The Quaternary Interpolating Scheme for Geometric Design

Abstract

A novel 4-point interpolating subdivision scheme is presented, which generates the family of 𝐶² limiting curves and its limiting function has support on [−70𝑥0002𝑓3,70𝑥0002𝑓3]. It behaves better than classical 4-point binary and ternary schemes with the same approximation order in many aspects that it has smaller support size, higher smoothness, and is computationally more efficient. The proposed nonstationary scheme can reproduce the functions of linear spaces spanned by {1,sin(𝛼𝑥),cos(𝛼𝑥)} for 0<𝛼<𝜋0𝑥0002𝑓3. Moreover, some examples are illustrated to show that the proposed scheme can also reproduce asteroids, cardioids, and conic sections as well.

1. Introduction

Subdivision schemes have become one of the most important, significant, and emerging modelling tools in computer applications, medical image processing, scientific visualization, reverse engineering, computer aided geometric designing, and so forth, because of their simple, elegant, and efficient ways to create smooth curves from initial control polygon by subdividing them according to some refining rules, recursively. These refining rules take the initial polygon to produce a sequence of finer polygons converging to a smooth limiting curve.

In the field of nonstationary subdivision schemes, binary and ternary schemes are present in the literature [1–6]. But the quaternary schemes are not discussed as others. The aim of this paper is to introduce a quaternary nonstationary scheme which reproduces the linear functions spanned by {1,sin(𝛼𝑥),cos(𝛼𝑥)}. Moreover, the proposed scheme can also generate asteroids, cardioids, and in particular conic sections like circles, ellipses, and so forth. Some properties of the proposed scheme (like support size and symmetry of basic limit function, approximation order, convergence, and smoothness) have also been investigated. Besides the convergence and approximation order, the two important and interesting properties of basic limit function of a subdivision scheme are the support size and the smoothness. They are commonly manipulated with each other in the sense that a higher degree of smoothness generally requires a larger support, thus leading to a more global influence of each initial data value on the limit function. Raising the arity of the subdivision scheme provides a way to overcome this problem to some extent. The proposed scheme has smaller support and higher smoothness than the classical binary 4-point scheme, and all other schemes reproduce cubic polynomials by construction whether they are stationary or nonstationary [1–4, 6, 7].

As interpolating subdivision schemes retain the points of stage 𝑘 as the subset of the points of next stage 𝑘+1, thus the structure of a quaternary 4-point interpolating scheme in general form can be written as 𝑓^𝑘+1_4𝑖=𝑓^𝑘_𝑖𝑓^𝑘+1_4𝑖+1=𝑎₀^(𝑘)𝑓^𝑘_𝑖−1+𝑎₁^(𝑘)𝑓^𝑘_𝑖+𝑎₂^(𝑘)𝑓^𝑘_𝑖+1+𝑎₃^(𝑘)𝑓^𝑘_𝑖+2𝑓^𝑘+1_4𝑖+2=𝑏₀^(𝑘)𝑓^𝑘_𝑖−1+𝑏₁^(𝑘)𝑓^𝑘_𝑖+𝑏₂^(𝑘)𝑓^𝑘_𝑖+1+𝑏₃^(𝑘)𝑓^𝑘_𝑖+2𝑓^𝑘+1_4𝑖+3=𝑎₃^(𝑘)𝑓^𝑘_𝑖−1+𝑎₂^(𝑘)𝑓^𝑘_𝑖+𝑎₁^(𝑘)𝑓^𝑘_𝑖+1+𝑎₀^(𝑘)𝑓^𝑘_𝑖+2, (1) where the coefficients {𝑎_𝑗^(𝑘)}_𝑗=0,1,2,3 and {𝑏_𝑗^(𝑘)}_𝑗=0,1,2,3 are chosen to satisfy the relations ∑³_𝑗=0𝑎_𝑗^(𝑘)=1 and ∑³_𝑗=0𝑏_𝑗^(𝑘)=1, respectively, for a large value of 𝑘.

This paper is organized as follows. In Section 2 the basic notion and definitions of the quaternary subdivision scheme are considered. The construction of the proposed scheme has been shown in Section 3. In Section 4, the convergence and smoothness 𝐶² of the scheme are being calculated using the theory of asymptotic equivalence. The basic limit function and its properties are being discussed in Section 5. The conclusion is drawn in Section 6.

2. Preliminaries

A quaternary subdivision scheme, to generate smooth curves or surfaces, is defined as the limit of refined control polygons or meshes, according to some refinement rules, recursively, in which the set of control points 𝑓^𝑘={𝑓^𝑘_𝑖∈𝑅∣𝑖∈𝑍} of polygon at 𝑘th level is mapped to a refined polygon to generate the new set of control points 𝑓^𝑘+1={𝑓_𝑖^𝑘+1∈𝑅∣𝑖∈𝑍} at the (𝑘+1)st level by applying the following subdivision rule: 𝑓_𝑖^𝑘+1=𝑆_𝑎(𝑘)𝑓^𝑘_𝑖=_𝑗∈𝑍𝑎^(𝑘)_𝑖−4𝑗𝑓^𝑘_𝑗∀𝑖∈𝑍 (2) with the sequence of finite sets of real coefficients {𝑎_𝑖^(𝑘)∶𝑘≥0}, constitute the so-called mask of the subdivision scheme. If the mask of a scheme is independent of 𝑘 (namely, if 𝑎^(𝑘)=𝑎 for all 𝑘≥0), then it is called stationary 𝑆_𝑎; otherwise it is called nonstationary 𝑆_𝑎(𝑘).

The convergent subdivision scheme 𝑆_𝑎 with the corresponding mask {𝑎_𝑖,𝑖∈𝑍} necessarily satisfies _𝑗∈𝑍𝑎_4𝑗+𝑙=1,𝑙=0,1,2,3. (3)

It follows that the symbol of a convergent subdivision scheme satisfies the conditions 𝑎(1)=4,𝑎𝑒^{((2𝑖𝜋/4)𝑗)}=0,𝑗=1,2,3. (4)

For this, we introduce a symbol called the Laurent polynomial 𝑎(𝑧)=_𝑖∈𝑍𝑎_𝑖𝑧^𝑖 (5) of a mask {𝑎_𝑖,𝑖∈𝑍} with finite support. In view of Dyn [10], the sufficient and necessary conditions for a uniform convergent scheme are defined as follows.

A quaternary subdivision scheme 𝑆_𝑎 is uniformly convergent if and only if there is an integer 𝐿≥1, such that 14𝑆_𝑎1^𝐿_∞<1, (6) and subdivision 𝑆_𝑎1 with symbol 𝑎₁(𝑧) is related to 𝑆_𝑎 with symbol 𝑎(𝑧), where 𝑎₁(𝑧)=(4𝑧³/(1+𝑧+𝑧²+𝑧³))𝑎(𝑧) satisfying the property 𝑑𝑓^𝑘=𝑆₁𝑑𝑓^𝑘−1,𝑘=1,2,…, (7) where 𝑓^𝑘=𝑆^𝑘_𝑎𝑓⁰={𝑓^𝑘_𝑖∶𝑖∈𝑍} and 𝑑𝑓^𝑘={(𝑑𝑓^𝑘)_𝑖=4^𝑘(𝑓^𝑘_𝑖+1−𝑓^𝑘_𝑖)∶𝑖∈𝑍}. The norm ‖𝑆_𝑎‖_∞ of a subdivision scheme 𝑆_𝑎 with a mask 𝑎_𝑖,𝑖∈𝑍, is defined by ‖𝑆_𝑎‖_∞ and 14𝑆_𝑎^𝐿_∞=max_𝛽𝑎^[𝐿]_{𝛾+4𝐿𝛽}∶𝛾=0,1,…,4^𝐿−1, (8) where 𝑎_𝑚^[𝐿](𝑧)=^𝐿−1_𝑗=0𝑎_𝑚𝑧^4𝑗. (9)

The notation of convergence for quaternary subdivision 𝐶^𝑚 scheme, similar to that given by Dyn and Levin in [11] for a binary scheme, is given in the following.

Definition 1. A quaternary subdivision scheme {𝑆_𝑎𝑘} is said to be convergent if for every initial data 𝑓⁰∈𝑙^∞ there exists a limit function 𝑓∈𝐶^𝑚(𝑅) such that lim_𝑘→∞sup_𝑖∈𝑍𝑓^𝑘_𝑖−𝑓4^−𝑘𝑖=0, (10) and 𝑓 is not identically zero for some initial data 𝑓⁰.

Definition 2. Two quaternary subdivision schemes {𝑆_𝑎(𝑘)} and {𝑆_𝑏^(𝑘)} are said to be asymptotically equivalent if ^∞_𝑘=1𝑆_𝑎(𝑘)−𝑆_𝑏^(𝑘)_∞<∞, (11) where 𝑆_{𝑎(𝑘)∞}=max_𝑗=𝑍𝑎^(𝑘)_4𝑖,_𝑖∈𝑍𝑎^(𝑘)_4𝑖+1,_𝑖∈𝑍𝑎^(𝑘)_4𝑖+2,_𝑖∈𝑍𝑎^(𝑘)_4𝑖+3. (12)

Theorem 3. Let {𝑆_𝑎𝑘} and {𝑆_𝑎} be quaternary nonstationary and stationary subdivision schemes, respectively, which are said to be asymptotically equivalent if they have finite masks of the same support. If stationary scheme {𝑆_𝑎} is 𝐶^𝑚 and ^∞_𝑘=04^mk𝑆_𝑎(𝑘)−𝑆_𝑎∞<∞, (13) then nonstationary scheme {𝑆_𝑎𝑘} is also 𝐶^𝑚.

3. The Nonstationary Subdivision Scheme

In this paper, an interpolating 4-point nonstationary scheme is presented which is associated with Lagrange-like cubic trigonometric interpolants. For this, consider a space 𝜏∶=span{1,sin(𝛼𝑥),cos(𝛼𝑥)} for some 𝛼 such that 0<𝛼<𝜋, for the data set, 𝐷={(𝑥₋₁,𝑓(𝑥₋₁)),(𝑥₀,𝑓(𝑥₀)),(𝑥₁,𝑓(𝑥₁)),(𝑥₂,𝑓(𝑥₂))}, and also consider the function, 𝐿(𝑥)=²_𝑗=−1𝑓𝑥_𝑗𝐿_𝑗(𝑥), (14) where 𝐿_𝑗(𝑥)=²_{𝑘=−1,𝑘≠𝑗}sin(𝛼/4)𝑥−𝑥_𝑘sin(𝛼/4)𝑥_𝑗−𝑥_𝑘, (15) is a unique function in 𝜏 which interpolates the data set with having minimum amplitude among other interpolants from 𝜏 [12], where the function 𝐿(𝑥) is called the trigonometric Lagrange-like interpolant of the previous data. Since the function 𝐿_𝑗(𝑥) is defined for 𝑥_𝑗=𝑗,𝑗=−1,0,1,2, where 𝐿_𝑗(𝑥) is represented by 𝐿^𝐷𝑚_𝑗(𝑥), 𝑚=1,2,3 for the following discussion.(i)If 𝐷¹={𝑥_𝑗=𝑗∶𝑗=−1,0,1,2and𝑥^′=1/4}, then 𝐿^𝐷1₋₁𝑥^′()()()=−sin𝛼/16sin3𝛼/16sin7𝛼/16()()(),𝐿sin𝛼/4sin2𝛼/4sin3𝛼/4_𝐷10𝑥^′=sin(3𝛼/16)sin(5𝛼/16)sin(7𝛼/16)sin²()(),𝐿𝛼/4sin2𝛼/4_𝐷11𝑥^′=sin(𝛼/16)sin(5𝛼/16)sin(7𝛼/16)sin²()(),𝐿𝛼/4sin2𝛼/4_𝐷12𝑥^′()()()=−sin𝛼/16sin3𝛼/16sin5𝛼/16()()().sin𝛼/4sin2𝛼/4sin3𝛼/4 (16)(ii)If 𝐷²={𝑥_𝑗=𝑗∶𝑗=−1,0,1,2and𝑥^′′=1/2}, then 𝐿^𝐷2₋₁𝑥^′′=𝐿^𝐷2₂𝑥^′′=−sin²(𝛼/8)sin(3𝛼/8)()()(),𝐿sin𝛼/4sin2𝛼/4sin3𝛼/4_𝐷20𝑥^′′=𝐿^𝐷2₁𝑥^′′=sin(𝛼/8)sin²(3𝛼/8)sin²()(),𝛼/4sin2𝛼/4 (17) and similarly, it can be obtained for the following.(iii)If 𝐷³={𝑥_𝑗=𝑗∶𝑗=−1,0,1,2and𝑥^′′′=3/4}, then 𝐿^𝐷1₋₁𝑥^′=𝐿^𝐷3₂𝑥^′′′,𝐿^𝐷1₀𝑥^′=𝐿^𝐷3₁𝑥^′′′,𝐿^𝐷1₁𝑥^′=𝐿^𝐷3₀𝑥^′′′,𝐿^𝐷1₂𝑥^′=𝐿^𝐷3₋₁𝑥^′′′. (18)

After that, adjust the previous values to the masks {𝑎_𝑗^(𝑘)}_𝑗=0,1,2,3 of the subdivision scheme (1) in the following manner: 𝑎₀^(𝑘)=sin𝛼/2^𝑘+4sin3𝛼/2^𝑘+4sin7𝛼/2^𝑘+4sin𝛼/2^𝑘+2sin2𝛼/2^𝑘+2sin3𝛼/2^𝑘+2=𝐿^𝐷1₋₁𝑥^′=𝐿^𝐷3₂𝑥^′′′𝑎₁^(𝑘)=sin3𝛼/2^𝑘+4sin5𝛼/2^𝑘+4sin7𝛼/2^𝑘+4sin²𝛼/2^𝑘+2sin2𝛼/2^𝑘+2=𝐿^𝐷1₀𝑥^′=𝐿^𝐷3₁𝑥^′′′𝑎₂^(𝑘)=sin𝛼/2^𝑘+4sin5𝛼/2^𝑘+4sin7𝛼/2^𝑘+4sin²𝛼/2^𝑘+2sin2𝛼/2^𝑘+2=𝐿^𝐷1₁𝑥^′=𝐿^𝐷3₀𝑥^′′′𝑎₃^(𝑘)=sin𝛼/2^𝑘+4sin3𝛼/2^𝑘+4sin5𝛼/2^𝑘+4sin𝛼/2^𝑘+2sin2𝛼/2^𝑘+2sin3𝛼/2^𝑘+2=𝐿^𝐷1₂𝑥^′=𝐿^𝐷3₋₁𝑥^′′′, (19) and also we have 𝑏₀^(𝑘)=𝑏₃^(𝑘)=sin²𝛼/2^𝑘+3sin3𝛼/2^𝑘+3sin𝛼/2^𝑘+2sin2𝛼/2^𝑘+2sin3𝛼/2^𝑘+2,𝑏₁^(𝑘)=𝑏₂^(𝑘)=sin𝛼/2^𝑘+3sin²3𝛼/2^𝑘+3sin²𝛼/2^𝑘+2sin2𝛼/2^𝑘+2. (20)

Thus, an interpolating 4-point nonstationary subdivision rule to refine the control polygon is defined in (21), where the mask of proposed scheme satisfies the affine properties, presented by the set of (19) and (20). So, we have 𝑓^𝑘+1_4𝑖=𝑓^𝑘_𝑖𝑓^𝑘+1_4𝑖+1=−𝑎₀^(𝑘)𝑓^𝑘_𝑖−1+𝑎₁^(𝑘)𝑓^𝑘_𝑖+𝑎₂^(𝑘)𝑓^𝑘_𝑖+1−𝑎₃^(𝑘)𝑓^𝑘_𝑖+2𝑓^𝑘+1_4𝑖+1=−𝑏₀^(𝑘)𝑓^𝑘_𝑖−1+𝑏₁^(𝑘)𝑓^𝑘_𝑖+𝑏₂^(𝑘)𝑓^𝑘_𝑖+1−𝑏₃^(𝑘)𝑓^𝑘_𝑖+2𝑓^𝑘+1_4𝑖+2=−𝑎₃^(𝑘)𝑓^𝑘_𝑖−𝑖+𝑎₂^(𝑘)𝑓^𝑘_𝑖+𝑎₁^(𝑘)𝑓^𝑘_𝑖+1−𝑎₀^(𝑘)𝑓^𝑘_𝑖+2. (21)

It may be noted that the points ((4𝑖+1)/4^𝑘+1,𝑓^𝑘+1_4𝑖+1), ((4𝑖+2)/4^𝑘+1,𝑓^𝑘+1_4𝑖+1), and ((4𝑖+3)/4^𝑘+1,𝑓^𝑘+1_4𝑖+1) lie on the Lagrange-like interpolant of the set of points {((𝑖−1)/4^𝑘,𝑓^𝑘_𝑖−1),(𝑖/4^𝑘,𝑓^𝑘_𝑖),((𝑖+1)/4^𝑘,𝑓^𝑘_𝑖+1), and ((𝑖+2)/4^𝑘,𝑓^𝑘_𝑖+2)}. Moreover, the mask of scheme (21) is bounded by the following coefficients: 𝑎^𝑘₀⟶7128,𝑎^𝑘₁⟶105,𝑎128_𝑘2⟶35128,𝑎^𝑘₃⟶5,𝑏128_𝑘0=𝑏^𝑘₃⟶116,𝑏^𝑘₁=𝑏^𝑘₂⟶9.16 (22)

The proof of the previous boundedness can be followed from Lemma 4.

4. Convergence and Smoothness

The theory of asymptotic equivalence is used to investigate the convergence and smoothness of the scheme following [11]. For this, some estimations of 𝑎_𝑖^(𝑘),𝑖=0,1,2,3 and 𝑏_𝑖^(𝑘),𝑖=0,1,2,3 used are given in the following lemmas. To prove the lemmas, the following three inequalities are used: sin𝑎≥𝑎sin𝑏𝑏𝜋for0<𝑎<𝑏<2,𝜋𝜃csc𝜃≤𝑡csc𝑡for0<𝜃<𝑡<2,cos𝑥≤sin𝑥𝑥𝜋for0<𝑥<2. (23)

Lemma 4. For 𝑘≥0 and 0<𝛼<𝜋/2, one has (i)7128≤𝑎₀^(𝑘)≤71128cos³3𝛼/2^𝑘+1,(ii)105128≤𝑎₁^(𝑘)≤1051128cos³𝛼/2^𝑘+1,(iii)35128≤𝑎₂^(𝑘)≤351128cos³𝛼/2^𝑘+1,()5iv128≤𝑎₃^(𝑘)≤51128cos³3𝛼/2^𝑘+1,(v)116≤𝑏₀^(𝑘)≤1116cos³𝛼/2^𝑘+1,()9vi16≤𝑏₁^(𝑘)≤9116cos³3𝛼/2^𝑘+1. (24)

Proof. To prove the inequality (i) 𝑎₀^(𝑘)=sin𝛼/2^𝑘+4sin3𝛼/2^𝑘+4sin7𝛼/2^𝑘+4sin𝛼/2^𝑘+2sin2𝛼/2^𝑘+2sin3𝛼/2^𝑘+2≥𝛼/2^𝑘+43𝛼/23^𝑘+47𝛼/2^𝑘+4𝛼/23^𝑘+22𝛼/2^𝑘+23𝛼/2^𝑘+2=7,𝑎128_0(𝑘)≤21𝛼³2^3𝑘+12𝛼csc2^𝑘+2csc2𝛼2^𝑘+2csc3𝛼2^𝑘+2≤21𝛼³2^3𝑘+1236csc³3𝛼2^𝑘+1≤21𝛼³2^3𝑘+121363𝛼/2^𝑘+13cos³3𝛼/2^𝑘+1≤71128cos³3𝛼/2^𝑘+1. (25)(26)
The proof of (ii), (iii), (iv), (v), and (vi) can be obtained similarly.

Lemma 5. For some constants 𝐶₀,𝐶₁,𝐶₂,𝐶₃,𝐶₄, and 𝐶₅ independent of 𝑘, one has the following: (i)𝑎₀^(𝑘)−7128≤𝐶₀14^𝑘,(ii)𝑎₁^(𝑘)−105128≤𝐶₁14^𝑘,(iii)𝑎₂^(𝑘)−35128≤𝐶₂14^𝑘,(iv)𝑎₃^(𝑘)−5128≤𝐶₃14^𝑘,(v)𝑏₀^(𝑘)−116≤𝐶₄14^𝑘,(vi)𝑏₁^(𝑘)−916≤𝐶₅14^𝑘. (27)

Proof. Inequality (i) can be proved using Lemma 4. So (i)𝑎₀^(𝑘)−7≤71281281−cos³3𝛼/2^𝑘+1cos³𝛼≤71283sin²3𝛼/2^𝑘+12cos³𝛼≤189𝛼²1024cos³𝛼14^𝑘. (28)
The proof of (ii), (iii), (iv), (v), and (vi) can be obtained similarly.

Lemma 6. The Laurent polynomial 𝑎^(𝑘)(𝑧) of the scheme {𝑆_𝑎(𝑘)} at the 𝑘𝑡ℎ level can be written as 𝑎^(𝑘)(𝑧)=((1+𝑧+𝑧²+𝑧³)/4)𝑐^(𝑘)(𝑧), where 𝑐^(𝑘)(𝑧)=4−𝑎₃^(𝑘)𝑧⁴+𝑎₃^(𝑘)−𝑏₀^(𝑘)𝑧³+𝑏₀^(𝑘)−𝑎₀^(𝑘)𝑧²+𝑎₀^(𝑘)𝑧+𝑎₂^(𝑘)−𝑎₃^(𝑘)+𝑏₁^(𝑘)−𝑏^(k)₀−𝑎₂^(𝑘)+𝑎₃^(𝑘)𝑧⁻¹+𝑎₁^(𝑘)−𝑎₀^(𝑘)−𝑏₁^(𝑘)+𝑏₀^(𝑘)𝑧⁻²+𝑎₂^(𝑘)−𝑎₃^(𝑘)𝑧⁻³+𝑎₀^(𝑘)𝑧⁻⁴+𝑏₀^(𝑘)−𝑎₀^(𝑘)𝑧⁻⁵+𝑎₃^(𝑘)−𝑏₀^(𝑘)𝑧⁻⁶−𝑎₃^(𝑘)𝑧⁻⁷. (29)

Proof. Since 𝑎^𝑘(𝑘)(𝑧)=−𝑎₃^(𝑘)𝑧⁷−𝑏₃^(𝑘)𝑧⁶−𝑎₀^(𝑘)𝑧⁵+𝑎₂^(𝑘)𝑧³+𝑏₂^(𝑘)𝑧²+𝑎₁^(𝑘)𝑧+1+𝑎₁^(𝑘)𝑧⁻¹+𝑏₁^(𝑘)𝑧⁻²+𝑎₂^(𝑘)𝑧⁻³−𝑎₀^(𝑘)𝑧⁻⁵−𝑏₀^(𝑘)𝑧⁻⁶−𝑎₃^(𝑘)𝑧⁻⁷, (30)
therefore using −𝑎₀^(𝑘)+𝑎₁^(𝑘)+𝑎₂^(𝑘)−𝑎₃^(𝑘)=1 and −𝑏₀^(𝑘)+𝑏₁^(𝑘)=1/2 following result can be established 𝑎^(𝑘)(𝑧)=((1+𝑧+𝑧²+𝑧³)/4)𝑏^(𝑘)(𝑧).

Lemma 7. The Laurent polynomial 𝑎(𝑧) of the scheme {𝑆_𝑎} can be written as 𝑎(𝑧)=((1+𝑧+𝑧²+𝑧³)/4)𝑐(𝑧), where 𝑐(𝑧)=132−5𝑧⁴−3𝑧³+𝑧²+7𝑧+30+34𝑧⁻¹+34𝑧⁻²+30𝑧⁻³+7𝑧⁻⁴+𝑧⁻⁵−3𝑧⁻⁶−5𝑧⁻⁷, (31) and the subdivision scheme {𝑆_𝑐} corresponding to the symbol 𝑐(𝑧) is 𝐶¹.

Proof. The mask defined by the scheme {𝑆_𝑐} converges and has the smoothness 𝐶¹ (it can be proved using Laurent polynomial method). If {𝑆_𝑐} is 𝐶¹, then {𝑆_𝑎} with the mask 1[]128−5,8,−7,0,35,72,105,128,105,72,35,0,−7,8,−5 (32) is 𝐶².

Theorem 8. The scheme defined by (21) converges and has smoothness 𝐶².

Proof. To prove the proposed scheme to be 𝐶², it is sufficient to show that the scheme corresponding to 𝑐^𝑘(𝑧) is 𝐶¹ (Theorem (8) given by Dyn and Levin [13]). Since {𝑆_𝑐} is 𝐶¹ by Lemma 7, then following Theorem 3, it has been shown that the schemes {𝑆_𝑐^(𝑘)} and {𝑆_𝑐} are asymptotically equivalent. So we have to show that ^∞_𝑘=04^𝑘𝑆_𝑐(𝑘)−𝑆_𝑐∞<∞, (33) where 𝑆_𝑐(𝑘)−𝑆_𝑐∞=max^∞_𝑗=𝑍𝑐^(𝑘)_𝑖+4𝑗−𝑐_𝑖+4𝑗∶𝑖=0,1,2,3. (34)
From Lemmas 6 and 7, it can be written as ^∞_𝑗∈𝑍𝑐^(𝑘)_4𝑗−𝑐_4𝑗=4𝑎^𝑘₃−5+324𝑎^𝑘₂−4𝑎^𝑘₃−30+324𝑎^𝑘₀−7𝑎32=4^𝑘₀−7𝑎128+4^𝑘₂−35𝑎128+8^𝑘₃−5=128_{∞𝑗∈𝑍}𝑐^𝑘_4𝑗+3−𝑐_4𝑗+3. (35)
Similarly, one can obtain ^∞_𝑗∈𝑍𝑐^(𝑘)_4𝑗+1−𝑐_4𝑗+1=4𝑎^𝑘₀−7𝑎128+4^𝑘₂−35𝑎128+8^𝑘₃−5𝑏128+12^𝑘₀−1𝑏16+4^𝑘₁−9,16_{∞𝑗∈𝑍}𝑐^(𝑘)_4𝑗+2−𝑐_4𝑗+2=8𝑎^𝑘₀−7𝑎128+4^𝑘₁−105𝑎128+4^𝑘₃−5𝑏128+12^𝑘₀−1𝑏16+4^𝑘₁−9.16 (36)
From Lemma 5, we have ∑^∞_𝑘=04^𝑘|𝑎₀^(𝑘)−7/128|<∞, ∑^∞_𝑘=04^𝑘|𝑎₁^(𝑘)−105/128|<∞,∑^∞_𝑘=04^𝑘|𝑎₂^(𝑘)−35/128|<∞, ∑^∞_𝑘=04^𝑘|𝑎₃^(𝑘)−5/128|<∞,∑^∞_𝑘=04^𝑘|𝑏₀^(𝑘)−1/16|<∞, and ∑^∞_𝑘=04^𝑘|𝑏₁^(𝑘)−9/16|<∞. So, it can be written as ^∞_𝑘=04^𝑘𝑆_𝑐(𝑘)−𝑆_𝑐∞<∞. (37)
Thus, the scheme corresponding to symbol 𝑐^(𝑘)(𝑧) is 𝐶¹ and hence the scheme (21) is 𝐶².

Figure 1 shows the reproduction of the unit circle and ellipse by the proposed scheme (21), taking regular hexagon as control points for circle.

(a)

(b)

Figure 1 

The reproduction of the unit circle and ellipse by the proposed scheme.

In Figure 2, twelve points have been taken as control points on the parametric curves of asteroid: 𝑥(𝑡)=3cos𝑡+cos3𝑡,𝑦(𝑡)=3sin𝑡−sin3𝑡 and cardioid: 𝑥(𝑡)=(1−cos𝑡)sin𝑡,𝑦(𝑡)=(1−cos𝑡)cos𝑡 to reconstruct the asteroid and cardioid, respectively, by the proposed scheme.

(a)

(b)

Figure 2 

The reconstruction of asteroid and cardioid by the proposed scheme.

Figure 3 shows the behavior of basic limit function for different values of 𝛼.

(a)

(b)

Figure 3 

Basic limit function of proposed scheme for different values of 𝛼.

5. Properties of the Scheme

In this section some properties (approximation order, reproduction property, support size, and symmetry of basic limit function) are being discussed. The results show that the proposed scheme with the same approximation order has the higher smoothness and smaller support than the existing 4-point interpolating subdivision schemes (see also Table 1).

Table 1 

Comparison of the 4-point interpolating subdivision schemes.

Another advantage (see also [14]) of the scheme is the computational cost. In one subdivision step, for a sequence of 𝑛 points the proposed scheme requires 12𝑛 multiplications and 9𝑛 additions. On the other hand, computing the same number of 4𝑛 refined points with the binary scheme requires two subdivision steps with a total cost of 18𝑛 multiplications and 9𝑛 additions, so the quaternary scheme is more efficient.

The basic limit function of the scheme is symmetric about 𝑌-axis for the data 𝑓⁰_𝑖=1,𝑖=0,0,𝑖≠0. (38)

Theorem 9 (see [3]). The basic limit function 𝐹 is symmetric about the 𝑌-axis.

Proof. Let 𝐹 denote the basic limit function and define 𝐷_𝑛∶={𝑖/4^𝑛∣𝑖∈𝑍} such that restriction of 𝐹 to 𝐷_𝑛 satisfies 𝐹(𝑖/4^𝑛)=𝑓^𝑛_𝑖 for all 𝑖∈𝑍. The symmetry of basic function is proved using induction on 𝑛.
It can be observed that 𝐹(𝑖)=𝐹(−𝑖) for all 𝑖∈𝑍; thus 𝐹(𝑖/4^𝑛)=𝐹(−𝑖/4^𝑛) for all 𝑖∈𝑍 and 𝑛=0. Assume that 𝐹(𝑖/4^𝑛)=𝐹(−𝑖/4^𝑛) for all 𝑖∈𝑍 and then 𝑓^𝑛_𝑖=𝑓^𝑛_−𝑖 for all 𝑖∈𝑍. It may be observed that 𝐹4𝑖4^𝑛+1=𝑓^𝑛+1_4𝑖=𝑓^𝑛+1_−4𝑖=𝐹−4𝑖4^𝑛+1,𝐹4𝑖+14^𝑛+1=𝑓^𝑛+1_4𝑖+1=−𝑎^𝑛₀𝑓^𝑛_𝑖−1+𝑎^𝑛₁𝑓^𝑛_𝑖+𝑎^𝑛₂𝑓^𝑛_𝑖+1−𝑎^𝑛₃𝑓^𝑛_𝑖+2=−𝑎^𝑛₃𝑓^𝑛_−𝑖−2+𝑎^𝑛₂𝑓^𝑛_−𝑖−1+𝑎^𝑛₁𝑓^𝑛_−𝑖−𝑎^𝑛₀𝑓^𝑛_−𝑖+1=𝑓^𝑛+1_−4𝑖−1=𝐹−4𝑖+14^𝑛+1. (39)
Similarly 𝐹4𝑖+24^𝑛+1=𝑓^𝑛+1_4𝑖+2=𝑓^𝑛+1_−4𝑖−2=𝐹−4𝑖+24^𝑛+1,𝐹4𝑖+34^𝑛+1=𝑓^𝑛+1_4𝑖+3=𝑓^𝑛+1_−4𝑖−3=𝐹−4𝑖+34^𝑛+1. (40)
Hence, 𝐹(𝑖/4^𝑛)=𝐹(−𝑖/4^𝑛) for all 𝑖∈𝑍 and 𝑛=𝑍₊; thus from the continuity of 𝐹, 𝐹(𝑥)=𝐹(−𝑥) holds for all 𝑥∈𝑅, which completes the required result.

In the following the support of basic limit function of the proposed scheme has been calculated generated by the control point 𝑓⁰_𝑖

Theorem 10. The basic limit function F defined by the scheme vanishes outside [−7/3,7/3].

Proof. Define 𝑡_𝑛 recursively by taking 𝑡_𝑛+1=𝑡_𝑛+1/4^𝑛+1 with 𝑡₀=0. It can be observed that 𝑡_𝑛∈𝐷¹_𝑛 but 𝑡_𝑛∉𝐷¹_𝑛−1. We claim that restriction of 𝐹 to 𝐷_𝑛 vanishes outside [−𝑡_𝑛,𝑡_𝑛]. Since 𝐹 is symmetric, it is enough to prove that 𝐹 vanishes outside [0,𝑡_𝑛]. Thus, after 𝑘th subdivision step, the furthest nonzero vertex on the left/right will be at (7/4)∑^𝑘−1_𝑗=0(1/4^𝑗). So, it can be concluded that the left or right influence of the original nonzero vertex will propagate along lim_𝑛→∞𝑡_𝑛=∑^∞_𝑗=0(7/4)(1/4)^𝑗=(7/4)(1/(1−(1/4)))=7/3. It can be concluded that [−𝑡_𝑛,𝑡_𝑛] is contained in [−7/3,7/3] for all 𝑛 and lim_𝑛→∞[−𝑡_𝑛,𝑡_𝑛]=[−7/3,7/3], which is the required result.

Theorem 11 11(see [15]). A convergent subdivision scheme that reproduces polynomial 𝜋_𝑛 has an approximation order 𝑛+1.

So in the light of the previous theorem, the proposed scheme has approximation order 4 because the construction of mask is formulated through the cubic trigonometric polynomials.

The polynomial reproduction property has its own importance, as the reproduction property of the polynomials up to certain degree 𝑑 implies that the scheme has 𝑑+1 approximation order. For this, polynomial reproduction can be made from initial data which has been sampled from some polynomial functions. So, in view of the previous theorem, scheme (32) has approximation order 4.

6. Conclusion

A 4-point novel quaternary interpolating nonstationary subdivision scheme has been developed which generates the family of 𝐶² limiting curves and its limiting function has support on [−7/3,7/3]. The nonstationary scheme is associated with trigonometric polynomials which reproduces the functions of linear spaces spanned by {1,sin(𝛼𝑥),cos(𝛼𝑥)} for 0<𝛼<𝜋/3. The comparison shows that, with existing stationary and nonstationary interpolating schemes, the proposed scheme is better. Moreover, it also can reconstruct or reproduce the conic sections, asteroids, and cardioids.