Master’s ThesisStability of nonlinear subdivision schemes and

multiresolutions

Stanislav Harizanov,Jacobs University, Bremen

December, 2007

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Contents

1 Introduction 3

2 Notation and basic definitions 6

2.1 Subdivision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Multiresolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 MR of sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.2 MR of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Survey of the existing literature 12

4 Linear subdivision 17

4.1 Basic representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.2.1 The univariate case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.2.3 The multivariate case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5 Univariate nonlinear subdivision. Examples 29

5.1 Cubic ENO and WENO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.2 PPH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.3 Quadratic, dyadic median-interpolating subdivision scheme . . . . . . . . . . . . . . 34

6 Stability analysis on a univariate nonlinear subdivision 37

6.1 Analysis on conditions (6.6) and (6.7) . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6.2 The Differential Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

7 Applications of Theorem 6.2 49

7.1 PPH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

7.2 WENO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

7.3 Quadratic, dyadic median-interpolating subdivision scheme . . . . . . . . . . . . . . 53

8 Conclusions 58

9 Appendix 63

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1 Introduction

Subdivision is a process of recursively refining discrete data using a set of subdivision rules (calledsubdivision scheme) which generates a continuous or even smooth limit. It has numerous applica-tions, such as image reconstruction, the design of curves and surfaces, shape preservation in dataand geometric objects, the approximation of arbitrary functions, etc. On the other hand, subdivi-sion lies in the core of multiresolution analysis (MRA) and wavelet transforms, and thus plays acentral role in data compression, noise removal, and so on. The great variety of applications as wellas the necessity of improving the performance of the existent algorithms lead to the invention of agreat variety of subdivision schemes. The main mathematical issues to be investigated for a subdi-vision scheme are convergence as the number of subdivision steps goes to infinity, the smoothnessof the limit objects, and the stability of the subdivision process and the associated multiresolutionrepresentations. In the linear, stationary case, due to standard techniques [CDM91], [Dyn92], thereis a complete theory about convergence and stability of both subdivision and multiresolution, aswell as about determining the smoothness of the first. In the nonlinear case, however, there are stillvery few results about stability and smoothness. This is due to the fact that the theorems fromthe linear setup can not be directly extended to the nonlinear one. For example, there are weaklynonlinear subdivision schemes such as the essentially non-oscillatory (ENO) scheme [CDM03] whichare uniformly convergent but not stable. Moreover, in linear subdivision, stability of the schemeguarantees stability of the corresponding multiresolution. However, in this paper we will show thatcertain convergent schemes for which subdivision is stable (e.g., the dyadic median-interpolatingscheme) do not possess a stable associated pyramid transform. In summary, analyzing subdivisionand the associated multiresolution is a very active research field with many open problems, that isdriven by both theory and applications.

Let us briefly introduce some notions and fix the notation. For details see Section 2. In everysubdivision scheme, a family of “nested” grids Γj, j ≥ 0 is given (i.e., Γi ⊂ Γi+1,∀i), the initialdata v0 is a uniformly bounded sequence on the coarsest grid (v0 ∈ l∞(Γ0)), and the subdivisionoperators S[j] : l∞(Γj)→ l∞(Γj+1) map sequences corresponding to two consecutive grids into eachother. The grids may be finite or infinite, and data can be arbitrary. We will denote by vj thedata sequence on Γj, generated by v0 and j subdivision steps. The subdivision process is local,respectively regular, respectively univariate if the operator S[j] is built from local rules, respectivelythe grids Γi∞i=0 are uniform, respectively univariate. It is stationary, if S[j] ≡ S, ∀j ∈ N, and linearif all operators S[j] are linear. We are interested only in stationary subdivision, and all the schemeswe consider are of that type. Moreover, if the subdivision operator is in addition shift-invariant theanalysis can be performed on l∞(Z). The locality of S assures that changing the initial data at oneplace will have a local effect on the limit function, i.e., the new limit will differ from the originalone only in a neighborhood of the initial point. In that case we can further restrict our analysis ofthe scheme to a finite-dimensional subspace of l∞(Z).

Multiresolution, as defined by Harten in [Har96], is a pyramid transform that, starting from fine-scale data, constructs data on coarser scales by restriction and so called “details” (analysis step)and conversely reconstructs the original data from their coarse-scale part and the details (synthesisstep). More precisely, for a given J ∈ N we define a set of grids ΓjJ0 , a set of restriction operatorsR[j] : l∞(Γj+1) → l∞(Γj), j = 0, . . . , J − 1, as well as subdivision operators S[j]J−1

0 such thatR[j]S[j] = Ij,∀j = 0, . . . , J − 1 where Ij is the identity operator on l∞(Γj). Thus, one can uniquely

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decompose every sequence vJ ∈ l∞(ΓJ) into its coarse-scale part v0 = R[0]R[1] · · ·R[J−1]vJ and asequence of details dj ∈ l∞(Γj), j = 1, . . . , J , such that there is a correspondence

vJ ←→ v0, d1, . . . , dJ.

Again, we are interested only in stationary shift-invariant multiresolution analysis, i.e., the sub-division and the restriction operators are not level-dependent and can be modeled as operatorson l∞(Z), and we denote them by S and R respectively. In most of the applications the detailsequences dj are linked with the sequence of coarse-scale representations vj via the formulasvj = Svj−1 + dj and vj = RJ−jvJ , j ≤ J . There are more general ways of defining these sequences,see for instance [DRS04] or [URDS+05], but we will not go into these generalizations for the momentbeing. We will concentrate at the synthesis step of the multiresolution analysis, and will refer toit as the multiresolution scheme associated to S. We introduce the coarse-to-fine grid operator Mby the formula vj = Mvj−1 := Svj−1 + dj. Note that, although M depends on two sequences vj−1

and dj, we have chosen not to indicate the argument dj to keep the exposition as easy to follow aspossible. M is linear if S is linear.

We define Lipschitz stability of a subdivision scheme, given by S, and its corresponding mul-tiresolution scheme, given by M as follows: There exists a constant C such that

‖SJv0 − SJ v0‖ ≤ C‖v0 − v0‖, (1.1)

‖MJv0 −MJ v0‖ ≤ C(‖v0 − v0‖+

J∑j=1

‖dj − dj‖), (1.2)

for all J ∈ N and all sequences involved.

The following example helps to better understand the above definitions. Consider the subdivisionscheme Sc based on central Lagrange cubic interpolation. More precisely, let Γj = 2−jZ and defineScv

j as follows: The values at the “old” grid points 2−jk belonging to Γj+1 ∩ Γj are inheritedfrom the corresponding entries of vj (i.e., Sc is interpolatory) while the value at “new” grid points2−(j+1)(2k+1) is given by pk(2

−(j+1)(2k+1)), where pk is a cubic polynomial that interpolates vj atthe four points 2−jl, l = k− 1, . . . , k+2 neighboring it (fig. 1a). Since 2−1Z ∼= Z and using dilation,we have defined an operator Sc : l∞(Z)→ l∞(Z). Direct computations lead to the following explicitformula

vj+12k = (Scv

j)2k = vjk

vj+12k+1 = (Scv

j)2k+1 = −1

16vjk−1 +

9

16vjk +

9

16vjk+1 −

1

16vjk+2,

(1.3)

where vjk is the value of vj at 2−jk for every j, k. Sc is linear and local, since the entries of vj+1

are finite linear combinations of those of vj; univariate and regular, because the grids Γj areequidistant subsets of R; stationary and shift-invariant, as the coefficients in (1.3) do not dependon the subdivision level j and the position k; and dyadic, because of dilation factor 2. Locality andshift-invariance enable us to represent the subdivision operator Sc by a single finitely supportedsequence a, called mask, such that

(Scv)k =∑l

ak−2lvl, k ∈ Z. (1.4)

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(a)

vjk−1

vjk

vj+12k+1

vjk+1

vjk+2

−5 −4 −3 −2 −1 0 1 2 3 4 5

(b)

Figure 1: (a) The action of Sc, (b) The range of the dependence of v00 after two subdivision steps.

The only nonzero entries of a are a−3 = a3 = − 116; a−1 = a1 = 9

16; a0 = 1. Fig. 1b shows that

changing the data at v00 affects the limit function only in the interval [−3, 3]. On the other hand,analyzing the behavior of the limit in [0, 1] characterizes it globally on R and thus, instead of l∞(Z)one can work with sequences on −2,−1, 0, 1, 2, 3.

There are many cases in which linear multiresolution gives unsatisfactory results, and nonlin-ear alternatives are necessary. Tentatively, we can group the so-far proposed nonlinear subdivisionschemes and MRAs into four: schemes that are shape preserving or capture singularities, normalmultiresolution, statistical and morphological pyramids, and manifold subdivision. The first groupcontains of schemes that deal with Gibbs-type phenomena that are typical for linear schemes nearjump singularities in the data [HEOC87], [LOC94], [SM04], or address shape preservation (mono-tonicity, convexity, etc.) without loss of smoothness in the limit [KvD99], [KvD97]. Normal meshesand multiresolution [GVSS00] deal with the efficient encoding of curves and surfaces, and nonlinear-ity in these schemes appears in the way details are computed, because they are of different type thanthe initial data. Examples from the third group have been proposed in connection with removingheavily-tailed (Cauchy) noise, nonlinearity here results mainly from the use of nonlinear robust es-timators, such as the median [DY00]. The nonlinearity in the last group comes from the restrictionof the control points to a manifold, surface, or a Lie group, this is a nonlinearly constrained set inthe ambient space [URDS+05], [WD05]. Although most of the above references provide analysisonly in the univariate case, most of the schemes have natural multivariate analogues and have beenintroduced with view towards multivariate applications.

Our master thesis develops a general theory for stability of univariate nonlinear subdivision andmultiresolution schemes. The paper is organized as follows: Section 2 sets the notation and containsall the definitions that we will use later on. Section 3 reviews the existing results, and providesreferences to the work of other authors in the field of nonlinear subdivision and multiresolution.Section 4 covers the theory in the linear setting and closely follows [Dyn92]. In Section 5 we introducein details the nonlinear univariate subdivision schemes, which we will analyze later. Section 6 is ourmain contribution. It contains our stability theorem, its joint spectral radius version, and analyzes

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the conditions in the theorem. The applications of this theory are in Section 7. Section 8 is forconclusions.

2 Notation and basic definitions

2.1 Subdivision

In this paper we deal only with stationary subdivision schemes, and thus, we will not write the word“stationary” anymore. Note that in the introduction we discussed that in order the subdivisionscheme S to be stationary, the subdivision process should be regular. Therefore we consider onlyregular subdivision as well. Moreover, we always use the space of bounded sequences l∞ in thedefinition of subdivision.

Let us give some basic definitions.

Definition 2.1 A subdivision scheme S is finite if there exists an integer B ≥ 0, such that forevery α ∈ Zs, (v1)α is a function of at most B elements of v0.

Definition 2.2 A subdivision scheme S reproduces constants if S1 = 1, where 1 ∈ l∞(Zs) isthe constant sequence 1.

Definition 2.3 A subdivision scheme S is affine invariant if S(av + b1) = aSv + b1 for everya, b ∈ R and v ∈ l∞(Zs).

In most of the papers concerning subdivision, the entries of the sequence v0 are interpreted as dataover the grid Zs. Intuitively, we think of (v0)α as a point in Rs+1 with coordinates (α, (v0)α), whereα ∈ Zs is a multi-index.

Definition 2.4 Let S : l∞(Z) → l∞(Z) be a subdivision rule. The associate subdivision scheme Sis shift-invariant if there exists an s× s integer matrix D with limn→∞D−n = 0, such that

S Tα = TDα S, α ∈ Zs, (2.1)

where (Tαv)β = vα+β is the translation operator.D is called the dilation matrix, or simply dilation factor, when s = 1, of S.

We will work only with diagonal dilation matrices D = diagr1, r2, . . . , rs, where ri ≥ 2, i =1, . . . , s, and we will say that “S has dilation factor r = (r1, . . . , rs) ∈ Zs” instead of “S has dilationmatrix D”. Note that, due to Tα+β = Tα Tβ , it suffices to check (2.1) only for a basis of Zs.Subdivision schemes with dilation factor 2 are called dyadic, and those with dilation factor 3 -triadic.

Due to avoiding technicalities, we will set s = 1 in the following definition. Its generalization inhigher dimensions is obvious and is left to the reader.

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Definition 2.5 Let S : l∞(Z) → l∞(Z) be a subdivision rule. The associate subdivision scheme Sis local, if it is finite, shift-invariant with dilation factor r, and for every i ∈ Z, (v1)i depends onlyon (v0)[i/r]+j, j ∈ J , where J = [a, b] : a, b ∈ Z, b− a ≤ 2B − 1 contains 0.

In this paper we consider only local subdivision schemes on Z. All the norms that we useare infinity-norms and hence we will simply denote them by ‖ · ‖ . Whether the norm is on anoperator, on a sequence, or on a function will be clear from the context. We denote by pj ∈ l∞(Z)the sequence that corresponds to the values of the polynomial p(x) on the grid Γj, for example ifΓj = r−jZ, then pjk = p(r−jk) ∀k ∈ Z. We denote by Πn(R) the space of all polynomials with realcoefficients of degree less or equal to n. We denote, by 4n the n-th order forward finite differenceoperator, i.e. for v ∈ l∞(Z)

(4nv)k =n∑

m=0

(−1)n−m

(n

m

)vk+m.

Now we are ready to define the basic notions of convergence and stability of a subdivision scheme.We will use the definitions in [Dyn92], and [CDM03].

Definition 2.6 Let S be regular, shift-invariant subdivision scheme, with dilation factor r ∈ Zs.For every v0 ∈ l∞(Zs) and every j ∈ Z+ ∪ 0 define f j ∈ C(Rs) as the piecewise linear functionthat interpolates the data vj = Sjv0 over Γj, and gj ∈ L∞(Rs) as the piecewise constant functionwith gj(x) = (vj)α, x ∈

∏si=1 r

−ji [αi, αi + 1].

Definition 2.7 Let S : l∞(Zs)→ l∞(Zs) be a regular, shift-invariant subdivision rule with dilationfactor r ∈ Zs.

(i) S is convergent if for every sequence v0 ∈ l∞(Zs), there exists a continuous functionf ∈ C(Rs) such that

limj<J→∞

|(SJv0)rJ−jα − f(r−jα)| = 0, α ∈ Zs, j ∈ Z+,

and such that the operator

S∞ : l∞(Zs)→ C(Rs), S∞v0 = f,

is not trivially zero.

(ii) S is uniformly convergent if for every sequence v0 ∈ l∞(Zs)

limJ→∞

‖SJv0 − f( ·rJ

)‖ = 0, (2.2)

where f(r−J ·) denotes the sequence f(α/rJ) : α ∈ Zs, and S∞ is a non-trivial map.

Here, by αβ for α, β ∈ Zs we understand γ ∈ Zs, such that γi = αiβi, i = 1 . . . s.

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Thus, a subdivision scheme is convergent iff gj∞j=0 converges pointwise in L∞(R) to a continuousfunction f , and uniformly convergent iff both f j∞j=0 and gj∞j=0 converge uniformly in L∞(R) tof (due to uniform convergence, the limit is automatically continuous). There are, of course, otherhj(x)∞j=0 based on vk that converge uniformly to f , but we will not use them in this paper. Acomplete classification of such functional sequences is done in Lemma 2.3. in [Dyn92]. f is calledlimit function, and in the special cases s = 1 (s = 2) - limit curve (surface). The definition foruniform convergence can be formulated in an epsilon-delta way:

Definition 2.8 S is uniformly convergent, if for any initial sequence v0 ∈ l∞(Zs), any boundeddomain Ω ⊂ Rs and ε > 0 there exists J(v0, ε,Ω) such that

|(Sjv0)α − S∞v0(r−jα)| < ε, j > J(v0, ε,Ω), α ∈ Zs ∩ rjΩ. (2.3)

For shift-invariant subdivision schemes S, it is enough to take Ω = [0, 1]s, since every boundeddomain in Rs can be covered by a finite union of shifts Ω + βi, βi ∈ Zs, i = 1, . . . , N of Ω. Theshift-invariance guarantees

J(v0, ε,Ω + βi1) = J(v0, ε,Ω + βi2) ∀i1, i2 ∈ 1, . . . , N.

Moreover, if S is local then we can take a finite subsequence of v0 and work only with it. Note,that in this case for every j, vj will be a finite sequence, but its length increases monotonically withrespect to j.

Definition 2.9 A subdivision scheme S is called (Lipschitz) stable if, there exists a positive con-stant C, such that for any v0, v0 ∈ l∞(Zs)

‖S∞v0 − S∞v0‖ ≤ C‖v0 − v0‖. (2.4)

The equivalent epsilon-delta definition for stability is:

Definition 2.10 S is stable, if there exists a positive, finite constant C such that for any ε > 0,any initial sequence v0 ∈ l∞(Zs), and any bounded domain Ω ⊂ Rs there exists J(v0, ε,Ω) with theproperty: for any v0 ∈ B(v0, ε/C) = v ∈ l∞(Zs) : ‖v − v0‖ < ε/C

|(Sjv0)α − (Sj v0)α| < ε j > J(v0, ε,Ω), α ∈ Zs ∩ rjΩ. (2.5)

Note that this is a very strong version of stability. Usually the notion of stability is local, i.e. canbe formulated as “small perturbation δ of the initial data leads to a small perturbation ε of thederived data”. The above definitions are more restrictive, since they are for arbitrary ε, not onlyfor a sufficiently small one and the ratio δ/ε is uniformly bounded by C. Thus, there are authorswho prefer weakened versions (see [CDM03]). In this paper, we will use (2.4), unless something elseis specified.

In theory, there are two main types of stability - asymptotic and Liapunov (see [Wig03]). If wethink of (v0, 0) and (v0, 0) as points in the space l∞(Zs) × t ≥ 0 then (vj, j) and (vj, j) arediscrete trajectories, and the corresponding definitions are:

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Definition 2.11 (Liapunov Stability.) Let S : l∞(Zs) → l∞(Zs) be a subdivision scheme. Letv := v0, v1, . . . be the discrete trajectory with respect to S, associated to the initial data v0 ∈l∞(Zs) (i.e., vj+1 = Svj, ∀j ∈ N). Then v is Liapunov stable, if for any ε > 0, there exists aδ = δ(ε) > 0 such that for any v = v0, v1, . . . satisfying ‖v0 − v0‖ < δ, max

J∈N‖vJ − vJ‖ < ε.

S is Liapunov stable itself, if all the discrete trajectories with respect to S are Liapunov stable.

Definition 2.12 (Asymptotic Stability.) v is said to be asymptotically stable if it is Lia-punov stable and for any other trajectory, v with respect to S, there exists a constant b > 0 suchthat, if ‖v0 − v0‖ < b, then lim

J→∞‖vJ − vJ‖ = 0.

Note that, since we are dealing with stationary subdivision schemes, our dynamical system isautonomous! A subdivision scheme S which reproduces constants, cannot be asymptotically stable,because if we take v0 = 1 and v0 = 0, then for every j ≥ 0 the distance between (vj, j) and(vj, j) is 1. (2.5) looks like Liapunov stability but the main drawback of this interpretation isthat, unlike the Liapunov stability, the trajectories of “close” initial points stay in a tube aroundthe initial trajectory only after some finite moment, that depends on the initial sequence and isnot uniformly bounded. Again, as with the uniform convergence, if S is shift-invariant it suffices totake Ω = [0, 1]s, and if S is local - we can work with finite subsequences of v0 and v0.

2.2 Multiresolution

The general framework for multiresolution representation (MR) of data was presented by Hartenin [Har96]. For the purposes of this paper, we will restrict ourselves to the one-dimensional dyadicmultiresolution, with Sj = V j = l∞(2−jZ) =: l∞(Γj), and F = BC(R) - the space of bounded andcontinuous real-valued functions on R.

2.2.1 MR of sequences

Let the linear operator Rj−1j and the subdivision rule Sj

j−1 satisfy

Rj−1j : l∞(Γj)→ l∞(Γj−1), (2.6a)

Sjj−1 : l∞(Γj−1)→ l∞(Γj), (2.6b)

Rj−1j Sj

j−1 = Ij−1, (2.6c)

where Ij−1 is the identity operator in l∞(Γj−1). Rj−1j is known as the restriction operator, and Sj

j−1

as the prediction operator. Since, we are dealing only with stationary subdivision, we need Sjj−1,

and thus Rj−1j (see (2.6c)) not to depend on the level j, i.e., for every j ∈ N

Sjj−1 = S, Rj−1

j = R. (2.7)

We will keep the lower and upper indexes of the two operators just to indicate on which resolutionlevel we are.

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Let vJ ∈ l∞(ΓJ). Then, using the restriction operator, we generate the sequence vjJj=0 :

vj = Rjj+1R

j+1j+2 . . . R

J−1J vJ ∈ l∞(Γj). (2.8)

For each 1 ≤ j ≤ J we denote by dj the detail

dj := vj − Sjj−1v

j−1. (2.9)

Note that although dj is a sequence on the j-th level of multiresolution, in the interpolatory dyadiccase it is more natural to relate it to the (j − 1)-st one, since (dj)2k = 0, ∀k ∈ Z, and the only“meaningful” entries are the odd ones.

Therefore we can represent any sequence vJ in a unique way as a sequence on the zero level ofresolution and a sequence of details on all the intermediate resolution levels.

vJ ←→M(vJ) := v0, d1, . . . , dJ.

We refer toM(vJ) as the MR of vJ . We will assign an operatorM to every multiresolution, namely

Mvj := vj+1 = Svj + dj+1. (2.10)

Unlike subdivision, in multiresolution we have two operators and two processes involved: thedirect MR transform (vJ → M(vJ)), and the inverse MR transform (M(vJ) → vJ). Therefore,in order to talk about stability of a multiresolution, we have to prove stability for both of thetransforms.

Definition 2.13 (stability of the direct MR) Let vJ , vJ ∈ l∞(ΓJ), and let M(vJ) and M(vJ)be the corresponding MRs. We say that the direct MR transform is stable if there exists C suchthat

‖v0 − v0‖ ≤ C‖vJ − vJ‖, ‖dj − dj‖ ≤ C‖vJ − vJ‖, ∀j = 1, . . . , J. (2.11)

Definition 2.14 (stability of the inverse MR) Let vJ , vJ ∈ l∞(ΓJ), and let M(vJ) and M(vJ)be the corresponding MRs. We say that the inverse MR transform is stable if there exists C suchthat

‖vJ − vJ‖ ≤ C(‖v0 − v0‖+

J∑j=1

‖dj − dj‖). (2.12)

2.2.2 MR of functions

Analogously, to the MR of discrete data, one can define a multiresolution representation of contin-uous data. Therefore, for any j ∈ N Harten introduced the linear operator Rj : BC(R) → l∞(Γj)and the operator S∞

j : l∞(Γj)→ BC(R) such that

RjS∞j = Ij. (2.13)

Rj is called a discretization operator, and S∞j a reconstruction operator.

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A classical example about the relation between Section 2.2.1 and Section 2.2.2 is the use of aninterpolatory, uniformly convergent subdivision rule S as the prediction operator (Sj

j−1 = S,∀j ∈ N).Then, a possible choice for the restriction operator is

(Rj−1j vj)k = vj2k, k ∈ Z, j ∈ N. (2.14)

We can take the discretization operator to be the projection of f on Γj

(Rjf)k = f(2−jk), f ∈ BC(R), , k ∈ Z, j ∈ N, (2.15)

and the reconstruction operator to be the limit action of S

S∞j v

j = S∞v0. (2.16)

We refer to the multiresolution operator defined by this algorithm, as the multiresolution asso-ciated to S. Most of the examples of multiresolution operators that we consider in this paper areexactly of this type.

To prove stability of a multiresolution M associated to an interpolatory subdivision scheme S,we need to prove stability of the direct and the inverse MR. Letting all the details be zero in (2.12),we obtain that a necessary condition for stability of the inverse MR is the stability of the subdivisionscheme S, itself. In this particular case of M this condition is enough for proving stability of thedirect MR.

Lemma 2.15 Let S be stationary, interpolatory, stable subdivision scheme. Let M be the multires-olution associated to S. Then the direct MR is stable!

Proof. First, note that since S is interpolatory, vi is a subsequence of vj for every j ≥ i.Therefore, ‖vi − vi‖ ≤ ‖vj − vj‖. In particular ‖v0 − v0‖ ≤ ‖vJ − vJ‖ for every J ∈ N and we onlyhave to prove that

‖dj − dj‖ ≤ C‖vj − vj‖ ∀j ∈ N,

where the constant C must be uniformly bounded.

‖vj − vj‖ = ‖Svj−1 + dj − Svj−1 − dj‖ ≥ ‖dj − dj‖ − ‖Svj−1 − Svj−1‖.

Now using that S is stable and that (vj−1 − vj−1) is a subsequence of (vj − vj), we derive

‖vj − vj‖ ≥ ‖dj − dj‖ − C1‖vj−1 − vj−1‖ ≥ ‖dj − dj‖ − C1‖vj − vj‖.

Since S is stationary, C1 and thus C := C1 + 1 do not depend on j and the lemma is proved.

Corollary 2.16 Let S be interpolatory subdivision rule. The multiresolution M associated to S isstable if and only if there exists a constant C such that for any J ∈ N

‖vJ − vJ‖ ≤ C(‖v0 − v0‖+

J∑j=1

‖dj − dj‖). (2.17)

11

3 Survey of the existing literature

In the context of image (curve, surface) representation as well as in the context of numericallycomputing weak solutions of nonlinear conservation laws, piecewise smooth data with jump dis-continuities appear. Due to the Gibbs phenomena, a linear multiresolution operator cannot simul-taneously localize these discontinuities and provide smooth approximation to the initial functionaway from the singularities. Another big problem is that convexity preserving linear subdivisionschemes are at most C1 smooth even when the data is strictly convex. These drawbacks motivatesthe investigation of non-linear subdivision schemes and their usage in a nonlinear multiresolutiontransforms.

Several subdivision schemes have been introduced and analyzed so far. Harten, Engquist, Osherand Chakravarthy [HEOC87] proposed the essentially non-oscillatory (ENO) subdivision scheme,which uses not only central cubic interpolation like Sc but also left and right cubic ones (i.e., theinterpolating nodes are shifted by one once to the left, and once to the right). If we denote by plk, p

ck,

and prk the corresponding interpolating polynomials, then for vj+12k+1 we use the value at 2

−(j+1)(2k+1)of the least oscillatory one (fig. 2a). This method is uniformly convergent [CDM03], behaves likeSc away from singularities, and prevents the Gibbs phenomena around them. Unfortunately, smallperturbations of data may lead to changing the interpolating polynomial, and thus ENO is unstable.Therefore, the weighted ENO (WENO) subdivision scheme was suggested [LOC94], where insteadof taking only one of the interpolating polynomials for imputation, one uses a convex combination

pk = α1plk + α2p

ck + α3p

rk, α1 + α2 + α3 = 1; α1, α2, α3 ≥ 0

of all the three polynomials with data-dependent weights αi31. Extensions of these schemes bytensor-product techniques to 2D edge-adaptive image representation have been considered andnumerically investigated in [AACD02], [AAC+01], [CDSB97], [CDSB00]. Both ENO and WENOare built on linear subdivision rules, but the data-dependence of these rules is what makes the wholeprocess nonlinear. In [CDM03] Cohen, Dyn, and Matei analyze the convergence, smoothness, andstability of a larger class of so called quasilinear datadependent subdivision schemes, defined by anoperator-valued map Φ, such that

Sv = Φ(v)v, ∀v ∈ l∞(Z), (3.1)

and Φ(v) : l∞(Z) → l∞(Z) is linear for every v. Using the developed theory, the authors provestability of the WENO scheme. However, due to the form of the weights αi31, the constant Cfrom (1.1) heavily depend on a fixed parameter ε. The question what the asymptotical behavior ofWENO is when ε → 0 as well as the one about stability of the associated multiresolution are stillopen.

The idea behind the so far introduced schemes is: take a set of “smooth enough” linear subdi-vision rules, choose one of them for smooth data and always apply it in such occasions, and useadaptivity around discontinuities. Thus the limit function has the same smoothness as the fixedsubdivision rule away from singularities, and lacks smoothness but removes the Gibs phenomenaaround them. An “opposite” approach is to use the simplest linear subdivision rule - midpointlinear interpolation SL, defined by (SLv)2i = vi, (SLv)2i+1 = vi+vi+1

2and to perturb it away from

discontinuities so that to increase the smoothness of the limit. Kuijt and van Damme introduced

12

(a) ENO

vjk−2 vj

k−1vj

k

vjk+1

vjk+2

vjk+3

(Slvj)

2k+1

(Scvj)

2k+1(S

rvj)

2k+1

(b) PPH

Figure 2: (a) ENO algorithm: the three candidates for imputation, and the less oscillatory choice;(b) one step of PPH and the limit function.

two classes of nonlinear subdivision schemes, based on that idea, which in addition are monotonicity[KvD99] and convexity [KvD97] preserving. The most exploited member of the second class is thepiecewise polynomial harmonic (PPH) scheme, introduced also by Floater and Micchelli in [FM98],which is interpolatory and dyadic with

(SPPHv)2k+1 =vk + vk+1

2− 1

8H(42vk−1,42vk), (3.2)

where H : R2 → R is the harmonic mean defined by

H(x, y) =

2xyx+y

, xy > 0

0, otherwise.

PPH is a second order perturbation of SL, i.e., SPPHv = SLv + F (42v), where F is a nonlinearoperator. In [AL05] stability of PPH and the corresponding multiresolution is proven. The idea ofthis proof stays in the core of our stability theorem. Proof for stability of a large subclass of theschemes proposed in [KvD97], including PPH, can be found in [KvD98b]. In [ADLT06] a bivariatescheme, based on tensor product on PPH is introduced and analyzed in terms of convergence andstability.

Experimentations with the ENO method have indicated several practical drawbacks, thereforeSerna and Marquina [SM04] introduce a new class of reconstruction procedures, the Power ENOmethods, which are based on an extended class of limiters

powerp(x, y) =(x+ y)

2

(1−

∣∣∣x− yx+ y

∣∣∣p),defined as a correction of classical ENO methods. This class contains both WENO and PPH (takep = 2 in the powerp mean), reduces smearing near discontinuities, and provides good resolutionof corners and local extrema. Moreover a new weighted ENO method (Weighted Power-ENO5) is

13

proposed, which has the highest possible order of polynomial reproduction - six. The convergenceof these methods and their extension to 2D are subject of [DL]. Stability is an open question.

A family of nonlinear adaptive 4-point interpolatory, data-dependent subdivision schemes forimproving image reconstruction is introduced in [MDL04]. These schemes are proven to uniformlyconverge and are experimentally shown to be C1. Stability is, again, an open question.

Normal meshes and multiresolution for curves and surfaces is a recent concept in efficient ge-ometry representation using local data-dependent coordinate systems adapted to tangential andnormal directions. Here the corresponding subdivision scheme is usually linear but the details arenot linearly added, i.e., instead of having Mv = Sv+ d we have Mv = F (v, d), where F is linear inv and nonlinear in d. More precisely,

(Mvj)2k+1 = (Svj)2k+1 + (σjdjnj)k, (Mvj)2k = vjk, vj = (xj, yj) ∈ l∞(Z)× l∞(Z)

where nj = 4vj,⊥

‖4vj‖2 := (−4yj ,4xj)‖4vj‖2 is the corresponding normal vector (fig. 3a). This idea was

originally introduced by Guskov and others in [GVSS00]. [DRS04] studies properties like regularity,convergence, and stability of normal multiresolution analysis for curves. The central idea of thepaper is to study normal approximation as a perturbation of a linear subdivision scheme. This worksonly under additional assumptions on the smoothness of the initial curve Γ, thus only Cβ(R2), β > 1curves are considered. Furthermore, the authors always assume that Γ can be parameterized by itsx-coordinate and interpret the normal multiresolution as a nonregular scalar-valued multiresolutionon the grid xj. To avoid difficulties, they assume that the new nodes are “well placed”, i.e., startingwith a monotonic sequence x0, to have monotonic xj’s for all j > 0, and the points of xj not to cluster(both this conditions depend on the parametrization of Γ, and one can usually obtain them viarotation of the coordinate system). Under these assumptions the authors manage to prove uniformexponential convergence, regularity (that depends on the regularity of S and the smoothness of Γ),geometrical decay of the details, and finally stability of the normal multiresolution. Moreover, theyproduce a counterexample that the results above do not hold for initial curves Γ ∈ Cβ, β ≤ 1. Thequestion about normal surfaces and how normal meshes work for less smooth spaces, particularlyspaces that are used to model natural images such as BV, and the Besov space B1,1 remains open.The other uninvestigated direction is using non-interpolatory linear schemes as a subdivision rule.

[KG03] implements the normal “remeshing” technique, based on the Butterfly subdivisionscheme [DLG90] introduced in [GVSS00] for irregular highly detailed meshes and implements itin the progressive geometry coding described in [KSS00]. This paper is an attempt for generalizingthe theory in [DRS04] from the curve to the surface case. The authors use both the Butterfly schemeagain, and the Loop [Loo87] scheme for the wavelet transform. Measuring the performance of theabove coding gives better compression rate of normal meshes obtained using Butterfly wavelets thanthe Loop ones, and a better compression rate of normal meshes versus other remeshes when theLoop wavelets are used. The first result is intuitive, but the second one is very interesting and isnot theoretically proven, yet. The smoothness of the normal mesh parametrization is expected, butis also an open question. Another open problem is whether using non-linear optimization approachrather than the Butterfly scheme for normal remeshing will significantly decrease the percentageof non-normal coefficients and thus will be worth investigating. (The non-normal coefficients comefrom locations where the piercing did not find any “valid” intersection point with the original sur-face, and for adaptive semi-regular meshes, produced by the Butterfly optimization and having thesame number of vertices as the original irregular mesh their number is below 10%.)

14

(a) Normal multiresolution

vjk

vjk+1

(Svj)2k+1

(Mvj)2k+1

−3 −1 0 1 3

(b) Median interpolation

v11

v00v1

0

p0(x)

v0−1

v01

Figure 3: (a) Normal multiresolution using the midpoint-interpolating subdivision scheme. (b) Onestep of the median-interpolating subdivision scheme

Another possible direction of generalization and improving the progressive geometry compres-sion is allowing irregular refinements, as well. Such kind of approach is investigated in [GKSS02],where each step of a regular refinement is followed by an irregular one. The irregular operations arelimited and they are as many as needed for topology changes, feature alignment, and stretching res-olution. The main problem of the regular multiresolution is that “spiky” features lead to a stretchedmap, which may cause bad aspect ratio polygons, poor approximation, and numerical problems.On the other hand, the irregular algorithms are more complex for multiresolution, smoothing, com-pression, editing, etc. The above construction combines the advantages of both the refinements,and “controls” the disadvantages. All the results in the paper are experimental, which leaves roomfor research. Moreover, representing high topological complexity models such as isosurfaces frommedical imaging or scientific computing, and approximating dynamically changing, topologicallycomplex geometries does not give good results with the novel approach, so further generalizationsare expected.

In the signal processing, linear multiresolution is a promising approach for successfully removingGaussian noise. Unfortunately, the linear case does not work with strongly non-Gaussian noise, ex-amples of which are typical in analogue telephony, radar signal processing, and laser radar imaging,since classical statistical models show that maximum likelihood estimators are often linear in theGaussian case, but highly nonlinear when Cauchy noise appears. The remedy is to use robust esti-mators. In [DY00] Donoho and Yu introduce a class of nonlinear, triadic subdivision schemes basedon median interpolation, one of which, the quadratic median-interpolating scheme Smed,3 (MI), al-lows a closed-form representation and is the main object of investigation. Its dyadic version Smed

[Osw04] is computationally easier to deal with, and although it does not have the same denoisingproperties as the original triadic scheme, it can be used for testing theoretical results.

Let f be real-valued continuous function on a bounded interval I with positive Lebesgue measure(m(I) > 0). Then the median of f on I is defined by

med(f ; I) := supα : m(x : f(x) < α) ≤ 1

2m(I)

. (3.3)

15

Smed,3 uses the coarse-level information on three consecutive intervals to construct the quadraticpolynomial p that has the same medians in this intervals as the given data, and use its medianson the intervals, produced from the central interval by the grid refinement, to impute data on thenext level (fig. 3b). This scheme is “closely” related to two linear schemes: the cubic midpoint-interpolating scheme Smid,3 and the average interpolating scheme. In [DY00] uniform convergenceand Holder-α regularity for Smed,3 is proven with α > 0.0997. Oswald [Osw04] improves the lowerbound to α > 0.8510, before Xie and Yu [XY05] to prove α > 1 − ε which is exactly the criticalHolder exponent for Smid,3, as conjectured in [DY00]. The question about stability remains open,and its proof is the first goal in our research. For Smed we can prove stability, up to one case forwhich strong numerical evidences are provided. After theoretically proving this remaining case, wewant to adjust the computations to the triadic setting. Using MI, Donoho and Yu [DY00] develop amedian-interpolating pyramid transform (MIPT) and analyze the transform coefficients. There areseveral distinguishable things in this construction. First, unlike in the types of the geometry-drivenapplications, it is better to associate a piece-wise constant function to the sequence vk ∈ l∞(Z)rather than the linearly interpolating one (fig 3b), i.e.,

gj(·) =∑k∈Z

vkj 1Ij,k(·), Ij,k = [k3−j, (k + 1)3−j).

Second, Smed,3 is triadic and non-interpolatory, thus the transform is expensive. Finally, the restric-tion operator (taking the discrete median of the values at the three fine-level intervals) is nonlinear.Despite all these “oddities”, the median-interpolating pyramid transform satisfies the typical prop-erties of a wavelet transform, such as coefficient localization, coefficient decay, and Gaussian-noiseremoval. Moreover, it removes Cauchy noise with the same efficiency as the Gaussian one. Stabilityof MIPT is crucial for applications, and is not proven yet. Our experience with the dyadic caseshows that without additional assumptions on the smoothness of the initial function f (as in normalmultiresolution) the transform is not stable.

In [GH00a] and [GH00b] Goutsias and Heijmans present a general theory for constructing linearas well as nonlinear pyramid and wavelet decomposition schemes for signal analysis and synthe-sis. The main assumptions correspond to the Harten’s approach and are: “perfect reconstruction”,which states that subdivision followed by restriction returns the original signal, and that the gridsare nested. The proposed theory unites different tools for constructing multiresolution signal de-composition schemes, such as pyramids, wavelets, morphological skeletons and granulometries, andgives a complete characterization of restriction and subdivision operators between two adjacentlevels.

The necessity of considering manifold-valued data arises from real-life examples like: headings,orientations, rigid motions, deformation tensors, distance matrices, projections, subspaces, etc. Ifthe control points of a subdivision are restricted to a certain manifold, surface, or a Lie group,then, due to these restrictions, even when the original subdivision scheme is linear, its modificationbecomes nonlinear. The questions about how to modify the linear subdivision scheme and which ofits properties can be inherited by the associated manifold-valued scheme are discussed in [WD05],[Wal06], [WP06], [WNYG07], [GW07], [Gro06]. The analysis of the latter is based on its proximityto the linear scheme it is derived from. This general principle has been used before, e.g. in theanalysis of non-stationary linear schemes in [DL95], and nonlinear schemes in [DRS04], [XY05],[XY06].

16

In [URDS+05] a generalized wavelet analysis on a Manifold-valued data is introduced. It isanalogous to the wavelet analysis in R, but in the generalized wavelet transform, like in normalmultiresolution, there is an important structural distinction between the coarse-scale information,which belong to the manifold M and the fine scale information (i.e., the details) which consists oftangent vectors to M . The main idea of the new approach is to project the manifold-valued dataonto a tangent space Tp0(M) via the Exp map, where p0 ∈M is suitably chosen, then to apply thecorresponding real-valued linear subdivision scheme, and finally to go back on M via the Log map.This procedure works only locally, i.e., when the data p(k) (this is the part of p which is necessaryfor obtaining p2k+1), and the imputed point p2k+1 can be mapped onto a single tangent plane.Due to different applications, both “point-concentrated” (based on Sc) and “interval-concentrated”(based on Average-Interpolating) pyramids are considered. This approach is shown to work well onvarious test and real-life examples, but no mathematical proofs are given. It is conjectured that thesmoothness of the new subdivision schemes coincide with the smoothness of the underlaying linearones.

In [WD05] proximity conditions are defined, and general theorems for convergence and C1

smoothness of a nonlinear scheme T that satisfy proximity condition with an affine-invariant, in-terpolatory, linear, univariate subdivision scheme S are proven. Then, since S can be expressed interms of repeated affine averages

avα(x, y) := (1− α)x+ αy, α ∈ R,

the authors suggest in order to restrict the data to a manifold or a group either to substitute theaffine averages by geodesic ones or to project the affine averages onto the manifold or the group.Finally, it is proven that (under some local restrictions) for a given S all the analogous geodesicschemes (both for a surface and for a matrix group of constant velocity), and all the analogousprojecting schemes fulfil proximity conditions, and hence inherit the C0 and C1 smoothness ofS. [Wal06] is a continuation of [WD05], where the proximity analysis is extended and appliedfor Ck, k ∈ N smoothness. Again, only the univariate case is considered, and the geodesic andprojective constructions from the previous article are shown to inherit C2 smoothness, as well.[WNYG07] deals with a particular log-exponential geodesic analogue of a univariate, linear, dyadicsubdivision in Lie groups. This analogue is shown to satisfy the generalized proximity conditionsfrom [Wal06] with k = 2, and thus to be C2 smooth. [GW07] studies an alternative log-exponentialanalogue, which allows arbitrary dilation factors, and is better suited for generalization to themultivariate setting. The authors prove that the proximity conditions are satisfied and give sufficientconditions for convergence, C1 and C2 smoothness of the introduced schemes. Finally, [Gro06]extends the theory from [WD05] to the multivariate case. It is proven that the proximity conditionsare satisfied for a large class of nonlinear multivariate schemes, namely the bivariate geodesic-average and projection-average analogues (defined in [WD05]), the “closest point projection” aswell as the Log-Exp analogue (defined in [GW07]), and thus (under some local restrictions on theinitial data) the above proximity schemes, defined on an arbitrary abstract Riemannian manifoldor an arbitrary abstract Lie group, have the same smoothness (up to C1) as the underlaying linearsubdivision scheme S.

17

4 Linear subdivision

4.1 Basic representations

Let us start with an example (see [Dyn92]). The centered B-spline Bn of degree n over Z in R(or arbitrary regular grid Γ) is a real-valued, (n − 1)-times differentiable, non-negative piecewisepolynomial function of degree n compactly supported on [−bn+1

2c, dn+1

2e]. Let

Sn,Z = f : f ∈ Cn−1, f |(k,k+1) ∈ Πn(R), k ∈ Z

be the space of all splines of degree n over Z. The union of all integral shifts of Bn Bn(x−k) : k ∈ Z(or d-shifts in general, where d is the step-size of Γ ) form a basis of Sn,Z (Sn,Γ) and partition ofunity of R: ∑

k∈Z

Bn(x− k) = 1, x ∈ R. (4.1)

The easiest way to define Bn(x) is by recursion. Take B0(x) = χ[0,1](x) - a piecewise constantfunction, that is one in [0, 1] and zero otherwise. Then

Bn+1(x) =

∫ x−εn+1

x−εn

Bn(t)dt, εn = n mod 2.

Let f ∈ Sn,Z. Then

f(x) =∑k∈Z

v0kBn(x− k), (4.2)

where v0k ∈ R, k ∈ Z are uniquely determined.

But for every j ∈ Z, Sn,2jZ ⊂ Sn,Z. Hence, f(x) ∈ Sn,2jZ, as well, where Bn(2jx − k) : k ∈ Z

form a basis. Thusf(x) =

∑k∈Z

vjkBn(2jx− k), (4.3)

with unique real coordinates vjk. In the special case, when f(x) = Bn(x) and j = 1 we obtain:

Bn(x) =∑l∈Z

al,nBn(2x− l). (4.4)

Note that, due to the compact support of the B-spline, only finitely many al,n’s are nonzero. There-fore the sum in (4.4) is finite.

Now, combining (4.3) and (4.4) we derive

f(x) =∑l∈Z

vjlBn(2jx− l) =

∑l∈Z

vjl∑k∈Z

ak,nBn(2j+1x− 2l − k) =

∑k∈Z

∑l∈Z

ak−2l,nvjlBn(2

j+1x− k).

Hence,

vj+1k =

∑l∈Z

ak−2l,nvjl , ∀k ∈ Z. (4.5)

18

0 1 2 3 40

0.2

0.4

0.6

0.8(a)

0 1 2 3 40

0.2

0.4

0.6

0.8(b)

Figure 4: Chaikin’s algorithm: (a) practical application; (b) theoretical setup.

It is well known (see [LR80]) that letting j tend to infinity, (2−jk, vjk) converges uniformly tof . Thus, using (4.4) and (4.5), we can construct a uniformly convergent, linear subdivision schemeS : l∞(Z)→ l∞(Z) such that

(Sv)k =∑l∈Z

ak−2l,nvl ∀k ∈ Z. (4.6)

In the literature, S is known as uniform B-spline subdivision scheme.

Note that we did everything “backwards”. We started with the limit curve, “projected” it onthe grids 2−jZ, and then “obtained” the linear subdivision rule.

If we take n = 1 and do explicitly all the above calculations, we get:

(SLv)2k = vk, (SLv)2k+1 =vk + vk+1

2. (4.7)

This scheme is interpolatory, since all the points at all levels stay on the limit curve and is knownas: the degree 1 centered Lagrange interpolation. The limit curve f coincides with f j for all j ∈ Z.

n = 2 gives:

(Sv)2k =3

4vk +

1

4vk−1, (Sv)2k+1 =

3

4vk +

1

4vk+1. (4.8)

This scheme is known as Chaikin’s algorithm ([Cha74]) and is example of a non-interpolatorysubdivision scheme.Chaikin’s algorithm provides an example that the theoretical setup and the practical application ofa subdivision scheme could sometimes differ. This is due to the different interpretations of the data.In theory, we think of vj as an R data over the regular grid 2−jZ (fig. 4b), while in practice, wethink of vj as a set of control points in R2 (fig. 4a). This ambiguity has no impact on mathematicalproperties of the subdivision scheme, such as convergence, stability and smoothness, since for anyv0 ∈ l∞(Z), at any level j ∈ Z f j

a is a shift of f jb by dj =

∑ji=0 2

−(1+j) and, thus, this is true for thetwo limit functions, as well!

There are several representations of a local, linear subdivision scheme S that are used:

1) By a 2s-infinite matrix S = (sα,β), α, β ∈ Zs.

19

2) By its mask a ∈ l∞(Zs).

3) By a finite collection of linear rules.

4) By its action on the Kronecker delta.

Since S is a linear operator from l∞(Zs) to itself, we can think of it as a 2s-infinite matrixS = (sα,β), such that for any v0 ∈ l∞(Zs)

(Sv0)α =∑β∈Zs

sα,βv0β. (4.9)

The locality of the subdivision scheme guarantees that the matrix is very sparse, i.e. it gives

sα,β = 0, |α− rβ| > B, (4.10)

where r is the dilation factor of S and B is as in definition 2.1. For example, for the Chaikin’salgorithm we have: s = 1, r = B = 2, and the non-zero entries of the matrix S are

s2k,k−1 = 1/4, s2k,k = 3/4, s2k+1,k = 3/4, s2k+1,k+1 = 1/4, k ∈ Z.

S =

. . . . ..

0 1/4 3/4 0 0 00 0 3/4 1/4 0 00 0 1/4 3/4 0 00 0 0 3/4 1/4 00 0 0 1/4 3/4 0

. .. . . .

.

Using 2s-infinite matrix is not comfortable, since the matrix is extremely sparse, infinite in alldirections, and it doesn’t use the shift-invariance property of the subdivision scheme. Therefore, itis more convenient to use the mask a ∈ l∞(Zs) of the subdivision scheme S for defining its action.In other words

(Sv)α =∑β∈Zs

aα−rβvβ v ∈ l∞(Zs), (4.11)

where, again due to locality, aα = 0 for |α| = |α1| + |α2| + · · · + |αs| > B. We will use the notion“mask” interchangeably for both a and its minimal finite subsequence, that contains all the non-zeroentries of a. Obviously, sα,β = aα−rβ. For the Chaikin’s algorithm the mask is

a = (a−1, a0, a1, a2) = (1/4, 3/4, 3/4, 1/4).

Using a system of linear rules∥∥∥(Sv)rα+γ =∑β∈Zs

arα+γ−rβvβ =∑β∈Zs

aγ+rβvα−β, γ ∈ Es, α ∈ Zs, v ∈ l∞(Zs) (4.12)

20

for defining S, helps visualizing and understanding the geometry of the subdivision scheme. Here,Es = γ : γi ∈ 0, 1, . . . , ri − 1, i = 1, 2, . . . , s. For the Chaikin’s algorithm this representation is(4.8).

Let δ = δα,0 be the Kronecker delta. Here 0 ∈ Zs, and δα,β is the Kronecker symbol. Let Sbe linear and local. Then, if we know the action of S on δ, we know the action of S on everyv0 ∈ l∞(Zs):

v1α =∑β∈Zs

v0β(Sδ)α−β. (4.13)

The sum is finite, since S is local.When S is uniformly convergent ϕ(x) = S∞δ is called the S-refinable function, and we have

S∞v0 =∑α∈Zs

v0αϕ(· − α), ∀v0 ∈ l∞(Zs). (4.14)

Note that ϕ(x) has compact support, since S is local and uniform convergent (for the Chaikin’salgorithm ϕ = B2). Every uniform convergent, linear subdivision scheme S should reproduceconstants (see [CDM91]), so if we take v0 = 1 in (4.14), we see that the integral shifts of ϕ form apartition of unity in Rs: ∑

α∈Zs

ϕ(x− α) = 1, x ∈ Rs. (4.15)

The above two properties of ϕ give l∞(Zs)-linear independence of Φ = ϕ(· − α) : α ∈ Zs, i.e., ifa (not necessary finite) linear combination of elements of Φ vanishes in Rs, this linear combinationmust be the trivial one.Finally, there is a relation between the mask and the refinable function for a uniform convergentscheme S. Plugging v = δ in (4.11) gives (Sδ)α = aα. Now, using (4.14) and the fact that δ on Zs

and Sδ on r−1Zs have the same limit functions, we derive

ϕ(x) =∑α∈Zs

aαϕ(rx− α), x ∈ Rs. (4.16)

By induction on j ∈ Z, we get∑α∈Zs

vjαϕ(rkx− α) =

∑α∈Zs

v0αϕ(x− α), ∀v0 ∈ l∞(Zs). (4.17)

Therefore, as in the case of B-splines, we can interpret vj as the coordinates of the limit functionS∞v0 in span ϕ(rjx − α) : α ∈ Zs. The representation of a uniformly convergent scheme S viaits refinable function is somehow dual to the previous three approaches. Instead of starting fromthe coarsest grid Zs and “refining” the initial data ((4.9), (4.11), (4.12)), one begins with the limitfunction and “projects” it on the corresponding grid ((4.17)).

We showed that to every uniformly convergent subdivision scheme S we can assign a uniquerefinable function ϕ. The converse is also true (see proposition 2.3 in [CDM91]) but only if ϕsatisfies (4.15), (4.16) with finite mask a, and the “stability hypothesis”

‖v0‖ ≤ c‖∑α∈Zs

v0αϕ(x− α)‖ (4.18)

21

for any v0 ∈ l∞(Zs). In our case (4.18) is not much stronger than Φ being l∞(Zs)-linearly indepen-dent and we can use the latter restriction, but the situation is different with Lp(Rs), 1 ≤ p < ∞.For example, the compact support of ϕ guarantees the l2(Zs)-linear independence, but L2-stabilityholds if and only if (see [CDM91])

x : ϕ(x+ 2πα) = 0 : α ∈ Zs = ∅

with ϕ being the Fourier transform of ϕ. For deeper analysis in this direction, one can see [JW93].

Hence, there is a one to one correspondence between the class of uniformly convergent subdi-vision schemes S : l∞(Zs) → l∞(Zs) and SC0(Rs)/∼: the class SC0(Rs) of compactly supported,continuous functions ϕ : Rs → R that satisfy (4.15), (4.16) with finite mask a, and (4.18), quotionedby the equivalence relation ∼, where

ϕ ∼ ψ ⇐⇒ ∃α ∈ Zs : ϕ = ψ(· − α).

The above bijection allows to attack the problems, concerning uniformly convergent schemes fromtwo opposite sides, and plays an important role in most of the proofs!

4.2 Stability analysis

In the linear case, the question about stability is completely solved (see [CDM91]). Here, we willrepeat the main results.

Lemma 4.1 A local, linear subdivision scheme S is stable, if and only if it is uniformly convergent.

Proof. The definition (2.4) explicitly implies that S is uniformly convergent.Now, let S be uniformly convergent, and let ϕ be its refinable function. Denote by C the L∞-normof ϕ:

‖ϕ‖ = C.

Let v0 ∈ l∞(Zs) be arbitrary, and fix x ∈ Rs. Denote by M = suppϕ∩Zs and by Ω(x) = α ∈ Zs :x ∈ suppϕ(· − α). M is finite, since ϕ has compact support, and |Ω(x)| ≤ M + s. Using (4.14),we obtain

|S∞v0(x)| = |∑α∈Zs

v0αϕ(x− α)| = |∑

α∈Ω(x)

v0αϕ(x− α)| ≤

≤ C∑

α∈Ω(x)

|v0α| ≤ (M + s)C‖v0‖.

Hence, ‖S∞v0‖ ≤ (M + s)C‖v0‖, and S is stable.

22

4.2.1 The univariate case

In this subsection, we consider only the case s = 1! Let us start with defining the important notionof polynomial reproduction.

Definition 4.2 Let S be a stationary subdivision scheme. S has order of polynomial repro-duction N ∈ N, if S reproduces constants, and N is the maximal number with the property: forevery n < N , and every monic polynomial p(x) of degree n, there exists a monic polynomial q(x)of degree n such that

Sp0 = q1.

We also say that S reproduces polynomials of degree n, for every n < N .

In other words, S reproduces polynomials of degree n, if Πn(R) \ Πn−1(R) stays invariant underthe action of S. There is a natural question: does polynomial reproduction of degree n guaranteepolynomial reproduction of degree m ∈ N, for every m < n? The answer is affirmative, but onlywhen in addition S reproduces constants. Different proofs can be found in Section 2 in [Han01] andin Proposition 2.4 in [JZ04].

Proposition 4.3 Let S be a linear, shift invariant (not necessary finite) subdivision scheme thatreproduces constants. Then there exists a linear subdivision scheme S1, such that

41 S = S1 41. (4.19)

S1 is called the first derived scheme of S.

Proof. The main trick in this proof is to use symbolic calculus.For v0 ∈ l∞(Z), the generating function V0(z) is given by the Laurent series

V0(z) =∑k∈Z

vkzk.

Let a(z) be the generating function of the mask a of S. We will refer to it as the symbol of thesubdivision scheme and, obviously, S is completely determined by its symbol. Then

Vj+1(z) =∑k∈Z

vj+1k zk =

∑k∈Z

(∑l∈Z

ak−rlvjl

)zk =

∑l∈Z

vjl zrl∑k∈Z

akzk = a(z)Vj(z

r). (4.20)

Here, r is the dilation factor of S.Denote by

Dj(z) =∑k∈Z

(41vj)kzk =

∑k∈Z

(vjk+1 − vjk)z

k = Vj(z)1− zz

(4.21)

the generating function of 41Sjv0. Using (4.20) and (4.21), we get

Dj+1(z) = Vj+1(z)1− zz

= a(z)Vj(zr)1− zz

=

= a(z)1− zz

zr

1− zrDj(z) = a(1)(z)Dj(z),

(4.22)

23

with

a(1)(z) =zr−1a(z)

zr−1 + zr−2 + · · ·+ 1. (4.23)

Thus, to finish our prove it suffices to show that a(1)(z) is a generating function. Let ξ be a primitiver-th root of 1. We need to prove that a(ξ) = 0. Indeed,

a(ξ) =∑i∈Z

aiξi =

r−1∑l=0

(∑j∈Z

al−rj

)ξl =

r−1∑l=0

ξl = 0 (4.24)

For the third equality above we used (4.11) and that S reproduces constants.

Remark 4.4 Note that we not only proved existence of a first derived scheme, but also gave analgorithm for its construction. Moreover, if S is finite (4.23) gives

|supp (a(1))| ≤ |supp (a)|. (4.25)

By induction, if N is the order of polynomial reproduction of S, for each n ≤ N there exists ann-th derived scheme Sn such that

4n S = Sn 4n.

Moreover, if the S-refinable function ϕ ∈ C l(R), we have

di

dxiS∞v0 = S∞

i 4iv0, v0 ∈ l∞(Z), ∀i : 1 ≤ i ≤ max(N − 1, l). (4.26)

Definition 4.5 Let S : l∞(Z)→ l∞(Z) be a linear and bounded (with respect to the operator norm)operator. The spectral radius of S is

ρ∞(S) = lim infj→∞

‖Sj‖1/j. (4.27)

Theorem 4.6 A linear, local subdivision scheme S is uniformly convergent, if and only if S repro-duces constants and ρ∞(S1) < 1.

Proof. We follow the proof from [Dyn92].Let S be uniformly convergent and v0 ∈ l∞(Z). Hence, S reproduces constants and S1 exists. Thenfor any k ∈ Z

|(Sj141v0)k| = |(41vj)k| = |vjk+1 − v

jk| ≤

≤ |vjk+1 − S∞v0(r−j(k + 1))|+ |vjk − S

∞v0(r−jk)|++ |S∞v0(r−j(k + 1))− S∞v0(r−jk)| −−−→

j→∞0

Since S converges uniformly, and the refinable function ϕ is compactly supported and thus, uni-formly continuous, S1 converges uniformly to the zero function for all initial data v0 ∈ l∞(Z). It

24

remains to show that the uniform convergence of S1 to zero is equivalent to ρ∞(S1) < 1.Take v0 ∈ l∞(Z) with ‖v0‖ = 1. For any ε > 0, any k ∈ Z, and j > J ′(ε) we have

|(Sj1v

0)k| =∣∣∣(∑

l∈Z

v0l Sj1δl

)k

∣∣∣ = ∣∣∣∑l∈Z

v0l (Sj1δl)k

∣∣∣ ≤∑l∈Z

|(Sj1δ)k−rj l| ≤M‖Sj

1δ‖ < ε,

where M = |supp (a)|, and J ′(ε) comes from the uniform convergence to zero of S1. We used (4.25)and that, due to (4.13) and induction, supp (Sj

1δ) ⊂ (rj − 1)supp (a(1)).Therefore, there exists a positive integer J , and µ ∈ (0, 1), such that for all v0 ∈ l∞(Z)

‖SJ1 v

0‖ < µ‖v0‖, (4.28)

which givesρ∞(S1) < µ < 1. (4.29)

On the other hand, suppose that ρ∞(S1) < 1. Let f j be as in Definition 2.6. Than, we have toshow that the sequence f j(x) : j ∈ Z+ converges uniformly in L∞(R), which, since L∞(R) is aBanach space, is equivalent to showing that the sequence is Cauchy.

Observe that f j is the limit curve of vj for the scheme SL,r with dilation factor r and refinablefunction - the B-spline B1(x) of degree 1. For example, the scheme defined via (4.7) is exactly SL,2.SL,r reproduces constants due to (4.1) and (4.15). Oswald has the following result:

Lemma 4.7 Let T : l∞(Z) → l∞(Z) be a bounded linear operator with local support, such thatT1 = 0. Then there exists, a bounded linear operator T with the same support as T which satisfies

Tv = T4v, ∀v ∈ l∞(Z).

The proof can be found in [Osw02].Now, the linear operator T = S − SL,r has local support and T1 = 0, since both S and SL,r arelocal and reproduce constants. Moreover, T is shift-invariant and thus - bounded. By Lemma 4.7

‖(S − SL,r)vj‖ ≤ E‖41vj‖ ≤ 2E‖vj‖, (4.30)

with some positive constant E <∞.

Using (4.17) and that B1(x) is the SL,r-refinable function, we derive

f j+1(x)− f j(x) =∑k∈Z

((S − SL,r)vj)kB1(r

j+1x− k). (4.31)

Now, combining the last two results with (4.1), the fact that B1 is non-negative, and (4.28), wederive

|f j+1(x)− f j(x)| ≤ ‖(S − SL,r)vj‖ ≤ 2E‖vj‖ ≤ 2E max

0≤i<J‖vi‖µ[j/J ]. (4.32)

The geometrical decay in (4.32) assures that f j(x) : j ∈ Z+ is a Cauchy sequence and completesthe proof.

Theorem 4.6 and Remark 4.4 give an explicit algorithm for checking uniform convergence ofa local, linear subdivision scheme. In order to prove uniform convergence, we need to find thefirst J ∈ Z such that ‖SJ

1 ‖ < 1. From theoretical point of view the number of iterations J is notuniformly bounded (for every J ∈ N one can always find a uniformly convergent scheme S for which‖Sj

1‖ ≥ 1, ∀j ≤ J), but from practical point of view, ifJ > 10 no convergence occurs in the actualperformance of the scheme, since only a small number of steps (j < 10) are carried out in practice.

25

4.2.2 Examples

We denote by Smid,r the midpoint-interpolating subdivision scheme with dilation factor r, where2 ≤ r ∈ N. More precisely, for any v0 ∈ l∞(Z) and any k ∈ Z take the unique polynomial pk ∈ Π2(R)that interpolates the data vk−1, vk, vk+1, i.e.

pk(k + l) = v0k+l, l ∈− 1, 0, 1

. (4.33)

Then

(Smid,rv0)rk+l = pk

(k +

2l + 1− 2r

2r

), l ∈

1− [r], . . . , [r]

. (4.34)

The last equation means, that we have divided the interval [k − 1/2, k + 1/2] into r equally longsubintervals and use the values of pk(x) in the midpoints for constructing v1. Using the Lagrangeformula for interpolation, we derive

p0(x) =x(x− 1)

2v0−1 + (1− x2)v00 +

x(x+ 1)

2v01. (4.35)

Obviously, Smid,r is linear, shift-invariant and local subdivision scheme with order of polynomialreproduction 3. (4.33) is usually known as the interpolation step, and (4.34) - as the imputation stepin defining subdivision. Most of the (not necessarily linear) schemes used in practise follow thesetwo steps: (W)ENO (see [CDM03]), PPH (see [AL05]), median-interpolating subdivision scheme(see [DY00]), etc.Amoung the family of subdivision schemes Smid,r : 2 ≤ r ∈ N, Smid,2 and Smid,3 have greatestimportance in applications.

Example 4.8 S = Smid,2. Using(4.33), (4.34), and (4.35) we get:

(Smid,2v0)2k =

5

32v0k−1 +

15

16v0k −

3

32v0k+1, (Smid,2v

0)2k+1 = −3

32v0k−1 +

15

16v0k +

5

32v0k+1. (4.36)

The symbol of Smid,2 is

a(z) = − 3

32z−2 +

5

32z−1 +

15

16+

15

16z +

5

32z2 − 3

32z3.

Therefore

a(1)(z) =z

z + 1a(z) = − 3

32z−1 +

1

4+

11

16z +

1

4z2 − 3

32z3,

and

(S1v0)2k =

1

4v0k−1 +

1

4v0k, (S1v

0)2k+1 = −3

32v0k−1 +

11

16v0k −

3

32v0k+1.

Since ρ∞(S1) ≤ ‖S1‖ = 1516< 1, Smid,2 is uniformly convergent.

For the higher-ordered derived schemes, we get

a(2)(z) =z

z + 1a(1)(z) = − 3

32+

11

32z +

11

32z2 − 3

32z3, ‖S2‖ =

7

16;

a(3)(z) =z

z + 1a(2)(z) = − 3

32z +

7

16z2 − 3

32z3, ‖S3‖ =

7

16.

26

−3 −2 −1 0 1 2 3 4−0.2

0

0.2

0.4

0.6

0.8

1

1.2(a)

−3 −2 −1 0 1 2 3 4−0.2

0

0.2

0.4

0.6

0.8

1

1.2(b)

Figure 5: The refinable functions for: (a) Smid,2, (b) Smid,3.

Example 4.9 S = Smid,3. Analogously to Example 4.8

(Sv0)3k−1 =2

9v0k−1 +

8

9v0k −

1

9v0k+1, (Sv0)3k = v0k, (Sv0)3k+1 = −

1

9v0k−1 +

8

9v0k +

2

9v0k+1. (4.37)

a(z) = −1

9z−4 +

2

9z−2 +

8

9z−1 + 1 +

8

9z +

2

9z2 − 1

9z4;

a(1)(z) =z2

z2 + z + 1a(z) = −1

9z−2 +

1

9z−1 +

2

9+

5

9z +

2

9z2 +

1

9z3 − 1

9z4;

(S1v0)3k−1 =

2

9v0k−1 +

1

9v0k, (Sv0)3k =

1

9v0k−1 +

2

9v0k, (Sv0)3k+1 = −

1

9v0k−1 +

5

9v0k −

1

9v0k+1;

‖S1‖ =7

9< 1.

Therefore, Smid,3 is uniformly convergent.For the higher-ordered derived schemes, we get

a(2)(z) =z2

z2 + z + 1a(1)(z) = −1

9+

2

9z +

1

9z2 +

2

9z3 − 1

9z4, ‖S2‖ =

1

3;

a(3)(z) =z2

z2 + z + 1a(2)(z) = −1

9z2 +

1

3z3 − 1

9z4, ‖S2‖ =

1

3.

4.2.3 The multivariate case

After establishing systematic theory for stability of local, linear, univariate subdivision schemes inSection 4.2.1, the main question is how far this theory can be extended. Two immediate candidatesare the nonlinear univariate subdivision schemes and the linear multivariate ones. In this section

27

we will show that for the second type of schemes there is a generalization of Theorem 4.6, which isvery natural but, unfortunately, not so applicable in practice.

Let e = (1, . . . , 1) ∈ Zs, and e(i) : e(i)j = δij, j = 1, . . . , s be the i-th coordinate vector in Zs.

Then we can extend the notion of a first derived difference in higher dimensions:

41 : l∞(Zs)→ (l∞(Zs))s, (41v)α =vα+e(i) − vα : 1 ≤ i ≤ s

.

A detailed analysis of the proof of Theorem 4.6 gives that it was based on the existence of a derivedscheme S1, Lemma 4.7 and (4.28).

In s > 1 uniform convergence again implies reproduction of constants, and, analogously tothe case s = 1, reproduction of constants implies existence of a local, linear subdivision schemeS1 : (l∞(Zs))s → (l∞(Zs))s (see Proposition 6.8. in [Dyn92]), such that

41 S = S1 41.

It is not difficult to prove the multivariate version of Lemma 4.7, either (see Proposition 6.7. in[Dyn92]). We only have to be careful with the fact that, unlike in 1D, 41 is not an endomorphismand the image space (l∞(Zs))s is equipped with the infinite-norm

‖41v‖ = supα∈Zs

‖41vα‖,

where ‖41vα‖ = max1≤i≤s

|(41vα)i|.

Now, we are ready the extend the stability analysis to the multivariate case.

Theorem 4.10 Let S : l∞(Zs) → l∞(Zs) be local, linear subdivision scheme that reproduces con-stants. Then S is uniformly convergent, if and only if there exist J ∈ Z+ and µ ∈ (0, 1) suchthat

‖SJ141v0‖ ≤ µ‖41v0‖, ∀v0 ∈ l∞(Zs). (4.38)

The proof follows that of Theorem 4.6.

Although Theorem 4.10 is nothing but “translating” all the definitions from the one-dimensionalcase, to the “language” of the multidimensional one, its application in practice is tremendouslysmaller. The problem is again in 41 and that, unlike the univariate case, this linear operator is notsurjective! For example, if v ∈ l∞(Zs), then 41v satisfies relations like

e(j) · [(41v)α+e(i) − (41v)α] = e(i) · [(41v)α+e(j) − (41v)α], α ∈ Zs,

for every i, j ∈ 1, . . . , s.This means, that we don’t have formulation of Theorem 4.10 involving the spectral radius of thesubdivision scheme S1 and thus, no lower bound for µ. (compare with (4.29)!) In other words, tocheck whether S1 satisfies (4.38) is sometimes very complicated, which makes (4.38) “bad sufficientcondition” for applications.

The latter can be improved by a one-level higher generalization on (4.28).

28

Definition 4.11 Let D : l∞(Zs)→ [0,∞) be a non-trivial function. We say that S is contractiverelative to D, if there exists µ ∈ (0, 1) and J ∈ Z+ such that for all v ∈ l∞(Zs)

D(SJv) ≤ µD(v). (4.39)

For example, both (4.28) and (4.38) mean that S1 is contractive relative to D(v) = ‖41v‖. Wewill see later, that this extension happens to play a role in the stability analysis on a univariate,nonlinear subdivision as well.

Theorem 4.12 Let S be local, linear subdivision scheme that reproduces constants and is contrac-tive relative to D. Let S be a uniformly convergent subdivision scheme with a mask a and S-refinablefunction, ϕ ∈ C(Rs), satisfying the “stability hypothesis”

c1‖v‖ ≤∥∥∥∥∑

α∈Zs

vαϕ(x− α)∥∥∥∥ ≤ c2‖v‖, v ∈ l∞(Zs). (4.40)

If‖(S − S)v‖ ≤ cD(v), v ∈ l∞(Zs), (4.41)

then S converges uniformly.

The proof follows directly from that of Theorem 4.6 with ϕ instead of B1.

Not fixing D a priori, makes the theorem more flexible and applicable. The only question is: dowe know enough appropriate uniformly convergent linear multivariate subdivision schemes? Dynin [Dyn92] shows that a tensor product of s copies of a uniformly convergent univariate linearsubdivision scheme is uniformly convergent (Proposition. 6.3.), as well as local, linear subdivisionschemes with positive mask (Theorem 6.4.).

5 Univariate nonlinear subdivision. Examples

5.1 Cubic ENO and WENO

The essentially non-oscillatory (ENO) and the weighted essentially non-oscillatory (WENO) subdi-vision schemes are interpolatory and dyadic, i.e.

(Svj)2k = vjk ∀j, k ∈ Z.

Thus, in order to determine these schemes, we only need to give a formula for the odd entries ofSvj.

Let us fix a cost function ϕ : Π3(R)→ [0,∞). Let Γj = 2−jZ, k ∈ Z, and vj ∈ l∞(Γj).

• (the interpolation step) Compute the cubic polynomials pjk,i, i = 0, 1, 2 that interpolate(2−j(k − 2 + i+ l); vk−2+i+l

), l = 0, . . . , 3.

29

• (the imputation step) Among these polynomials take the least oscillatory one, i.e. the onethat minimizes ϕ. Let us denote it by pjk. Then

(SENOvj)2k+1 = pjk(2

−(j+1)(2k + 1)).

The ENO scheme was originally proposed by Harten, Engquist, Osher and Chakravarthy in [HEOC87].It is one of the most robust algorithms for avoiding oscillations of a continuous approximation ofdata, near points of discontinuity (the so called Gibbs phenomena). A simple example is given infig. 6.

The set of grid points, used for constructing pjk is called stencil. The interpolation step givesthat for a fixed k ∈ Z there are three candidates for stencil at k, namely

2−j(k − 1− l), 2−j(k − l), 2−j(k + 1− l), 2−j(k + 2− l), l = −1, 0, 1,

to which we will refer as right, central, and left stencils. It is obvious, that ENO has order ofpolynomial reproduction 4 and, although defined in terms of the grid Γj, doesn’t depend on thelevel of subdivision, i.e. is stationary. If for some k the least oscillatory polynomial is not unique,there is no prescribed rule about which one should be taken in the imputation step.

Although globally nonlinear, ENO is locally linear, i.e. for every k, (SENOv)2k+1 is a linearcombination of the values of v on its stencil. More precisely, for v ∈ l∞(Z) let

(Slv)2k = vk (Slv)2k+1 =1

16vk−2 −

5

16vk−1 +

15

16vk +

5

16vk+1 (5.1)

(Scv)2k = vk (Scv)2k+1 = −1

16vk−1 +

9

16vk +

9

16vk+1 −

1

16vk+2 (5.2)

(Srv)2k = vk (Srv)2k+1 =5

16vk +

15

16vk+1 −

5

16vk+2 +

1

16vk+3 (5.3)

be the linear subdivision schemes, that at every k ∈ Z take: (5.1) the left, (5.2) the central, or (5.3)the right stencil. Then we can represent ENO by a map Φ : l∞(Z)→ L := Sl, Sc, Sr, such that

SENOv = Φ(v)v. (5.4)

Remark 5.1 The above representation holds in general, i.e. for any nonlinear subdivision schemeS, one can find a set L of linear subdivision rules with the same (or smaller) support as S, suchthat (5.4) is satisfied. Φ is sometimes called quasi-linear map (see [CDM03]). The minimal orderof polynomial reproduction of the elements of L is called order of polynomial reproduction of Φ.From (5.4) it follows that the order of polynomial reproduction of Φ is less or equal to that of S.For a large class of subdivision schemes the two orders coincide (ENO, WENO, etc.), but thereare some schemes (PPH), for which the “quasi-linear order” is strictly smaller than the subdivisionone. Since for every v ∈ l∞(Z) the action of Φ(v) is known a priori only on v, Φ is by no meansunique! The questions about finding the set L with minimal cardinality, and the quasi-linear mapΦ with maximal order of reproduction depend heavily on S and are still open.

30

−5 0 5

0

0.2

0.4

0.6

0.8

1

(a)

−5 0 5

0

0.2

0.4

0.6

0.8

1

(b)

Figure 6: The action of: (a) ENO, (b) Sc, on the step function.

Let S1l , S

1c , and S

1r be the corresponding first derived schemes. From proposition 4.3 and (5.4)

it follows that there exists an operator-valued function Φ : l∞(Z)→ L′ := S1l , S

1c , S

1r, such that

41(SENOv) = Φ(v)41v, v ∈ l∞(Z). (5.5)

In [CDM03] it is proved that ENO converges for any cost function ϕ. On the other hand sinceit uses only one of the candidate stencils and small perturbation of the initial data could changethe stencil, ENO is unstable. This problem can be solved by a slight modification of the ENOinterpolation technique - WENO.

WENO uses the same idea as ENO, but instead of fixing the stencil, one works with a con-vex combination of the interpolating polynomials pjk,i3i=1, i.e. the polynomial pjk, used in theimputation step is of the form

pjk = αk,1pjk,1 + αk,2p

jk,2 + αk,3p

jk,3, αk,1 + αk,2 + αk,3 = 1, (5.6)

with nonnegative weights that depend on the initial data and the position k, but not on the levelof subdivision.

This scheme was originally proposed by Liu, Osher, and Chan in [LOC94]. There, they used

ϕ(pjk,i) = 2−j

∫ 2−j(k+1)

2−jk

((pjk,i)

′)2dx+ 2−3j

∫ 2−j(k+1)

2−jk

((pjk,i)

′′)2dx (5.7)

as cost function, and

αk,i =ak,i

ak,1 + ak,2 + ak,3, i = 1, 2, 3,

where

ak,i :=di

(ε+ bk,i)2, bk,i = ϕ(pjk,i), (5.8)

31

with positive constants di3i=1 and ε.

Since the interpolating polynomials depend on the grid Γj, to make WENO a stationary subdi-vision scheme, one should use normalization factors in the definition of the cost function (see (5.7)).(5.8) implies that, in order WENO to be shift-invariant, ϕ(pjk,i) must be a function of the first

derivative of pjk,i rather than of the polynomial itself. Thus, the constant functions lie in the kernelof ϕ and the additional parameter ε in (5.8) is necessary for defining ak,i correctly. On the otherhand, introducing ε makes WENO not affine-invariant: the scheme is still translation invariant, butdirect computation gives

pjk,i(λv) = λpjk,i(v)

and

αk,i(λv) =

di(λ−2ε+bk,i(v))2

d1(λ−2ε+bk,1(v))2

+ d2(λ−2ε+bk,2(v))2

+ d3(λ−2ε+bk,3(v))2

6= αk,i(v),

where λ ∈ R! Therefore, the limiting behavior of WENO when ε tends to zero is an importantquestion, but, up to the knowledge of the author, still without an answer.The bk,i’s are called “smoothness indicators” and they preserve the non-oscillatory property of theweighted ENO, i.e. αk,i is “big” when ϕ(pjk,i) is “small” and vice versa. Note that, due to theprevious remarks, bk,i and hence, αk,i are functions on 41v rather than on v!The positive constants di3i=1 are, in general, arbitrary. The order of polynomial reproduction ofWENO is at least the one of ENO - 4, but it can be increased by one using a special choice of thedi’s (see [LOC94] for details).

We can modify (5.4) and derive the corresponding representation for WENO. We just need totake L to be the “convex hull” of Sl, Sc, Sr, i.e.

Φ(v) = αlSl + αcSc + αrSr, v ∈ l∞(Z), (5.9)

where αl : l∞(Z) → l∞(Z) is the nonlinear operator with respect to α·,1: (αlv)2k+1 := αk,1(v),(αlv)2k := 1/3, and αlSl is their “inner product”

((αlSl)v

)k:= (αlv)k(Slv)k. (Analogously for αc,

and αr). Moreover, due to the special type of the cost function, there exists a Φ : l∞(Z)→ L, suchthat Φ(v) = Φ(41v):

SWENOv = Φ(v)v = Φ(41v)v. (5.10)

Cohen, Dyn, and Matei proved in [CDM03] that, unlike ENO, the weighted essentially non-oscillatory scheme is stable (in a weakened version of stability)! The main drawback of this proofis that it heavily depends on ε and does not provide us with any information about the limit caseε→ 0.

5.2 PPH

The piecewise polynomial harmonic (PPH) subdivision scheme is again interpolatory and dyadic,with

(SPPHv)2k+1 =vk + vk+1

2− 1

8H(42vk−1,42vk), (5.11)

32

where H : R2 → R is the harmonic mean defined by

H(x, y) =

2xyx+y

, xy > 0

0, otherwise.

(5.11) is the explicit result of the following algorithm:

Let Γj = 2−jZ, k ∈ Z, and vj ∈ l∞(Z).

• (the interpolation step) Compute the cubic polynomial pjk that interpolates the data

x 2−j(k − 1) 2−jk 2−j(k + 1) 2−j(k + 2)

pjk(x) vjk−1 vjk vjk+1 vjk+1 + vjk − vjk−1 + 2H(42vk−1,42vk)

• (the imputation step)(SPPHv)2k+1 = pjk(2

−j−1(2k + 1)) (5.12)

The PPH subdivision scheme was introduced by Kuijt and van Damme in [KvD98a]. PPH isstationary and affine invariant, due to (5.11), and has order of polynomial reproduction 3, due to(5.12), the fact that 42

hp(x) = 2h2a0 for any p ∈ Π2(R) with p(x) = a0x2 + . . . and any x, h ∈ R,

the trivial equalityH(x, x) = A(x, x), ∀x ∈ R,

where

A(x, y) =x+ y

2is the arithmetic mean, and the observation

vjk+2 = vjk+1 + vjk − vjk−1 + 2A(42vk−1,42vk). (5.13)

Let us prove that every quasi-linear operator-valued function Φ with respect to PPH has strictlylower order of polynomial reproduction than PPH. The general form of Φ is

(Φ(v)w)2k = wk; (Φ(v)w)2k+1 =k+2∑

l=k−1

a2k+1,l(v)wl.

In order Φ(v) to reproduce constants, linear and quadratic polynomials, as well as to satisfy SPPHv =Φ(v)v, the following 4x4 linear system for the unknown coefficients a2k+1,lk+2

l=k−1 should be satisfied:∣∣∣∣∣∣∣∣a2k+1,k−1 + a2k+1,k + a2k+1,k+1 + a2k+1,k+2 = 1−a2k+1,k−1 + a2k+1,k+1 + 2a2k+1,k+2 = 1/2a2k+1,k−1 + a2k+1,k+1 + 4a2k+1,k+2 = 1/4

vk−1a2k+1,k−1 + vka2k+1,k + vk+1a2k+1,k+1 + vk+2a2k+1,k+2 =vk+vk+1

2− 1

8H(42vk−1,42vk)

Using Gaussian elimination, we derive

(241vk−1 + 541vk −41vk+1)a2k+1,k+2 =41vk−1 + 241vk +H(42vk−1,42vk)

4,

33

which doesn’t have solution for vk−1 = 0; vk = −1; vk+1 = 0; vk+2 = 3.

There are plenty of quasi-linear maps Φ corresponding to PPH with order of polynomial repro-duction 2. Among them we would like to mention

(Φ(v)w)2k+1 = −Ck(v)wk−1+

(1

2+ 2Ck(v)

)wk+

(1

2− Ck(v)

)wk+1, (5.14)

where

Ck(v) =sgn(42vk−142vk) + 1

8

42vk42vk−1 +42vk

.

Here, sgn(x) = 1 if x > 0, and sgn(x) = −1, otherwise.The function, defined in (5.14) has two important properties. First, the support of Φ(v) is strictlysmaller then the support of PPH, and moreover this is the minimal support for a linear operator toreproduce quadratic polynomials. Second, Ck and thus Φ itself are functions not on v but on 42vwhich stresses the fact that PPH is a second order perturbation of the degree 1 centered Lagrangeinterpolation SL (see (5.11)). Hence, we have

SPPHv = Φ(v)v = Φ1(41v)v = Φ2(42v)v, (5.15)

where the functions Φ1 and Φ2 can be explicitly computed from Φ.

The main property of PPH is that it is convexity preserving, i.e., if the initial data is convex(concave), then it remains convex (concave) at any level of subdivision, and in particular, the limitfunction is also convex (concave).

Indeed, for any k ∈ Z direct computations give

42vj+12k−2 =

1

4H(42vjk−2,4

2vjk−1);

42vj+12k−1 =

(42vjk−1)2

2

( 1

42vjk−2 +42vjk−1

+1

42vjk−1 +42vjk

);

42vj+12k =

1

4H(42vjk−1,4

2vjk).

(5.16)

Hence, if vjk+i2i=−2 is convex, i.e., (42vj)k−l > 0, l = 0, 1, 2, then so is vj+12k+i2i=−2. (If the data is

concave, take vj = −vj.)Furthermore, (5.16) shows that if the initial data v0k−1, v

0k, v

0k+1, v

0k+2 is neither convex nor

concave, than on the interval [k, k + 1] the PPH subdivision coincides with SL, hence it is purelylinear. On the contrary, if the initial data v0k−1, v

0k, v

0k+1, v

0k+2 is either convex or concave, the PPH

subdivision on the interval [k, k + 1] is always nonlinear (see fig. 2). This “separation” of linearityand nonlinearity, along with the fact that SL is convergent and stable due to Section 4.2.1, allowsus to work only with convex data, in order to prove convergence and stability of PPH.

The piecewise polynomial harmonic subdivision scheme is convergent and stable. The latter wasproved by Amat and Liandrat in [AL05] and their proof was the starting point of our research.

34

5.3 Quadratic, dyadic median-interpolating subdivision scheme

Let f be real-valued continuous function on a bounded interval I with positive Lebesgue measure(m(I) > 0). Then the median of f on I is defined by

med(f ; I) := supα : m(x : f(x) < α) ≤ 1

2m(I)

. (5.17)

The quadratic, dyadic median-interpolating subdivision scheme is given by the following interpolation-imputation algorithm:Let Γj = 2−jZ, k ∈ Z, and vj ∈ l∞(Z).

• (the interpolation step) For the interval Ijk = [2−jk, 2−j(k + 1)] find the quadratic polynomialpjk such that

med(pjk; Ijk+l) = vjk+l, l = −1, 0, 1. (5.18)

• (the imputation step)

(Smedvj)2k+l = med(pjk; I

j+12k+l), l = 0, 1. (5.19)

This scheme is a member of a family of subdivision schemes introduced by Donoho and Yu in[DY00]. In our paper we chose to work exactly with this median-interpolating subdivision scheme,because there is an explicit formula for the action of the scheme, and there are less parameters in thedyadic case than in the r-adic one, which simplifies the computations. From now on we will skipthe words “quadratic” and “dyadic” and will refer to (5.19) as median-interpolating subdivisionscheme.

The median-interpolating scheme is stationary, finite, and shift-invariant with order of polyno-mial reproduction 3. It is affine invariant, as well, due to

med(af + b; I) = a.med(f ; I) + b. (5.20)

For any data v ∈ l∞(Z) there exists a unique polynomial pk that satisfies (5.18). Unfortunately thenormal form of the polynomial a0x

2 + a1x + a2 is messy and bad for applications. Therefore (see[Osw04]), we prefer to write pk in the form

pk(x) = a(x− c)2 + b. (5.21)

Note that all quadratic polynomials but the linear ones can be written in such a way. The linearpolynomials are monotonic and for monotone functions the median coincides with the value ofthe function in the midpoint of the interval. Thus, in this case the median-interpolating schemecoincides with the midpoint-interpolating scheme Smid,2 defined via (4.36) on Z+ 1/2 instead of Z.We have already proven stability on the latter in Section. 4.2.2, therefore the linear polynomialsdoesn’t play any role in the stability analysis on the median-interpolating subdivision scheme andwithout loss of generality, we can assume that the interpolating polynomials pk(x) are of the form(5.21).

35

Since Smed is shift-invariant, we can derive a formula only for p0, which we will denote by pmed.In order to use symmetry and to simplify the computations and the final formulas, we will worknot on the grid Z but on 2Z− 1, i.e.,

vk = med(pmed; [2k − 1, 2k + 1]), k = −1, 0, 1.

We will denote by δ := 42v−1, by dk := 41vk, k = −1, 0, and by pmid the corresponding midpoint-interpolating polynomial (see (4.35)):

pmid(x) =x(x− 2)

8v−1 +

4− x2

4v0 +

x(x+ 2)

8v1.

It is a trivial exercise to show that for any p = a(x− c)2+ b and any bounded interval I = [α, β]

med(p; I) =

p(α+β

2),

∣∣∣c− α+β2

∣∣∣ ≥ |I|4,

p(c± |I|

4

)= p(α+β

2) + a

4

(|I|24− 4(α+β

2− c)2

),∣∣∣c− α+β

2

∣∣∣ < |I|4.

(5.22)

In particular,

vk = pmed(2k) +a

4ε2k = a(2k − c)2 + b+

a

4ε2k, k = −1, 0, 1, (5.23)

where εn := (1−4(n−c)2)+ ∈ [0, 1] depend on the center c of pmid and vanish for c /∈ (n−1/2, n+1/2).Therefore, using that 42

2pmed(−2) = 8a, we obtain

a =4δ

32 + ε−2 − 2ε0 + ε2. (5.24)

Equations (5.24) and (5.23) imply that one needs only the center c in order to determine themedian-interpolating polynomial pmed. Oswald suggested the following approximation for c.

Lemma 5.2 Let v = [v−1, v0, v1] be an arbitrary triple with 42v 6= 0.

a) We have a canonical representation

v = v0 +42v

2[1 + γ, 0, 1− γ], γ =

v−1 − v142v

. (5.25)

b) The center c of the polynomial pmed associated with v is a continuous, piecewise smooth(w.r.t. −∞ < −5

2< −3

2< −1

2< 1

2< 3

2< 5

2< +∞) strictly monotone function of γ. More

precisely, its inverse function γ = γ(c) is explicitly given by

γ(c) =32c+ ε−2 − ε2

32 + ε−2 − 2ε0 + ε2. (5.26)

Proof. The first part follows from (5.20), and the second part - from (5.23).

36

−4 −3 −2 −1 0 1 2 3 4−0.2

0

0.2

0.4

0.6

0.8

1

1.2(a)

S

med

Smid,2

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3(b)

c

x

Figure 7: Smid,2 vs. Smed: (a) limit functions on Dirac sequence (b)γ as a function on c

Remark 5.3 Note that, γ is the center of the corresponding polynomial pmid. Due to (5.22), pmed

and pmid, and thus Smed and Smid,2 are closely related (they even coincide, unlessv−1−v142v

∈ (−5/2,−3/2)∪ (−1/2, 1/2)∪ (3/2, 5/2)). Hence, we can think of the median-interpolatingsubdivision scheme as slightly perturbed midpoint-interpolating subdivision scheme.

Using (5.26), we can derive the inverse formula:

c(γ) =

0, γ = 0,4−√

16−15γ2

2γ, γ ∈ (−1

2, 0) ∪ (0, 1

2),

4(γ−1)+√

33γ2−30γ+1

2(1+γ), γ ∈ (3

2, 52),

4(γ+1)−√

33γ2+30γ+1

2(1−γ), γ ∈ (−5

2,−3

2),

γ, otherwise.

For the refined medians, (5.22) gives

v10 = pmed(−1

2) +

a

16ε−1/2, ε−1/2 := (1− 4(−1− 2c)2)+,

v11 = pmed(1

2) +

a

16ε1/2, ε1/2 := (1− 4(1− 2c)2)+.

(5.27)

Hence, using (5.22), (4.36) and (5.24)

v10 = pmid(−1

2)− α0,0δ =

5

32v−1 +

15

16v0 −

3

32v1 − α0,0δ,

v11 = pmid(1

2)− α1,0δ = −

3

32v−1 +

15

16v0 +

5

32v1 − α1,0δ,

(5.28)

37

whereas

α0,0 :=30ε0 + 5ε−2 − 3ε2 − 8ε−1/2

32(32− 2ε0 + ε−2 + ε2), α1,0 :=

30ε0 − 3ε−2 + 5ε2 − 8ε1/232(32− 2ε0 + ε−2 + ε2)

. (5.29)

The second index stays for the stencil, i.e., due to shift-invariance the corresponding formulas forv12k and v12k+1 will be the same as (5.28) with αi,k, i = 0, 1 instead of αi,0.

The median-interpolating subdivision scheme is convergent (see [DY00] for proof), but there areno results about stability, yet. Later in this paper we will provide strong numerical evidences thatthe scheme is stable.

6 Stability analysis on a univariate nonlinear subdivision

The following result is contained in the Bachelor’s thesis of Christian Kuhn:

Theorem 6.1 ( Christian Kuhn, see [Kuh05] ) Let S : l∞(Z)→ l∞(Z) be a (nonlinear) subdi-vision scheme. Decompose S into the ‘sum’ of a linear subdivision scheme Slin and a nonlinear subdi-vision scheme Snlin, in particular Svj = Slinv

j +Snlinvj. Let v0, d1, . . . , dJ and v0, d1, . . . , dJ be

two multiresolution decompositions for vJ , vJ ∈ l∞(Z). Let vj+1 :=Mvj = Svj +dj+1, where we de-note by M the above multiresolution. Assume that for some nonnegative constants C0, C1, K ∈ R,some ρ ∈ (0, 1), finite k, n ∈ N and all j ∈ N:

‖Slin‖ ≤ 1 (6.1)

‖Snlinvj − Snlinv

j‖ ≤ C0‖4k(vj − vj)‖ (6.2)

‖4k(vj+n − vj+n)‖ ≤ ρ‖4k(vj − vj)‖+ C1

n∑i=1

‖4k(dj+i − dj+i)‖ (6.3)

‖4k(vj+1 − vj+1)‖ ≤ K(‖4k(vj − vj)‖+ ‖4k(dj+1 − dj+1)‖). (6.4)

Then one has the following inequality

‖vJ − vJ‖ ≤ C2‖v0 − v0‖+ C3

J∑j=1

‖dj − dj‖ (6.5)

with constants C2, C3 which depend on k, n, ρ, C0, C1 and K but not on J . In particular the mul-tiresolution M associated to the subdivision scheme S is stable.

It is an attempt for generalizing the proof of the stability of PPH given by Amat and Liandrat[AL05], for a larger class of subdivision schemes (those that satisfy conditions (6.1)-(6.4)).

At first glance there are two main problems with Theorem 6.1:

• ‖Slin‖ ≤ 1

38

In section 4.2, we showed that in the linear case S ≡ Slin, uniform convergence implies stability,and that S is uniformly convergent only if it reproduces constants. The latter means that ‖S‖ ≥1. Moreover ‖S‖ = 1 if and only if all the non-trivial coefficients of S are positive. Therefore,Theorem 6.1 doesn’t work as a necessary condition for stability even in the linear case. E.g., since(4.36) gives ‖Smid,2‖ = 19/16 > 1, it does not apply to many schemes of interest.

• The nonuniqueness of the splitting of S into a linear and a nonlinear part.

There are many ways of splitting an operator S. For example, one could always take the trivialsplitting Slin = 0 and Snlin = S. This makes the theorem dubious, since one can never know whichsplitting is the most optimal one for a specific scheme.

These two obstacles are the main reason for putting efforts into improving the above theoremand finding the optimal conditions for it, i.e., those conditions that will (eventually) lead to an “ifand only if” statement for stability. A step in this direction is the following

Theorem 6.2 Let S : l∞(Z)→ l∞(Z) be a stationary (nonlinear) subdivision scheme. Letv0, d1, . . . , dJ and v0, d1, . . . , dJ be two arbitrary multiresolution decompositions forvJ , vJ ∈ l∞(Z). Let vj+1 := Mvj = Svj + dj+1, where we denote by M the above multiresolution.Assume that for some nonnegative constants C0, C1 ∈ R, some ρ ∈ (0, 1), and finite k, n ∈ N:

‖Sv0 − Sv0‖ ≤ ‖v0 − v0‖+ C0‖4k(v0 − v0)‖ (6.6)

‖4k(vn − vn)‖ ≤ ρ‖4k(v0 − v0)‖+ C1

n∑i=1

‖di − di‖. (6.7)

Then one has the following inequality

‖vJ − vJ‖ ≤ C2‖v0 − v0‖+ C3

J∑j=1

‖dj − dj‖ (6.8)

with constants C2, C3 which depend on k, n, ρ, C0 and C1 but not on J . In particular the subdivisionscheme S, and the multiresolution M associated to it are stable.

Proof. Since S is stationary, (6.6) and (6.7) give rise to

‖Svj − Svj‖ ≤ ‖vj − vj‖+ C0‖4k(vj − vj)‖

‖4k(vj+n − vj+n)‖ ≤ ρ‖4k(vj − vj)‖+ C1

n∑i=1

‖dj+i − dj+i‖

for every j ∈ N. Fix 0 ≤ j ≤ J − 1.

‖vj+1 − vj+1‖ = ‖Svj + dj+1 − (Svj + dj+1)‖ ≤ ‖Svj − Svj‖+ ‖dj+1 − dj+1‖≤ ‖vj − vj‖+ C0‖4k(vj − vj)‖+ ‖dj+1 − dj+1‖.

39

Applying this result iteratively J times we derive

‖vJ − vJ‖ ≤ ‖vJ−1 − vJ−1‖+ C0‖4k(vJ−1 − vJ−1)‖+ ‖dJ − dJ‖

≤ ‖v0 − v0‖+ C0

J−1∑i=0

‖4k(vi − vi)‖︸ ︷︷ ︸:=A

+J∑

i=1

‖di − di‖. (6.9)

Now to prove our theorem, it suffices to show that A can be estimated by the expression in theright-hand side of (6.8). Then from (6.7) it follows that

‖4k(vi − vi)‖ ≤ ρ‖4k(vi−n − vi−n)‖+ C1

n−1∑t=0

‖di−t − di−t‖.

Let s := bi/nc. Then by induction

‖4k(vi − vi)‖ ≤ ρs‖4k(vi−sn − vi−sn)‖+ C1

s−1∑r=0

ρr(r+1)n−1∑

t=rn

‖di−t − di−t‖, (6.10)

and after summation and using ρ < 1

A ≤ C(ρ)( n−1∑

j=0

‖4k(vj − vj)‖+J∑

i=1

‖di − di‖).

To estimate ‖4k(vj − vj)‖ for j = 1, . . . , n− 1, we use (6.6):

‖Svj − Svj‖ ≤ ‖vj − vj‖+ C0‖4k(vj − vj)‖ ≤ (2kC0 + 1)‖vj − vj‖,

which, applied j times, gives rise to

‖4k(vj − vj)‖ ≤ 2k‖Svj−1 − Svj−1‖+ 2k‖dj − dj‖≤ 2k(2kC0 + 1)‖vj−1 − vj−1‖+ 2k‖dj−1 − dj−1‖

≤ 2k(2kC0 + 1)j‖v0 − v0‖+ 2kj−1∑i=0

(2kC0 + 1)i‖dj−i − dj−i‖.

Thus,

‖4k(vj − vj)‖ ≤ 2k(2kC0 + 1)n−1‖v0 − v0‖+ 2k(2kC0 + 1)n−2

n−1∑i=1

‖di − di‖. (6.11)

Combining (6.9), (6.10), (6.11) and

J−1∑i=0

ρs ≤∞∑i=0

ρs =∞∑i=0

ρbi/nc = n∞∑i=0

ρi =n

1− ρ

40

we derive

‖vJ − vJ‖ ≤ ‖v0 − v0‖+ C0

J−1∑i=0

ρs(2k(2kC0 + 1)n−1‖v0 − v0‖+ 2k(2kC0 + 1)n−2

n−1∑j=1

‖dj − dj‖)

+ C1

J∑i=1

s−1∑r=0

ρr(r+1)n−1∑

t=rn

‖di−t − di−t‖+J∑

i=1

‖di − di‖

≤(1 +

2kC0(2kC0 + 1)n−1n

1− ρ

)‖v0 − v0‖+ C3

J∑j=1

‖dj − dj‖.

C3 is finite and doesn’t depend on J , because one can easily check that for any fixed 1 ≤ j ≤ Jthe coefficient in front of ‖dj − dj‖ is of the same type as the coefficient in front of ‖v0 − v0‖, i.e.,a finite sum of geometric series with respect to ρ times some uniformly bounded constants. Hence,for every j we can do the same trick as for the coefficient in front of ‖v0 − v0‖ and declare themaximal of them to be C3.

Remark 6.3 During the proof of Theorem 6.2 we have obtained an upper bound for the constantC2, namely

C2 ≤ 1 +2kC0(2

kC0 + 1)n−1n

1− ρ. (6.12)

Remark 6.4 If we set all the details dj and dj to be zero, Theorem 6.2 gives stability on thesubdivision scheme S itself.

Remark 6.5 If either S is linear or n = 1, stability of the subdivision scheme implies stability ofthe corresponding multiresolution.

This is obvious due to the following proposition.

Proposition 6.6 If a stationary linear subdivision scheme S is stable, then so is the multiresolutionM associated to it.

Proof. Since S is stable, there exists C0 ∈ R+, such that for any J ∈ N and any v0, v0 ∈ l∞(Z)

‖SJv0 − SJ v0‖ ≤ C0‖v0 − v0‖.

By induction it follows that

MJv0 = SJv0 +J∑

j=1

SJ−jdj.

Therefore

‖MJv0 −MJ v0‖ ≤ ‖SJv0 − SJ v0‖+J∑

j=1

‖SJ−jdj − SJ−j dj‖ ≤ C0

(‖v0 − v0‖+

J∑j=1

‖dj − dj‖).

41

On the other hand, Remark 6.5 cannot be extended to a nonlinear subdivision scheme S and n ≥ 2,because

M2v0 = S(Sv0 + d1) + d2 6= S2v0 + Sd1 + d2.

Remark 6.7 Theorem 6.2 is a generalization of Theorem 6.1.

Indeed, Theorem 6.1 is a corollary of Theorem 6.2 due to the following proposition.

Proposition 6.8 Let M be a multiresolution, associated to a stationary subdivision rule S thatsatisfies (6.1)-(6.4). Then M satisfies (6.6) and (6.7), as well.

Proof. Combining (6.1) and (6.2) and using triangle inequality for the norm, we have:

‖Svj − Svj‖ = ‖Slinvj + Snlinv

j − (Slinvj + Snlinv

j)‖ ≤ ‖Slin(vj − vj)‖+ ‖Snlinv

j − Snlinvj‖

≤ ‖Slin‖‖vj − vj‖+ C0‖4k(vj − vj)‖ ≤ ‖vj − vj‖+ C0‖4k(vj − vj)‖.

Now taking into account that S is stationary, we obtain that for every v0, v0 ∈ l∞(Z)

‖Sv0 − Sv0‖ ≤ ‖v0 − v0‖+ C0‖4k(v0 − v0)‖,

which is exactly (6.6).

Hence (6.1) ∪ (6.2)⇒(6.6).

Again, since S is stationary and ‖4kd‖ ≤ 2k‖d‖, ∀d ∈ l∞(Z), we have that (6.3)⇒(6.7). On the other hand, (4.28) and (4.30) give that any stable linear subdivision scheme satisfy (6.6)and (6.7) with k = 1. Therefore, our theorem is “stronger” than Kuhn’s one.

The proof of Theorem 6.2 shows that, analogously to the multivariate linear case (see (4.39)), inthe stability analysis on univariate nonlinear subdivision we can increase the level of abstractnessand use more general functions than the infinity norm of the k-th order divided difference.

Definition 6.9 Let D : l∞(Z)→ [0,∞) be bounded, i.e., there exists B > 0 such thatD(v) ≤ B‖v‖,∀v ∈ l∞(Z). We say that S is D-Lipschitz contractive, if there exists C > 0,ρ ∈ (0, 1) and n ∈ Z+ such that for all v, v ∈ l∞(Z)

‖Sv − Sv‖ ≤ ‖v − v‖+ CD(v − v), (6.13)

D(Snv − Snv) ≤ ρD(v − v). (6.14)

Corollary 6.10 A stationary subdivision scheme S is stable, if S is D-Lipschitz contractive forsome D : l∞(Z)→ [0,∞).

42

6.1 Analysis on conditions (6.6) and (6.7)

In this section we will try to analyze (6.6) and (6.7). We will prove that k from theorem 6.2 cannotbe bigger than the order of polynomial reproduction of the subdivision scheme S. We will provethat (6.6) with k ≤ 2 always holds in case of a linear subdivision scheme S with order of polynomialreproduction at least 2. Finally, we will give a counter example of a stable linear scheme S thatreproduces polynomials up to degree 2, for which (6.6) doesn’t hold for k = 3. With the help ofthese results, we will conclude that (6.6) and (6.7) are independent, i.e. no one implies the other.For example, (6.1)-(6.4) are not independent, as (6.4) follows from (6.1) and (6.2).

Although the problem about stability of linear schemes is completely solved, and Theorem 6.2 isexpected to help in the nonlinear case, a reasonable test for it is exactly the linear case. Therefore,in this section S will be a linear subdivision scheme.

Lemma 6.11 Let k ∈ N and v ∈ l∞(Z). Then 4kv ≡ 0 if and only if for any j ∈ N there existsp ∈ Πk−1(R) depending on j, such that v = pj, where Γj = 2−jZ.

Proof. ⇐) Fix j ∈ N. Let v = pj for some p ∈ Πk−1(R). Then from the standard course inNumerical Methods (see[Boy98]) we have that

4kvi = 4k(pj)i = k!2−jkp[xji , . . . , xji+k] = 2−kjp(k)(ξ) = 0

where p[x0, . . . , xn] denotes the divided difference of p with respect to the nodes x0, . . . , xn andxji < ξ < xji+k. Hence 4kv ≡ 0.

⇒) Fix j ∈ Z. Fix i ∈ Z. Then there exists a unique polynomial p ∈ Πk(R) such that

p((i+ l)2−j) = p(xji+l) = vji+l ∀l = 0, 1, . . . , k.

From the Newton’s formula (see [Boy98]) we have:

p(x) =k∑

l=0

p[xji , . . . , xji+l](x− x

ji )(x− x

ji+1) . . . (x− x

ji+l−1).

Let us calculate the leading coefficient ak of p(x):

ak = p[xji , . . . , xji+k] =

2jk4kvik!

= 0.

Hence, p ∈ Πk−1(R).Analogously, if q, r ∈ Πk(R) are the interpolation polynomials of v with respect to the nodes

xji+1, . . . , xji+k+1 and xji−1, . . . , x

ji+k−1, then q, r ∈ Πk−1(R). Now

p(xji+1+l) = vi+1+l = q(xji+1+l) ∀l = 0, . . . , k − 1 =⇒ p(x) ≡ q(x).

Analogously, p(x) ≡ r(x), and applying induction in both the positive and the negative directions,we obtain that pj ≡ v.

43

Remark 6.12 The above lemma is not true for a general grid! Take the nodes x0 = 0, x1 = 2, x2 =3, v = (0, 2, 3) = x0, v = (0, 2, 4). Then 42v = −1 6= 0 and 42v = 0, but v 6= p0 for any p ∈ Π1(R).

Proposition 6.13 Let S be a linear stationary subdivision scheme S that satisfies (6.6) and (6.7)with order of polynomial reproduction N . Then

0 ≤ k ≤ N. (6.15)

Proof. Take v0 ≡ 0. Hence for (6.7) we have

‖4kSnv‖ ≤ ρ‖4kv‖. (6.16)

Take v = p0 for some p ∈ Πk−1(R). Then, from Lemma 6.11, it follows that 4kv ≡ 0. (6.16) impliesthat 4kSnv ≡ 0 and again Lemma 6.11 gives Snv = q1 for some q ∈ Πk−1(R).

Therefore, Sn reproduces polynomials of degree k− 1. Finally, due to Proposition 2.4 in [JZ04],and that for every n ∈ N the order of polynomial reproduction of Sn and S coincides, we obtaink ≤ N .

Having proved the above statement, one could dream of proving (6.6) with N instead of k.Then, since ‖4Nv‖ ≤ 2N−k‖4kv‖, (6.6) will be true for any k ≤ N , and thus (6.7) will imply (6.6).Unfortunately, as it is shown below, this is not true even for linear subdivision schemes and k ≥ 3.

Lemma 6.14 Let S be a linear stationary subdivision scheme with order of polynomial reproductionat least 2. Then, there exists a linear scheme S with positive coefficients, such that

Spj = Spj

for all p ∈ Π1(R) and all j ∈ N, where Γj = 2−jZ.

Proof. Since S reproduces polynomials of degree 1 S(x)0 = (x− c)1 for some c ∈ R. Take S tobe:

(Sv)2k = cvk−[c]−1 + (1− c)vk−[c]

(Sv)2k+1 = cvk−[c]−1 + vk−[c]

2+ (1− c)

vk−[c] + vk−[c]+1

2

Obviously S has positive coefficients. Let f = (x)0, i.e., fk = k ∀k ∈ Z. Then:

(Sf)2k = c(k − [c]− 1) + (1− c)(k − [c]) = k − [c]− c = k − c = (Sf)2k

(Sf)2k+1 =f2k + f2k+2

2= k +

1

2− c = (Sf)2k+1.

Hence Sf = Sf . Due to the linearity of S, it follows that for every p ∈ Π1(R) Sp0 = Sp0 and, sinceboth S and S are stationary, the Lemma is proved. Now we are ready to prove that for linear schemes, (6.6) with k = 2 always holds.

44

Proposition 6.15 Let S be a linear stationary subdivision rule with order of polynomial reproduc-tion at least 2. Then there exists K ∈ N such that for all v ∈ l∞(Z)

‖Sv‖ ≤ ‖v‖+K‖42v‖. (6.17)

Proof. Let S be as in Lemma 6.14. Then (S− S)1 = 0 and Lemma 4.7 implies that there existsa bounded linear operator T such that for each v ∈ l∞(Z)

(S − S)v = T4v.

Moreover, from S(x)0 = S(x)0 and Lemma 6.11 it follows that 0 = (S − S)(x)0 = T4(x)0 = T1.

Applying Lemma 4.7 one more time, there exists a bounded linear operator T with

(S − S)v = T4v = T42v

for every v ∈ l∞(Z).

‖Sv‖ = ‖Sv + (S − S)v‖ ≤ ‖Sv‖+ ‖(S − S)v‖≤ ‖S‖‖v‖+ ‖T‖‖42v‖ ≤ ‖v‖+K‖42v‖.

For the last step, we use that T is bounded and that since S1 = 1 and S is positive, ‖S‖ = 1.

Remark 6.16 The above statement is not true for k ≥ 3. For example, take Smid,2, fix ε > 0 andlet v ∈ l∞(Z) be symmetric with respect to −1/2 (i.e., v−n = vn−1, ∀n ∈ N) and such that

vi = −(i+1

2)2, i = −1, 0, 1.

Let vn ≤ vn−1,∀n ∈ N and ‖43v‖ = ε. To achieve that, we can follow the following algorithm:

• Step 1: Start with n = 0.

• Step 2: While (42vn−1 < 0) do42vn = min(42vn−1 + ε, 0);n = n+ 1

.

• Step 3: While (41vn < 0) do41vn+1 = 41vn+2 = min(41vn + ε, 0);n = n+ 2

.

The case n < 0 follows by symmetry. Now let A = A(ε) := ‖v‖ and w := A+v. Then wi ≥ 0, ∀i ∈ Z,‖w‖ = w0 = A− 1

4, and ‖43w‖ = ‖43v‖ = ε.

On the other hand,

‖Smid,2w‖ ≥ |(Smid,2w)0| = |A+ (Smid,2v)0| = A− 1

16.

Hence,

‖Smid,2w‖ − ‖w‖ ≥3

16=

3

16ε‖43w‖,

and letting ε→ 0 we see that (6.6) doesn’t hold for Smid,2 and k = 3 with uniformly bounded constantC0.

45

Note that the above algorithm can be applied to any linear subdivision scheme that reproduces con-stants and has at least one negative coefficient! Thus, we can conclude that the only “meaningful”values of k are 1 and 2.

Corollary 6.17 (6.6) does not imply (6.7)!

Proof. In [AL05] it is shown that (6.7) with k = 1 does not hold for the PPH subdivision schemefor any n ∈ N. On the other hand, PPH is a second-order perturbation of a linear subdivisionscheme (see [AL05]). Hence, Proposition 6.15 is applicable, and (6.6) holds for k = 1.

Corollary 6.18 (6.7) does not imply (6.6)!

Proof. Follows directly from Remark 6.16 and Example 4.8 with S = Smid,2, k = 3, and n = 1.

6.2 The Differential Approach

In this section we work only with dyadic schemes and we try to develop an algorithm for proving(6.7). In general, (6.7) is much harder to prove than (6.6). It is a generalization of the contractingproperty (4.28) from the linear case, which we interpreted as:“The spectral radius of the matrix, corresponding to the first derived scheme, is less than 1”.

Remark 5.1 indicates that the “matrix” approach may be extended to the nonlinear case as well.The main problem is, that due to nonlinearity, we are no longer interested in the behavior of thesingle sequence

4kSJv0

∞J=0

, but we are interested in the difference between this sequence and4kSJ v0

∞J=0

. In other words, we need to study the properties of the “derivative” of the derivedscheme, rather than the properties of the scheme itself.In order to be able to differentiate the k-th derived scheme, the quasilinear map Φ should be afunction of 4kv instead of v. For k = 1 (5.10) and (5.15) show that this is true. Moreover, we cancompletely characterize the class of subdivision schemes S with the above property.

Lemma 6.19 Let S be a (nonlinear) offset invariant subdivision scheme (i.e., S(v + c) = Sv +c,∀v ∈ l∞(Z), c ∈ R), with quasilinear representation

Sv = Φ(v)v.

Then, there exists a quasilinear map Φ such that

Sv = Φ(41v)v. (6.18)

Proof. From the definition of quasilinearity it follows that the family of linear operators Φ(v) :v ∈ l∞(Z) is quasilinear with respect to S if and only if for every v ∈ l∞(Z) Φ(v) has support notbigger than the one of S and the action of S and Φ(v) on v coincide. Let us define the relation ∼in the space of all bounded real sequences by

v ∼ w ⇐⇒ ∃c ∈ R : v = w + c. (6.19)

46

Obviously this is an equivalence relation, and let us denote the equivalence class of a given sequencev by [v]. Then 41v = 41w ⇔ w ∈ [v]. Now, from each equivalence class in l∞(Z) take onerepresentative vα and define

Φ(41w) := Φ(vα), ∀w ∈ [vα]. (6.20)

Since S is offset invariant, (6.18) is satisfied. Assume that S is dyadic, finite, offset and shift invariant. Let Φ(v) and thus Φ(v) reproducesconstants for any v ∈ l∞(Z). (We can assume this, because the offset invariance of S impliesS1 = 1.) Proposition 4.3 gives rise to

41Sv = Φ1(41v)41v, v ∈ l∞(Z). (6.21)

Let us denote by xj := 41Sjv. We have

Φ1(xj−1)xj−1

i =∑l∈I

ai−2l(xj−1)xj−1

l , (6.22)

where I is the support of S and ai−2l is a function only of xj−1l , l ∈ I.

Assume for simplicity that ai−2l are differentiable. Then

d(ai−2l(x

j−1))=∑s∈I

∂ai−2l

∂xj−1s

(xj−1)dxj−1s . (6.23)

Hence,

dxji = d(Φ1(x

j−1)xj−1i

)= d

(∑l∈I

ai−2l(xj−1)xj−1

l

)=∑l∈I

(d(ai−2l(x

j−1))xj−1l + ai−2l(x

j−1)dxj−1l

)=∑l∈I

(∑s∈I

∂ai−2l

∂xj−1s

(xj−1)xj−1l dxj−1

s + ai−2l(xj−1)dxj−1

l

)=∑l∈I

(∑s∈I

∂ai−2s

∂xj−1l

(xj−1)xj−1s + ai−2l(x

j−1)

)dxj−1

l

=∑l∈I

ti,l(xj−1)dxj−1

l .

We showed that there exists a linear, local (but not shift invariant!) data dependent operatorT : l∞(Z)→ l∞(Z), such that

d(xj)= T (xj−1)d

(xj−1

), ∀j ∈ N. (6.24)

Therefore, by recursion it follows that

d(xj)= T (xj−1)T (xj−2) . . . T (x0)d

(x0), ∀j ∈ N. (6.25)

47

Remark 6.20 The functions ai−2l : l∞(Z) → R and, thus the operator T do not depend on thequasilinear representation Φ and are uniquely determined by S!

Now let v0, v0 ∈ l∞(Z). Define x(t)j : [0, 1]→ l∞(Z)

x(t)j := 41Sj(tv0 + (1− t)v0). (6.26)

Then,

‖41(Snv0 − Snv0)‖ = ‖x(1)n − x(0)n‖ = ‖∫ 1

0

d(x(t)n

)‖

= ‖∫ 1

0

T (x(t)n−1)T (x(t)n−2) . . . T (x(t)0)d(x(t)0

)‖

≤ supv∈l∞(Z)

‖T (41Sn−1v)T (41Sn−2v) . . . T (41v)‖‖41(v0 − v0)‖.

(6.27)

For the last result, we needed T to be only Lipschitz continuous, and thus the coefficients ai−2l notto be differentiable, but just in the Sobolev space W 1

∞(R). Hence, we proved the following:

Theorem 6.21 Let S be local, offset and shift invariant, stationary subdivision scheme withai−2l ∈ W 1

∞(R), where ai−2l are defined by (6.22). Let S1 be the first derived scheme of S. Thena sufficient condition for (6.7) to hold for S with k = 1 and dj = dj ≡ 0 is: the joint spectralradius of T with respect to S1 is less than 1, i.e.,

ρ(T, S1) := lim infj→∞

supv∈l∞(Z)

‖T (Sj−11 v)T (Sj−2

1 v) . . . T (v)‖1/j < 1. (6.28)

Furthermore, we can extend the above result also for the multiresolution M associated to S:

Theorem 6.22 Let S be local, offset and shift invariant, stationary subdivision scheme withai−2l ∈ W 1

∞(R), where ai−2l are defined by (6.22). Then a sufficient condition for (6.7) to holdfor the multiresolution operator M associated to S with k = 1 is: the joint spectral radius of Twith respect to M1 is less than 1, i.e.,

ρ(T,M1) := lim infj→∞

supv,d1,...,dj−1∈l∞(Z)

‖T (M j−11 v)T (M j−2

1 v) . . . T (v)‖1/j < 1, (6.29)

where Mk1 v = S1(M

k−11 v) +41dk, ∀k ≤ j − 1.

Proof. It is a standard fact in functional analysis that (6.29) is equivalent to existence of aconstant C > 1 and ρ ∈ (0, 1) such that for arbitrary j ≥ 0 one has

‖T (M j−11 v)T (M j−2

1 v) . . . T (v)‖ ≤ Cρj. (6.30)

Let v0, v0 ∈ l∞(Z). Denote by yj := 41M jv. Then (6.24) gives rise to

d(yj)= T (yj−1)d

(yj−1

)+ d

(41dj

)∀j ∈ N, (6.31)

48

and thus, by recursion

d(yj)= T (yj−1)T (yj−2) . . . T (y0)d

(y0)+ T (yj−2) . . . T (y0)d

(41dj

)+ · · ·+ d

(41d1

). (6.32)

Now, define y(t)j := 41M j(tv0 + (1− t)v0) and, analogously to (6.27), using the additive propertyof the integral, (6.30), and that x0 ≡ y0 we derive

‖41(Mnv0 −Mnv0)‖ ≤ µ‖41(v0 − v0)‖+ Cn∑

j=1

‖41(dj − dj)‖

≤ µ‖41(v0 − v0)‖+ 2Cn∑

j=1

‖dj − dj‖.(6.33)

Here n := minj ∈ Z : Cρj < 1 and µ := Cρn ∈ (0, 1). Proposition 6.21 and Proposition 6.22 can be straightforwardly generalized for arbitrary k ∈ Z andr-adic subdivision scheme. Let us denote by Ωk the class of nonlinear, shift-invariant subdivisionschemes S that allow representation

Sv = Φ(4kv)v =⇒ 4kSv = Sk4kv = Φ(4kv)4kv =: Ψ(4kv),

where Φ is an operator-valued function, and Ψ has uniformly bounded generalized partial derivatives(i.e., Ψ ∈ W 1

∞).

Corollary 6.23 (Spectral radius version of Theorem 6.2) Let S ∈ Ωk such that

‖Sv − Sv‖ ≤ ‖v − v‖+ C0‖4k(v − v)‖, ∀v, v ∈ l∞(Z).

Let T be the linear data dependent operator that satisfies

d(4kSv

)= T (4kv)d

(4kv

).

Then if Sk is the kth derived scheme of S and Mk the kth derived scheme of the correspondingmultiresolution operator M

(i) S is stable if ρ(T, Sk) < 1.

(ii) M is stable if ρ(T,Mk) < 1.

Sometimes (PPH for example) the spectral radius of T

ρ(T ) := lim infj→∞

sup(u0,u1,...,uj−1)∈(l∞(Z))j

‖T (uj−1)T (uj−2) . . . T (u0)‖1/j (6.34)

is smaller than 1 and we do not need to consider Sk or Mk. On the other hand, Mk depends on thedetails and

ρ(T,Mk) = ρ(T )

unless we put some restrictions on the dj’s. Section 2.2 suggests that, in general, the details cannotbe arbitrary, but the question what is the admissible class of details for a given mutliresolution isstill open and heavily depends on the corresponding subdivision scheme S.

49

7 Applications of Theorem 6.2

7.1 PPH

One of the easiest applications of the newly developed machinery is the PPH scheme. The beginningof the proof of Theorem 1 in [AL05] says that for any v0, v0 ∈ l∞(Z)

‖SPPHv0 − SPPH v

0‖ ≤ ‖v0 − v0‖+ 1

4‖42(v0 − v0)‖, (7.1)

and thus, (6.6) holds.

(5.15), together with the fact that the function

Ck(v) =sgn(42vk−142vk) + 1

8

42vk42vk−1 +42vk

defined in (5.14) is continuous and almost everywhere differentiable in R2 allows us to use the

“differential” approach. For convenience, we will denote by 4k := 42vk, by χk := sgn(4k−14k)+1

2,

and by 4k := (42SPPHv)k. Note that χk is 1 for convex (concave) data, and 0 otherwise.Direct computations give:

42k =χk

2

4k−14k

4k−1 +4k

(7.2)

42k+1 =4k

2− χk

4

4k−14k

4k−1 +4k

− χk+1

4

4k4k+1

4k +4k+1

. (7.3)

Therefore, for the entries of the differential matrix T (v) = ti,j(v) we obtain

d(42k) =χk

2

(4k)2

(4k−1 +4k)2d(4k−1) +

χk

2

(4k−1)2

(4k−1 +4k)2d(4k), (7.4)

d(42k+1) =

(1

2− χk

4

(4k−1)2

(4k−1 +4k)2− χk+1

4

(4k+1)2

(4k +4k+1)2

)d(4k)

− χk

4

(4k)2

(4k−1 +4k)2d(4k−1)−

χk+1

4

(4k)2

(4k +4k+1)2d(4k+1)

(7.5)

Hence, for any k ∈ Z the only nontrivial entries of T are t2k,k−1, t2k,k, t2k+1,k, t2k+1,k−1 and t2k+1,k+1.Moreover,

0 ≤ t2k,k−1, t2k,k, t2k+1,k ≤1

2, − 1

4≤ t2k+1,k−1, t2k+1,k+1 ≤ 0, (7.6)

and∑i∈Z

|t2k,k+i| =χk

2

((4k)

2

(4k−1 +4k)2+

(4k−1)2

(4k−1 +4k)2

)<

1

2(7.7a)

∑i∈Z

|t2k+1,k+i| =1

2− χk

4

(4k−1)2

(4k−1 +4k)2+χk

4

(4k)2

(4k−1 +4k)2− χk+1

4

(4k+1)2

(4k +4k+1)2+χk+1

4

(4k)2

(4k +4k+1)2

< 1.

(7.7b)

50

This is not enough for claiming stability of PPH, since the infinity norm of T (v) is 1. Indeed,take ε > 0, 4k−1 = 4k+1 = ε, and 4k = 1− ε. Then∑

i∈Z

|t2k+1,k+i| = 1− ε −−→ε→0

1.

On the other hand, it suffices to consider two iterations of T :

Proposition 7.1 For any v, w ∈ l∞(Z)

‖T (w)T (v)‖ ≤ 3

4. (7.8)

Proof. Let us, for simplicity, denote the entries of T (v) and T (w) by ti,j and ti,j, respectively.Then

‖T (w)T (v)‖ = max0≤i≤3

∑j∈Z

∑l∈Z

|t4k+i,ltl,j| ≤ max0≤i≤3

∑l∈Z

|t4k+i,l|(∑

j∈Z

|tl,j|). (7.9)

For even i = 2i′ (7.9) and (7.7a) give∑l∈Z

|t4k+i,l|(∑

j∈Z

|tl,j|)≤∑l∈Z

|t2(2k+i′),l| ≤1

2, (7.10)

while for odd i = 2i′ + 1∑l∈Z

|t4k+i,l|(∑

j∈Z

|tl,j|)≤ |t2(2k+i′)+1,2k+i′−1|+

1

2|t2(2k+i′)+1,2k+i′|+ |t2(2k+i′)+1,2k+i′+1| ≤

3

4. (7.11)

For the last inequality we used the estimate

|t2k+1,k−1|+1

2|t2k+1,k|+ |t2k+1,k+1| ≤

3

4.

Its proof follows directly from (7.6). Therefore, the PPH multiresolution satisfies conditions (6.6) and (6.7) with constants

k = 2; n = 2; ρ =3

4,

and thus, is stable!

7.2 WENO

In this section we will sketch a proof of stability of WENO subdivision scheme. We will use thenotation developed in Section 5.1. Due to practical reasons and the fact that SWENO is not affineinvariant, we use a weaker version of stability for WENO (see [CDM03]), namely we allow theconstants C2 and C3 in (6.8) to depend in a continuous nondecreasing way on max ‖v0‖, ‖v0‖.

51

Lemma 7.2 Let v0, v0 ∈ l∞(Z). Then

‖SWENOv0 − SWENOv

0‖ ≤ ‖v0 − v0‖+ C0‖41(v0 − v0)‖, (7.12)

where C0 depends in a continuous nondecreasing way on max ‖v0‖, ‖v0‖.

Proof. In [CDM03] it is proved that SWENO continuously depends on data, i.e., with respect tothe notation in (5.10) ‖Φ(v0)−Φ(v0)‖ ≤ B‖v0−v0‖, where B depends in a continuous nondecreasingway on max ‖v0‖, ‖v0‖. Therefore,

‖Φ(41v0)− Φ(41v0)‖ ≤ C‖41(v0 − v0)‖, (7.13)

where, again, C depends in a continuous nondecreasing way on max ‖v0‖, ‖v0‖.Now, (5.10) and (7.13) give

‖SWENOv0 − SWENOv

0‖ = ‖Φ(41v0)v0 − Φ(41v0)v0‖≤ ‖Φ(41v0)v0 − Φ(41v0)v0‖+ ‖Φ(41v0)v0 − Φ(41v0)v0‖≤ ‖Φ(41v0)v0 − Φ(41v0)v0‖+ C‖v0‖‖41(v0 − v0)‖.

(7.14)

On the other hand, the linear schemes Sl, Sc, and Sr, defined in Section 5.1 reproduce constants,and due to (4.30) there exist Cl, Cc, Cr, such that

‖Slv0 − Slv

0‖ ≤ ‖v0 − v0‖+ Cl‖41(v0 − v0)‖;‖Scv

0 − Scv0‖ ≤ ‖v0 − v0‖+ Cc‖41(v0 − v0)‖;

‖Srv0 − Srv

0‖ ≤ ‖v0 − v0‖+ Cr‖41(v0 − v0)‖.

Let D := max(Cl, Cc, Cr). Hence, if Φ(41v0) = αlSl + αcSc + αrSr, with αl, αc, αr ≥ 0 andαl + αc + αr = 1, then

‖Φ(41v0)v0 − Φ(41v0)v0‖ ≤ αl‖Slv0 − Slv

0‖+ αc‖Scv0 − Scv

0‖+ αr‖Srv0 − Srv

0‖≤ ‖v0 − v0‖+D‖41(v0 − v0)‖.

(7.15)

Finally, combining (7.14) and (7.15) we derive (7.12) with C0 := D + C‖v0‖. (5.9) shows that 41SWENOv = (αlS

1l + αcS

1c + αrS

1r )41v =: SWENO,141v. For the first derived

schemes of Sl, Sc, and Sr we have the explicit formulas

(S1l v)2k = −

1

16vk−2 +

1

4vk−1 +

5

16vk (S1

l v)2k+1 =1

16vk−2 −

1

4vk−1 +

11

16vk

(S1c v)2k =

1

16vk−1 +

1

2vk −

1

16vk+1 (S1

c v)2k+1 = −1

16vk−1 +

1

2vk +

1

16vk+1

(S1rv)2k =

11

16vk −

1

4vk+1 +

1

16vk+2 (S1

rv)2k+1 =5

16vk +

1

4vk+1 −

1

16vk+2.

Let us denote by x := 41v, and by x1 := 41SWENOv. Then

x12k = −αk,1

16xk−2 +

4αk,1 + αk,2

16xk−1 +

5αk,1 + 8αk,2 + 11αk,3

16xk −

αk,2 + 4αk,3

16xk+1 +

αk,3

16xk+2

x12k+1 =αk,1

16xk−2 −

4αk,1 + αk,2

16xk−1 +

11αk,1 + 8αk,2 + 5αk,3

16xk +

αk,2 + 4αk,3

16xk+1 −

αk,3

16xk+2.

(7.16)

52

Note that the weights αk,i3i=1 depend only on vk+j, j = −2, . . . 3. Moreover, since WENO is offsetinvariant, Lemma 6.19 says that the weights are functions on xk+j, j = −2, . . . , 2. Assume thatfor every i αk,i ∈ W 1

∞ (for example this is true, when the cost function φ is differentiable). Thenwe can apply the “differential approach” with k = 1 and for the “even” entries of the matrix T weobtain

t2k,k−2 =∂αk,1

∂xk−2

[− xk−2

16+xk−1

4+

5xk16

]+∂αk,2

∂xk−2

[xk−1

16+xk2− xk+1

16

]+∂αk,3

∂xk−2

[11xk16− xk+1

4+xk+2

16

]− αk,1

16

t2k,k−1 =∂αk,1

∂xk−1

[− xk−2

16+xk−1

4+

5xk16

]+∂αk,2

∂xk−1

[xk−1

16+xk2− xk+1

16

]+∂αk,3

∂xk−1

[11xk16− xk+1

4+xk+2

16

]+

4αk,1 + αk,2

16

t2k,k =∂αk,1

∂xk

[− xk−2

16+xk−1

4+

5xk16

]+∂αk,2

∂xk

[xk−1

16+xk2− xk+1

16

]+∂αk,3

∂xk

[11xk16− xk+1

4+xk+2

16

]+

5αk,1 + 8αk,2 + 11αk,3

16

t2k,k+1 =∂αk,1

∂xk+1

[− xk−2

16+xk−1

4+

5xk16

]+∂αk,2

∂xk+1

[xk−1

16+xk2− xk+1

16

]+∂αk,3

∂xk+1

[11xk16− xk+1

4+xk+2

16

]− αk,2 + 4αk,3

16xk+1

t2k,k+2 =∂αk,1

∂xk+2

[− xk−2

16+xk−1

4+

5xk16

]+∂αk,2

∂xk+2

[xk−1

16+xk2− xk+1

16

]+∂αk,3

∂xk+2

[11xk16− xk+1

4+xk+2

16

]+αk,3

16.

The “odd” entries are similar due to symmetry and we will not write them explicitly.

αk,1 =ak,1

ak,1 + ak,2 + ak,3=⇒ ∂αk,1

∂xj=

∂ak,1∂xj

ak,2 +∂ak,1∂xj

ak,3 − ak,1 ∂ak,2∂xj− ak,1 ∂ak,3∂xj

(ak,1 + ak,2 + ak,3)2.

Using (5.8) we derive

∂ak,i∂xj

=−2∂bk,i/∂xjε+ bk,i

ak,i, ∀i = 1, 2, 3, ∀j = k − 2, . . . , k + 2

=⇒ ∂αk,1

∂xj=

[∂bk,2/∂xj

ε+bk,2− ∂bk,1/∂xj

ε+bk,1

]2ak,1ak,2 +

[∂bk,3/∂xj

ε+bk,3− ∂bk,1/∂xj

ε+bk,1

]2ak,1ak,3

(ak,1 + ak,2 + ak,3)2.

53

All these formulas are very “ugly” and difficult to analyze. Therefore, it is practically impossibleto estimate the joint spectral radius of T and to investigate stability of the associated multiresolutionoperator. However, since the WENO scheme is uniformly convergent, we have that

41SjWENOv −−−→

j→∞0, ∀v ∈ l∞(Z).

This, together with our assumption that all the partial derivatives of the weights are uniformlybounded (αk,i ∈ W 1

∞) implies that T (SjWENOv) −−−→

j→∞SWENO,1(S

jWENOv) and thus

ρ(T, SWENO,1) = ρ(SWENO,1). (7.17)

In [CDM03] Cohen, Dyn, and Matei have proven that

supu,w∈l∞(Z)

‖SWENO,1(u)SWENO,1(w)‖ < 1,

which is enough for claiming stability of the WENO subdivision scheme.

7.3 Quadratic, dyadic median-interpolating subdivision scheme

The aim in this section is to sketch the proof of the stability of median subdivision in the dyadiccase. We heavily rely on the explicit formulas from [Osw04], see also Section 5.3 above.

To set notation, let x = 41v, y = 41Smedv, i.e., y = Smed,1x, and δ = 42v = 41x. Also,denote by c = ck the sequence of centers of median-interpolating polynomials associated withvk−1, vk, vk+1, defined via (5.21). (Recall that ck = ∞ is acceptable and corresponds to the caseδk−1 = 0.) Finally, let γ = γk be the sequence of centers of midpoint-interpolating polynomialsassociated with vk−1, vk, vk+1 (see Lemma 5.2), i.e., γk = −xk+xk−1

δk−1.

We will apply Theorem 6.2 with k = 1.

Lemma 7.3‖Smedv − Smedv‖ ≤ ‖v − v‖+K‖41(v − v)‖,

for any v, v ∈ l∞(Z), where K is a uniform bounded constant that does not depend on v and v.

Proof. The proof is in the appendix. For (6.7), we will apply the “differential” approach. From (5.28) and (5.29) it follows that for

any k ∈ Z

y2k =1

4(xk−1 + xk) + δk−1α2(ck), (7.18)

y2k+1 =−3xk−1 + 22xk − 3xk+1

32+ δk−1α1(ck)− δkα0(ck+1), (7.19)

54

−2 0 2−0.01

0

0.01

0.02

0.03

α0(c)

−2 0 2−0.01

0

0.01

0.02

0.03

α1(c)

Figure 8: α0(c) and α1(c).

where αl(ck) := αl,k, l = 0, 1 and α2(ck) := α0(ck)− α1(ck).Hence, the nonlinear terms in (7.18) and (7.19) are of the form δk−1α(ck) for some piecewise smoothα. Formal differentiation gives

d(δk−1α(ck)

)= α(ck)dδk−1 + δk−1α

′(ck)c′(γk)dγk

= α(ck)(dxk − dxk−1) + δk−1α′(ck)

γ′(ck)dγk

= α(ck)(dxk − dxk−1) + δk−1α′(ck)

γ′(ck)

(− dxk + dxk−1

δk−1

+xk + xk−1

δ2k−1

dδk−1

)=(− α(ck)−

α′(ck)

γ′(ck)+α′(ck)

γ′(ck)γ(ck)

)dxk−1 +

(α(ck)−

α′(ck)

γ′(ck)− α′(ck)

γ′(ck)γ(ck)

)dxk.

This leads to the following explicit formulas for the entries tk,l of T (x):

dy2k =

[1

4−(α2 +

α′2

γ′(1− γ)

)(ck)

]dxk−1+

[1

4+(α2 −

α′2

γ′(1 + γ)

)(ck)

]dxk (7.20)

dy2k+1 =

[− 3

32−(α1 +

α′1

γ′(1− γ)

)(ck)

]dxk−1+

[− 3

32−(α0 −

α′0

γ′(1 + γ)

)(ck+1)

]dxk+1

+

[11

16+(α1 −

α′1

γ′(1 + γ)

)(ck) +

(α0 +

α′0

γ′(1− γ)

)(ck+1)

]dxk.

(7.21)

Lemma 7.4 For any k ∈ Z ∑l∈Z

∣∣∣t2k,l(x)∣∣∣ ≤ 0.6250 =: Ceven,∑l∈Z

∣∣∣t2k+1,l(x)∣∣∣ ≤ 1.1430 =: Codd.

(7.22)

55

Proof. From (5.29) and fig. 8 we see that α1(−c) = α0(c), and from Lemma 5.2 and fig. 7(b) wesee that γ is an odd function. Now, using all these symmetries and the small values of the α’s, itis not hard to observe that∑

l∈Z

∣∣∣t2k,l(x)∣∣∣ =(1

4−(α2 +

α′2

γ′(1− γ)

)(ck)

)+

(1

4+(α2 −

α′2

γ′(1 + γ)

)(ck)

)

≤ supc

(1

2− 2α′

2(c)

γ′(c)

)=

1

2− 2 inf

c

(α′2(c)

γ′(c)

)= 0.6250,

and ∑l∈Z

∣∣∣t2k+1,l(x)∣∣∣ =( 3

32+(α1 +

α′1

γ′(1− γ)

)(ck)

)+

(3

32+(α0 −

α′0

γ′(1 + γ)

)(ck+1)

)

+

(11

16+(α1 −

α′1

γ′(1 + γ)

)(ck) +

(α0 +

α′0

γ′(1− γ)

)(ck+1)

)

≤ supc,c′

(7

8+ 2(α1 −

α′1

γ′γ)(c) + 2

(α0 −

α′0

γ′γ)(c′)

)=

7

8+ 4 sup

c

((α0 −

α′0

γ′γ)(c)

)= 1.1430.

The tedious analytical verification is left as an exercise. From this Lemma we see, first, that ‖T (x)‖ = Codd > 1 and Theorem 6.2 doesn’t hold for n = 1.

On the other hand, since CevenCodd < 1, it is obvious that (see (7.10))∣∣(T (x′)T (x)w)2k

∣∣ ≤ CevenCodd‖w‖, x′, x, w ∈ l∞(Z). (7.23)

Furthermore, (7.23) holds for 4k + 1, as well. To show this, we need the following fact.

Lemma 7.5 For any x,w ∈ l∞(Z) and k ∈ Z

|(T (x)w)2k − (T (x)w)2k−1| = |41(T (x)w)2k−1| ≤ 1.1965‖w‖. (7.24)

Proof. First, let us observe some trivial properties of the entries of T (x). For each k ∈ Z,t2k,k−1, t2k,k, t2k+1,k−1 are functions only on ck, t2k+1,k+1 is a function only on ck+1 and one can verifythe following relations:

t2k,k−1(−ck) = t2k,k(ck); t2k+1,k−1(−ck) = t2k−1,k(ck); t2k+1,k+1(−ck+1) = t2k+3,k(ck+1).

Moreover, we can split t2k+1,k into two parts:

t−2k+1,k(ck) :=11

32+ (α1 −

α′1

γ′(1 + γ))(ck); t+2k+1,k(ck+1) :=

11

32+ (α0 +

α′0

γ′(1− γ))(ck+1).

Simple Matlab routines give

t2k,k−1, t2k,k ∈ [0.1684, 0.3636]; t2k+1,k−1, t2k+1,k+1 ∈ [−0.2946,−0.02]; t2k+1,k ∈ [0.4687, 1.0669].

56

−2 0 2

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

(dα2/dγ)(c)

−2 0 2

−0.04

−0.02

0

0.02

0.04

0.06

(α0−γdα

0/dγ)(c)

Figure 9:α′2

γ′ (c) and(α0 − α′

0

γ′ γ)(c).

Now, let us denote by w1 := T (x)w. Then (7.20) and (7.21), together with the above estimations,give rise to

|w12k − w1

2k−1| = | − t2k−1,k−2wk−2 + (t2k,k−1 − t2k−1,k−1)wk−1 + (t2k,k − t2k−1,k)wk|≤(− t2k−1,k−2 + t2k−1,k−1 − t2k,k−1 + t2k,k − t2k−1,k

)‖w‖

≤(supck−1

(t−2k−1,k−1 − t2k−1,k−2) + supck

(t+2k−1,k−1 − t2k,k−1 + t2k,k − t2k−1,k))‖w‖

≤ (0.5715 + 0.6250)‖w‖ = 1.1965‖w‖.

(7.25)

Therefore

w24k+1 = (T (x′)T (x)w)4k+1 = t4k+1,2k−1w

12k−1 + t4k+1,2kw

12k + t4k+1,2k+1w

12k+1. (7.26)

Lemma 7.4 and the fact that t4k+1,2k is positive and t4k+1,2k−1, t4k+1,2k+1 are negative implies that|w2

4k+1|/‖w‖ ≤ CevenCodd whenever w12k doesn’t have the opposite sign of both w1

2k−1 and w12k+1 (the

details are left to the reader). Now, without loss of generality, let us assume that w12k is positive

and w12k−1, w

12k+1 are negative. Let w1

2k = a‖w‖, a ∈ [0, 0.625]. Applying Lemma 7.5 we derive

|w24k+1| ≤

(− t4k+1,2k−1(1.1965− a) + t4k+1,2ka− t4k+1,2k+1(1.1965− a)

)‖w‖

≤((t4k+1,2k − t4k+1,2k−1 − t4k+1,2k+1)a+ (−t4k+1,2k−1 − t4k+1,2k+1)(1.1965− 2a)

)‖w‖

≤ maxCoddCeven, (0.5892)(1.1965)‖w‖ = CevenCodd‖w‖.(7.27)

Unfortunately, such kind of contracting property does not hold for 4k+3 even for the subdivisioncase, where computer simulations give that (see fig. 10) there exist x,w ∈ l∞(Z) such that

|(T (Smed,1x)T (x)w)4k+3| = 1.0669‖w‖. (7.28)

However, there are strong numerical evidences that the “differential” approach works with n = 3.Indeed, due to Lemma 7.4, (7.23) and (7.27) we have:

57

Corollary 7.6 Let w3 = T (x′′)T (x′)T (x)w, where x′′, x′, x, w ∈ l∞(Z) are arbitrary sequences.Then,

|w38k|, |w3

8k+2|, |w38k+4|, |w3

8k+6| ≤ Ceven‖w2‖ ≤ CevenC2odd‖w‖ = 0.8165‖w‖

|w38k+1|, |w3

8k+5| ≤ CevenCodd‖w1‖ ≤ CevenC2odd‖w‖ = 0.8165‖w‖

|w38k+3| =

∣∣∣ 4k+2∑l=4k

t8k+3,l(x′′)w2

l

∣∣∣ ≤ Codd maxl=0,1,2

|w24k+l| ≤ CevenC

2odd‖w‖ = 0.8165‖w‖.

Hence, the only “bad” case is |w38k+7|.

v0 ≡

v0−1

v00v01v02

S−→ v1 ≡

v10v11v12v13

S−→ v2 ≡

v22v23v24v25

S−→ v3 ≡

v36v37v38v39

c0

↓ 41 c1

↓ 41 c2

↓ 41 c3

↓ 41

x0 ≡

x0−1

x00x01

S1−→ x1 ≡

x10x11x12

S1−→ x2 ≡

x22x23x24

S1−→ x3 ≡

x36x37x38

Since Smed is local and shift-invariant, we reduce the problem of stability of the median-

interpolating subdivision scheme to the study of 3 × 3 matrix products. This can be seen bythe above diagram, where S are nonlinear maps on R4 (restrictions of Smed to the indicated coordi-nates) and S1, T are nonlinear maps on R3 (restrictions of Smed,1, respectively of T ). Each of thesemaps depends on the underlying centers cj ∈ R2. In particular, T = T (cj), and we prefer to workwith the more explicit expressions via cj.In the multiresolution case, we use the same notation but we do not assume that cj are producedby subdivision from one “seed” v0, i.e., we consider them unrelated.

The following result has only been verified by numerical methods (see fig. 11) and needs furtherscrutiny:

Proposition 7.7 Let v0 = (v0−1, v00, v

01, v

02) and w = (w−1, w0, w1). Then∣∣∣(T (c2)T (c1)T (c0)w)

15

∣∣∣ ≤ 0.81‖w‖. (7.29)

Thus, we can conclude:

Theorem 7.8 (Stability of median-interpolating subdivision scheme) The dyadic median-interpolating subdivision scheme is stable!

Proof. Corollary 7.6 and Proposition 7.7 give

supx∈l∞(Z)

‖T (S2med,1x)T (Smed,1x)T (x)‖ ≤ 0.8165 < 1,

which, together with Proposition 6.21 and Lemma 7.3, implies Theorem 6.2.

Remark 7.9 In the multiresolution case, without additional restriction on the class of the admis-sible details, we do not expect stability, since there exists v0 ∈ l∞(Z) such that

ρ(T (41v0)) > 1.

58

8 Conclusions

In this thesis we presented a new stability theorem for univariate subdivision and multiresolution,showed that this theorem covers all the existing results for stability (WENO, PPH), and provedstability of a new subdivision scheme - the median-interpolating one. The joint spectral radiusversion of our theorem shows that for a large class of subdivision schemes the conditions of thetheorem are “close to optimal”, i.e., they are not only sufficient but also “almost” necessary (weuse the word “almost” because we still do not know what happens when the joint spectral radiusis exactly one).

59

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9 Appendix

Lemma 7.3:

Proof. We will use the notation from Section 5.3. Let us denote by v1, resp. v1 the refinementSmedv of v, resp. Smedv of v. Due to symmetry, locality, and shift-invariance, it is enough to showthat

|v10 − v10| ≤ maxi=−1,0,1

|vi − vi|+K|42(v − v)−1|. (9.1)

Indeed, then ‖v1 − v1‖ ≤ ‖v − v‖+K‖42(v − v)‖ ≤ ‖v − v‖+ 2K‖41(v − v)‖.Hence, we only need the finite subsequences v := [v−1, v0, v1] and ¯v := [v−1, v0, v1]. We will denoteby 42v := 42v−1, and by 42 ¯v := 42v−1. (5.28) gives rise to

v10 − v10 =1

32

(5(v−1 − v−1) + 30(v0 − v0)− 3(v1 − v1)

)+ α042 ¯v − α042v

=1

4(v−1 − v−1) +

3

4(v0 − v0) +

3

3242(¯v − v) + α042 ¯v − α042v,

(9.2)

where, for simplicity we have denoted α0,0 by α0, resp. α0,0 by α0.Using that α0(c) = 0 = α0(c) if c, c /∈ (−5/2,−3/2)∪ (−1/2, 1/2)∪ (3/2, 5/2) (see (5.29)), we derivefrom (9.2) that in this case (9.1) holds with K = 3/32.

From now on, w.l.o.g., we will assume that c ∈ (−5/2,−3/2) ∪ (−1/2, 1/2) ∪ (3/2, 5/2).Let us first deal with the degenerate case 42v = 0 (i.e., c = ∞). (9.2), together with the obvious|α0| ≤ 1/32, implies that (9.1) holds with K = 3/32 + 1/32 = 1/8.

Thus from now on c ∈ R, in addition to the restrictions on c. Using that Smed is affine-invariantand the canonical representation (5.25), we can reduce our investigations to

v = [1 + γ, 0, 1− γ], γ = γ(c);¯v = b+

(1 + d

2

)[1 + γ, 0, 1− γ], γ = γ(c).

(9.3)

This scaling, together with (5.26), leads to

42v = 2, 42 ¯v = 2 + d, 42(¯v − v) = d, γ ∈ R, |γ| ≤ 5

2.

Plain substitution in (9.2) gives

v10 − v10 = −b+ 1

4(γ − γ) + 2(α0 − α0)︸ ︷︷ ︸

A

+d(α0 +

3

32− 1 + γ

8

)︸ ︷︷ ︸

B

. (9.4)

From |α0| ≤ 1/32 and |γ| ≤ 5/2, it follows that −1232≤ B ≤ 10

32, and thus, in order to prove (9.1) it

remains to show that |A| ≤ ‖v − ¯v‖+K1|d|.

Lemma 9.1 ‖γ′(c)‖ ≥ 1116

and ‖α′0(c)‖ ≤ 2

15.

64

Proof. Lemma 5.2 and (5.29) imply that γ, α0 ∈ W 1∞. Moreover, γ′ ≡ 1, c /∈ (−5/2,−3/2) ∪

(−1/2, 1/2) ∪ (3/2, 5/2), and α′0 ≡ 0, c /∈ (−5/2,−3/2) ∪ (−3/4, 1/2) ∪ (3/2, 5/2).

Let us start with γ(c).

γ′(c) =(32 + ε′−2 − ε′2)(32 + ε−2 − 2ε0 + ε2)− (ε′−2 − 2ε′0 + ε′2)(32c+ ε−2 − ε2)

(32 + ε−2 − 2ε0 + ε2)2.

Case I. c ∈ [−1/2, 1/2]. Then

γ′(c) =32(30 + 8c2)− 16c32c

(30 + 8c2)2=

32(30− 8c2)

(30 + 8c2)2∈[78,16

15

].

We used that γ′(c) is even and monotonically decreasing in [0, 1/2].

Case II. c ∈ [−5/2,−3/2] ∪ [3/2, 5/2]. Due to central symmetry (see fig 7b), the two cases areequivalent and w.l.o.g. c ∈ [−5/2,−3/2].

γ′(c) =(32− 8(c+ 2))(33− 4(c+ 2)2) + 8(c+ 2)(32c+ 1− 4(c+ 2)2)

(33− 4(c+ 2)2)2

=32(4(c+ 2)2 − 24(c+ 2) + 33)

(33− 4(c+ 2)2)2∈[γ′(− 3

2

), γ′(− 5

2

)]=[1116,23

16

].

We used that γ′(c) is monotonically decreasing in [−5/2,−3/2].For α0(c) we proceed in the same way, but we have to consider more cases:

α′0(c) =

(30ε′0 + 5ε′−2 − 3ε′2 − 8ε′−1/2)(32 + ε−2 − 2ε0 + ε2)− (ε′−2 − 2ε′0 + ε′2)(30ε0 + 5ε−2 − 3ε2 − 8ε−1/2)

32(32 + ε−2 − 2ε0 + ε2)2.

Case I. c ∈ [−5/2,−3/2] ∪ [−1/4, 1/2] ∪ [3/2, 5/2]. Using (5.29) and that ε−2 and ε2 are justshifts of ε0, we derive that (after shifting the argument, if necessary)

α0 =C1ε0

32(32 + C2ε0)=⇒ α′

0 = C1ε′0

(32 + C2ε0)2,

where C1 equals to 5, 30, resp. −3 and C2 equals to−1, 2, resp. −1, when c lies in [−5/2,−3/2], [−1/4, 1/2],resp. [3/2, 5/2]. In all the three cases, direct computations show that

ε′0(32+C2ε0)2

is monotonically

decreasing, and∣∣ ε′0(32+C2ε0)2

∣∣ ≤ 1162

. Thus

|α′0(c)| ≤

5/162, c ∈ [−5/2,−3/2]30/162, c ∈ [−1/4, 1/2]3/162, c ∈ [3/2, 5/2]

.

Case II. c ∈ [−3/4,−1/2]. Then

α′0(c) =

1 + 2c

8∈[− 1

16, 0].

65

Case III. c ∈ [−1/2,−1/4]. Then

α′0(c) =

−8c2 − 3c+ 30

(15 + 4c2)2∈[α′0

(− 1

2

), α′

0

(− 1

4

)]⊂[0,

2

15

].

We used that α′0(c) is positive and monotonically decreasing, as well as that

α′0

(− 1

4

)=

2(15 + 1

8

)(15 + 1

4

)(15 + 1

4

) < 2

15.

Now using Lemma 9.1, and that both α0(c) and c(γ) are continuous and piecewise differentiable,we derive

|2(α0 − α0)| ≤ ‖α′0‖|c− c| ≤ ‖α′

0‖‖c′‖|γ − γ| ≤4

15

16

11|γ − γ| ≤ 2

5|γ − γ|, (9.5)

which gives rise to|A| ≤ max| − b+ (γ − γ)|, | − b− (γ − γ)|. (9.6)

On the other hand (9.3) implies

‖v − ¯v‖ =∥∥[−b+ (γ − γ), 0,−b− (γ − γ)]− d

2[1 + γ, 0, 1− γ]

∥∥≥ max| − b+ (γ − γ)|, | − b− (γ − γ)| − |d|

2max|1 + γ|, |1− γ|.

(9.7)

Finally, combining (9.6) and (9.7), and recalling that |γ| ≤ 5/2, we conclude that

|A| ≤ ‖v − ¯v‖+ 7

4|d|. (9.8)

Remark 9.2 In order to simplify the exposition, as well as to decrease the number of cases, we usedvery rough inequalities, and thus the actual value of K is much smaller than the above estimation!However, the exact value of K is of no interest for proving stability of Smed.

Remark 9.3 The same approach works also for the triadic median-interpolating subdivision scheme.There (being more precise in the estimations) we have proven that

‖Smed,3v − Smed,3v ≤ ‖v − v‖+1

3‖42(v − v)‖.

Again, 1/3 is not the exact lower bound for the constant K, but numerical simulations show thatK > 0.29.

66

0 500500.5

0.6

0.7

0.8

0.9

Case [1 0 x y]

0 500500.4

0.6

0.8

1

Case [1 x 0 y]

0 500500.4

0.6

0.8

1

Case [1 x y 0]

0 500500

0.5

1

1.5

Case [x 1 0 y]

Figure 10: sup‖w‖=1

|T (Smed,1x)T (x)w7| as a function of the centers c0 and c1 of v, where x = 41v.

67

0 500500.2

0.4

0.6

0.8

1

Case [1 0 x y]

0 500500.2

0.4

0.6

0.8

1

Case [1 x 0 y]

0500

500.2

0.4

0.6

0.8

1

Case [1 x y 0]

0 500500.2

0.4

0.6

0.8

1

Case [x 1 0 y]

Figure 11: sup‖w‖=1

|T (S2med,1x)T (Smed,1x)T (x)w15| as a function of the centers c0 and c1 of v, where

x = 41v.

68