On a Nonlinear Three-Point Subdivision Scheme Reproducing Piecewise Constant Functions

Abstract

In this article, a nonlinear binary three-point non-interpolatory subdivision scheme is presented. It is based on a nonlinear perturbation of the three-point subdivision scheme: A new three point approximating $C^2$ subdivision scheme, where the convergence and the stability of this linear subdivision scheme are analyzed. It is possible to prove that this scheme does not present Gibbs oscillations in the limit functions obtained. The numerical experiments show that the linear scheme is stable even in the presence of jump discontinuities. Even though, close to jump discontinuities, the accuracy is loosed. This order reduction is equivalent to the introduction of some diffusion. Diffusion is a good property for subdivision schemes when the discontinuities are numerical, i.e., they appear when discretizing a continuous function close to high gradients. On the other hand, if the initial control points come from the discretization of a piecewise continuous function, it can be interesting that the subdivision scheme produces a piecewise continuous limit function. For instance, in the approximation of conservation laws, real discontinuities appear as shocks in the solution. The nonlinear modification introduced in this work allows to attain this objective. As far as we know, this is the first subdivision scheme that appears in the literature with these properties.

1. Introduction

Subdivision schemes have attracted a lot of attention for applications in CAGD since Dyn et al. introduced, in [1], a four point interpolatory subdivision scheme in 1987. These schemes can be considered an efficient way to generate curves and surfaces from an initial set of control points. In past years, a great number of works have been published on linear subdivision schemes. See, for example, Refs. [2,3,4,5,6,7,8,9,10,11,12,13] and the references therein for an incomplete list of relevant publications. Nonlinear subdivision schemes have also attracted great attention. See, for example, Refs. [14,15,16,17,18,19,20,21,22,23,24,25,26,27] and the references therein for an incomplete list of publications about the subject.

Subdivision schemes can be classified into stationary or nonstationary in determining their dependence on the level k of application of the scheme. If the scheme is stationary, the masks have an invariant expression independently of the level of subdivision k. Otherwise, the scheme is said to be nonstationary. It is possible to find many stationary subdivision schemes in the state of the art literature. Among them, some binary schemes have been also proposed in the past years [1,28,29,30,31,32]. All of them suffer from diffusion close to jump discontinuities and some of them present Gibbs phenomenon [33].

As mentioned in the abstract, if the control points are obtained from the sampling of a continuous function with high gradients, numerical discontinuities can appear due to low sampling frequencies that do not manage to capture the regularity of the function at those high gradients. If this is the case, the diffusion introduced in the limit function is not a problem and can even be considered a good quality of the subdivision scheme. If the control points are obtained from the sampling of a piecewise continuous function, the discontinuities should also appear in the limit function.

Thus, the main objective of this paper is to introduce a new nonlinear, noninterpolatory, stationary and binary family of subdivision schemes converging to a piecewise continuous function and that do not oscillate close to jump discontinuities. In fact, close to the discontinuities, the new family of schemes do not introduce diffusion. The properties of the family of subdivision schemes will be analyzed. In particular, convergence and stability of the family of subdivision schemes are also established. As far as we know, this is the first time that a binary, stable and convergent subdivision scheme without Gibbs oscillations close to jump discontinuities and reproducing piecewise constant functions has appeared in the literature.

The paper is organized as follows. In Section 2, we introduce a new nonlinear, noninterpolatory family of subdivision schemes based on the one studied in [31]. Writing the scheme as a perturbation of a linear scheme and establishing a contractivity property of this perturbation, we deduce the convergence (studied in Section 3) and stability (studied in Section 4) of the subdivision scheme, that, due to the nonlinear nature of the scheme, is not a consequence of the convergence. Section 5 is devoted to numerical examples. Section 5.1 analyzes the numerical regularity of some members of the family of schemes presented. Section 5.2 presents some experiments of subdivision for piecewise smooth functions. Section 5.3 is dedicated to the analysis of the order of accuracy of the schemes close to the discontinuities. Finally, Section 6 presents our conclusions.

2. A Nonlinear Family of Binary Subdivision Schemes

This section is devoted to the introduction of a nonlinear binary subdivision scheme as a perturbation of the linear scheme introduced in [31]. In this work, S. Shahid and A. Nadeem prove that the scheme that they proposed converges to a limit function with $C^2$ regularity. They analyze the following binary subdivision scheme:

$$\begin{array}{ccc}\hfill {\left(S_w{f}^k\right)}_{2n}& =& a{f}_{n-1}^k+b{f}_n^k+c{f}_{n+1}^k,\hfill \\ \hfill {\left(S_w{f}^k\right)}_{2n+1}& =& c{f}_{n-1}^k+b{f}_n^k+a{f}_{n+1}^k,\hfill \end{array}$$

(1)

where $\left\{f_i^k\right\}$ is a set of initial control points with $c=\frac{1}{2}{w}^2$, $b=\frac{1}{2}(1+2w-2w^2)$ and $a=\frac{1}{2}{(w-1)}^2$ and $w\in [0,\frac{1}{2}]$. The theory presented in [19] assures the absence of Gibbs oscillations close to jump discontinuities.

Thus, the only drawback that the previous linear noninterpolatory scheme can present is that diffusion appears in the limit function when the scheme starts from data that might arise from the sampling of a discontinuous function. The mentioned linear subdivision scheme is based on a quadratic B-spline parametrized in the interval $[x_i,x_{i+1}]$. Due to the convex hull property of B-splines, this scheme cannot present Gibbs phenomenon [34]. The authors prove, in [31], that, for $w=\frac{1}{4}$, the scheme presents a limit function with $C^2$ regularity. This result can also be expected because the scheme is based on a binary B-spline.

In order to obtain a nonlinear family of schemes from (1), let us define ${\mathbb{R}}_{+}:=(0,\infty ),{\mathbb{R}}_{+}^2:={\mathbb{R}}_{+}\times {\mathbb{R}}_{+},$ and consider the function $M:{\mathbb{R}}_{+}^2\to {\mathbb{R}}_{+}$. The nonlinear scheme we analyze in this paper follows [18,35], and it is based on the substitution of the arithmetic mean with a M function. We first propose a new formulation of the scheme (1):

$$\begin{array}{cc}\hfill {\left(S_w{f}^k\right)}_{2n}& =f_n^k-ad{f}_n^k+cd{f}_{n+1}^k,\hfill \\ \hfill {\left(S_w{f}^k\right)}_{2n+1}& =f_n^k-cd{f}_n^k+ad{f}_{n+1}^k,\hfill \end{array}$$

(2)

where $d{f}_n^k=f_n^k-f_{n-1}^k$ denotes the first order differences. The nonlinear scheme $S_{NL}$ that we propose is given by:

• If $|d{f}_n^k|\ge |d{f}_{n+1}^k|$,

  $$\begin{array}{cc}\hfill {\left(S_{NL}{f}^k\right)}_{2n}& =f_n^k+(c+a)d{f}_{n+1}^k-2aM(d{f}_n^k,d{f}_{n+1}^k),\hfill \\ \hfill {\left(S_{NL}{f}^k\right)}_{2n+1}& =f_n^k+(c+a)d{f}_{n+1}^k-2cM(d{f}_n^k,d{f}_{n+1}^k).\hfill \end{array}$$

  (3)
• If $|d{f}_n^k|<|d{f}_{n+1}^k|$,

  $$\begin{array}{cc}\hfill {\left(S_{NL}{f}^k\right)}_{2n}& =f_n^k-(c+a)d{f}_n^k+2cM(d{f}_n^k,d{f}_{n+1}^k),\hfill \\ \hfill {\left(S_{NL}{f}^k\right)}_{2n+1}& =f_n^k-(c+a)d{f}_n^k+2aM(d{f}_n^k,d{f}_{n+1}^k).\hfill \end{array}$$

  (4)

For $M_{amean}(x,y)=\frac{x+y}{2}$, we recover the original linear scheme in (1). In the numerical experiments, we use the harmonic mean, properly introduced in Section 5.

In [36,37,38,39], we use a similar technique to adapt subdivision schemes to the presence of discontinuities. Before analyzing in detail the properties of the new scheme $S_{NL}$, in the following definition we summarize the most important properties that the function M must verify (see [23] for more details).

Definition 1. 

$\left(M1\right)$

The function M is a mean on ${\mathbb{R}}_{+}^2$, and thus:

$\left(M1a\right)$

$M(x,y)\le \max \{x,y\}.$

$\left(M1b\right)$

$M(x,y)\ge \min \{x,y\}.$

$\left(M1c\right)$

$M(x,x)=x.$

$\left(M2\right)$

$M(x-h,x+h)=x+O\left(h^2\right)$ as $h\to 0$ uniformly with respect to x

on compact subintervals on ${\mathbb{R}}_{+}.$

$\left(M3\right)$

$M(tx,ty)=tM(x,y)$$(x,y,t\in {\mathbb{R}}_{+}).$

$\left(M4\right)$

$M(x+o(h),y+o(h\left)\right)=M(x,y)+o\left(h\right)$.

$\left(M5\right)$

$M(x,y)\le \frac{x+y}{2}$$(x,y\in {\mathbb{R}}_{+}).$

$\left(M6\right)$

$M(x,y)\le \gamma min(x,y)$. $(x,y\in {\mathbb{R}}_{+})$ for some constant $\gamma >1.$

$\left(M7\right)$

If $x=O\left(1\right)$, $y=O\left(1\right)$, $|y-x|=O\left(h\right)$ and $xy>0$, then

$$\left|\frac{x+y}{2}-M(x,y)\right|=O\left(h^2\right).$$

$\left(M8\right)$

for all $x,x^{\prime },y,y^{\prime }\in {\mathbb{R}}_{+}$ there exist non-negative numbers α and β (depending on these points) such that

$$M(x,y)-M(x^{\prime },y^{\prime })=\alpha (x-x^{\prime })+\beta (y-y^{\prime }),$$

and $\alpha +\beta \le \gamma$ with a constant $\gamma >1$ not depending on $x,x^{\prime },y,y^{\prime }.$

As a consequence of $\left(M5\right)$, the authors presented, in [23], the following proposition:

Proposition 1. 

Suppose that M is continuous on $R_{+}^2$. If $\left(M5\right)$ holds, then the extension of M by

$$M(x,y):=\left\{\begin{array}{cc}\mathit{sign}\left(x\right)M\left(\right|x|,|y\left|\right)\hfill & \mathit{if} xy>0,\hfill \\ 0\hfill & \mathit{if} xy\le 0,\hfill \end{array}\right.$$

(5)

is continuous on ${\mathbb{R}}^2$.

The proof can be found in Proposition 2 of [23]. We use this extension to ${\mathbb{R}}^2$ in the rest of the paper.

For the linear family of schemes presented in this section, the elimination of the Gibbs phenomenon in the presence of discontinuities is not needed to be studied, as the family of schemes present positive masks that assure the absence of oscillations close to discontinuities [19]. Indeed, in [19], we prove a theorem that assures the absence of Gibbs oscillations for stationary subdivision schemes with positive masks. They follow the notation used by Dyn and Levin in [3], where a univariate stationary subdivision scheme with a mask $\mathbf{a}={\left\{a_j\right\}}_{j\in \mathbb{Z}}$ with finite support is defined as beginning with an initial sequence of finite data $f^0={\left\{f_i^0\right\}}_{i\in J_0}$. New values are obtained through refinement at level $k+1$, denoted by $f^{k+1}={\left\{f_i^{k+1}\right\}}_{i\in J_{k+1}}$. The subdivision scheme is the maximal set obtained by applying the rule:

$${\left(S_{\mathbf{a}}{f}^k\right)}_i=f_i^{k+1}:=\sum _{j\in J_k}{a}_{i-2j}{f}_j^k,\mathrm{then},J_{k+1}=2J_k+supp\left(a\right)=\{i\in \mathbb{Z}:i=2j+\gamma ,j\in J_k,\gamma \in supp\left(a\right)\}.$$

There are two rules to define the points on the level $k+1$:

$$\begin{array}{cc}\hfill & {\left(S_{\mathbf{a}}{f}^k\right)}_{2i}=f_{2i}^{k+1}=\sum _{\gamma \in \mathbb{Z}}{a}_{2\gamma }{f}_{i-\gamma }^k,i\in \mathbb{Z},\hfill \\ \hfill & {\left(S_{\mathbf{a}}{f}^k\right)}_{2i+1}=f_{2i+1}^{k+1}=\sum _{\gamma \in \mathbb{Z}}{a}_{2\gamma +1}{f}_{i-\gamma }^k,i\in \mathbb{Z}.\hfill \end{array}$$

(6)

We can represent these rules using an algebraic formalism in terms of z-transforms. The symbol of the mask $\mathbf{a}={\left\{a_j\right\}}_{j\in \mathbb{Z}}$ is defined as $a\left(z\right)={\sum }_{j\in \mathbb{Z}}{a}_j{z}^j$. If $a^{\left[k\right]}$ is denoted as the k iterated symbol (see [3]), then, it is known that $a^{\left[k\right]}\left(z\right)={\prod }_{l=1}^{k}a\left(z^{2^{l-1}}\right)$. Therefore, if $0\le l<2^k$, then

$$f_{2^{k}i+l}^k={\left(S_{\mathbf{a}}^k{f}^0\right)}_{2^{k}i+l}={\left(S_{{\mathbf{a}}^{\left[k\right]}}{f}^0\right)}_{2^{k}i+l}=\sum _{\gamma \in \mathbb{Z}}{a}_{2^k\gamma +l}^{\left[k\right]}{f}_{i-\gamma }^0,$$

(7)

with

$$a_j^{\left[k\right]}=\sum _{i\in \mathbb{Z}}{a}_i^{[k-1]}{a}_{j-2i},$$

(8)

being $a_j^{\left[1\right]}=a_j$, $\forall j$. As in [19], in this paper, we impose to the coefficients of the mask the following conditions:

• The mask $\mathbf{a}={\left\{a_j\right\}}_{j\in \mathbb{Z}}$ is defined as:

  $$\left\{\begin{array}{cc}-1\le a_j\le 1,\hfill & U\le j\le U+V,\hfill \\ a_U,a_{U+V}\ne 0,\hfill \\ a_j=0,\hfill & j>U+V\mathrm{or}j<U,\hfill \end{array}\right.$$

  (9)

  with $U<0$ and V fixed integers with $V\ge 2$ and $U+V>0$.
• The scheme $S_{\mathbf{a}}$ is convergent. Therefore, one necessary condition is

  $$\sum _{\gamma \in \mathbb{Z}}{a}_{2\gamma }=\sum _{\gamma \in \mathbb{Z}}{a}_{2\gamma +1}=1.$$

  (10)
• For any $f\in {\mathcal{C}}^n$, $n\ge 2$ and any $h>0$, with

  $$\left\{\begin{array}{cc}{f}_i^0=f\left(ih\right),i\in \mathbb{Z},\hfill & \mathrm{if}\mathrm{the}\mathrm{scheme}\mathrm{has}\mathrm{primal}\mathrm{parametrization},\hfill \\ f_i^0=f\left(\left(i-\frac{1}{2}\right)h\right),i\in \mathbb{Z},\hfill & \mathrm{if}\mathrm{the}\mathrm{scheme}\mathrm{has}\mathrm{dual}\mathrm{parametrization},\hfill \end{array}\right.$$

  and then

  $$\underset{x\in \mathbb{R}}{max}|\left(S_{\mathbf{a}}^{\infty }{f}^0\right)\left(x\right)-f\left(x\right)|\le K{h}^n,$$

  being K a constant, which depends on f, and it does not depend on h.

With this notation, the theorem that assures the absence of Gibbs phenomenon for subdivision schemes with positive masks is the following.

Theorem 1. 

Given $0\le \xi \le h$, let f be any function defined by

$$\begin{array}{ccc}& \forall x\le \xi ,& f\left(x\right)=f_{-}\left(x\right),f_{-}\in {\mathcal{C}}^n\left(\right]-\infty ,\xi \left]\right),\hfill \\ & \forall x\ge \xi ,& f\left(x\right)=f_{+}\left(x\right),f_{+}\in {\mathcal{C}}^n\left(\right[\xi ,+\infty \left[\right),\hfill \end{array}$$

with $n\ge 2$ and $f_{-}\left(\xi \right)>f_{+}\left(\xi \right)$. Let $S_{\mathbf{a}}$ be a univariate stationary subdivision scheme with:

$${\lambda }_l^{\left[k\right]}\left(i\right)=\left\{\begin{array}{cc}\sum _{\tau \le i}{a}_{2^k\tau +l}^{\left[k\right]},\hfill & i<0,\hfill \\ 0,\hfill & i=0,\hfill \\ \sum _{\tau \ge i}{a}_{2^k\tau +l}^{\left[k\right]},\hfill & i>0,\hfill \end{array}\right.$$

(11)

$0\le l<2^k$, being $a^{\left[k\right]}$ defined in Equation (8). Then, if ${\lambda }_l^{\left[k\right]}\left(i\right)\ge 0$, $\forall i,k$; and if h is sufficiently small (meaning an h for which the differences between adjacent data at smooth zones are significantly smaller than the differences close to discontinuities), we have:

1 . 

If $\left|x\right|\ge max\left\{\right|\frac{U-1}{2}|,|\frac{U+V}{2}+1\left|\right\}h$, then

$$|f\left(x\right)-\left(S_{\mathbf{a}}^{\infty }{f}_h\right)\left(x\right)|=O\left(h^n\right),$$

with $n\ge 2$.

2 . 

If $\left|x\right|\le max\left\{\right|\frac{U-1}{2}|,|\frac{U+V}{2}+1\left|\right\}h$, there exists ${\alpha }_h=O\left(h\right)$ such that

$$f_{1,h}-{\alpha }_h\le f_{+}\left(h\right)-{\alpha }_h\le \left(S_{\mathbf{a}}^{\infty }{f}_h\right)\left(x\right)\le f_{-}\left(0\right)+{\alpha }_h=f_{0,h}+{\alpha }_h.$$

(12)

The interested reader can refer to [19] for the proof of this theorem.

3. Convergence of the New Family of Subdivision Schemes

Let us recall the following definitions that will be useful for our purposes,

Definition 2. 

A binary subdivision scheme S is said to be convergent if

$$\forall f\in l^1\left(\mathbb{Z}\right),\exists S^{\infty }f\in L^1\left(\mathbb{R}\right)such that\underset{j\to +\infty }{lim}\left|\right|A{S}^{j}f-S^{\infty }f{\left|\right|}_{L^1\left(\mathbb{R}\right)}=0,$$

where $A{S}^j$ denotes the piecewise affine interpolation of $S^{j}f$ at the grid points.

Definition 3. 

A convergent subdivision scheme is stable if

$$\exists C<+\infty such that\forall f^0,g^0\in l^1\left(\mathbb{Z}\right)\left|\right|S^{\infty }f-S^{\infty }{g\left|\right|}_{L^1\left(\mathbb{R}\right)}\le C\left|\right|f^0-g^0{\left|\right|}_{l^1\left(\mathbb{Z}\right)}.$$

Let us first rewrite the scheme $S_{NL}$ (3) and (4) as a particular perturbation of the linear subdivision scheme $S_L$ (projection), which is defined as,

$${\left(S_L{f}^k\right)}_{2n}={\left(S_L{f}^k\right)}_{2n+1}=f_n^k.$$

(13)

We can see that the nonlinear scheme in (3) and (4) can be written in terms of the linear scheme $S_L$. For all $f\in l^1\left(\mathbb{Z}\right)$, we have

$$S_{NL}f=S_{L}f+F\left(df\right),$$

where ${\left(S_{L}f\right)}_{2n}={\left(S_{L}f\right)}_{2n+1}=f_n$, and F is the nonlinear operator

$$\begin{array}{c}\hfill F{\left(df\right)}_{2n}=\left\{\begin{array}{ccc}(c+a)d{f}_{n+1}-2aM(d{f}_n,d{f}_{n+1}),\hfill & \mathrm{if}& \hfill |d{f}_n|\ge |d{f}_{n+1}|,\\ -(c+a)d{f}_n+2cM(d{f}_n,d{f}_{n+1}),\hfill & \mathrm{if}& \hfill |d{f}_n|<|d{f}_{n+1}|.\end{array}\right.\end{array}$$

(14)

$$\begin{array}{c}\hfill F{\left(df\right)}_{2n+1}=\left\{\begin{array}{ccc}(c+a)d{f}_{n+1}-2cM(d{f}_n,d{f}_{n+1}),\hfill & \mathrm{if}& \hfill |d{f}_n|\ge |d{f}_{n+1}|,\\ -(c+a)d{f}_n+2aM(d{f}_n,d{f}_{n+1}),\hfill & \mathrm{if}& \hfill |d{f}_n|<|d{f}_{n+1}|.\end{array}\right.\end{array}$$

(15)

We also need to recall the following result established in [14]. Let $S_{NL}$ be a subdivision scheme defined by

$$\forall f\in l^1\left(\mathbb{Z}\right),\forall n\in \mathbb{Z}{\left(S_{NL}f\right)}_n={\left(Sf\right)}_n+F{\left(\delta f\right)}_n,$$

(16)

where F is a nonlinear operator defined on $l^1\left(\mathbb{Z}\right)$, $\delta$ is a linear and continuous operator on $l^1\left(\mathbb{Z}\right)$ and S is a linear subdivision scheme convergent in $l^1\left(\mathbb{Z}\right)$.

Theorem 2. 

If S is a linear convergent subdivision scheme and if $S_{NL}$, F and δ verify

$$\begin{array}{ccc}& & \exists K>0such that\forall d\in l^1{\left(\mathbb{Z}\right)\left|\right|F\left(d\right)\left|\right|}_{l^1\left(\mathbb{Z}\right)}\le {K\left|\right|d\left|\right|}_{l^1\left(\mathbb{Z}\right)},\hfill \end{array}$$

(17)

$$\begin{array}{ccc}& & \exists L>0,\exists c<1such that\forall f\in l^1\left(\mathbb{Z}\right)\left|\right|\delta S_{NL}^L{\left(f\right)\left|\right|}_{l^1\left(\mathbb{Z}\right)}\le {c\left|\right|\delta f\left|\right|}_{l^1\left(\mathbb{Z}\right)},\hfill \end{array}$$

(18)

then the subdivision scheme $S_{NL}$ defined by (16) is convergent.

Using Theorem 2, we can prove the following theorem:

Theorem 3. 

The nonlinear subdivision scheme $S_{NL}$ is $l^1\left(\mathbb{Z}\right)$ convergent for all $w\in [0,\frac{1}{2}]$.

Proof. 

The perturbation F defined in (14), (15) can be bounded using that $\left|M(x,y)\right|\le \left|\frac{x+y}{2}\right|$.

$$\begin{array}{cc}\hfill |F{\left(d{f}^k\right)}_{2n}|& \le (a+2c)|d{f}_n^k|+(2a+c)|d{f}_{n+1}^k|,\hfill \\ \hfill |F{\left(d{f}^k\right)}_{2n+1}|& \le (2a+c)|d{f}_n^k|+(a+2c)|d{f}_{n+1}^k|.\hfill \end{array}$$

(19)

Then

$$\begin{array}{cc}\hfill \parallel F\left(d{f}^k\right){\parallel }_{l^1\left(\mathbb{Z}\right)}& =h_k\sum _{n\in \mathbb{Z}}|F{\left(d{f}^k\right)}_n|\le 6(a+c){\parallel d{f}^k\parallel }_{l^1\left(\mathbb{Z}\right)},\hfill \end{array}$$

that satisfies precisely hypothesis (17).

Now, we consider hypothesis (18), which is related to the contraction of first order differences $\left(d{f}^k\right)$.

We have 16 different cases (see the Appendix A at the end of the paper).

In all of the cases, we obtain:

$$\begin{array}{c}\forall n\in \mathbb{Z},\forall k\in \mathbb{N},\exists C_n^k\in ]0,2[:\hfill \\ |dS{f}_{2n-2}^k|+|dS{f}_{2n-1}^k|+|dS{f}_{2n}^k|+|dS{f}_{2n+1}^k|+|dS{f}_{2n+2}^k|\le C_n^k\left|d{f}_n^k\right|+\mathrm{other}\mathrm{terms}.\hfill \end{array}$$

Thus, for all $f\in l^{\infty }\left(\mathbb{Z}\right)$:

$$\begin{array}{cc}\hfill \parallel d(S_{NL}(f^k{)\parallel }_{l^1}=h_{k+1}\sum _{n\in \mathbb{Z}}\left|d\right(S_{NL}\left(f^k{)}_n\right|& \le \frac{1+2a+2c}{2}{\parallel d{f}^k\parallel }_{l^1},\hfill \\ & \le \frac{2w^2-2w+1}{2}{\parallel d{f}^k\parallel }_{l^1}.\hfill \end{array}$$

If $w\in [0,\frac{1}{2}]$, then ${\displaystyle \frac{2w^2-2w+1}{2}}<1$.

Thus, hypothesis (18) is satisfied, and $S_{NL}$ is $l^1\left(\mathbb{Z}\right)$ convergent. □

Remark 1. 

Other terms in the last theorem means that they do not depend on $d{f}_n^k$.

In this paper, we prove the stability only at the regularity parts using the $l^{\infty }\left(\mathbb{Z}\right)$ norm; thus, we also have to prove the convergence at these parts using the $l^{\infty }\left(\mathbb{Z}\right)$ norm.

Theorem 4. 

($l^{\infty }\left(\mathbb{Z}\right)$ convergence)

If F and δ verify:

$$\begin{array}{ccc}& & \exists M>0such that\forall d\in l^{\infty }{\left(\mathbb{Z}\right)\parallel F\left(\delta \right)\parallel }_{l^{\infty }}\le M{\parallel \delta \parallel }_{l^{\infty }\left(\mathbb{Z}\right)}.\hfill \end{array}$$

(20)

$$\begin{array}{ccc}& & \forall f\in l^{\infty }\left(\mathbb{Z}\right)\parallel \delta S_{NL}{\left(f\right)\parallel }_{l^{\infty }\left(\mathbb{Z}\right)}\le \frac{1}{2}{\parallel \delta f\parallel }_{l^{\infty }\left(\mathbb{Z}\right)}+O\left(h\right).\hfill \end{array}$$

(21)

Then, the subdivision scheme $S_{NL}$ is uniformly convergent in $l^{\infty }\left(\mathbb{Z}\right)$.

Proof. 

Let $\phi$ be the limit function of the linear subdivision S. $\forall j>0,\forall f\in l^{\infty }\left(\mathbb{Z}\right)$, we consider the continuous function $f^j(.)$ as:

$$f^j\left(x\right)=\sum _n{f}_n^j\phi (2^{j}x-n)\forall x\in \mathbb{R}$$

where $f^j$ is the sequence $S_{NL}^{j}f$. We have

$$\begin{array}{cc}\hfill f^{j+1}\left(x\right)-f^j\left(x\right)=& \sum _n{f}_n^{j+1}\phi (2^{j+1}x-n)-\sum _n{f}_n^j\phi (2^{j}x-n),\hfill \end{array}$$

Using lemma 2.3 in [29], the definition of $S_{NL}$ and hypotheses (20) and (21), we find:

$$\begin{array}{cc}\hfill f^{j+1}\left(x\right)-f^j\left(x\right)=& \sum _n(f_n^{j+1}-{\left(S{f}^j\right)}_n)\phi (2^{j+1}x-n).\hfill \\ \hfill \parallel f^{j+1}-f^j{\parallel }_{L^{\infty }}\le & M\parallel \delta \left(S_{NL}{f}^{k-1}\right){\parallel }_{l^{\infty }},\hfill \\ \hfill \le & M[\frac{1}{2}\parallel \delta \left(f^{k-1}\right){\parallel }_{l^{\infty }},+o\left(h_{k-1}\right)],\hfill \\ \hfill \le & \frac{M}{2^k}{\parallel \delta \left(f^0\right)\parallel }_{l^{\infty }}+M\sum _{i=0}^{k-1}\frac{o\left(h_i\right)}{2^{k-1-i}}\hfill \\ \hfill \le & \frac{M}{2^k}{\parallel \delta \left(f^0\right)\parallel }_{l^{\infty }}+{\displaystyle \frac{M}{2^{k-1}}}\sum _{i=0}^{k-1}{2}^{i}o\left(h_i\right)\hfill \\ \hfill \le & \frac{M}{2^k}{\parallel \delta \left(f^0\right)\parallel }_{l^{\infty }}+M{\displaystyle \frac{k}{2^{k-1}}}o\left(h_0\right)\hfill \end{array}$$

Then, $\left(f^k(.)\right)$ is a Cauchy Sequence in $l^{\infty }\left(\mathbb{Z}\right)$ and, hence, converges uniformly to a continuous function. Using Proposition II.1 of [40], we conclude that the subdivision scheme $S_{NL}$ is $l^{\infty }\left(\mathbb{Z}\right)$ convergent. □

Theorem 5. 

Let $g\in C^1\left([a_i,b_i]\right)$, $\forall i$ (this function can be discontinuous at some points),

$f^0=g{\left(nh\right)}_{n\in \mathbb{Z}}$. Then, the nonlinear subdivision scheme $S_{NL}$ is $l^{\infty }\left(\mathbb{Z}\right)$ convergent in $[a_i,b_i]$, $\forall i$.

Proof. 

The inequalities (A1) imply that $\parallel F\left(d{f}^k\right){\parallel }_{l^{\infty }}\le (3a+3c){\parallel d{f}^k\parallel }_{l^{\infty }}$.

In the regularity zones, we have that $\frac{d{f}^k}{h_k}=f^{\prime }+o\left(h_k\right)$ and then $d{f}^{k+1}=\frac{d{f}^k}{2}+o\left(h\right)$. Thus, hypothesis (21) is satisfied, and $S_{NL}$ is $l^{\infty }$ convergent at the regularity zones. □

Theorem 6. 

Let g a continuous function and strictly monotonic or constant in $[a,b]\in \mathbb{R}$, $f^0=g{\left(nh\right)}_{n\in \mathbb{Z}}$. If the linear subdivision scheme $S_w$ is $C^1$ convergent in $[a,b]$, then:

$$\forall k>k_0,|{\left(f_w^k\right)}_n-{\left(f_{NL}^k\right)}_n|=o\left(h_k^2\right),\mathrm{in}[a,b].$$

Proof. 

Let $k_0$ the first integer when we define the nonlinear subdivision scheme as:

• If $|d{\left(f_w\right)}_n^{k_0}|\ge |d{\left(f_w\right)}_{n+1}^{k_0}|$,

  $$\begin{array}{ccc}\hfill {\left(f_{NL}\right)}_{2n}^{k_0+1}& ={\left(S_{NL}{f}^{k_0}\right)}_{2n}\hfill & \hfill ={\left(f_w\right)}_n^{k_0}+(c+a)d{\left(f_w\right)}_{n+1}^{k_0}-2aM(d{\left(f_w\right)}_n^{k_0},d{\left(f_w\right)}_{n+1}^{k_0}),\\ \hfill {\left(f_{NL}\right)}_{2n+1}^{k_0+1}& ={\left(S_{NL}{f}^{k_0}\right)}_{2n+1}\hfill & \hfill ={\left(f_w\right)}_n^{k_0}+(c+a)d{\left(f_w\right)}_{n+1}^{k_0}-2cM(d{\left(f_w\right)}_n^{k_0},d{\left(f_w\right)}_{n+1}^{k_0}).\end{array}$$
• If $|d{\left(f_w\right)}_n^{k_0}|<|d{\left(f_w\right)}_{n+1}^{k_0}|$,

  $$\begin{array}{ccc}\hfill {\left(f_{NL}\right)}_{2n}^{k_0+1}& ={\left(S_{NL}{f}^{k_0}\right)}_{2n}\hfill & \hfill ={\left(f_w\right)}_n^{k_0}-(c+a)d{\left(f_w\right)}_{n+1}^{k_0}+2cM(d{\left(f_w\right)}_n^{k_0},d{\left(f_w\right)}_{n+1}^{k_0}),\\ \hfill {\left(f_{NL}\right)}_{2n+1}^{k_0+1}& ={\left(S_{NL}{f}^{k_0}\right)}_{2n+1}\hfill & \hfill ={\left(f_w\right)}_n^{k_0}-(c+a)d{\left(f_w\right)}_{n+1}^{k_0}+2aM(d{\left(f_w\right)}_n^{k_0},d{\left(f_w\right)}_{n+1}^{k_0}).\end{array}$$

Using the properties $\left(M4\right)$ and $\left(M7\right)$, we find

• If $|d{\left(f_w\right)}_n^{k_0}|\ge |d{\left(f_w\right)}_{n+1}^{k_0}|$,

  $$\begin{array}{cc}\hfill |{\left(f_w^{k_0+1}\right)}_{2n}-{\left(f_{NL}^{k_0+1}\right)}_{2n}|& =2a|{\displaystyle \frac{d{\left(f_w\right)}_n^{k_0}+d{\left(f_w\right)}_{n+1}^{k_0}}{2}}-M(d{\left(f_w\right)}_n^{k_0},d{\left(f_w\right)}_{n+1}^{k_0})|=o\left(h^2\right),\hfill \\ \hfill |{\left(f_w^{k_0+1}\right)}_{2n+1}-{\left(f_{NL}^{k_0+1}\right)}_{2n+1}|& =2c|{\displaystyle \frac{d{\left(f_w\right)}_n^{k_0}+d{\left(f_w\right)}_{n+1}^{k_0}}{2}}-M(d{\left(f_w\right)}_n^{k_0},d{\left(f_w\right)}_{n+1}^{k_0})|=o\left(h^2\right).\hfill \end{array}$$
• If $|d{\left(f_w\right)}_n^{k_0}|<|d{\left(f_w\right)}_{n+1}^{k_0}|$,

  $$\begin{array}{cc}\hfill |{\left(f_w^{k_0+1}\right)}_{2n}-{\left(f_{NL}^{k_0+1}\right)}_{2n}|& =2c|{\displaystyle \frac{d{\left(f_w\right)}_n^{k_0}+d{\left(f_w\right)}_{n+1}^{k_0}}{2}}-M(d{\left(f_w\right)}_n^{k_0},d{\left(f_w\right)}_{n+1}^{k_0})|=o\left(h^2\right),\hfill \\ \hfill |{\left(f_w^{k_0+1}\right)}_{2n+1}-{\left(f_{NL}^{k_0+1}\right)}_{2n+1}|& =2a|{\displaystyle \frac{d{\left(f_w\right)}_n^{k_0}+d{\left(f_w\right)}_{n+1}^{k_0}}{2}}-M(d{\left(f_w\right)}_n^{k_0},d{\left(f_w\right)}_{n+1}^{k_0})|=o\left(h^2\right).\hfill \end{array}$$

Using an induction strategy (see Appendix A at the end of the paper), we arrive at

$$\forall k>k_0,|{\left(f_w^k\right)}_n-{\left(f_{NL}^k\right)}_n|=o\left(h_k^2\right),\mathrm{in}[a,b].$$

 □

Corollary 1. 

Under the same hypothesis of the above theorem, the nonlinear subdivision scheme $S_{NL}$ is convergent in $l^{\infty }\left([a,b]\right)$.

4. Stability of the New Family of Subdivision Schemes

Let us start by recalling the following definition,

Definition 4. 

A convergent nonlinear subdivision operator is called stable if there exists a constant C such that, for every pair of initial data f, $g\in l^{\infty }\left(\mathbb{Z}\right)$,

$$\parallel S^{\infty }f-S^{\infty }{g\parallel }_{L^{\infty }\left(\mathbb{R}\right)}\le C{\parallel f-g\parallel }_{l^{\infty }\left(\mathbb{Z}\right)}.$$

In order to prove the stability at the smooth zones, we present, only for theoretical reasons, the following subdivision scheme, which we denote by $\overline{S}$:

$$\overline{S}{f}_n^k=\left\{\begin{array}{cc}{f}_n^k,\hfill & |f_n^k-f_{n-1}^k|=O\left(1\right)\mathrm{or}|f_n^k-f_{n+1}^k|=O\left(1\right),\hfill \\ {\left(S_L{f}^k\right)}_n,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Due to the fact that the linear subdivision scheme $S_L$ is convergent with $C^2$ regularity, $\overline{S}$ is convergent; moreover, if the initial points come from the discretization of a piecewise continuous function, the limit function of the scheme $\overline{S}$ has $C^2$ regularity at the regularity zones.

Theorem 7. 

Let $g,\tilde{g}\in C^1\left([a_i,b_i]\right)$, $\forall i$ ( these functions can be discontinuous at some points), $h_0=x_{n+1}^0-x_n^0\forall n$. $f^0$ and ${\tilde{f}}^0$ are two sequences defined as $f^0=g{\left(nh\right)}_{n\in \mathbb{Z}},{\tilde{f}}^0=\tilde{g}{\left(nh\right)}_{n\in \mathbb{Z}}$. If $\exists C$ such that $h_0\le C{\parallel f^0-{\tilde{f}}^0\parallel }_{l^{\infty }\left(\mathbb{Z}\right)}$, then $S_{NL}$ is stable in $[a_i,b_i]$, $\forall i$.

Proof . 

We start by proving the following formula

$$\forall f^0\in l^{\infty }\left(\mathbb{Z}\right),h_0>0,{\parallel S_{NL}^{\infty }{f}^0-{\overline{S}}_{NL}^{\infty }{f}^0\parallel }_{l^{\infty }\left(\mathbb{Z}\right)}=O\left(h_0\right)$$

Using the definition of $S_{NL}$,${\overline{S}}_{NL}$ we find:

• In the fist iteration $\parallel f^1-{\overline{f}}^1{\parallel }_{l^{\infty }\left(\mathbb{Z}\right)}={\parallel S_{NL}{f}^0-{\overline{S}}_{NL}{f}^0\parallel }_{l^{\infty }\left(\mathbb{Z}\right)}=O\left(h_0\right)$.
• In the next iterations, we use the stability of the linear scheme:

  $$\begin{array}{cc}\hfill \parallel S_{NL}{f}^1-{\overline{S}}_{NL}{\overline{f}}^1{\parallel }_{l^{\infty }\left(\mathbb{Z}\right)}\le & \parallel S_{NL}{f}^1-{\overline{S}}_{NL}{f}^1{\parallel }_{l^{\infty }\left(\mathbb{Z}\right)}+{\parallel {\overline{S}}_{NL}{f}^1-{\overline{S}}_{NL}{\overline{f}}^1\parallel }_{l^{\infty }\left(\mathbb{Z}\right)}.\hfill \\ \hfill \le & O\left(h_1\right)+O\left(h_0\right)\hfill \\ \hfill ⋮\\ \hfill \parallel S_{NL}^{\infty }{f}^0-{\overline{S}}_{NL}^{\infty }{f}^0{\parallel }_{l^{\infty }\left(\mathbb{Z}\right)}\le & C\sum _i{h}_i=O\left(h_0\right)\hfill \end{array}$$
• Using the stability of the linear subdivision, we obtain the result:

  $$\begin{array}{ccc}\hfill \parallel S_{NL}^{\infty }\left(f^0\right)-S_{NL}^{\infty }\left({\tilde{f}}^0\right){\parallel }_{l^{\infty }\left(\mathbb{Z}\right)}& \le & \parallel S_{NL}^{\infty }\left(f^0\right)-{\overline{S}}_{NL}^{\infty }\left(f^0\right)\parallel +\parallel {\overline{S}}_{NL}^{\infty }\left(f^0\right)-{\overline{S}}_{NL}^{\infty }\left({\tilde{f}}^0\right)\parallel +\parallel {\overline{S}}_{NL}^{\infty }\left({\tilde{f}}^0\right)-S_{NL}^{\infty }\left({\tilde{f}}^0\right)\parallel \hfill \\ & \le & O\left(h_0\right)+C_1{\parallel f^0-{\tilde{f}}^0\parallel }_{l^{\infty }\left(\mathbb{Z}\right)}\hfill \\ & \le & (C+C_1){\parallel f^0-{\tilde{f}}^0\parallel }_{l^{\infty }\left(\mathbb{Z}\right)}.\hfill \end{array}$$

 □

5. Numerical Experiments

In Section 3, we studied that the contraction of the differences in the $l^1\left(\mathbb{Z}\right)$ norm implies convergence. If the contraction of the differences occurs in the $l^{\infty }\left(\mathbb{Z}\right)$ norm, the limit function is at least continuous. As it occurs in the $l^1\left(\mathbb{Z}\right)$ norm, the limit function can be discontinuous. In fact, the linear version of the scheme has $C^2$ limit functions. The nonlinear version of the scheme should keep this property at smooth zones. In this section, we investigate the validity of these assertions through numerical experiments. In all the experiments that we perform, we set $w=1/4$.

In order to obtain a nonlinear version of the scheme, we particularize the mean M used in the previous sections to the harmonic mean, defined as,

$$(x,y)\in {\mathbb{R}}^2\mapsto H(x,y):=\frac{xy}{x+y}(\mathrm{sgn}\left(xy\right)+1),$$

(22)

with $\mathrm{sgn}\left(x\right)=1$ if $x\ge 0$ and $\mathrm{sgn}\left(x\right)=-1$ if $x<0$.

The most important properties of this mean are included in the following proposition (see [18] for more details) and are related to the ones presented in Definition 1.

Proposition 2. 

For all $(x,y)\in {\mathbb{R}}^2$, the harmonic mean $H(x,y)$ satisfies

1 . 

$H(x,y)=H(y,x)$.

2 . 

$H(x,y)=0ifxy\le 0$.

3 . 

$H(-x,-y)=-H(x,y)$.

4 . 

$H(x,y)=\frac{sign\left(x\right)+sign\left(y\right)}{2}min\left(\right|x|,|y\left|\right)\left[1+\left|\frac{x-y}{x+y}\right|\right].$

5 . 

$\left|H\right(x,y\left)\right|\le max\left(\right|x|,|y\left|\right)$.

6 . 

$\left|H\right(x,y\left)\right|\le 2min\left(\right|x|,|y\left|\right)$.

7 . 

For $x,y>0$, $min(x,y)\le H(x,y)\le \frac{x+y}{2}$.

8 . 

If $x=O\left(1\right)$, $y=O\left(1\right)$, $|y-x|=O\left(h\right)$ and $xy>0$, then

$$\left|\frac{x+y}{2}-H(x,y)\right|=O\left(h^2\right).$$

9 . 

For all $x,x^{\prime },y,y^{\prime }\in {\mathbb{R}}_{+}$, there exist non-negative numbers α and β (depending on these points) such that

$$H(x,y)-H(x^{\prime },y^{\prime })=\alpha (x-x^{\prime })+\beta (y-y^{\prime })$$

and $\alpha +\beta \le \gamma$ with a constant $\gamma =2$ not depending on $x,x^{\prime },y,y^{\prime }.$

Remark 2. 

Property 2 of Proposition 2 implies that the use of the harmonic mean (22) produces order reduction when the arguments have different sign. This problem can be solved translating both arguments (such that both are positive or negative), obtaining the mean and then translating back the result by the same amount. A typical value for the translation is two times the minimum of the absolute value of the arguments, as proposed in [41]. This technique was used in all the experiments in order to avoid order reduction close to the discontinuities.

5.1. Numerical Regularity

Following [42], the regularity of a limit function can be evaluated numerically. We use the scheme $S_{NL}$ defined in (3–4) for M equal to the arithmetic mean or the harmonic mean (22) and $w=1/4$. Then, the subdivision schemes for the differences of order $k=1$ and $k=2$ associated with $S_{NL}$ (which can be derived due to the specific definition of $S_{NL}$), allow to estimate the regularity constant ${\beta }_k$ for $k=1,2$ using different number of subdivision scales j:

$${\beta }_k=-{log}_2\left(2^k\frac{\left|\right|{\left(S_k^{j+1}f\right)}_{n+1}-{\left(S_k^{j+1}f\right)}_n{\left|\right|}_{\infty }}{\left|\right|{\left(S_k^{j}f\right)}_{n+1}-{\left(S_k^{j}f\right)}_n{\left|\right|}_{\infty }}\right).$$

This expression provides an estimate for ${\beta }_1$ and ${\beta }_2$ such that the limit functions belong to $C^{1+{\beta }_1-}$ and $C^{2+{\beta }_2-}$. In

Table 1. Numerical estimation of the limit functions regularity $C^{1+{\beta }_1-}$ and $C^{2+{\beta }_2-}$ for $S_{NL}$ in (3–4) using the arithmetic mean and the harmonic mean. The number of subdivision scales is represented by j.

	j	5	6	7	8	9	10
${\beta }_1$	Arithmetic	9.997$\times {10}^{-01}$	1.000$\times {10}^{+00}$	1.000$\times {10}^{+00}$	1.000$\times {10}^{e+00}$	1.000$\times {10}^{+00}$	1.000$\times {10}^{+00}$
	Harmonic	7.648$\times {10}^{-01}$	9.450$\times {10}^{-01}$	9.773$\times {10}^{-01}$	9.987$\times {10}^{-01}$	9.999$\times {10}^{-01}$	1.000$\times {10}^{+00}$
${\beta }_2$	Arithmetic	4.150$\times {10}^{-01}$	4.150$\times {10}^{-01}$	4.150$\times {10}^{-01}$	4.150$\times {10}^{-01}$	4.150$\times {10}^{-01}$	4.150$\times {10}^{-01}$
	Harmonic	−2.430$\times {10}^{-01}$	−1.566$\times {10}^{-02}$	−3.817$\times {10}^{-03}$	−1.161$\times {10}^{-03}$	−6.646$\times {10}^{-05}$	−3.798$\times {10}^{-06}$

, we present some numerical estimations of the regularity constant for the nonlinear scheme defined in (3–4) with M equal to the harmonic mean and for the linear scheme defined in (2). The table was obtained using from $j=5$ to $j=10$ levels of subdivision, taking as initial data the function $f\left(x\right)=e^x$ in $[-1,1]$ discretized with 16 equidistant points.

From this table, we can see that the numerical estimation of the regularity constant for the nonlinear scheme is close to the one obtained for the linear scheme [31], i.e., $C^{2-}$, independently of the mean that we choose. Thus, the nonlinear perturbation introduced by the substitution of the arithmetic mean by the harmonic mean in (3–4), has a weak influence on the regularity at smooth zones.

5.2. Subdivision of Univariate Functions

In this subsection, we analyze the performance of the scheme $S_{NL}$ when subdividing univariate functions using the arithmetic mean and the harmonic mean. In all cases, we start from an initial data vector of length equal to 128, and we apply 10 scales of subdivision.

Example 1. 

Let us start by the function in (23),

$$g\left(x\right)=\left\{\begin{array}{c}-sin\left(5x\right),0\le x\le 0.5,\hfill \\ 1+{sin}^2\left(3x\right),0.5<x\le 1.\hfill \end{array}\right.$$

(23)

This is a piecewise continuous function with a jump discontinuity. Figure 1 top left presents the limit function obtained by the nonlinear version of the three points binary algorithm in (3-4). The hollow green circles represent the original sampled function, and the blue points are the function obtained using subdivision. Figure 1 at the top right presents a zoom around the discontinuity. At the bottom of the same figure, we can observe the results obtained by the linear algorithm in (2). None of the results show Gibbs oscillations; however, the nonlinear algorithm manages to remove the diffusion close to the discontinuity. The numerical effect that we can appreciate is not due to diffusion but to the reduction of the order of accuracy to $O\left(h\right)$ close to the discontinuity.

Example 2. 

Let us continue with the function in (24),

$$g\left(x\right)=\left\{\begin{array}{c}3x^6-x^4-2x^2-3x,0\le x\le 0.5,\hfill \\ -3x^6+x^4+x^2+3x,0.5<x\le 1,\hfill \end{array}\right.$$

(24)

which is a piecewise polynomial with a jump discontinuity. Figure 2 shows the results obtained for the nonlinear algorithm (top) and for the linear one (bottom). The conclusions are similar: none of the algorithms introduce Gibbs oscillations, and the nonlinear algorithm removes the diffusion close to the discontinuity.

Example 3. 

Let us finish with the piecewise constant function shown in (25),

$$g\left(x\right)=\left\{\begin{array}{c}2,0\le x\le 0.5,\hfill \\ 5,0.5<x\le 1.\hfill \end{array}\right.$$

(25)

In this experiment, our aim is to show that the new non linear scheme is capable of reproducing piecewise constant functions. In Figure 3, we show the result obtained for the nonlinear scheme (top) and for the linear one (bottom). It is clear that the new new nonlinear algorithm reproduces piecewise constant functions while the linear algorithm introduces diffusion.

5.3. Analysis of the Order of Approximation Close to the Discontinuities

To say that a scheme is convergent means that the numerical approximation of the data becomes closer to the true data as the grid is refined. In Section 3, we analyzed that choosing $w=1/4$ in (3–4) with M equal to the harmonic mean (22), the nonlinear scheme is convergent in the $l^1\left(\mathbb{Z}\right)$ norm. As mentioned before, this means that the limit function can be discontinuous. In this case, the mentioned convergence implies that the diffusion in the numerical results obtained by the nonlinear algorithm in the previous subsection results from approximation errors and is due to the fact that the nonlinear scheme loses its order of accuracy to $O\left(h\right)$ close to the discontinuities.

In this subsection, we investigate whether this hypothesis is true through a grid refinement analysis. If the errors that we observe close to the discontinuity are due to the diffusion introduced by the scheme, they should not disappear as the grid is refined. If we start from data obtained from the discretization of a piecewise continuous function, we know that the greatest errors will be located close to the discontinuity, and thus we can choose the $l^{\infty }\left(\mathbb{Z}\right)$ norm of the error to perform a grid refinement analysis.

In this way, we are sure that the conclusions that we reach about the order of accuracy of the scheme will be limited by the numerical artifacts that we observe close to the discontinuity.

Table 2. Grid refinement analysis in the $L^{\infty }$ norm for the numerical estimation of the order of approximation using j scales of subdivision. By i, we represent the number of initial control points that we use. We used the nonlinear scheme (with M equal to the harmonic mean in (3–4)) and the function in (24). For this piecewise continuous function, the greatest error is located at the discontinuity. Thus, the numerical errors presented in the table correspond to the errors at the discontinuity. We can see that they reduce as we refine the grid, reaching $O\left(h\right)$ accuracy.

	j	1		2		3		4		5		6
i		${\mathit{e}}_{\mathit{i}}$	${log}_{\mathbf{2}}\left(\frac{{\mathit{e}}_{\mathit{i}}}{{\mathit{e}}_{\mathit{i}+\mathbf{1}}}\right)$	${\mathit{e}}_{\mathit{i}}$	${log}_{\mathbf{2}}\left(\frac{{\mathit{e}}_{\mathit{i}}}{{\mathit{e}}_{\mathit{i}+\mathbf{1}}}\right)$	${\mathit{e}}_{\mathit{i}}$	${log}_{\mathbf{2}}\left(\frac{{\mathit{e}}_{\mathit{i}}}{{\mathit{e}}_{\mathit{i}+\mathbf{1}}}\right)$	${\mathit{e}}_{\mathit{i}}$	${log}_{\mathbf{2}}\left(\frac{{\mathit{e}}_{\mathit{i}}}{{\mathit{e}}_{\mathit{i}+\mathbf{1}}}\right)$	${\mathit{e}}_{\mathit{i}}$	${log}_{\mathbf{2}}\left(\frac{{\mathit{e}}_{\mathit{i}}}{{\mathit{e}}_{\mathit{i}+\mathbf{1}}}\right)$	${\mathit{e}}_{\mathit{i}}$	${log}_{\mathbf{2}}\left(\frac{{\mathit{e}}_{\mathit{i}}}{{\mathit{e}}_{\mathit{i}+\mathit{1}}}\right)$
64		0.21777	-	0.34569	-	0.4637	-	0.57448	-	0.6794	-	0.77926	-
128		0.1087	1.0025	0.17446	0.98656	0.23653	0.97119	0.29614	0.95596	0.35393	0.9408	0.41022	0.9257
256		0.054299	1.0014	0.087619	0.99359	0.11941	0.98607	0.15028	0.97859	0.18054	0.97112	0.21034	0.96366
512		0.027136	1.0007	0.043905	0.99687	0.05999	0.99315	0.075694	0.98945	0.091168	0.98574	0.10649	0.98204
1024		0.013565	1.0004	0.021976	0.99845	0.030066	0.9966	0.037985	0.99476	0.045808	0.99292	0.053575	0.99107

shows a grid refinement analysis performed for the piecewise continuous function in (24) using M equal to the harmonic mean in (3–4). Fixing the number of scales of subdivision j, we refine the grid spacing and check the $l^{\infty }\left(\mathbb{Z}\right)$ norm of the error. We define the order of accuracy of the reconstruction as,

$$order=lo{g}_2\left(\frac{e_i}{e_{i+1}}\right),$$

with $e_i$ the error obtained with a grid spacing h and $e_{i+1}$ the error obtained with a grid spacing $h/2$. We use the discrete $l_{\infty }$ norm,

$$\begin{array}{ccc}\hfill \left|\right|f^j{\left|\right|}_{\infty }& =& \underset{n\in \mathbb{Z}}{max}\left\{\right|f_n^j\left|\right\}.\hfill \end{array}$$

In Table 2 and

Table 3. Grid refinement analysis in the $L^{\infty }$ norm for the numerical estimation of the order of approximation using j scales of subdivision. By i, we represent the number of initial control points that we use. We used the linear scheme (with M equal to the arithmetic mean in (3–4)) and the function in (24). For this piecewise continuous function, the greatest error is located at the discontinuity. Thus, the numerical errors presented in the table correspond to the errors at the discontinuity. We can see that they do not reduce as we refine the grid, reaching $O\left(1\right)$ accuracy.

	j	1		2		3		4		5		6
i		${\mathbf{e}}_{\mathbf{i}}$	${log}_{\mathbf{2}}\left(\frac{{\mathit{e}}_{\mathit{i}}}{{\mathit{e}}_{\mathit{i}+\mathbf{1}}}\right)$	${\mathit{e}}_{\mathit{i}}$	${log}_{\mathbf{2}}\left(\frac{{\mathit{e}}_{\mathit{i}}}{{\mathit{e}}_{\mathit{i}+\mathbf{1}}}\right)$	${\mathit{e}}_{\mathit{i}}$	${log}_{\mathbf{2}}\left(\frac{{\mathit{e}}_{\mathit{i}}}{{\mathit{e}}_{\mathit{i}+\mathbf{1}}}\right)$	${\mathit{e}}_{\mathit{i}}$	${log}_{\mathbf{2}}\left(\frac{{\mathit{e}}_{\mathit{i}}}{{\mathit{e}}_{\mathit{i}+\mathbf{1}}}\right)$	${\mathit{e}}_{\mathit{i}}$	${log}_{\mathbf{2}}\left(\frac{{\mathit{e}}_{\mathit{i}}}{{\mathit{e}}_{\mathit{i}+\mathbf{1}}}\right)$	${\mathit{e}}_{\mathit{i}}$	${log}_{\mathbf{2}}\left(\frac{{\mathit{e}}_{\mathit{i}}}{{\mathit{e}}_{\mathit{i}+\mathbf{1}}}\right)$
64		1.0834	-	1.5284	-	1.7328	-	1.8316	-	1.8803	-	1.9045	-
128		1.0733	0.013469	1.5137	0.013987	1.7158	0.014222	1.8135	0.014329	1.8616	0.014379	1.8856	0.014403
256		1.0684	0.0066745	1.5064	0.0069506	1.7074	0.0070774	1.8045	0.0071358	1.8524	0.0071632	1.8763	0.0071764
512		1.0659	0.0033221	1.5028	0.0034644	1.7032	0.0035301	1.8001	0.0035606	1.8478	0.0035749	1.8716	0.0035818
1024		1.0647	0.0016572	1.501	0.0017295	1.7011	0.0017629	1.7978	0.0017784	1.8456	0.0017858	1.8693	0.0017893

, we represent, by i, the number of initial control points that we use and, by j, the number of subdivision scales.

The results presented in Table 2 show that the nonlinear scheme works like a zoom of first order near the discontinuities and preserving jumps. The table shows that the artifacts that we can see close to the discontinuity in the results obtained by the nonlinear scheme are not due to diffusion. Instead, they are approximation errors of first order. Table 3 shows the results obtained by the linear algorithm, i.e., with M equal to the arithmetic mean in (3–4). We can see how the errors do not reduce as the grid is refined. In this case, the linear algorithm is affected by diffusion. At the moment, we are working in high order schemes in order to increase the accuracy of the nonlinear schemes presented in this work.

5.4. 2D Experiments: Subdivision of Bivariate Functions

Let us consider the next bivariate function,

$$f\left(x\right)=\left\{\begin{array}{cc}cos\left(\pi x\right)cos\left(\pi y\right),\hfill & \mathrm{if}{(x+\frac{1}{2})}^2+{(y-\frac{1}{2})}^2<1\hfill \\ 1-cos\left(\pi x\right)sin\left(\pi y\right),\hfill & \mathrm{if}{(x+\frac{1}{2})}^2+{(y-\frac{1}{2})}^2\ge 1.\hfill \end{array}\right.$$

(26)

This is represented in Figure 4. The subdivision of this data is represented in Figure 5.

6. Conclusions

In this paper, we analyzed the behavior of a new family of stationary nonlinear, noninterpolatory subdivision schemes in the presence of strongly varying data and, in particular, the possible apparition of diffusion. A way to construct subdivision schemes with improved diffusion close to jump discontinuities was introduced. The convergence and stability of the new family was also analyzed. The components of the family do not present Gibbs oscillations as the masks are considered positive.

The numerical experiments presented agree with the theoretical results obtained. As far as we know, this is the first time that a binary, stable and convergent subdivision scheme without Gibbs oscillations close to jump discontinuities and reproducing piecewise constant functions has appeared in the literature.