies that dk+13i > 0. Similarly, we see that dk+1

3i+1 > 0, and dk+13i+2 > 0

for 0.2 ≤ λ ≤ 0.9. Now we prove that 1λ

≤ Gk+1 ≤ λ, first we show that

gk+13i − λ ≤ 0. Since

gk+13i − λ =

dk+13i+1

dk+13i

− λ =ξ1ξ2,

where

ξ1 ={1− 31gki−2 + 789gki−1g

ki−2 − 31gki g

ki−1g

ki−2 + gki+1g

ki g

ki−1g

ki−2 + 14λ− 164λgki−2

−699λgki−1gki−2 + 133λgki g

ki−1g

ki−2 − 13λgki+1g

ki g

ki−1g

ki−2

},

and

ξ2 = {−14 + 164gki−2 + 699gki−1gki−2 − 133gki g

ki−1g

ki−2 + 13gki+1g

ki g

ki−1g

ki−2}.

This implies that

gk+13i − λ ≤ {134λ4 − 13λ3 + 789λ2 − 716λ− 163}

{13λ4 + 699λ2 + 151λ− 14}.

91

Since {134λ4 − 13λ3 + 789λ2 − 716λ− 163} is less than zero and {13λ4 + 699λ2 +

151λ− 14} is greater than zero for 0.2 ≤ µ ≤ 0.9.

This implies that

gk+13i − λ ≤ 0.

This further implies gk+13i ≤ λ. Now we show that 1

gk+13i

− λ ≤ 0.

For this consider

1

gk+13i

− λ =dk+13i

dk+13i+1

− λ.

This implies that

1

gk+13i

− λ ≤dki−2

2187{44λ4 − λ3 + 730λ2 − 759λ− 14}

dki2187

{2λ2 − 62 1λ+ 789}

.

Since dki−2

2187{44λ4−λ3+730λ2−759λ−14} is less than zero and dki

2187{2λ2−62 1

λ+789}

is greater than zero for 0.2 ≤ µ ≤ 0.9.

This implies that

1

gk+13i

− λ ≤ 0.

This further implies 1

gk+13i

≤ λ. In the same way, we see that gk+13i+1 ≤ λ, gk+1

3i+2 ≤ λ,

1

gk+13i+1

≤ λ and 1

gk+13i+2

≤ λ. SoGk+1 ≤ λ. SinceGk+1 = maxi{gk+1i , 1

gk+1i

}, it is obvious

that Gk+1 ≥ 1λ

.


Similarly, we can prove the following theorem.


i < f 0i+1,

i ∈ Z. Let

dki = fki+1 − fki , gki =dki+1

dki, Gk = max

i{gki ,

1

gki}, k ≥ 0, k ∈ Z, i ∈ Z.

92

(a) (b)

Figure 4.2: The curves (a)and (b) are produced by schemes (4.5) and (4.6) respectively

by using monotone data set.

Furthermore, let 0.1 ≤ λ ≤ 1 , ξ ∈ R. If 1λ

≤ G0 ≤ λ, {fki } is defined by the

subdivision scheme (4.6), then

dki > 0,1

λ≤ Gk ≤ λ, k ≥ 0, k ∈ Z, i ∈ Z. (4.12)

4.2.2 Numerical Examples

Figure 4.2 is produced by using monotone data set given in Table 4.1. Monotone

curves shown in Figure 4.2(a) and 4.2(b) are produced by schemes (4.5) and (4.6)

respectively. In Figure 4.3, the initial control polygons are shown by dotted lines

and solid lines show the limit curves. Limit curves presented in 4.3(a) and 4.3(b)

are obtained by proposed schemes (4.5) and (4.6) respectively.

93

(a) (b)

Figure 4.3: Both (a) and (b) show limit curves of the schemes (4.5) and (4.6) respective-

ly.

4.3 Conclusion

We have presented a simple and well-organized three-step algorithm which

generates a family of 6-point n-ary interpolating subdivision scheme. Smooth-

ness analysis of some proposed schemes has been carried out. Some important

properties of proposed ternary and quaternary schemes like degree of polyno-

mial generation, polynomial reproduction and approximation order have been

discussed. Shape preserving property that is monotonicity preservation of data

fitting has also been derived. Visual performance of proposed scheme is shown

by several examples.

94

Chapter 5

3n-point quaternary shape

preserving subdivision schemes

In this chapter, an algorithm to construct 3n-point quaternary approximating

subdivision schemes is presented. It is to be observed that the proposed schemes

have bell-shaped mask, satisfying the basic sum rules, go up to a convergent

monotonicity preserving subdivision scheme. We analyze the shape-preserving

properties such that convexity and concavity of proposed schemes. In the end,

we show that the quaternary schemes associated with certain refinable function-

s with dilation 4 have shape preserving properties of higher order. The visual

quality of schemes with several examples is also demonstrated.

5.1 Shape preserving subdivision schemes of higher

order

Subdivision schemes preserving shape with higher order can be expressed as

aβ,N = {ai,β,N}1≤i≤3(N+1), (5.1)

95

where

ai,β,N =β

4

{bi,N−1 +

(4

β− 2

)bi−1,N−1 + bi−2,N−1

}for 0 ≤ β ≤ 2 is a shape parameter and N = 3, 7, 11, 15, .... The subdivision

scheme Saβ,Nconverges and generate CN−1 limit function for 0 ≤ β ≤ 2. The

divided difference masks of the symbol

aβ,N(z) =1

4N(1 + z + z2 + z3)N(βz2 + (4− 2β)z + β), (5.2)

is

brβ,N(z) = (1 + z + z2 + z3)−r4raβ,N(z), 1 ≤ r ≤ N,

are all bell-shaped. Thus, the subdivision scheme Saβ,N has shape-preserving

properties of order N, i.e. Saβ,N preserves the sign of the l-order difference ∆rf

for 1 ≤ r ≤ N , where ∆rf = ∆(∆r−1f). The subdivision scheme Saβ,N has

optimal shape-preserving properties in the sense that it preserves the highest -

w.r.t. to the mask support - possible order differences. The Laurent polynomials

of 3-point (N=3) and 6-point (N=7) quaternary schemes will be

aβ,3(z) =1

43{β + (4 + β)z + (12 + β)z2 + (24 + β)z3 + (40− 2β)z4

+(48− 2β)z5 + (48− 2β)z6 + (40− 2β)z7 + (24 + β)z8

+(12 + β)z9 + (4 + β)z10 + βz11}, (5.3)

and

aβ,7(z) =1

47{β + (4 + 5β)z + (28 + 15β)z2 + (112 + 35β)z3 + (336 + 63β)z4

+(812 + 91β)z5 + (1652 + 105β)z6 + (2912 + 85β)z7 + (4512 + 26β)z8

+(6216− 62β)z9 + (7672− 154β)z10 + (8512− 210β)z11

+(8512− 210β)z12 + (7672− 154β)z13 + (6216− 62β)z14

+(4512 + 26β)z15 + (2912 + 85β)z16 + (1652 + 105β)z17

+(812 + 91β)z18 + (336 + 63β)z19 + (112 + 35β)z20

+(28 + 15β)z21 + (4 + 5β)z22 + βz23}, (5.4)

96

The mask of 3-point quaternary approximating scheme in Siddiqi and Younis

(2013) coincide with aβ,3 when β = 1/2 and this scheme gives C2 continuity. The

mask aβ,7 gives C6 continuity. Similarly aβ,11 gives C10 continuity. we present

general explicit formulae to construct the mask of 3n-point quaternary subdivi-

sion schemes which are bell-shaped compactly supported.

We show that any convergent, linear, uniform, and stationary subdivision scheme

reproduces linear functions with respect to an appropriately chosen parametriza-

tion. It is obvious that all convergent subdivision schemes reproduce constants.

Theorem 5.1.1. A convergent quaternary subdivision scheme reproduces polynomials

of degree m with respect to the parametrization defined in (1.2) if and only if

a(k)(1) = 4k−1∏l=0

(τ − l) and a(k)(e2niΠ

4 ) = 0,

for k = 0, 1, ...,m and n = 1, 2, 3.

Proof. The induction over m can be performed to prove this theorem follow-

ing Conti and Hormann (2011).

In view of Conti and Hormann (2011), the following proposition helps to find

the necessary conditions defined in (5.5).

Proposition 5.1.2. Let m ∈ N and τ ∈ R. Then a subdivision symbol a(z) satisfies

a(k)(1) = 4k−1∏l=0

(τ − l) for k = 0, 1, ...,m (5.5)

iff b(z) = a(z4)z−4τ satisfies b(1) = 4 and b(k)(1) = 0 for k = 0, 1, ...,m.

Proposition 5.1.3. Let a quaternary subdivision scheme that reproduces polynomial

up to degree m. Then the smoothed scheme Sb with the symbol

b(z) =1 + z + z2 + z3

4a(z),

satisfies the conditions b(1) = 4 and b(k)(e2niΠ

4 ) = 0 for k = 0, 1, ...,m + 1 and hence

generates polynomial of degree m+ 1, but it has only linear reproduction.

97

Proof. For some symbol b(z) with b(1) = 14m

, we have

a(z) = (1 + z + z2 + z3)m+1b(z).

The first derivative of b(z) is

b′(z) =1 + z + z2 + z3

4a′(z) +

1 + 2z + 3z2

4a(z),

since

τb =b′(1)

4=

1

4a′(1) +

3

8a(1) = τa +

3

2.

The second derivative is

b′′(z) =1 + z + z2 + z3

4a′′(z) +

1 + 2z + 3z2

2a′(z) +

2 + 6z

4a(z)

and

b′′(1) = a′′(1) + 3a′(1) + 2a(1) = 4τa(τa − 1) + 12τa + 8,

simplifying, we get

b′′(1)− 4τb(τb − 1) = 0.

So, the proposed scheme reproduces polynomial of degree m = 1.

5.1.1 Convexity preservation

We prove the convexity preservation of the subdivision schemes Saβ,3 with uni-

form initial control points.


strictly convex, i.e. d0i > 0, for all i ∈ Z. Let


dki,

Y k = maxi

{yki ,1

yki}, ∀k ≥ 0, k ∈ Z, i ∈ Z.

98

Furthermore, let 0.1 ≤ β ≤ 0.9 and δ = 5 + 6β, δ ∈ R. Then for 1δ≤ Y 0 ≤ δ,

dki > 0,1

δ≤ Y k ≤ δ, ∀k ≥ 0, k ∈ Z, i ∈ Z. (5.6)

That is, the limit function generated by the subdivision scheme Saβ,3is strictly convex.



Suppose that (5.6) holds for k. i.e. dki > 0, 1δ≤ Y k ≤ δ, next we will prove that

(5.6) hold for k + 1. since

dk+14i = 42k(2)−1(fk+1

4i−1 − 2fk+14i + fk+1

4i+1).

This implies that

dk+14i =

β

4dki−1 +

(1− β

4

)dki .

Similarly

dk+14i+1 = dki ,

dk+14i+2 = dki ,

dk+14i+3 =

(1− β

4

)dki−1 +

β

4dki .

Next we show that

dk+14i > 0, dk+1

4i+1 > 0, dk+14i+2 > 0 and dk+1

4i+3 > 0.

Since

dk+14i = dki

{β

4

1

yki−1

+

(1− β

4

)}.

99

Then

dk+14i = dki

{β

4

1

δ+

(1− β

4

)}.

As we know that dki > 0 and{β

4

1

δ+

(1− β

4

)}> 0, for 0.1 ≤ β ≤ 0.9 and δ = 5 + 6β.

This implies that dk+14i > 0. Similarly, we can prove that dk+1

4i+1 > 0, dk+14i+2 > 0 and

dk+14i+3 > 0 for 0.1 ≤ β ≤ 0.9 and δ = 5 + 6β.

Now we prove that 1δ≤ Y k+1 ≤ δ, we first show that yk+1

4i − δ ≤ 0. Since

yk+14i − δ =

dk+14i+1

dk+14i

− δ =dki −

β4dki−1δ −

(1− β

4

)dki δ

β4dki−1 +

(1− β

4

)dki

.

This implies that

yk+14i − δ ≤

{1− β

4−(1− β

4

)δ}

{β4δ +

(1− β

4

)}

.

Since{1− β

4−(1− β

4

)δ}

is less than zero and{β4δ +

(1− β

4

)}is greater than

zero for 0.1 ≤ β ≤ 0.9 and δ = 5 + 6β.

This implies that

yk+14i − δ ≤ 0.

Further implies that yk+14i ≤ δ. Now we show that 1

yk+14i

− δ ≤ 0. Since

1

yk+14i

− δ =dk+14i

dk+14i+1

− δ =β4dki−1 +

(1− β

4

)dki − dki δ

dki.

This implies that

1

yk+14i

− δ ≤{(

β

4− 1

)δ + 1− β

4

}.

100

Since{(β

4− 1

)δ + 1− β

4

}< 0, for 0.1 ≤ β ≤ 0.9 and δ = 5 + 6β.

This implies that

1

yk+14i

− δ ≤ 0.

Further implies that 1

yk+14i

≤ δ. In the same way, we see that yk+14i+1 ≤ δ, yk+1

4i+2 ≤ δ,

yk+14i+3 ≤ δ, 1

yk+14i+1

≤ δ, 1

yk+14i+2

≤ δ and 1

yk+14i+3

≤ δ. So Y k+1 ≤ δ. Since Y k+1 =

maxi{yk+1i , 1

yk+1i

}, it is obvious that Y k+1 ≥ 1δ.


5.1.2 Concavity preservation

We prove the concavity preservation of subdivision schemes Saβ,3 with uniform

initial control points.


strictly concave, i.e. D0i < 0, for all i ∈ Z. Let


Dki+1

Dki

,

Qk = maxi

{qki ,1

qki}, ∀k ≥ 0, k ∈ Z, i ∈ Z.

Furthermore, let 0.1 ≤ β ≤ 0.9 and λ = 5 + 6β, λ ∈ R. Then for 1λ≤ Q0 ≤ λ,

Dki < 0,

1

λ≤ Qk ≤ λ, ∀k ≥ 0, k ∈ Z, i ∈ Z. (5.7)

That is, the limit function generated by the subdivision scheme Saβ,3is strictly concave.

Proof. (5.7) will be proved by mathematical induction. When k = 0, D0i < 0,

1λ≤ Q0 ≤ λ, then (5.7) is true.

101

Suppose that (5.7) holds for k. i.e. Dki < 0, 1

λ≤ Qk ≤ λ, next we will prove that

(5.7) hold for k + 1. since

Dk+14i = 42k(2)−1(fk+1

4i−1 − 2fk+14i + fk+1

4i+1).

This implies that

Dk+14i =

β

4Dki−1 +

(1− β

4

)Dki .

Similarly

Dk+14i+1 = Dk

i ,

Dk+14i+2 = Dk

i ,

Dk+14i+3 =

(1− β

4

)Dki−1 +

β

4Dki .

Next we show that

Dk+14i < 0, Dk+1

4i+1 < 0, Dk+14i+2 < 0 and Dk+1

4i+3 < 0.

Since

Dk+14i = Dk

i

{β

4

1

qki−1

+

(1− β

4

)}.

Then

Dk+14i ≤ Dk

i

{β

4λ+

(1− β

4

)}.


β

4δ +

(1− β

4

)}> 0, for 0.1 ≤ β ≤ 0.9 and λ = 5 + 6β.

102

This implies that Dk+14i < 0. Similarly, we can prove that Dk+1

4i+1 < 0, Dk+14i+2 < 0

and Dk+14i+3 < 0 for 0.1 ≤ µ ≤ 0.9 and λ = 5 + 6β.

Now we prove that 1λ≤ Qk+1 ≤ λ, we first show that qk+1

4i − λ ≤ 0. Since

qk+14i − λ =

Dk+14i+1

Dk+14i

− λ =Dki −

β4Dki−1λ−

(1− β

4

)Dki λ

β4Dki−1 +

(1− β

4

)Dki

.

This implies that

qk+14i − λ ≤

{1− β

4−(1− β

4

)λ}{

β4λ+

(1− β

4

)} .

Since{1− β

4−(1− β

4

)λ}

is less than zero and{β4λ+

(1− β

4

)}is greater than

zero for 0.1 ≤ β ≤ 0.9 and λ = 5 + 6β.

This implies that

qk+14i − λ ≤ 0.

Further implies that qk+14i ≤ λ. Now we show that 1

qk+14i

− λ ≤ 0.

1

qk+14i

− λ =Dk+1

4i

Dk+14i+1

− λ =β4Dki−1 +

(1− β

4

)Dki −Dk

i λ

Dki

.

This implies that

1

qk+14i

− λ ≤{(

β

4− 1

)λ+ 1− β

4

}.

Since{(β

4− 1

)λ+ 1− β

4

}< 0, for 0.1 ≤ β ≤ 0.9 and λ = 5 + 6β.

This implies that

1

qk+14i

− λ ≤ 0.

103

Further implies that 1

qk+14i

≤ λ. In the same way, we see that qk+14i+1 ≤ λ, qk+1

4i+2 ≤ λ,

qk+14i+3 ≤ λ, 1

qk+14i+1

≤ λ, 1

qk+14i+2

≤ λ and 1

qk+14i+3

≤ λ. So Qk+1 ≤ δ. Since Qk+1 =

maxi{qk+1i , 1

qk+1i

}, it is obvious that Qk+1 ≥ 1λ

.


Table 5.1: Convex data set

x 1 2 4 5 10

y 10 2.5 0.625 0.4 0.1

Table 5.2: Concave data set

x 1 2 3 4 5 6 7

y 1 8 27 64 125 216 343

5.2 Numerical examples and comparison

Figure 5.1 and Figure 5.2 are produced by using convex and concave data set

given in Table 5.1 and Table 5.2 respectively. Convex and concave curves are

produced by the schemes Saβ,3 and Saβ,7 are shown in Figure 5.1(a) , 5.1(b) and

Figure 5.2(a), 5.2(b) respectively at β = 0.5. Figure 5.3(a) and 5.3(b) show the

behavior of proposed schemes Saβ,3 and Saβ,7 at the different values of tension

parameter β. Table 5.3 shows the comparison of shape preserving quaternary

approximating schemes with existing schemes.

104

x

y

x

y

(a) (b)

Figure 5.1: (a) and (b) are the convex curves generated by schemes Saβ,3and Saβ,7

respectively.

x

y

x

y

(a) (b)

Figure 5.2: (a) and (b) are the concave curves generated by schemes Saβ,3and Saβ,7

respectively.

105

Table 5.3: Smoothness of proposed schemes with existing schemes.

Schemes Support continuity

4-point binary approximating of Dyn et al. (2005) 7 2

3-point ternary approximating of Mustafa et al. (2011) 4 2

4-point ternary approximating of Ko et al. (2007) 5.5 2

4-point quaternary of Ko (2009) 5 2

4-point quaternary of Mustafa and Khan (2009) 5 3

3-point quaternary proposed 3.6 2

6-point quaternary proposed 7.6 6

9-point quaternary proposed 13 10

(a) (b)

Figure 5.3: (a) and (b) Shows the increase in tightness of the curve with decreasing β.

106

5.3 Conclusions

A family of 3n-point quaternary shape preserving subdivision scheme with a

tension parameter has been discussed which generate smooth limiting curves.

The main objective is to introduce quaternary schemes with smaller support and

higher smoothness, comparing to binary and ternary schemes. The polynomial

reproduction, convexity, concavity and visual smoothness of proposed schemes

are also discussed.

107

Chapter 6

Univariate approximating schemes

and their non-tensor product

generalization

This chapter deals with univariate binary approximating subdivision schemes

and their generalization to non-tensor product bivariate subdivision schemes.

The two algorithms are presented with one tension and two integer parame-

ters which generate families of univariate and bivariate schemes. The tension

parameter control the shape of the limit curve and surface while integer param-

eters identify the members of the family. It is demonstrated that the proposed

schemes preserve monotonicity of initial data. Moreover, continuity, polynomi-

al reproduction and generation of the schemes are also discussed. Comparison

with existing schemes is also given.

108

6.1 Algorithm for univariate schemes

In this section, we present an algorithm for the construction of a family of binary

approximating subdivision schemes.

For this, we consider the odd sub-symbol of cubic B-spline scheme of Zheng et

al. (2014a).

αodd(z) =1 + z

2. (6.1)

Similarly even sub-symbol of 4-point binary interpolating scheme Dyn et al.

(1987) is

βeven(z) =

(1 + z

2

)(−1

8z2 +

10

8z − 1

8

). (6.2)

The symbol of the three point scheme of Mustafa et al. (2013) is given by

γµ(z) =

(1 + z

2

)3 (8µz2 + (2− 16µ)z + 8µ

). (6.3)

Let us denote the family of the binary approximating subdivision scheme by

fam,n,µ , where general member of proposed family has the symbol of the form

am,n,µ(z) = (αodd(z))m(βeven(z))

nγµ(z). (6.4)

Substituting (6.1), (6.2) and (6.3) in (6.4), we get symbol of the scheme fam,n,µ

am,n,µ(z) =

(1 + z

2

)m+n+3(−1

8z2 +

10

8z − 1

8

)n (8µz2 + (2− 16µ)z + 8µ

), (6.5)

where m and n are non-negative integers. As it is apparent that symbol of the

scheme fam,n,µ is dependent on the parameter µ and two other parameters m

and n. The parameter µ controls the shape of limit curves of the schemes while

m and n characterizes the elements of the scheme fam,n,µ .

109

6.1.1 Smoothness analysis of univariate schemes

In this section, we discuss the continuity and Hölder continuity of the schemes.

We use the theory of generating function Dyn and Levin (2002) for continuity

and Rioul’s (1992) method for Hölder continuity.

In the following theorem, we examine the convergence and smoothness of the

scheme fam,0,µ .

Theorem 6.1.1. The scheme fam,0,µ is Cm+2 for µ ∈ (0, 0.125).

Proof. Symbol of the scheme fam,0,µ is given by

am,0,µ(z) =

(1 + z

2

)ma(z), (6.6)

where

a(z) =

(1 + z

2

)3

b(z), (6.7)

and

b(z) = 8µz2 + (2− 16µ)z + 8µ.

Let Sb be the scheme corresponding to the symbol b(z). Since

∥∥∥∥12Sb∥∥∥∥∞

= max

{1

2

∑j∈Z

|b2j|,1

2

∑j∈Z

|b2j+1|

},

then for µ ∈ (0, 0.125), we have∥∥∥∥12Sb∥∥∥∥∞

= max

{∣∣∣∣8µ2∣∣∣∣+ ∣∣∣∣8µ2

∣∣∣∣ , ∣∣∣∣2− 16µ

2

∣∣∣∣} < 1.

Hence Sb is contractive. Therefore, by Corollary 4.17 of Dyn and Levin (2002),

the scheme Sa is C2 for µ ∈ (0, 0.125). So by (6.6) scheme fam,0,µ is Cm+2 for

µ ∈ (0, 0.125).

110

Similarly, we can easily find out continuity of other members of the scheme

fam,n,µ by taking into account the same formalism. Order of continuity of some

proposed univariate subdivision schemes fam,0,µ , fam,1,µ , fam,2,µ and fam,3,µ for

certain ranges of parameter is shown in Table 6.1.

Table 6.1: The order of continuity O(C) of proposed binary approximating schemes for

certain ranges of parameter.

n Scheme Ranges O(C) n Scheme Ranges O(C)

0 fam,0,µ −0.375 < µ < 0.625 Cm+0 2 fam,2,µ −0.195 < µ < 0.445 Cm+0

. . . −0.125 < µ < 0.375 Cm+1 . . . −0.194 < µ < 0.442 Cm+1

. . . 0 < µ < 0.125 Cm+2 . . . −0.034 < µ < 0.282 Cm+2

. . . −0.026 < µ < 0.235 Cm+3

. . . 0.045 < µ < 0.09 Cm+4

1 fam,1,µ −0.275 < µ < 0.525 Cm+0 3 fam,3,µ −0.356 < µ < 0.618 Cm+0

. . . −0.075 < µ < 0.3 Cm+1 . . . −0.131 < µ < 0.380 Cm+1

. . . −0.068 < µ < 0.295 Cm+2 . . . −0.128 < µ < 0.375 Cm+2

. . . 0.025 < µ < 0.104 Cm+3 . . . −0.002 < µ < 0.235 Cm+3

. . . 0.003 < µ < 0.191 Cm+4

. . . 0.006 < µ < 0.081 Cm+5

Hölder continuity is extension to the notion of continuity. In the following

theorem, we compute the Hölder continuity of the scheme fam,0,µ .

Theorem 6.1.2. The Hölder continuity of the scheme fam,0,µ is 3.

Proof. From (6.7), let b0 = 8µ, b1 = 2− 16µ, b2 = 8µ, then M0, M1 are the matrices

111

with elements (M0)ij = b2+i−2j,

(M1)ij = b2+i−2j+1,

where i, j = 1, 2, this implies

M0 =

2− 16µ 0

8µ 8µ

, M1 =

8µ 8µ

0 2− 16µ

. (6.8)

From (6.8) and Rioul (1992), the spectral radius λ of the metrics M0 and M1 can

be express as follows

max {2− 16µ, 2− 16µ} ≤ λ ≤ max {2− 16µ, 2− 16µ} .

Since the largest eigenvalue and the max-norm of the metrics is 1 for µ = 0.0625,

where µ ∈ (0, 0.125), so the Hölder continuity h = 2 − log2(1) = 3. So by (6.6),

Hölder continuity of the scheme fam,0,µ is Cm+3.

Table 6.2: Continuity of some members of the family of schemes

n µ Continuity Lower bound on Upper bound on

Hölder continuity Hölder continuity

0 0.0625 Cm+2 Cm+3 Cm+3

1 0.0375 Cm+3 Cm+3.255 Cm+3.2603

2 0.0676 Cm+4 Cm+4.478 Cm+5

Similarly, we can compute Hölder continuity of other members of the family.

If the largest eigenvalue and the max-norm of the metrics are not equal then we

calculate lower and upper bound of Hölder continuity. Lower bound of Hölder

112

continuity is h = 2− log2(∥b∥l)/l for some integer l and upper bound of Hölder

continuity is h = 2− log2(λ). It is clear from Table 6.2 that as we increase n, level

of continuity and Hölder continuity of the schemes fam,n,µ increase.

6.1.2 Response of univariate schemes to polynomial and mono-

tone data

In this section, we examine the response of schemes to polynomial data by tak-

ing into account the polynomial generation and reproduction. Moreover the

behavior of the schemes for monotone data is also part of this section. We use

the techniques developed by Romani (2015) to discuss polynomial generation

and polynomial reproduction.

Polynomial generation

The polynomial generation of degree d is the ability of subdivision scheme to

generate the full space of polynomials up to degree d denoted by πd. The gen-

eration degree of a subdivision scheme is the maximum degree of polynomials

that can potentially be generated by the scheme.

Theorem 6.1.3. The subdivision scheme fam,n,µ generates πm+n+2 for all m,n ∈ N .

Moreover, if µ = 116

, fam,n,µ generates πm+n+4.

Proof. Since conditions

am,n,µ(1) = 2, am,n,µ(−1) = 0, D(k)am,n,µ(−1) = 0, k = 1, 2, . . . ,m+ n+ 2,

are verified by am,n,µ(z) for all µ ∈ R and D(k) denote the kth derivative. Thus,

in view of Proposition 2.1 of Romani (2015) degree of polynomial generation is

m+n+2 for all µ ∈ R. Moreover, by setting µ = 116

two more terms (1+z) can be

113

factored out from am,n,µ(z), then we have D(k+1)am,n,µ(−1) = D(k+2)am,n,µ(−1) =

0. So the degree of polynomial generation is m+ n+ 4.

Polynomial reproduction

The polynomial reproduction is an attractive property for a subdivision scheme.

A subdivision scheme reproduce πd is that it must be able to generate polyno-

mials of the same degree as limit functions for some initial data. The degree

of polynomial reproduction can never exceed the degree of polynomial genera-

tion.

Theorem 6.1.4. If applying the parameter shift τ = 5+m+3n2

, the subdivision scheme

fam,n,µ reproduces π1 with respect to the parametrization in Romani (2015) for all

m,n ∈ N and µ ∈ R. Moreover, if µ = −3+m32

, fam,n,µ reproduces π3 for all m,n ∈ N.

Proof. Since the condition D(1)am,n,µ(1) = 5 +m + 3n is verified by the symbol

am,n,µ(z) for all µ ∈ R. Thus polynomial reproduction of fam,n,µ is π1 with the

parameter shift τ = 5+m+3n2

. We observe that when µ = −3+m32

, the following

two more conditions

D(2)am,n,µ(z)|z=1 = 2τ(τ − 1), D(3)am,n,µ(z)|z=1 = 2τ(τ − 1)(τ − 2),

are satisfied for all m,n ∈ N thus reproduction of fam,n,µ is π3.


Monotonicity preserving play a key role in shape preserving properties of sub-

division schemes.

In the following, we examine monotonicity preservation of binary scheme fa1,0,µ .

114

Theorem 6.1.5. Let {f 0i }i∈Z satisfy

. . . f 0−1 < f 0

0 < f 01 < . . . < f 0

n−1 < f 0n < f 0

n+1 . . . .

Denote

dki = fki+1 − fki , rki =dki+1

dki, Rk = max

i{rki ,

1

rki}, k ≥ 0, k ∈ Z, i ∈ Z.

Furthermore, let 0.1 ≤ µ ≤ 0.9 and ξ = − 1µ

, ξ ∈ R. If 1ξ≤ R0 ≤ ξ, {fki } is defined by

the subdivision scheme fa1,0,µ , then

dki > 0,1

ξ≤ Rk ≤ ξ, k ≥ 0, k ∈ Z, i ∈ Z. (6.9)

Proof. We use mathematical induction to prove (6.9). When k = 0,

d0i = f 0i+1 − f 0

i > 0, 1ξ≤ R0 ≤ ξ, then (6.9) is true.

Suppose that (6.9) holds for k, dki = fki+1 − fki > 0, 1ξ≤ Rk ≤ ξ, next we will

prove that (6.9) holds for k + 1. Since

dk+12i = fk+1

2i+1 − fk+12i =

{(1

8+

1

2µ

)dki +

(3

8− µ

)dki+1 +

(1

2µ

)dki+2

},

dk+12i+1 = fk+1

2i+2 − fk+12i+1 =

{(1

2µ

)dki +

(3

8− µ

)dki+1 +

(1

8+

1

2µ

)dki+2

}.

Now we show that

dk+12i > 0 and dk+1

2i+1 > 0.

Consider

dk+12i =

{(1

8+

1

2µ

)dki +

(3

8− µ

)dki+1 +

(1

2µ

)dki+2

}.

This implies

dk+12i = dki

{(1

8+

1

2µ

)+

(3

8− µ

)dki+1

dki+

(1

2µ

)dki+2

dki

}.

115

Again implies

dk+12i = dki

{(1

8+

1

2µ

)+

(3

8− µ

)rki +

(1

2µ

)rki+1r

ki

}.


dk+12i ≥ dki

{(1

8+

1

2µ

)+

(3

8− µ

)1

ξ+

(1

2µ

)1

ξ2

}.


8+

1

2µ

)+

(3

8− µ

)1

ξ+

(1

2µ

)1

ξ2

}> 0, for 0.1 ≤ µ ≤ 0.9 and ξ = − 1

µ.

This implies that dk+12i > 0. Now Consider

dk+12i+1 =

{(1

2µ

)dki +

(3

8− µ

)dki+1 +

(1

8+

1

2µ

)dki+2

}.

This implies

dk+12i+1 = dki

{(1

2µ

)+

(3

8− µ

)dki+1

dki+

(1

8+

1

2µ

)dki+2

dki

}.

Again implies

dk+12i+1 = dki

{(1

2µ

)+

(3

8− µ

)rki +

(1

8+

1

2µ

)rki+1r

ki

}.


dk+12i+1 ≥ dki

{(1

2µ

)+

(3

8− µ

)1

ξ+

(1

8+

1

2µ

)1

ξ2

}.


2µ

)+

(3

8− µ

)1

ξ+

(1

8+

1

2µ

)1

ξ2

}> 0, for 0.1 ≤ µ ≤ 0.9 and ξ = − 1

µ.

This implies that dk+12i+1 > 0.

Now we prove that 1ξ≤ Rk+1 ≤ ξ, first we show that rk+1

2i − ξ ≤ 0. Since

rk+12i =

dk2i+1

dk2i=

{(12µ)dki +

(38− µ

)dki+1 +

(18+ 1

2µ)dki+2

}{(18+ 1

2µ)dki +

(38− µ

)dki+1 +

(12µ)dki+2

} .116

This implies that

rk+12i − ξ =

1

dki+1

{(18+ 1

2µ)

1rki

+(38− µ

)+(12µ)rki+1

}dki {(1

2µ

)+

(3

8− µ

)rki

+

(1

8+

1

2µ

)rki r

ki+1 −

(1

8+

1

2µ

)ξ −

(3

8− µ

)ξrki −

(1

2µ

)ξrki r

ki+1

}.


rk+12i − ξ ≤

dki{(

18+ 1

2µ)ξ2 +

(14− 2µ

)ξ +

(−3

8+ 3

2µ)}

dki+1

{(18+ µ)ξ +

(38− µ

)} .

Since dki{(

18+ 1

2µ)ξ2 +

(14− 2µ

)ξ +

(−3

8+ 3

2µ)}

is greater than zero and

dki+1

{(18+ µ)ξ +

(38− µ

)}is less than zero for 0.1 ≤ µ ≤ 0.9 and ξ = − 1

µ.

This implies that

rk+12i − ξ ≤ 0.

Further implies that rk+12i ≤ ξ. Now we show that 1

rk+13i

− ξ < 0.

1

rk+12i

=dk2idk2i+1

=

{(18+ 1

2µ)dki +

(38− µ

)dki+1 +

(12µ)dki+2

}{(12µ)dki +

(38− µ

)dki+1 +

(18+ 1

2µ)dki+2

} .This implies

rk+12i − ξ =

1

dki+1

{(12µ)

1rki

+(38− µ

)+(18+ 1

2µ)rki+1

}dki {(1

8+

1

2µ

)+

(3

8− µ

)rki

+

(1

2µ

)rki r

ki+1 −

(1

2µ

)ξ −

(3

8− µ

)ξrki −

(1

8+

1

2µ

)ξrki r

ki+1

}.

Further implies that

1

rk+12i

− ξ ≤dki {1

2µξ2

(14− 2µ

)ξ + (3

2µ− 1

4)}

dki+1{(18 + µ)ξ + (38− µ)}

.

Since dki {12µξ2

(14− 2µ

)ξ+(3

2µ−1

4)} is greater than zero and dki+1{(18+µ)ξ+(3

8−µ)}

is less than zero for 0.1 ≤ µ ≤ 0.9 and ξ = − 1µ

.

This implies that

1

rk+12i

− ξ ≤ 0.

117

Further implies 1

rk+12i

≤ ξ. In the same way, we can get rk+12i+1 ≤ ξ and 1

rk+12i+1

≤ ξ.

So Rk+1 ≤ ξ. Since Rk+1 = maxi{rki , 1rki}, it is obvious that Rk+1 ≥ 1

ξ. Which

completes the proof.

6.1.4 Numerical experiments of univariate schemes

In this section, we present the performance, geometrical behavior and effect of

parameter on the limit curves of the schemes. we also present the response of

the limit curves produced by the schemes towards the initial data.


i 1 2 3 4 5 6 7 8 9 10 11

xi 0.1 4 6.5 10 15 25 40 50 62 65 66

yi 1 1 2 3.5 5.5 5.5 10 10 12.5 18 20

Figure 6.1 is produced by using monotone data set given in Table 6.3 bor-

rowed by Abbas et al. (2014). Figure 6.1(a)-6.1(d) are monotone curves obtained

by the schemes fa1,0,µ , fa1,1,µ , fa1,2,µ and fa1,3,µ respectively.

The Figures 6.2-6.5 show the comparison of proposed schemes with the exist-

ing schemes of Romani (2015). Dashed dotted lines indicate the initial polygon.

Solid lines show the most expanded curves and dashed lines show the most

shrinked curves. Arrow shows the distance between most expanded and most

shrinked curves. Figure 6.2(a)-6.2(c) show that the most expanded and most

shrinked curves are obtained by the schemes fa2,0,µ , fa1,2,µ and fa2,2,µ at different

parametric values and Figure 6.2(d) shows the behavior of existing scheme of

Romani (2015).

We can see that the Figure 6.3(a)-6.3(b) represent the interpolating behavior

118

(a) (b)

(c) (d)

Figure 6.1: The curves (a), (b), (c) and (d) are generated by the schemes fa1,0,µ , fa1,1,µ ,

fa1,2,µ and fa1,3,µ by using monotone data set.

119

(a) m = 2, n = 0 (b) m = 1, n = 2

(c) m = 2, n = 2 (d) n = 2

Figure 6.2: Most expanded and most shrinked curves: The curves (a), (b), (c) and (d)

are generated by the schemes fa2,0,µ , fa1,2,µ , fa2,2,µ and Romani (2015) respectively.

120

(a) m = 2, n = 0 (b) m = 1, n = 2 (c) n = 2

Figure 6.3: Interpolating behavior: The curves (a) , (b) and (c) are generated by the

schemes fa2,0,µ , fa1,2,µ and Romani (2015) respectively.

(a) m = 1, n = 0 (b) m = 1, n = 1 (c) n = 1

Figure 6.4: Most expanded and most shrinked curves: The curves (a), (b) and (c) are

generated by the schemes fa1,0,µ , fa1,1,µ and Romani (2015) respectively.

121

(a) m = 1, n = 0 (b) m = 1, n = 1 (c) n = 1

Figure 6.5: Interpolating behavior: The curves (a), (b) and (c) are generated by the

schemes fa1,0,µ , fa1,1,µ and Romani (2015) respectively.

of proposed scheme fa2,0,µ , fa1,2,µ respectively. Figure 6.3(c) shows the non-

interpolating behavior of Romani (2015) at any parametric value. The proposed

scheme fa2,0,µ and fa1,2,µ shows the approximating behavior as well as interpo-

lating behavior at different values of parameter.

The Figure 6.4(a)-6.4(c) shows the most expanded and most shrinked curves are

generated by the schemes fa1,0,µ , fa1,1,µ and Romani (2015) at different paramet-

ric values respectively. The limit curves presented in Figure 6.5(a)-6.5(c) shows

the interpolating behavior by the schemes fa1,0,µ , fa1,1,µ and Romani (2015) re-

spectively.

The schemes fa1,0,µ and fa1,1,µ have both approximating and interpolating behav-

ior while scheme in Romani (2015) gives only interpolating behavior.

6.2 Algorithm for non-tensor product schemes

By generalizing the algorithm as devised in Section 2, we get a family of non-

tensor product approximating schemes with tension parameter µ for quadrilat-

122

eral meshes. Let fam,n,µ be the family of non-tensor product bivariate subdivision

schemes then we propose the symbol of this family as

am,n,µ(z1, z2) = (αodd(z1))m(βeven(z2))

nγµ(z1)γµ(z2). (6.10)

By substituting m = 1 and n = 0 in (6.10), we get symbol of the scheme fa1,0,µ as

follows:

a1,0,µ(z1, z2) =

(1 + z1

2

)4(1 + z2

2

)3 (8µz21 + (2− 16µ)z1 + 8µ

)× (6.11)(

8µz22 + (2− 16µ)z2 + 8µ).

The bivariate subdivision scheme fa1,0,µ has the mask

a1,0,µ(z1, z2) =

12µ

2 12µ

2 + 18µ −µ2 + 3

8µ

µ2 + 18µ µ2 + 3

8µ+ 132 2µ2 + 1

2µ+ 332

− 12µ

2 + 12µ −1

2µ2 + 3

8µ+ 18 µ2 − 11

8 µ+ 38

−2µ2 + 34µ −2µ2 + 1

4µ+ 316 4µ2 − 3µ+ 9

16

− 12µ

2 + 12µ −1

2µ2 + 3

8µ+ 18 µ2 − 11

8 µ+ 38

µ2 + 18µ µ2 + 3

8µ+ 132 −2µ2 + 1

2µ+ 332

12µ

2 12µ

2 + 18µ −µ2 + 3

8µ

123

−µ2 + 38µ

12µ

2 + 18µ

12µ

2

−2µ2 + 12µ+ 3

32 µ2 + 38µ+ 1

32 µ2 + 18µ

µ2 − 118 µ+ 3

8 −12µ

2 + 38µ+ 1

8 − 12µ

2 + 12µ

4µ2 − 3µ+ 916 −2µ2 + 1

4µ+ 316 −2µ2 + 3

4µ

µ2 − 118 µ+ 3

8 −12µ

2 + 38µ+ 1

8 − 12µ

2 + 12µ

−2µ2 + 12µ+ 3

32 µ2 + 38µ+ 1

32 µ2 + 18µ

−µ2 + 38µ

12µ

2 + 18µ

12µ

2

. (6.12)

By substituting m = 1 and n = 1 in (6.10), we get symbol of the scheme fa1,1,µ

as follows:

a1,1,µ(z1, z2) = − 1

512(1 + z1)

4 (1 + z2)4 (z22 − 10z2 + 1

) (4µz21 + (1− 8µ)z1 + 4µ

)(4µz22 + (1− 8µ)z2 + 4µ

). (6.13)

The bivariate subdivision scheme fa1,1,µ has the mask

124

a1,1,µ(z1, z2) =

− 132µ

2 − 1128µ+ 1

4µ2 3

64µ+ 58µ

2 −14µ

2 + 33128µ

− 116µ

2 − 1128µ − 1

512 + 364µ+ 1

2µ2 3

256 + 14µ+ 5

4µ2 −1

2µ2 + 33

512 + 2964µ

132µ

2 − 132µ − 1

128 + 33128µ− 1

4µ2 3

64 + 3764µ− 5

8µ2 1

4µ2 + 33

128 − 65128µ

18µ

2 − 364µ − 3

256 + 1332µ− µ2 9

128 + 34µ− 5

2µ2 µ2 + 99

256 − 4532µ

132µ

2 − 132µ − 1

128 + 33128µ− 1

4µ2 3

64 + 3764µ− 5

8µ2 1

4µ2 + 33

128 − 65128µ

− 116µ

2 − 1128µ − 1

512 + 364µ+ 1

2µ2 3

256 + 14µ+ 5

4µ2 −1

2µ2 + 33

512 + 2964µ

− 132µ

2 − 1128µ+ 1

4µ2 3

64µ+ 58µ

2 −14µ

2 + 33128µ

−1916µ

2 + 1332µ −19

16µ2 + 13

32µ −14µ

2 + 33128µ

364µ+ 5

8µ2

3364µ+ 13

128 − 198 µ2 33

64µ+ 13128 − 19

8 µ2 −12µ

2 + 33512 + 29

64µ3

256 + 14µ+ 5

4µ2

−5132µ+ 13

32 + 1916µ

2 −5132µ+ 13

32 + 1916µ

2 14µ

2 + 33128 − 65

128µ364 + 37

64µ− 58µ

2

−10932 µ+ 39

64 + 194 µ2 −109

32 µ+ 3964 + 19

4 µ2 µ2 + 99256 − 45

32µ9

128 + 34µ− 5

2µ2

−5132µ+ 13

32 + 1916µ

2 −5132µ+ 13

32 + 1916µ

2 14µ

2 + 33128 − 65

128µ364 + 37

64µ− 58µ

2

3364µ+ 13

128 − 198 µ2 33

64µ+ 13128 − 19

8 µ2 −12µ

2 + 33512 + 29

64µ3

256 + 14µ+ 5

4µ2

−1916µ

2 + 1332µ −19

16µ2 + 13

32µ −14µ

2 + 33128µ

364µ+ 5

8µ2

125

− 1128µ+ 1

4µ2 − 1

32µ2

− 1512 + 3

64µ+ 12µ

2 − 116µ

2 − 1128µ

− 1128 + 33

128µ− 14µ

2 132µ

2 − 132µ

− 3256 + 13

32µ− µ2 18µ

2 − 364µ

− 1128 + 33

128µ− 14µ

2 132µ

2 − 132µ

− 1512 + 3

64µ+ 12µ

2 − 116µ

2 − 1128µ

− 1128µ+ 1

4µ2 − 1

32µ2

. (6.14)

In the same way by taking different values ofm and n in (6.10), we can easily get

mask of other non-tensor product subdivision schemes for surface generation.

6.2.1 Smoothness analysis of bivariate proposed schemes

Here, we use the theory of generating function Dyn and Levin (2002) to derive

continuity of non-tensor product schemes.

Theorem 6.2.1. If µ ∈ (−0.2215, 0.4785) then the subdivision scheme fa1,0,µconverges

to a continuous surface when starting from any regular quadrilateral mesh. Moreover,

if µ ∈ (−0.05178, 0.3017) and µ ∈ (−0.0517, 0.25), the limit surfaces generated by

scheme fa1,0,µhave C1 and C2-continuous respectively.

Proof. From (6.11), we have

b1,0,µ(z1, z2) =(8µz21 + (2− 16µ)z1 + 8µ

) (8µz22 + (2− 16µ)z2 + 8µ

).

126

In view of Dyn and Levin (2002), (Theorem 4.30), we can determine the range

of the parameter µ which guarantees the convergence of the scheme fa1,0,µ by

checking the contractivity of the scheme. Since the scheme with symbol 12

(1+z12

)3(1+z22

)3b1,0,µ(z1, z2), 1

2

(1+z12

)4 (1+z22

)2b1,0,µ(z1, z2) is contractive for µ ∈ (−0.2215,

0.4785) and then scheme fa1,0,µ is convergent for µ ∈ (−0.2215, 0.4785). In the

same way, the scheme with symbol 12

(1+z12

)2 (1+z22

)3b1,0,µ(z1, z2), 1

2

(1+z12

)3 (1+z22

)2b1,0,µ(z1, z2), 1

2

(1+z12

)4 (1+z22

)b1,0,µ(z1, z2) is contractive for µ ∈ (−0.05178, 0.3017)

therefore the scheme fa1,0,µ is C1-continuous. Again since, the scheme with sym-

bol 12

(1+z12

) (1+z22

)3b1,0,µ(z1, z2), 1

2

(1+z12

)2 (1+z22

)2b1,0,µ(z1, z2), 1

2

(1+z12

)3 (1+z22

)b1,0,µ(z1, z2), 1

2

(1+z12

)4b1,0,µ(z1, z2) is contractive for µ ∈ (−0.0517, 0.25), so the

scheme fa1,0,µ is C2-continuous.

Theorem 6.2.2. If µ ∈ (−0.322, 0.572) the subdivision scheme fa1,1,µconverges to a

continuous surface when starting from any regular quadrilateral mesh. Moreover, if

µ ∈ (−0.1724, 0.4412), µ ∈ (−0.093, 0.332) and µ ∈ (−0.0301, 0.1646), the limit

surface generated by scheme fa1,1,µis C1, C2 and C3 continuous respectively.

Proof. From (6.13), we have

b1,0(z1, z2) = −1

2

(z22 − 10z2 + 1

) (4µz21 + (1− 8µ)z1 + 4µ

) (4µz22 + (1− 8µ)z2 + 4µ

).

In view of Dyn and Levin (2002), (Theorem 4.30), we can determine the range

of the parameter µ which guarantees the convergence of the scheme fa1,1,µ by

checking the contractivity of the scheme with symbol 12(1+z1

2)3(1+z2

2)4b1,0(z1, z2),

12(1+z1

2)4(1+z2

2)3b1,0(z1, z2). This yields for µ ∈ (−0.322, 0.572) scheme fa1,1,µ is

convergent. In the same spirit, we can easily check continuous limit surface

is C1, C2 and C3 for µ ∈ (−0.1724, 0.4412), µ ∈ (−0.093, 0.332) and µ ∈

(−0.0301, 0.1646) respectively.

127

Table 6.4: The order of continuity O(C) of proposed non-tensor product schemes with

some existing non-tensor product schemes.

Scheme Type O(C)

Binary non-tensor product Romani (2015) Interpolating C1

Binary non-tensor product Romani (2015) Approximating C1

Binary non-tensor product Khan and Mustafa (2013) Approximating C1

Proposed binary non-tensor product fa1,0,µ Approximating C2

Proposed binary non-tensor product fa1,1,µ Approximating C3

In Table 6.4, we compare the continuity of proposed non-tensor product schemes

with some existing binary non-tensor product schemes. It is observed that the

continuity of proposed schemes is better than the continuity of existing schemes.

6.2.2 Response of non-tensor product schemes to polynomial

and monotone data

In this section, we investigate the capability of the non-tensor product approx-

imating subdivision schemes fa1,0,µ and fa1,1,µ of generating and reproducing

polynomials as well as monotonicity preservation of the data.

Theorem 6.2.3. The subdivision scheme fa1,0,µgenerates π2 for all µ ∈ R and generates

π4 for all µ = 116

.

Proof. Let w1 = (1,−1), w2 = (−1, 1), w3 = (−1,−1) and let Dj with j ∈ N2,

128

denote a directional derivative. Since a1,0,µ(1, 1) = 4 and

D(1,0)a1,0,µ(w1) = 0, D(1,0)a1,0,µ(w2) = 0, D(1,0)a1,0,µ(w3) = 0,

D(0,1)a1,0,µ(w1) = 0, D(0,1)a1,0,µ(w2) = 0, D(0,1)a1,0,µ(w3) = 0,

then scheme fa1,0,µ generates π1 for all µ ∈ R. Again since

D(1,1)a1,0,µ(w1) = 0, D(1,1)a1,0,µ(w2) = 0, D(1,1)a1,0,µ(w3) = 0,

D(2,0)a1,0,µ(w1) = 0, D(2,0)a1,0,µ(w2) = 0, D(2,0)a1,0,µ(w3) = 0,

D(0,2)a1,0,µ(w1) = 0, D(0,2)a1,0,µ(w2) = 0, D(0,2)a1,0,µ(w3) = 0,

then the scheme fa1,0,µ generates π2 for all µ ∈ R. Further

D(2,1)a1,0,µ(w1) = 0, D(2,1)a1,0,µ(w2) = 0, D(2,1)a1,0,µ(w3) = 0,

D(1,2)a1,0,µ(w1) = 0, D(1,2)a1,0,µ(w2) = 0, D(1,2)a1,0,µ(w3) = 0,

D(3,0)a1,0,µ(w1) = 0, D(3,0)a1,0,µ(w2) = 0, D(3,0)a1,0,µ(w3) = 0,

D(0,3)a1,0,µ(w1) = 48µ− 3, D(0,3)a1,0,µ(w2) = 0, D(0,3)a1,0,µ(w3) = 0,

so the scheme fa1,0,µ generates π3 for µ = 116

. Further more

D(2,2)a1,0,µ(w1) = 0, D(2,2)a1,0,µ(w2) = 0, D(2,2)a1,0,µ(w3) = 0,

D(3,1)a1,0,µ(w1) = 0, D(3,1)a1,0,µ(w2) = 0, D(3,1)a1,0,µ(w3) = 0,

D(1,3)a1,0,µ(w1) = 144µ− 9, D(1,3)a1,0,µ(w2) = 0, D(1,3)a1,0,µ(w3) = 0,

D(4,0)a1,0,µ(w1) = 0, D(4,0)a1,0,µ(w2) = 96µ− 6, D(4,0)a1,0,µ(w3) = 0,

D(0,4)a1,0,µ(w1) = 48µ− 3, D(0,4)a1,0,µ(w2) = 0, D(0,4)a1,0,µ(w3) = 0,

so the scheme fa1,0,µ generates π4 for µ = 116

. Which completes the proof.

Theorem 6.2.4. For the parameter shift (τ1, τ2) = (124, 10

4), the subdivision scheme

fa1,0,µreproduces π1 with respect to the parametrization defined in Romani (2015) for

all µ ∈ R.

129

Proof. Let Dj with j ∈ N2, denote a directional derivative. Since the symbol

a1,0,µ(z1, z2) satisfies the conditions in Theorem 6.2.3. Since a1,0,µ(1, 1) = 4 and

D(1,0)a1,0,µ(1, 1)− 4τ1 = 0, D(0,1)a1,0,µ(1, 1)− 4τ2 = 0,

then the scheme fa1,0,µ produced π1 for all µ ∈ R.

Theorem 6.2.5. The subdivision scheme fa1,1,µgenerates π3 for all µ ∈ R and generates

π4 for µ = 116

.


denote a directional derivative. Since a1,1,µ(1, 1) = 4 and

D(1,0)a1,1,µ(w1) = 0, D(1,0)a1,1,µ(w2) = 0, D(1,0)a1,1,µ(w3) = 0,

D(0,1)a1,1,µ(w1) = 0, D(0,1)a1,1,µ(w2) = 0, D(0,1)a1,1,µ(w3) = 0.

The scheme fa1,1,µ generates π1 for all µ ∈ R. Again since

D(1,1)a1,1,µ(w1) = 0, D(1,1)a1,1,µ(w2) = 0, D(1,1)a1,1,µ(w3) = 0,

D(2,0)a1,1,µ(w1) = 0, D(2,0)a1,1,µ(w2) = 0, D(2,0)a1,1,µ(w3) = 0,

D(0,2)a1,1,µ(w1) = 0, D(0,2)a1,1,µ(w2) = 0, D(0,2)a1,1,µ(w3) = 0.

The scheme fa1,1,µ generates π2 for all µ ∈ R. Further

D(2,1)a1,1,µ(w1) = 0, D(2,1)a1,1,µ(w2) = 0, D(2,1)a1,1,µ(w3) = 0,

D(1,2)a1,1,µ(w1) = 0, D(1,2)a1,1,µ(w2) = 0, D(1,2)a1,1,µ(w3) = 0,

D(3,0)a1,1,µ(w1) = 0, D(3,0)a1,1,µ(w2) = 0, D(3,0)a1,1,µ(w3) = 0,

D(0,3)a1,1,µ(w1) = 0, D(0,3)a1,1,µ(w2) = 0, D(0,3)a1,1,µ(w3) = 0.

130

The scheme fa1,1,µ generates π3 for all µ ∈ R. Furthermore

D(2,2)a1,1,µ(w1) = 0, D(2,2)a1,1,µ(w2) = 0, D(2,2)a1,1,µ(w3) = 0,

D(3,1)a1,1,µ(w1) = 0, D(3,1)a1,1,µ(w2) = 0, D(3,1)a1,1,µ(w3) = 0,

D(1,3)a1,1,µ(w1) = 0, D(1,3)a1,1,µ(w2) = 0, D(1,3)a1,1,µ(w3) = 0,

D(4,0)a1,1,µ(w1) = 0, D(4,0)a1,1,µ(w2) = 96µ− 6, D(4,0)a1,1,µ(w3) = 0,

D(0,4)a1,1,µ(w1) = −144µ+ 9, D(0,4)a1,1,µ(w2) = 0, D(0,4)a1,1,µ(w3) = 0.

The scheme fa1,1,µ generates π4 for µ = 116

. Which completes the proof.

Theorem 6.2.6. If applying the parameteric shift (τ1, τ2) = (3, 4), the subdivision

scheme fa1,1,µreproduces π1 with respect to the parametrization in Romani (2015) for

all µ ∈ R.


a1,1,µ(z1, z2) satisfies the conditions in Theorem 6.2.5. Since the conditions

a1,1,µ(1, 1) = 4

D(1,0)a1,1,µ(1, 1)− 4τ1 = 0, D(0,1)a1,1,µ(1, 1)− 4τ2 = 0,

the scheme fa1,1,µ produced π1 for all µ ∈ R.

Now, we examine monotonicity preservation of binary non-tensor product

approximating subdivision scheme fa1,0,µ .

Definition 6.2.1. Hussain et al. (2012). "A bivariate data (xi, yj, fi,j), i = 0, 1, 2, . . . , n

and j = 0, 1, 2, . . . ,m where x1 < x2 < . . . < xn and y1 < y2 < . . . < ym is said to

be monotonically increasing if fi,j < fi+1,j and fi,j < fi,j+1 ∀ i = 0, 1, 2, . . . , n and

j = 0, 1, 2, . . . ,m the derivative at the data points obey the condition di,j > 0 ∀

i = 0, 1, 2, . . . , n and ∀ i = 0, 1, 2, . . . , n."

131

Theorem 6.2.7. Suppose that the initial data {f 0i,j} = (x0i , y

0j , f

0i,j) is strictly monoton-

ically increasing for all i, j ∈ Z.

Denote

dki,j = fki+1,j+1 − fki+1,j − fki,j+1 + fki,j,

yki,j+t =dki+1,j+t

dki,j+t, yki+1,j+t =

dki+2t,j+t+1

dki+1,j+t

,

Y ki,j+t = max

i,j{yki,j+t,

1

yki,j+t}, Y k

i+1,j+t = maxi,j

{yki+1,j+t,1

yki+1,j+t

},

where t = 0, 1 and k ≥ 0, k ∈ Z, i, j ∈ Z.

Furthermore, let 0.1 ≤ µ ≤ 0.9 and δ = − 1µ

, δ ∈ R. If 1δ≤ Y 0

i,j+t, Y0i+1,j+t ≤ δ, {fki,j}

is defined by the subdivision scheme fa1,0,µ, then

dki,j > 0,1

δ≤ Y k

i,j+t, Yki+1,j+t ≤ δ, k ≥ 0, k ∈ Z, i, j ∈ Z. (6.15)

Proof. (6.15) will be proved by mathematical induction. When k = 0, d0i,j > 0,

1δ≤ Y 0

i,j+t, Y0i+1,j+t ≤ δ, then (6.15) is true.

Suppose that (6.15) holds for k i.e. dki,j > 0, 1δ≤ Y k

i,j+t, Yki+1,j+t ≤ δ, next we will

prove that (6.15) holds for k + 1.

dk+12i,2j = fk+1

2i+1,2j+1 − fk+12i+1,2j − fk+1

2i,2j+1 + fk+12i,2j,

dk+12i+1,2j = fk+1

2i+2,2j+1 − fk+12i+2,2j − fk+1

2i+1,2j+1 + fk+12i+1,2j,

dk+12i,2j+1 = fk+1

2i+1,2j+2 − fk+12i+1,2j+1 − fk+1

2i,2j+2 + fk+12i,2j+1,

dk+12i+1,2j+1 = fk+1

2i+2,2j+2 − fk+12i+2,2j+1 − fk+1

2i+1,2j+2 + fk+12i+1,2j+1.

Now we show that

dk+12i,2j > 0, dk+1

2i+1,2j > 0, dk+12i,2j+1 > 0 and dk+1

2i+1,2j+1 > 0.

132

First we show that dk+12i,2j > 0. Consider

dk+12i,2j = fk+1

2i+1,2j+1 − fk+12i+1,2j − fk+1

2i,2j+1 + fk+12i,2j.

After some simplification and substituting δ = − 1µ

, we get

dk+12i,2j = dki,j+3

{−27

2µ11 +

153

8µ10 − 363

16µ9 +

781

32µ8 − 831

32µ7 +

881

32µ6 − 711

64µ5

+377

64µ4 − 105

32µ3 + 2µ2 − 19

32µ+

5

32

}.

As we know that dki,j+3 > 0 and{−27

2µ11 +

153

8µ10 − 363

16µ9 +

781

32µ8 − 831

32µ7 +

881

32µ6 − 711

64µ5

+377

64µ4 − 105

32µ3 + 2µ2 − 19

32µ+

5

32

}> 0.

This implies that dk+12i,2j > 0. Similarly, we see that dk+1

2i+1,2j > 0, dk+12i,2j+1 > 0 and

dk+12i+1,2j+1 > 0 for 0.1 ≤ µ ≤ 0.9 and δ = − 1

µ.

Now we prove that 1δ≤ Y k

i,j+t, Yki+1,j+t ≤ δ, first we show that yk+1

2i,2j − δ ≤ 0.

For this consider

yk+12i,2j − δ =

dk+12i+1,2j

dk+12i,2j

− δ.


, we get

yk+12i,2j − δ ≤ ψ1

ψ2

,

where

ψ1 =

{−9

8µ3 +

549

32µ2 − 287

8µ+

2885

64+

5

16µ8− 53

32µ7+

285

32µ6− 341

16µ5+

1543

32µ4

− 821

16µ3+

3461

64µ2− 1679

32µ

},

and

ψ2 =

{−9

8µ3 +

513

32µ2 − 635

32µ+

1615

64− 5

32µ7+

3

4µ6− 69

16µ5+

41

4µ4− 783

32µ3

+859

32µ2− 1743

64µ

}.

133

The denominator is negative and numerator is positive of the above inequality

for 0.1 ≤ µ ≤ 0.9.

This implies that

yk+12i,2j − δ ≤ 0.

Further this implies that yk+12i,2j ≤ δ. Now we show that 1

yk+12i,2j

− δ < 0.

For this consider

1

yk+12i,2j

− δ =dk2i,2jdk2i,2j+1

− δ.


, we get

1

yk+12i,2j

− δ ≤ χ1

χ2

,

where

χ1 =

{−9

8µ3 +

549

32µ2 − 287

8µ+

2885

64

5

32µ9+

29

32µ8− 19

4µ7+

189

16µ6− 449

16µ5+

1111

32µ4

− 821

16µ3+

3461

64µ2− 1679

32µ

},

and

χ2 =

{−9

8µ3 +

513

32µ2 − 635

32µ+

1615

64+

5

32µ8− 29

32µ7+

147

32µ6− 177

16µ5+

95

4µ4− 783

32µ3

+859

32µ2− 1743

64µ

}.

The denominator is positive and numerator is negative of the above inequality

for 0.1 ≤ µ ≤ 0.9.

This implies that

1

yk+12i,2j

− δ ≤ 0.

134

Further this implies that 1

yk+12i,2j

≤ δ. In the same way, we can get yk+12i,2j+1 ≤ δ,

yk+12i+1,2j ≤ δ, yk+1

2i+1,2j+1 ≤ δ, 1

yk+12i,2j+1

≤ δ, 1

yk+12i+1,2j

≤ δ and 1

yk+12i+1,2j+1

≤ δ. So

Y ki,j+t, Y

ki+1,j+t ≤ δ. Since Y k

i,j+t = maxi,j{yki,j+t, 1yki,j+t

} and Y ki+1,j+t = maxi,j{yki+1,j+t,

1yki+1,j+t

}, it is obvious that Y ki,j+t, Y

ki+1,j+t ≥ 1

δ, which completes the proof.

6.2.3 Numerical experiments of non-tensor product schemes

In this section, we show the performance, geometrical behavior and effect of

parameter on the limit surfaces of the schemes fa1,0,µ and fa1,1,µ .


x/y 1 100 200 300

1 0.6931 9.2104 10.5967 11.4076

100 9.2104 9.9035 10.8198 11.5129

200 10.5967 10.8198 11.2898 11.7753

300 11.4076 11.5129 11.7753 12.1007

Monotone data set given in Table 6.5 borrowed by Hussain and Hussain

(2007) has been used to produce monotone surfaces. Figure 6.6(a) is the ini-

tial mesh of monotone data. Figure 6.6(b) is monotone surface generated by the

scheme fa1,0,µ for µ = 0.5. In Figures 6.7 and 6.8, we show the performance of

our schemes fa1,0,µ and fa1,1,µ by setting the shape parameter to various values,

which illustrate how this parameter effect the shape of the limit surface. Figures

6.7(a) and 6.8(a) are the initial control meshes while Figures 6.7(b)-6.7(d) are the

limit surfaces at µ = 0.1, 0.15, and 0.22 respectively and Figures 6.8(b)-6.8(d) are

the limit surfaces at µ = −0.03, 0.1, and 0.15 respectively.

135

0

100

200

300

0100

200300

0

2

4

6

8

10

12

14

(a) (b)

Figure 6.6: (a) Initial monotone data. (b) A monotonicity preserving surface obtained

by the proposed scheme fa1,0,µ.

6.2.4 Conclusion

In this chapter, we have proposed two algorithms to generate the families of

univariate and bivariate approximating subdivision schemes with one tension

and two integer parameters. The integer parameters identify members of pro-

posed family. It has been shown that the proposed schemes have higher conti-

nuity and Hölder continuity comparative to existing schemes. Comparison for

continuity of proposed non-tensor product schemes with some of the existing

non-tensor schemes has also been given. It has been demonstrated through sev-

eral examples that geometrical behavior of the univariate and bivariate subdivi-

sion schemes depends on the tension parameter. Monotonicity preservation of

proposed univariate and bivariate schemes has been proved. Moreover, poly-

nomial reproduction and generation of the proposed schemes have also been

discussed.

136

(a) (b)

(c) (d)

Figure 6.7: (a) Control mesh. (b)-(d) Limit surfaces obtained by the proposed schemes

fa1,0,µafter 5 steps of refinement.

137

(a) (b)

(c) (d)

Figure 6.8: (a) Control mesh. (b)-(d) Limit surfaces obtained by the proposed schemes

fa1,1,µafter 5 steps of refinement.

138

Chapter 7

Generalization of binary tensor

product schemes depending upon

four parameters

This chapter deal with two general formulae of parametric and non parametric

bivariate subdivision schemes. The generalization of bivariate schemes depend-

s upon four parameters. By assigning specific values to those parameters, we get

some special cases of existing tensor product schemes as well as new proposed

scheme. The behavior of schemes produced by the general formula are inter-

polating, approximating and relaxed. Comparison of polynomial reproduction,

polynomial generation and continuity of existing and proposed schemes has al-

so been established. Some numerical examples are also presented to show the

behavior of bivariate schemes.

139

7.1 Algorithm for tensor product schemes

We are going to construct the general formulas for bivariate approximating, in-

terpolating and relaxed subdivision schemes.

7.1.1 Univariate schemes

General formula of univariate binary subdivision scheme by Zheng et al. (2014b)

is fk+12i =

l+1∑q=0

α2q+1fki+q,

fk+12i+1 =

l+2∑q=0

α2qfki+q,

(7.1)

where the coefficients in equation (7.1) from Zheng et al. (2014b) are as follows

α0 = aC02l+1,

α1 = aC12l+1 + bC0

2l+1,

α2 = aC22l+1 + bC1

2l+1 + cC02l+1,

α3 = aC32l+1 + bC2

2l+1 + cC12l+1 + dC0

2l+1,

...

α2l = aC2l2l+1 + bC2l−1

2l+1 + cC2l−22l+1 + dC2l−3

2l+1 ,

α2l+1 = aC2l+12l+1 + bC2l

2l+1 + cC2l−12l+1 + dC2l−2

2l+1 ,

α2l+2 = bC2l+12l+1 + cC2l

2l+1 + dC2l−12l+1 ,

α2l+3 = cC2l+12l+1 + dC2l

2l+1,

α2l+4 = d.

(7.2)

a, b, c, d are the parameters, a + b + c + d = 12t

, t = 2l, where t ∈ Z, l = 2, 4, 6, . . .

and Cr2l+1 =

(2l+1)!r!(2l+1−r)! , r = 0, 1, 2, . . . , 2l + 1.

140

7.1.2 Bivariate schemes

By changing the notation i by j, q by p in (7.1) and using the tensor product

procedure, we can derive the general formula of bivariate schemes

fk+12i,2j =

l+1∑q=0

l+1∑p=0

α2q+1α2p+1fki+q,j+p,

fk+12i,2j+1 =

l+1∑q=0

l+2∑p=0

α2q+1α2pfki+q,j+p,

fk+12i+1,2j =

l+2∑q=0

l+1∑p=0

α2qα2p+1fki+q,j+p,

fk+12i+1,2j+1 =

l+2∑q=0

l+2∑p=0

α2qα2pfki+q,j+p.

(7.3)

By Substituting l = 1 in (7.3), we get tensor product scheme as follows:

fk+12i,2j = (2a+ b)2fki−1,j−1 + (2a+ b)(b+ 2c+ d)fki−1,j + (2a+ b)dfki−1,j+1

+(b+ 2c+ d)(2a+ b)fki,j−1 + (b+ 2c+ d)2fki,j + (b+ 2c+ d)d

fki,j+1 + (2a+ b)fki+1,j−1 + d(b+ 2c+ d)fki+1,j + d2fki+1,j+1,

fk+12i+1,2j = a(2a+ b)fki−1,j−1 + a(b+ 2c+ d)fki−1,j + adfki−1,j+1 + (b+ 2c+ d)

(2a+ b)fki,j−1 + (a+ 2b+ c)(b+ 2c+ d)fki,j + (a+ 2b+ c)dfki,j+1

+(c+ 2d)(2a+ b)fki+1,j−1 + (c+ 2d)(b+ 2c+ d)fki+1,j + (c+ 2d)

dfki+1,j+1,

fk+12i,2j+1 = a(2a+ b)fki−1,j−1 + (2a+ b)(a+ 2b+ c)fki−1,j + (2a+ b)(c+ 2d)

fki−1,j+1 + (b+ 2c+ d)afki,j−1 + (a+ 2b+ c)(b+ 2c+ d)fki,j +

(b+ 2c+ d)(c+ 2d)fki,j+1 + dafki+1,j−1 + d(a+ 2b+ c)fki+1,j

+(c+ 2d)dfki+1,j+1,

141

fk+12i+1,2j+1 = a2fki−1,j−1 + a(a+ 2b+ c)fki−1,j + a(c+ 2d)fki−1,j+1 + (a+ 2b+ c)

afki,j−1 + (a+ 2b+ c)2fki,j + (a+ 2b+ c)(c+ 2d)fki,j+1 + (c+ 2d)

afki+1,j−1 + (c+ 2d)(a+ 2b+ c)fki+1,j + (c+ 2d)2fki+1,j+1. (7.4)

• By letting a = − 112

, b = 14, c = 1

8and d = − 1

24(7.4). We get a new 16-point

tensor product interpolating scheme.

• By substituting a = d = 132

, b = c = 332

in (7.4). We get 16-point relaxed

tensor product approximating scheme of Zheng et al. (2014b).

• By taking a = − 332+µ, b = 7

16−2µ, c = − 3

32+µ, d = 0 in (7.4). We get 9-point

tensor product approximating scheme of Siddiqi and Rehan (2010).

By a slight variation on the scheme presented in (7.1), such that by replacing l+1

by l+2 in first equation of (7.1) and adopting the same procedure of tensor prod-

uct approach, we get another general formula of bivariate schemes mentioned

below

fk+12i,2j =

l+2∑q=0

l+2∑p=0

α2q+1α2p+1fki+q,j+p,

fk+12i,2j+1 =

l+2∑q=0

l+2∑p=0

α2q+1α2pfki+q,j+p,

fk+12i+1,2j =

l+2∑q=0

l+2∑p=0

α2qα2p+1fki+q,j+p,

fk+12i+1,2j+1 =

l+2∑q=0

l+2∑p=0

α2qα2pfki+q,j+p.

(7.5)

where the coefficients in (7.5) can be calculated by replacing l by l+ 12

in (7.2), we

get a system of equations with free parameters a, b, c, d such that a+b+c+d = 12t

,

t = 2l+1, where t ∈ Z, l = 0, 1, 2, . . . and Cr2l+2 =

(2l+2)!r!(2l+2−r)! , r = 0, 1, 2, . . . , 2l+

2. By substituting l = 0 in equation (7.5), we get general tensor product scheme

as follows:

142

fk+12i,2j = (3a+ b)2fki−1j−1 + (3a+ b)(a+ 3b+ 3c+ d)fki−1,j + (3a+ b)(c+ 3d)

fki−1,j+1 + (a+ 3b+ 3c+ d)(3a+ b)fki,j−1 + (a+ 3b+ 3c+ d)2fki,j

+(a+ 3b+ 3c+ d)(c+ 3d)fki,j+1 + (3a+ b)(c+ 3d)fki+1,j−1

+(c+ 3d)(a+ 3b+ 3c+ d)fki+1,j + (c+ 3d)2fki+1,j+1,

fk+12i+1,2j = a(3a+ b)fki−1,j−1 + a(a+ 3b+ 3c+ d)fki−1,j + a(c+ 3d)fki−1,j+1

+(3a+ 3b+ c)(3a+ b)fki,j−1 + (3a+ 3b+ c)(a+ 3b+ 3c+ d)fki,j

+(3a+ 3b+ c)(c+ 3d)fki,j+1 + (b+ 3c+ 3d)(3a+ b)fki+1,j−1

+(b+ 3c+ 3d)(a+ 3b+ 3c+ d)fki+1,j + (b+ 3c+ 3d)(c+ 3d)fki+1,j+1

+d(3a+ b)fki+2,j−1 + d(a+ 3b+ 3c+ d)fki+2,j + d(c+ 3d)fki+2,j+1,

fk+12i,2j+1 = a(3a+ b)fki−1,j−1 + (3a+ b)(3a+ 3b+ c)fki−1,j + (3a+ b)

(b+ 3c+ 3d)fki−1,j+1 + d(3a+ b)fki,j−1 + a(a+ 3b+ 3c+ d)

fki,j + (3a+ 3b+ c)(a+ 3b+ 3c+ d)fki,j+1 + (a+ 3b+ 3c+ d)

(b+ 3c+ 3d)fki+1,j−1 + (a+ 3b+ 3c+ d)dfki+1,j + a(c+ 3d)

fki+1,j+1 + (c+ 3d)(3a+ 3b+ c)fki+2,j−1 + (b+ 3c+ 3d)

(c+ 3d)fki+2,j + d(c+ 3d)fki+2,j+1,

fk+12i+1,2j+1 = a2fki−1,j−1 + a(3a+ 3b+ c)fki−1,j + a(b+ 3c+ 3d)fki−1,j+1

+dafki−1,j+2 + (3a+ 3b+ c)afki,j−1 + (3a+ 3b+ c)2fki,j

+(3a+ 3b+ c)(b+ 3c+ 3d)fki+1,j+1 + (3a+ 3b+ c)dfki+1,j+2

+(b+ 3c+ 3d)afki+1,j−1 + (b+ 3c+ 3d)(3a+ 3b+ c)fki+1,j

+(b+ 3c+ 3d)2fki+1,j+1 + (b+ 3c+ 3d)dfki+1,j+2 + dafki+2,j−1

+d(3a+ 3b+ c)fki+2,j + d(b+ 3c+ 3d)fki+2,j+1 + d2fki+2,j+2. (7.6)

143

• By substituting a = b = 0, c = d = 14

in (7.6). We get 4-point tensor product

approximating scheme of Chaikin (1974).

• By substituting a = d = 116

, b = c = 316

in (7.6). We get 9-point tensor

product approximating scheme of Ghaffar et al. (2012).

• By substituting a = d = − 332

, b = c = 1132


product approximating scheme of Siddiqi and Ahmad (2007).

• By substituting a = d = − 132

, b = c = 732


product approximating scheme of Hormann and Sabin (2008).

• By substituting a = 18, c = 3

8and b = d = 1

4, in (7.6). We get 9-point tensor

product interpolating scheme of Shen and Huang (2007).

• By substituting a = c = 18, b = 1

4, d = 0 in (7.6). We get 9-point tensor

product approximating scheme of Zheng et al. (2014b).

• By taking a = d = µ16

, b = c = 4−µ16

, in (7.6). We get 9-point tensor product

approximating scheme of Ghaffar et al. (2012).

• By taking a = d = −3ω, b = c = 14+ 3ω, in (7.6). We get 9-point tensor

product approximating scheme of Daniel and Shunmugaraj (2008).

By substituting l = 1 in (7.5) we get another general tensor product scheme

and after substituting different values of parameter, different existing schemes

becomes the special case of general tensor product scheme:

• If we set a = c = − 116

, b = 14

and d = 0 in (7.5). We get 16-point tensor

product interpolating scheme of Dyn et al. (1987).

144

• For a = d = 1+6ω96

and b = c = 5−6ω96


approximating scheme of Mustafa et al. (2009).

• For a = d = 164

and b = c = 364


approximating scheme of Ghaffar et al. (2013a).

• For a = d = ω16

and b = c = 1−ω16


approximating scheme of Ghaffar et al. (2013a).

• For a = d = − 5128

and b = c = 13128


approximating scheme of Hormann and Sabin (2008).

• For , a = d = 1384

and b = c = 23384


approximating scheme of Siddiqi and younis (2013).

• For a = d = u064

and b = c = 4−u064


approximating scheme of Ghaffar (2013).

7.2 Polynomial generation and reproduction of bi-

variate schemes

In this section, we will investigate the capability of the tensor product approxi-

mating subdivision scheme (7.6) by polynomial generation and polynomial re-

production.

The Laurent polynomial of the scheme (7.6) for a = d = 116

and b = c = 316

is

given by

a(z1, z2) =1

256(1 + z1)

5(1 + z2)5.

145

Theorem 7.2.1. If a = d = 116

and b = c = 316

, then the subdivision scheme (7.6)

generates polynomial of degree 4.


denote a directional derivative. Since a(1, 1) = 4 and

D(1,0)a(w1) = 0, D(1,0)a(w2) = 0, D(1,0)a(w3) = 0,

D(0,1)a(w1) = 0, D(0,1)a(w2) = 0, D(0,1)a(w3) = 0,

then scheme (7.6) generates polynomial of degree 1. Again since

D(1,1)a(w1) = 0, D(1,1)a(w2) = 0, D(1,1)a(w3) = 0,

D(2,0)a(w1) = 0, D(2,0)a(w2) = 0, D(2,0)a(w3) = 0,

D(0,2)a(w1) = 0, D(0,2)a(w2) = 0, D(0,2)a(w3) = 0,

then the scheme (7.6) generates polynomial of degree 2. Further

D(2,1)a(w1) = 0, D(2,1)a(w2) = 0, D(2,1)a(w3) = 0,

D(1,2)a(w1) = 0, D(1,2)a(w2) = 0, D(1,2)a(w3) = 0,

D(3,0)a(w1) = 0, D(3,0)a(w2) = 0, D(3,0)a(w3) = 0,

D(0,3)a(w1) = 0, D(0,3)a(w2) = 0, D(0,3)a(w3) = 0,

so the scheme (7.6) generates polynomial of degree 3. Further more

D(2,2)a(w1) = 0, D(2,2)a(w2) = 0, D(2,2)a(w3) = 0,

D(3,1)a(w1) = 0, D(3,1)a(w2) = 0, D(3,1)a(w3) = 0,

D(1,3)a(w1) = 0, D(1,3)a(w2) = 0, D(1,3)a(w3) = 0,

D(4,0)a(w1) = 0, D(4,0)a(w2) = 0, D(4,0)a(w3) = 0,

D(0,4)a(w1) = 0, D(0,4)a(w2) = 0, D(0,4)a(w3) = 0,

146

so the scheme (7.6) generates polynomial of degree 4.


Theorem 7.2.2. For the parameter shift (τ1, τ2) =(104, 10

4

), the subdivision scheme

(7.6) for a = d = 116

and b = c = 316

reproduces polynomial of degree 1 with respect to

the parametrization defined in Romani (2015).


a(z1, z2) satisfies the conditions in Theorem 7.2.1. Since a(1, 1) = 4 and

D(1,0)a(1, 1)− 4τ1 = 0, D(0,1)a(1, 1)− 4τ2 = 0,

then the scheme (7.6) produced polynomial of degree 1.

7.3 Numerical examples and comparison

Table 7.1 shows the continuity, polynomial generation and polynomial repro-

duction of existing bivariate schemes and proposed 16-point bivariate interpo-

lating scheme. In Figure 7.1, we show the performance of bicubic 9-point bivari-

ate subdivision scheme. Figure 7.1(a) is the initial control mesh. Figure 7.1(b)-

7.1(c) showing the subdivision at first and second iteration. In Figure 7.1(d), we

get the limit surface after successive number of refinements. In Figure 7.2, we

show the performance of proposed 16-point bivariate scheme. In Figure 7.2(a),

we design an initial control mesh. Figure 7.2(b)-7.2(c) showing the subdivision

at first and second iteration. In Figure 7.2(d), we get the limit surface after suc-

cessive number of refinements.

147

0

1

2

3

0

1

2

3

0

1

2

3

4

5

6

7

8

9

10

xy

z

00.5

11.5

22.5

3

0

1

2

3

0

1

2

3

4

5

6

7

8

9

10

xy

z

(a) (b)

00.5

11.5

22.5

3

0

1

2

3

0

1

2

3

4

5

6

7

8

9

10

xy

z

0

1

2

3

0

1

2

3

0

2

4

6

8

10

xy

z

(c) (d)

Figure 7.1: (a) Show the initial mesh. (b)-(d) Show the different refinement steps

148

−1

0

1

−1

0

11

1.5

2

2.5

xy

z

−1

0

1

−1

0

11

1.5

2

2.5

xy

z

(a) (b)

−1

0

1

−1

0

11

1.5

2

2.5

xy

z

−10

1

−1

0

11

1.2

1.4

1.6

1.8

2

2.2

2.4

(c) (d)

Figure 7.2: (a) Show the initial mesh. (b)-(d) Show the different refinement steps

149

7.4 Conclusion

This chapter contributes towards the general bivariate parametric subdivision

scheme for the surface modeling on the regular quad meshes. Some of the prop-

erties such that polynomial generation and polynomial reproduction of existing

schemes are also calculated. We can observe from the Table 7.1 that the ap-

proximating schemes have polynomial generation 6 and continuity 5 but the

interpolating scheme have polynomial generation 3 and continuity 1. The gen-

eral formula provides a variety of schemes to control the shape of initial mesh

according to our own choice. By adjusting suitable value of parameters, we

can get suitable schemes to handle the initial mesh. Most of the existing tensor

product schemes are the special case of proposed general bivariate schemes. D-

ifferent snapshots show the geometrical appearance of initial meshes after the

subdivision approach of bivariate schemes.

150

Table 7.1: Show the Continuity (C), polynomial generation (P. G) and polyno-

mial reproduction (P. R) of bivariate schemes

l Scheme Type C P. G P. R

1 9-point bivariate of Siddiqi and Ahmad (2007) Approximating 1 2 2

0 9-point bivariate of Hormann and Sabin (2008) Approximating 2 2 1

0 9-point bivariate of Zheng et al. (2014b) Approximating 2 3 2

0 4-point bivariate of Chaikin (1974) Approximating 1 2 1

0 9-point bivariate of Shen and Huang (2007) Interpolating 1 2 2

0 9-point bivariate of Ghaffar et al. (2012) Approximating 2 2 1

0 9-point bivariate of Daniel and Shunmugaraj (2008) Approximating 2 2 1

1 16-point bivariate of Zheng et al. (2014b) Approximating 4 5 1

1 9-point bivariate of Siddiqi and Rehan (2010) Approximating 1 2 2

1 16-point bivariate of Dyn et al. (1987) Interpolating 1 3 3

1 16-point bivariate of Ghaffar et al. (2013a) Approximating 5 6 3

1 16-point bivariate of Hormann and Sabin (2008) Approximating 2 4 3

1 16-point bivariate of Siddiqi and younis (2013) Approximating 4 4 1

1 16-point bivariate of Ghaffar et al. (2013a) Approximating 5 4 1

1 16-point bivariate of Ghaffar (2013) Approximating 5 4 1

1 16-point bivariate of Mustafa et al. (2009) Approximating 4 4 1

1 proposed 16-point bivariate Interpolating 1 2 1

151

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163

Publications of Robina Bashir

1. Four-point n-ary interpolating subdivision schemes, International Journal

of Mathematics and Mathematical Sciences, vol. 2013, Article ID 893414,

08 pages, 2013.

2. 3n-point quaternary shape preserving subdivision schemes, Mehran Uni-

versity Research Journal of Engineering and Technology, volume 36, no. 3,

pp. 489-500, 2017.

3. Generalization of binary tensor product schemes depends upon four pa-

rameters, Mehran University Research Journal of Engineering and Tech-

nology, 2017. (Accepted)

4. A class of shape preserving 5-point n-ary approximating schemes. (Sub-

mitted)

5. Univariate approximating schemes and their non-tensor product general-

ization. (Submitted)

6. A family of 6-point n-ary interpolating subdivision schemes. (Submitted)

164

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