Approximation Theory in Manifoldsgrohs/files/habil.pdf · sion schemes. Journal of Approximation Theory, 162:1085{1094, 2010. 1.3 Approximation Theory in Manifolds 1.3.1 Linear Theory

Habilitation Thesis

Approximation Theory inManifolds

submitted by Dipl. Ing. Dr. techn. Philipp Grohs

June 29, 2010

2

To Marlene.

Contents

1 Introduction 51.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 List of included articles . . . . . . . . . . . . . . . . . . . . . . . 61.3 Approximation Theory in Manifolds . . . . . . . . . . . . . . . . 6

1.3.1 Linear Theory . . . . . . . . . . . . . . . . . . . . . . . . 61.3.2 Multiscale Transforms for manifold data . . . . . . . . . . 91.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 [G1]: A general proximity analysis 212.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 Some results on linear subdivision . . . . . . . . . . . . . . . . . 23

2.2.1 Well-known facts . . . . . . . . . . . . . . . . . . . . . . . 232.2.2 New results . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3 Proximity conditions . . . . . . . . . . . . . . . . . . . . . . . . . 272.3.1 Nonlinear subdivision in manifolds . . . . . . . . . . . . . 272.3.2 Notation and statement of the main result . . . . . . . . . 292.3.3 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4 Riemannian center of mass . . . . . . . . . . . . . . . . . . . . . 352.4.1 The center of mass . . . . . . . . . . . . . . . . . . . . . . 352.4.2 Proof of the Proximity condition for the Riemannian ana-

logue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.4.3 Smooth Splines in CH-Manifolds . . . . . . . . . . . . . . 37

2.5 Generalization to the multivariate case . . . . . . . . . . . . . . . 38

3 [G2]: Interpolatory wavelets 473.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.1.1 Linear stationary subdivision rules . . . . . . . . . . . . . 483.1.2 Linear interpolating wavelet transforms . . . . . . . . . . 493.1.3 Subdivision rules and wavelet transforms in manifolds . . 50

3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.2.1 Wavelet coefficient decay and smoothness . . . . . . . . . 513.2.2 Proximity results . . . . . . . . . . . . . . . . . . . . . . . 52

3.3 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3.1 Proof of the proximity inequality . . . . . . . . . . . . . . 523.3.2 Wavelet coefficient decay in the linear case . . . . . . . . 553.3.3 Proof of Theorem 3.2.1 . . . . . . . . . . . . . . . . . . . 563.3.4 Remarks on the reconstruction process . . . . . . . . . . . 58

3

4 CONTENTS

4 [G3]: Stability 634.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.2 Notation and preliminaries . . . . . . . . . . . . . . . . . . . . . 65

4.2.1 Subdivision schemes . . . . . . . . . . . . . . . . . . . . . 654.2.2 Multiscale Data Representation using Subdivision . . . . 674.2.3 Lipschitz classes . . . . . . . . . . . . . . . . . . . . . . . 67

4.3 Stability from Proximity . . . . . . . . . . . . . . . . . . . . . . . 684.4 Stability of the Multiscale Decomposition . . . . . . . . . . . . . 74

4.4.1 Definition of the multiscale decomposition associated witha nonlinear subdivision scheme . . . . . . . . . . . . . . . 74

4.4.2 Well-definedness of the reconstruction procedure . . . . . 754.4.3 Stability of the reconstruction procedure . . . . . . . . . . 78

4.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.5.1 The log-exp analogue in Lie groups . . . . . . . . . . . . . 824.5.2 The log-exp analogue in Riemannian manifolds . . . . . . 844.5.3 The projection analogue . . . . . . . . . . . . . . . . . . . 86

4.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . 87

5 [G4]: Approximation order 935.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.2 Notation and Definitions . . . . . . . . . . . . . . . . . . . . . . . 945.3 Main theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.4 Weakening the assumptions . . . . . . . . . . . . . . . . . . . . . 975.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.5.1 Manifold-Valued Subdivision . . . . . . . . . . . . . . . . 995.5.2 Triadic Median Interpolation . . . . . . . . . . . . . . . . 1005.5.3 Power-p Scheme . . . . . . . . . . . . . . . . . . . . . . . 1015.5.4 Dyadic Median Interpolation . . . . . . . . . . . . . . . . 101

5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Chapter 1

Introduction

1.1 Preface

The ongoing desire in science and technology to assemble massive amounts ofdata is posing new problems for mathematicians and computer scientists. Newmethods of aquisition are being developed, producing more data than ever be-fore. Often one is confronted with nonstandard data types like for exampleDiffusion Tensor MRI, where the data points are positive definite matrices mea-suring the direction-dependent diffusion rate of water molecules travelling be-tween cell membranes [14].

This thesis is concerned with understanding such data types.Conventional signal processing deals mostly with signals that are given as

numerical arrays [26]. But in recent years it has become apparent that tools needto be developed to also handle data streams that obey nonlinear constraints.We mention the following incomplete list of important such types:

• Diffusion Tensor Imaging, where the data points consist of elements ofthe symmetric space of positive definite symmetric matrices. The data ismodeled as a function R3 → SPD(3).

• Positions of rigid bodies, where the data points consist of elements ofthe Lie group of rigid body motions. The data is modeled as a functionR→ SE(3).

• Orientations, where the data points consist of elements of the Lie groupof orthogonal matrices. The data is modeled as a function R→ O(3)

• Subspaces, where the data points consist of elements of a Grassmanianmanifold. The data is modeled as a function R→ G(k, n).

All these cases have in common that the data points lie in a manifold and thesignal model is given by functions defined in a linear space and assuming itsvalues in a nonlinear manifold. The handling of this information is a seriousproblem. While many classical methods can be (although not trivially) adaptedto the case of data which is modeled as functions defined in a manifold andvalues in a linear space (still forming a linear signal space), for the case ofnonlinear values virtually all methods from linear signal processing break down.

5

6 CHAPTER 1. INTRODUCTION

It is the purpose of this thesis to develop and study methods that can dealwith nonlinear data.

Outline. This text is organized as follows: Chapter 1 gives an introductionto the scope of problems that are adressed. After listing the submitted papersin Section 1.2, we describe the subject of approximation theory in manifolds inSection 1.3, starting with the linear theory in Subsection 1.3.1 and continuingwith the geometric case in Subsection 1.3.2. In Subsection 1.3.3. we brieflyoutline the results that make up the thesis. The later chapters consist of thepublished research papers which form the cumulative habilitation.

Acknowledgements

I am grateful to Johannes Wallner and Helmut Pottmann for their constantsupport and for providing me with a perfect working climate. I have beenvery lucky to be able to work in such an optimal environment to develop as aresearcher.

Furthermore, I thank Nira Dyn, Hans-Georg Feichtinger, Anton Gfrerrer,Stanislav Harizanov, Gitta Kutyniok, Niloy Mitra, Peter Oswald, Martin Peter-nell, Tomas Sauer, Andreas Weinmann, Thomas Yu, . . . for useful discussionsand support.

During all my work I have received financial support from the austrian sci-ence fund FWF through projects P19780, P18575 and S9209.

My deepest thanks go to Marlene for sharing her life with me.

1.2 List of included articles

[G1] Philipp Grohs. A general proximity analysis of nonlinear subdivisionschemes. SIAM Journal on Mathematical Analysis, 42:729–750, 2010.

[G2] Philipp Grohs and Johannes Wallner. Interpolatory wavelets for manifold-valued data. Applied and Computational Harmonic Analysis, 27(3):325–333, 2009.

[G3] Philipp Grohs. Stability of manifold-valued subdivision schemes and mul-tiscale transformations. Constructive Approximation, 2010.

[G4] Philipp Grohs. Approximation order from stability of nonlinear subdivi-sion schemes. Journal of Approximation Theory, 162:1085–1094, 2010.

1.3 Approximation Theory in Manifolds

1.3.1 Linear Theory

A large part of approximation theory is concerned with approximating a contin-uous signal by a sequence of discrete numbers (see e.g. [6] for a comprehensiveintroduction into the field of approximation theory). Let us say that our contin-uous signal is a function f : R→ R lying in some function space F . This signalmight be a piece of music or a seismic signal. In order to be able to process thissignal with a computer, one usually transforms the signal into a discrete numbersequence. The usual way to do this is to approximate the infinite dimensional

1.3. APPROXIMATION THEORY IN MANIFOLDS 7

function space F by a filtration V 0 ⊂ V 1 ⊂ · · · ⊂ F of finite-dimensional spacesV i, i ∈ N. One may think of F as some smoothness space and the spaces V i aspiecewise polynomials on intervals [2−ik, 2−i(k+ 1)], k ∈ Z. Let us define someprojection operators P i : F → V i. Now the signal f may be transformed into adiscrete signal by choosing bases of V i and representing P if in the correspond-ing basis. It is natural to ask how well f is approximated by the sequence ofapproximations P if , i.e. how fast we have

‖P if − f‖ → 0

in some norm ‖ · ‖? Another central question to ask in approximation theory isif the rate of convergence of the above expression characterizes the membershipof f in a function space.

Example 1.3.1. This example has its origins already in the work of Faber inthe year 1909 [13]. Let F be the space Lip α for some α > 0. Furthermore, welet ϕ be a continuous function which satisfies the refinability condition

ϕ(·) =∑j∈Z

ajϕ(2 · −j) (1.1)

for some finite sequence a = (aj)j∈Z ∈ l0. Let us also assume that ϕ is cardinal,meaning that ϕ

∣∣Z = δ0, the Dirac sequence. Define the space V 0 as the shift-

invariant space generated by ϕ in L∞ and V i :=ϕ(2·) : ϕ ∈ V i−1

for i > 1.

The refinability condition ensures that the spaces V i are nested. We also needa way to project a function f ∈ Lip α onto our spaces V i. In this example wechoose to do this by interpolation, i.e. we define

P if :=∑j∈Z

f(2−ij)ϕ(2i · −j).

Note that by our assumption on ϕ to be cardinal, the projection P if interpolatesthe initial function f on the grid 2−iZ. A classical theorem in approximationtheory states that

‖P if − f‖∞ . 2−αi.

A more general statement is true: with Phf :=∑j∈Z f(hj)ϕ(h−1 ·−j), we have

‖Phf − f‖∞ . hα. (1.2)

We say that the approximation process has approximation order α. Note thatP i = P 2−i .

There is another way to define the projection operators Ph using subdivisionschemes [3]. Define the subdivision scheme Sa to be the operator from l∞ to itselfmapping a sequence f = (fj)j∈Z to the sequence (Safj)j∈Z via

Safj =∑k∈Z

aj−2kfk, (1.3)

where we require that ∑j∈Z

ai−2j = 1 for all i ∈ Z. (1.4)


A subdivision scheme is meant to be iterated. It is called convergent if for allnonzero initial data f there exists a nonzero continuous function S∞a f with

limi→∞

∥∥S∞a f ∣∣2−iZ − Siaf∥∥∞ = 0.

It turns out that ϕ = S∞a δ0 and consequently we can write Phf = S∞a f∣∣hZ

(h−1·

)and formulate (1.2) in terms of subdivision schemes.

We also have an inverse result: if ϕ is sufficiently smooth, then the approxi-mation order α characterizes the membership of a function f in Lip α. In orderto ensure that ϕ is smooth enough one needs to study the smoothness of thesubdivision scheme Sa.

It is actually more economical to not approximate f by its projections P ifdirectly but to compute one coarse approximation P 0f and the approximationerrors that arise between consecutive levels. This is the basic idea of Multireso-lution Analysis [25]. We do this as follows: Choose a basis of V 0 and bases of thedetail spaces W i which are (deliberately) vaguely defined as V i+1 = V i + W i,i ∈ N. We can now consider the union of these bases as a multiscale basisof F and decompose the signal f in this discrete basis. Having a certain ap-proximation order implies that ’fine scale detail coefficients’ will quickly becomenegligible. Often the continuous function norm on F is equivalent to a discretenorm on the space of basis coefficients.

Example 1.3.2 (Continuation of Example 1.3.1). We now require our subdi-vision scheme to be interpolatory, which can be characterized by a(2·) = δ0(·).With ψ(·) := ϕ(2 ·+1) and W 0 the shift invariant space generated by ψ in L∞,we have V 1 = V 0 +W 0 in the sense that

ϕ(2·) = ϕ(·)−∑j∈Z

a2j−1ψ(· − j).

With W i :=f(2·) : f ∈W i−1

we obtain a sequence of detail spaces and as-

sociated bases. It is straightforward to see that

f = P 0f +∑i≥0

∑j∈Z

difjψ(2i − j),

and difj := f(2−i(2j + 1)) −(Saf

∣∣2−(i−1)Z

)2j+1

. Thus, to every continuous

function we can assign the discrete sequence of coefficients(c0f := (f(j))j∈Z, d

0f, . . . ,).

The following theorem shows that this so-called interpolating wavelet transformcharacterizes the membership of f in Besov spaces in terms of simple normequivalences:

Theorem 1.3.3 ([7]). Under some technical assumptions the discrete norm onthe interpolating wavelet coefficients given by

‖(c0, d0, d1, . . . )‖ := ‖c0‖p + ‖D‖lq(N),

whereDi := 2i(σ−p

−1)‖di‖p


characterizes the Besov space Bσp,q: We have

‖f‖Bσp,q ∼ ‖(c0f, d0f, d1f, . . . )‖

Note that, by the compact support of ϕ, the computation of the wavelet co-efficients is a purely local procedure, that is, unlike simple Fourier transformthis wavelet transform is also capable of describing pointwise smoothness prop-erties (This is one particularily useful feature of wavelet transforms in general,see for example the beautiful result [22]). A general framework for interpolatingmultiscale transforms is given in [20].

One of the technical assumptions in Theorem 1.3.3 is that the Besov spaceembeds into a space where point evaluation is defined. In particular the inter-polating wavelet coefficients do not characterize L2. For some applications thisis not desirable. However, there are many applications, like for example PDEsolver [23], where L2-stability is not needed.

1.3.2 Multiscale Transforms for manifold data

Previous Work. The study of nonlinear multiscale transformations is anactive topic. Daubechies et. al. [5] study nonlinear normal multiresolutionapproximations of curves. Their results are extended in [19, 28] to more generalsettings such as approximation of surfaces. The paper [32] presents a veryinteresting application of these schemes to wavefront tracking problems. A largepart of the theoretical work in these papers deals with the analysis of nonlinearsubdivision schemes. The papers [35, 34] use similar methods in the study ofmanifold-valued subdivision schemes. The paper [38] deals with applications ofsuch schemes. The paper [39] is concerned with data defined on irregular meshes.Approximation of polynomial loops in Lie groups is the subject of [29, 30]. In[10, 11, 27] a nonlinear median interpolating transform is introduced with thepurpose of dealing with nongaussian noise. Other data dependent nonlineartransforms are studied in [1, 2, 4]. These are similar to so-called EssentiallyNon Oscillating schemes which are used for the numerical solution of hyperbolicconservation laws [21]. A nice summary on recent work in nonlinear multiscaleanalysis is [12]

In this section we show how the linear approximation processes which we justintroduced can also be defined for data in a manifold. Note that the problem ofdealing with manifold-valued data is entirely different from that of linear datain that the main tools which are used in the linear case (functional analysis,harmonic analysis) are not available in manifolds. Certainly, one could makelinear computations in charts, but that way one would sacrifice all geometricstructure provided by the manifold. We are seeking a genuinely geometric analogwhich is intrinsically defined on the manifold.

We now show how to carry over the constructions in the Examples 1.3.1 and1.3.2 to a geometric setting.

Affine averages in manifolds. The key idea is the following: while linearcombinations cannot sensibly be defined in manifolds, there are notions of affineaverages. Indeed, given any smooth vector bundle E

π→ M over the manifoldM and two mappings ⊕ : E → M, (p, v) 7→ p ⊕ v, : M ×M → E, (p, q) 7→qp ∈ π−1 (p) satisfying the consistency relation p⊕ qp = q, we can definean affine average of the points pj with coefficients λj ,

∑j λj = 1 by picking a


basepoint p and computing

avp ((pj), (λj)) := p⊕

∑j

λjpj p

.

This is a perfectly natural analog of an affine average: if we replace M by avector space and define ⊕ and to be the usual addition and subtraction, weget the usual definition of affine average, independent of the choice of p. Oneparticular choice of p will later turn out to be especially convenient: pick p ∈Mso, that ∑

j

λjpj p = 0. (1.5)

One should regard equation (1.5) as an equillibrium condition: The vectorspointing from p to pj , weighted with λj cancel each other out. It is clear thatwith this choice of basepoint we have

avp ((pj), (λj)) = p.

What is the motivation for (1.5)? The answer lies in the follwing characteriza-tion of linear averages:

p∗ =∑j

λjpj(i)⇔ p∗ = argminp

∑j

λjd(p, pj)2 (ii)⇔

∑j

λj(p∗ − pj) = 0. (1.6)

Of course it it not at all clear if it is possible at all to find p such that (1.5)holds. But there is the following important example:

Example 1.3.4 (Riemannian center of mass). We let the vector bundle E bethe tangent bundle over M and define p⊕ v := expp(v) and q p := logp(q). Inthis case the point p with (1.5) is called the Riemannian center of mass. Thefollowing holds:

Theorem 1.3.5 ([24]). Locally, the Riemannian center of mass is well-defined.It can also be characterized by

p∗ = argminp∑j

λjd(p, pj)2,

i.e. equivalence (ii) in (1.6) holds true. If the underlying manifold is a CartanHadamard manifold, then the Riemannian center of mass is globally defined.

We write R-av ((pj), (λj)) for the Riemannian center of mass.

Geometric subdivision schemes. Having manifold versions of affineaverages at our disposal, we can now define subdivision schemes which operateon manifold-data. By (1.3) and (1.4), the action of a subdivision operator on apoint sequence p = (pj)j∈Z is given by an affine average and therefore in principlewe can simply replace a linear affine average by a geometric affine average inorder to define a subdivision scheme operating on manifold data. One subteltyin doing this is the choice of the basepoints. To that end we define a basepointmapping which assigns to a sequence p = (pj)j∈Z a basepoint sequence p =


(pj)j∈Z, cf. [G1]. Now we can define the geometric analog Ta = Ta(E,⊕,, ·)of a subdivision scheme Sa by defining

Tapj := avpj ((pk), (aj−2k)) (1.7)

Observe that the Riemannian analog of Sa, which is defined by replacing lin-ear affine averages with Riemannian averages, occurs as a special case of thisdefinition.

The natural question to ask at this point is if these nonlinear constructionscan be shown to converge, and if so, if they also produce limit curves as smoothas their linear counterparts. In the next section below we will describe someresults which settle this question. Let us assume for now that Ta is convergent,and call T∞a p the limit function with initial data p. Then we can immediatelydefine a nonlinear analog PhNL of the approximation operators Ph for manifold-valued functions f by

PhNLf := T∞a f∣∣hZ

(h−1·

).

Now, the natural question is how fast these approximations converge to f ifh→ 0, or in other words, what is the approximation order of Ta?

If M is a Lie group and the ⊕ and operations are defined via the exp andlog mappings, the previous construction with pj := pb j2 c

is essentially due to

Donoho [9]. This scheme is usually referred to as log-exp analog. At the timeof its introduction, the importance of the correct choice of basepoint mappinghas not been realized. In [40] Xie and Yu introduced the so-called g − f analogwhich falls into our previous definition of the geometric analog with p = T pwhere T is the log-exp analog of some interpolatory scheme. We would arguethat the most natural choice for a nonlinear analog of a linear subdivison schemeis the Riemannian analog. This is because it does not depend on any artificialchoices of basepoints and in addition it automatically preserves symmetries ofthe linear schemes. This last fact becomes especially important when treatingmultivariate schemes. There are also other advantages to which we will get backin the next section. As we have already seen, also the Riemannian analog occursas a special instance of the geometric analog defined above.

Geometric multiscale decompositions. With our machinery we can alsocarry over the multiscale decomposition from example 1.3.2 to the geometricsetting. With an interpolatory subdivision scheme Sa and its geometric analogTa we define the interpolating wavelet decomposition

(c0NLf, d0NLf, d

1NLf, . . . )

of a continuous function f : R→M via

c0NLfi := f(i), and diNLfj := f(2−i(2j + 1))(Taf

∣∣2−(i−1)Z

)2j+1

.

Observe that the coarse scale information c0NLf has values in M , while the’wavelet coefficients’ live in the vector bundle. For example if M is a Lie group,the wavelet coefficients live in the corresponding Lie algebra. For the log-expanalog this construction is due to [9, 31]

It is easy to see that (the values on dyadic rationals of) f can be recon-structed by iteratively adding back the wavelet coefficients to the prediction viaTa from the coarser scale. This reconstruction procedure is nonlinear, so an


important question to ask is what happens if we perturb the wavelet coeffiecintsslightly, or in other words, is the nonlinear multiscale decomposition stable?

In the spirit of Theorem 1.3.3 we might also ask what can be said for therelation between smoothness of a function and the decay rate of its waveletcoefficients.

1.3.3 Results

In this section we present the papers [G1-G4] which comprise this thesis andwhich provide an answer the questions stated above.

Smoothness [G1]. We begin by considering the question of smoothnessfor manifold-valued subdivision schemes. The question of smoothness has firstbeen asked in 2001 by Donoho for the log-exp scheme. Since then, partial resultscould be obtained, most of which were limited to interpolatory schemes or tolow orders of smoothness [35, 34, 36, 17]. The basic idea of all this work isto consider the nonlinear scheme Ta as a perturbation of the linear scheme Saand to estimate the difference. For example convergence and C1-smoothnessof Ta follows from convergence/smoothness of Sa together with the proximitycondition of order (0, 2)

‖Tap− Sap‖∞ . ‖∆p‖2∞, (1.8)

where ∆ is the forward differencing operator. This is the main result of [35].It is straightforward to establish (1.8) for Ta the geometric analog of Sa and ·virtually any basepoint mapping. In [34] it is shown that for a linear C2 schemeSa one can deduce C2 smoothness of the nonlinear scheme, provided that aproximity condition of order (1, 3) holds, i.e.

‖∆ (Sap− Tap)‖∞ . ‖∆p‖∞‖∆2p‖∞ + ‖∆p‖3∞.

In order to infer Cn smoothness for Ta from smoothness of Sa it would benecessary to show a higher order proximity condition (of order (n − 1, n + 1);for the precise definition of proximity conditions we refer to [G1]). However, itturns out that such an inequality does not hold for larger n (for example n = 4)and the log-exp analog.

The first general proof of smoothness equivalence for arbitrary orders andarbitrary schemes Sa is contained in [16] where Ta is defined for embeddedsubmanifolds as Tapj := P Sapj and P is a smooth retraction onto the manifold.This is known as the Projection analog and it also ocurs as a special case ofthe geometric analog defined above. In the previous section we have stated thatwe are searching for intrinsic constructions, and for that reason, the projectionapproach is not fully satisfying.

In [40] Xie and Yu constructed a modification of the log-exp analog by choos-ing as basepoint mapping the mapping p 7→ p := Tbp, where Tb is the log-expanalog of an interpolatory scheme Sb. Furthermore, they showed that withthis modified construction proximity conditions of arbitrary order, and hencesmoothness equivalence of arbitrary order, holds.

There are three problems with this construction: first, it depends on anartificial choice of an interpolatory subdivisions scheme Sb. Second, it does notpreserve symmetries. Third, all the results are contingent on the convergenceof Ta for the initial data p. Unfortunately for this construction convergence can


only be shown to hold for initial data p with ‖∆p‖∞ very small, see for instance[15].

By defining Ta to be the Riemannian analog of Sa, all these three problemsvanish automatically. There are no artificial choices, symmetries are preservedautomatically (even in higher dimensions and in the vincinity of irregular ver-tices), and furthermore if M is a Cartan-Hadamard manifold and Sa satisfiessome additional conditions (which are satisfied e.g. by the family of B-splineschemes), then Ta converges for all initial data p [37].

The paper [G1] consists of two main results. The first one is a completecharacterization of the proximity conditions in terms of algebraic conditionsof the mask a of the linear scheme Sa and decay conditions on the basepointmapping ·. From this result it is for instance readily seen that we can only expectC2 smoothness for the log-exp analog. Second, we use this first result togetherwith a bootstrapping argument to verify proximity conditions of arbitrary orderfor the Riemannian analog. All these results are shown to remain valid inarbitrary dimensions with arbitrary dilation.

If we are interested in data living in a Cartan-Hadamard manifold (the sym-metric space of symmetric positive definite matrices is one such manifold withmany applications in image processing) these results enable us to construct afamily of nonlinear subdivision schemes which converge for all input data andwhich produce arbitrarily smooth limit functions.

Interpolating Wavelets [G2]. We have seen in Section 1.3.2 that theinterpolating wavelet construction from Example 1.3.2 can be modified so as tobe applicable to data in manifolds. In the paper [G2] we show that also the normequivalences for the spaces Bα∞,∞ carry over to that setting. Again it turns outthat proximity inequalities are useful. The ’direct theorem’ which states thatthe wavelet coefficients of smooth functions decay fast follows pretty easily fromsuch inequalities. The converse direction turns out to be more involved. Alsospecial care needs to be taken in order to handle the spaces Bn∞,∞, n ∈ N,also known as Zygmund classes. We chose to use interpolation methods for thisissue, see e.g. [33]. This article was written jointly with Johannes Wallner.The proofs for all results except Theorem 3.3.4 are due to the author who alsooriginally had a different proof for Theorem 3.3.4. The proof in its present formhas been obtained jointly with J. Wallner.Stability [G3]. In the paper [G3] we consider the important question of(Lipschitz) stability for nonlinear subdivision schemes and nonlinear multiscaledecompositions. In general this question is not easy. In the linear case stabilityis a trivial consequence of convergence of the subdivision operator but in the nonlinear case this implication turns out to be false as we could show in [G3], see also[19] for other results of this type. In order to study the stability properties ofnonlinear multiscale transforms in general it is necessary to understand certainnonlinear spectral quantities which are difficult to handle at present. Recentlythere has been some progress [19] but we are still far from understanding thestability properties of nonlinear multiscale transformations. Fortunately in themanifold-valued case described in Section 1.3.2, we were able to obtain a generalresult. It turns out that in this special setting convergence does imply stability.The methods of proof in this paper are quite different from the usual proximityarguments, albeit still based on perturbation arguments. The inequality that


makes the technical machinery work is a differential proximity condition

dTa∣∣c

= Sa for constant sequences c

which holds for manifold-valued schemes. A typical theorem of [G3] is thefollowing: Define the distance d(v, w) between two tangent vectors by

d(v, w) := ‖v − Ptπ(v)π(w)(w)‖π(v),

where π is the bundle projection of the tangent bundle, ‖ · ‖p is the norm intro-duced by the metric tensor in a point p ∈ M and Ptqp is the parallel transportfrom the tangent space in p ∈M to the tangent space q ∈M along the shortestgeodesic connecting p and q, whenever defined. For two points p, q ∈ M welet d(p, q) be the usual geodesic distance. We have for the nonlinear multiscaledecomposition constructed from the Riemannian analog:

Theorem 1.3.6. For all M -valued functions f ∈ Lip γ there exist δ0, ε0 > 0such that for all initial data

(c0NL, d0NL, . . . )

with

supj∈Z

d((c0NL)j , c0NLfj) ≤ δ0 and sup

j∈Zd((diNL)j , d

iNLfj

)∞ ≤ ε0/2

γi, i ∈ N

the reconstruction yields a well-defined continuous function f with

supj∈Z

d(f(2−ij), f(2−ij)) . supj∈Z

d(c0NLfj , (c0NL)j) +

i∑l=1

supj∈Z

d(dlNLfj , (d

lNL)j

).

The constants δ0, ε0 are uniform for data values in a compact set. In some cases(Cartan Hadamard manifold, specific schemes) the constants δ0, εi can be takento be ∞.

The reader might wonder whether it is possible to use other projections thanthose based on interpolation. This is possible to a limited extent: In [18] theso-called average interpolating wavelets [8], where point-samples are replaced byaverages over dyadic cubes, are generalized to a manifold-valued setting. In [18]we show that also this decomposition is stable. We further show that we cannotexpect to be able to easily transfer more general biorthogonal transforms to themanifold-valued setting.Approximation Order [G4]. The paper [G4] proves approximation orderproperties of nonlinear subdivision schemes. This is important if we want toapproximate a function by the nonlinear approximation operators PhNLf and ifwe want those to converge to a function f depending on the norm of f in somesmoothness space. We show that the convergence is as fast as for the corre-sponding linear approximation operator, provided that the underlying subdivi-sion scheme Ta is stable and some other technical conditions are satisfied. Weapply this general result to show approximation order equivalence properties forlinear subdivision schemes and their geometric analogs but also for other caseslike for instance the dyadic and triadic median interpolating schemes.

1.4. OUTLOOK 15

1.4 Outlook

In the previous section we answered several questions about nonlinear multiscaletransforms. Some essential questions remain. Maybe the most pressing one isthe problem of defining more general wavelet transforms in the manifold-valuedsetting. As stated before, there are certain limitations in what one can expectby simply replacing linear affine averages with nonlinear affine averages. Still,using different, more complicated constructions it might well be possible to alsodefine e.g. Daubechies wavelet – like constructions in manifolds. In future workwe will investigate such constructions.

Another question is the problem of convergence. Exactly for which initialdata does (say) the Riemannian analog of a linear scheme converge? We knowthat for a Cauchy Hadamard manifold and under some conditions on the linearsubdivision scheme the answer is ’for all initial data’. How far can one generalizethis result?

Finally we mention an interesting direction that is currently pursued by PeterOswald’s group at Jacobs University Bremen. They study generalizations of theby now classical normal multiresolution representation of curves. While this isnot directly related to manifold-valued approximation theory, the methods andtools used there are very similar to the ones we present here.


Bibliography

[1] Sergio Amat, Katrin Dadourian, and Jaques Liandrat. Analysis of a class ofnonlinear subdivision schemes and associated multi-resolution transforms.Technical report.

[2] Sergio Amat and Jaques Liandrat. On the stability of PPH nonlinearmultiresolution. Applied and Computational Harmonic Analysis, 18:198–206, 2005.

[3] Albert Cavaretta, Wolfgang Dahmen, and Charles Micchelli. StationarySubdivision. American Mathematical Society, 1991.

[4] Albert Cohen, Nira Dyn, and Basarab Matei. Quasilinear subdivisionschemes with applications to ENO interpolation. Applied and Computa-tional Harmonic Analysis, 15:89–116, 2001.

[5] Ingrid Daubechies, Olof Runborg, and Wim Sweldens. Normal multiresolu-tion approximation of curves. Constructive Approximation, 20(3):399–463,2004.

[6] Ronald DeVore and George Lorentz. Constructive Approximation.Springer, 1993.

[7] David Donoho. Interpolating wavelet transforms. Technical report, De-partment of Statistics, Stanford University, 1992.

[8] David Donoho. Smooth wavelet decompositions with blocky coefficientkernels. Recent Advances in Wavelet Analysis, 308, 1993.

[9] David Donoho. Multiscale representation of equispaced data taking valuesin a Lie group, 2001. Talk at Approximation Theory at 60: Conference inHonor of R. A. DeVore, Charleston SC.

[10] David Donoho and Thomas Yu. Deslariers-Dubuc: Ten years after. InS. Dubuc and G. Deslauriers, editors, Spline Functions and the The-ory of Wavelets, pages 355–369, 1999. CRM (Centre de RecherchesMathematiques, Universite de Montreal) Proceedings & Lectures Notes,Volume 18.

[11] David Donoho and Thomas Yu. Nonlinear pyramid transforms based onmedian-interpolation. SIAM Journal of Math. Anal., 31(5):1030–1061,2000.

17

18 BIBLIOGRAPHY

[12] Nira Dyn and Peter Oswald. Univariate subdivision and multiscale trans-forms: The nonlinear case. In Multiscale, Nonlinear, and Adaptive Approx-imation (R.A. DeVore, A. Kunoth eds.). Springer, 2009.

[13] Georg Faber. Uber stetige funktionen. Mathematische Annalen, 66:81–94,1909.

[14] Aaron Filler. MR neurography and diffusion tensor imaging: Origins, his-tory and clinical impact. Neurosurgery, 65:29–43, 2009.

[15] Philipp Grohs. Smoothness analysis of subdivision schemes on regular gridsby proximity. SIAM Journal on Numerical Analysis, 46(4):2169, 2008.

[16] Philipp Grohs. Smoothness equivalence properties of univariate subdi-vision schemes and their projection analogues. Numerische Mathematik,113(2):163–180, 2009.

[17] Philipp Grohs. Smoothness of interpolatory multivariate subdivision in liegroups. IMA Journal of Numerical Analysis, 29:760–772, 2009.

[18] Philipp Grohs and Johannes Wallner. Definability and stability of multi-scale decompositions for manifold-valued data. Technical report, TU Graz,2009.

[19] Stanislav Harizanov, Peter Oswald, and Tatiana Shingel. Normal multi-scale transforms for curves. Foundation of Computational Mathematics.

[20] Ami Harten. Multiresolution representation of data: A general framework.SIAM Journal on Numerical Mathematic, 33:1205–1256, 1996.

[21] Ami Harten and Stanley Osher. Uniformly high-order accurate nonoscilla-tory schemes. SIAM Journal on Numerical Analysis, pages 279–309, 1987.

[22] M. Holschneider and Ph. Tchamitchian. Pointwise analysis of riemann’s”nondifferentiable” function. Inventiones Mathematicae, 105:157–175,1991.

[23] Mats Holstrom. Solving hyperbolic pdes using interpolating wavelets.SIAM Journal on Scientific Computing, 21:405–420, 2000.

[24] Hermann Karcher. Mollifier smoothing and riemannian center of mass.Communications on Pure and Applied Mathematics, 30:509–541, 1977.

[25] Stephan Mallat. Multiresolution approximations and wavelet orthonor-mal bases of L2(R). Transactions of the American Mathematical Society,315(1):69–87, 1989.

[26] Stephan Mallat. A wavelet tour of signal processing. Academic Press, 2003.

[27] Peter Oswald. Smoothness of nonlinear median-interpolation subdivision.Advances in Computational Mathematics, 20:401–423, 2004.

[28] Peter Oswald. A normal multiscale transform for surfaces. 2010.

[29] Peter Oswald and Tatiana Shingel. Splitting methods for SU(n) loop ap-proximation. Journal of Approximation Theory, 161:174–186, 2009.

BIBLIOGRAPHY 19

[30] Peter Oswald and Tatiana Shingel. Close-to-optimal bounds for SU(n) loopapproximation. Journal of Approximation Theory, 2010.

[31] Iman Ur Rahman, Iddo Drori, Victoria C. Stodden, David Donoho, andPeter Schrder. Multiscale representations for manifold-valued data. Multi-scale modeling and Simulation, 4(4):1201–1232, 2006.

[32] Olof Runborg. Fast interface tracking via a multiresolution representationof curves and surfaces. Communications in Mathematical Sciences, 7:365–398, 2009.

[33] Hans Triebel. Theory of Function Spaces. Birkhuser, 1983.

[34] Johannes Wallner. Smoothness analysis of subdivision schemes by proxim-ity. Constructive Approximation, 24:289–318, 2006.

[35] Johannes Wallner and Nira Dyn. Convergence and c1 analysis of subdivisionschemes on manifolds by proximity. Computer Aided Geometric Design,22:593–622, 2005.

[36] Johannes Wallner, Esfandiar Navayazdani, and Philipp Grohs. Smooth-ness properties of lie group subdivision schemes. Multiscale Modeling andSimulation, 6:493–505, 2007.

[37] Johannes Wallner, Esfandiar Navayazdani, and Andreas Weinmann. Con-vergence and smoothness analysis of subdivision rules in riemannian andsymmetric spaces. Advances in Computational Mathematics, 2010.

[38] Johannes Wallner and Helmut Pottmann. Intrinsic subdivision with smoothlimits for graphics and animation. ACM Transactions on Graphics, 25:356–274, 2006.

[39] Andreas Weinmann. Nonlinear subdivision schemes on irregular meshes.Constructive Approximation, 31:395–415, 2010.

[40] Gang Xie and Thomas Yu. Smoothness equivalence properties of generalmanifold-valued data subdivision schemes. Multiscale Modeling and Simu-lation, 7(3):1073–1100, 2008.

20 BIBLIOGRAPHY

Chapter 2

[G1]: A general proximityanalysis of nonlinearsubdivision schemes

Abstract

In recent work nonlinear subdivision schemes which operate on manifold-valueddata have been successfully analyzed with the aid of so-called proximity condi-tions bounding the difference between a linear scheme and the nonlinear one.The main difficulty with this method is the verification of these conditions. Inthe present paper we obtain a very clear understanding of which properties anonlinear scheme has to satisfy in order to fulfill proximity conditions. To thisend we introduce a novel polynomial generation property for linear subdivisionschemes and obtain a characterization of this property via simple multiplicativ-ity properties of the moments of the mask coefficients. As a main applicationof our results we prove that the Riemannian analogue of a linear subdivisionscheme which is defined by replacing linear averages by the Riemannian centerof mass satisfies proximity conditions of arbitrary order. As a corollary we con-clude that the Riemannian analogue always produces limit curves which are atleast as smooth as those of the linear scheme it has been derived from. If themanifold under consideration is a Cartan-Hadamard manifold, this result for thefirst time yields a manifold-valued subdivision scheme which converges for allinput data and produces arbitrarily smooth limit curves. We also generalize ourresults to the case of multivariate subdivision schemes with arbitrary dilationmatrix.

2.1 Introduction

The aim of this paper as well as our previous work is to provide a frameworkfor the analysis of functions which take values in a differentiable manifold. Thefundamental problem in doing this is the absence of any linear structure formanifold-valued functions. This fact makes it hard or even impossible to apply

21

22 CHAPTER 2. [G1]: A GENERAL PROXIMITY ANALYSIS

most tools from linear approximation theory. One tool from the linear theorywhich does however generalize nicely to the manifold-valued setting is subdivi-sion. Our approach to the problem stated above is therefore the following:

1. Formulate the linear tools in terms of subdivision schemes.

2. Generalize these tools to the manifold-valued setting via manifold-valuedsubdivision.

3. Prove that these generalized tools satisfy the same properties as theirlinear counterparts.

It is usually the third point that is the most difficult. In fact it has taken sometime and several attempts until manifold-valued subdivision schemes could bedeveloped which produce limit curves enjoying the same smoothness as theirlinear counterparts [114, 68].

To establish statements of the third kind it has proven fruitful to first localizeto a chart and then regard the manifold-valued subdivision scheme as a pertur-bation of the linear scheme it has been derived from operating in the domainof definition of the chart. In particular it has turned out that a special kind ofperturbation inequalities, called proximity conditions, are especially useful: forinstance to prove that a manifold-valued subdivision scheme is as smooth as thelinear scheme it has been derived from, but also to show other properties. Forthis task establishing the proximity conditions is usually the most difficult part.

We mention a few results which rely on proximity inequalities: Proximity⇒smoothness has been established in [61, 60, 68, 65], proximity⇒ approximationorder in [127, 50], proximity ⇒ characterization of smoothness of a functionw.r.t. interpolatory wavelet coefficients [116] and proximity ⇒ stability of thesubdivision scheme and associated multiscale transforms [52].

In the present paper we obtain a very clear understanding of the conditionsneeded for proximity inequalities to hold. As a main application of our resultswe show that the Riemannian analogue of a linear scheme, which we definebelow, satisfies proximity conditions of arbitrary high order and thus produceslimit curves which are at least as smooth as the limit curves of the correspondinglinear scheme.

If the manifold under consideration is a Cartan-Hadamard-manifold, thisresult for the first time shows how to adapt a linear scheme to the manifold-valued setting such that it converges for all input data and produces limit curvesof arbitrarily high smoothness.

The outline of our paper is as follows: In Section 2.2 we first review somebasic facts and definitions from the theory of linear subdivision schemes. Wealso introduce a new polynomial generation property for linear schemes whichwill turn out to characterize the degree of the proximity conditions which can beachieved. Section 2.3 first introduces the class of nonlinear subdivision schemesthat we are interested in. Then, after localizing to a chart, we prove our mainresult, namely a proximity condition for the previously introduced class of sub-division schemes. In Section 2.4 we apply the general result of Section 2.3 tothe Riemannian analogue of a linear scheme using a bootstrapping argument.In Section 2.5 we extend our results to the multivariate setting with arbitrarydilation matrix.

2.2. SOME RESULTS ON LINEAR SUBDIVISION 23

2.2 Some results on linear subdivision

2.2.1 Well-known facts

We start by defining our notion of subdivision schemes.

Definition 2.2.1. An m-dimensional subdivision operator (m ∈ Z+) is a map-ping T : (Rm)Z → (Rm)Z that is local and has dilation factor N > 1, meaningthat

σN T = T σ,

where σ denotes the right shift on Z. Locality means that the value (T p)i ∈Rm, i ∈ Z, p ∈ (Rm)Z depends only on a finite number of points.

Denote by Fl(T lp) the piecewise linear function that interpolates the dataT lp on the grid 1

N lZ. Then T is called convergent if the functions Fl(T lp)

uniformly converge to a continuous nonzero limit function T ∞p for all initialdata p 6= (. . . , 0, 0, 0, . . . ). If for all initial data p the limit function T ∞p ∈Cn, then T is called a Cn subdivision scheme. Of special interest are linearsubdivision schemes which we shall always denote by S. They can be writtenas

(Sp)i =∑j∈Z

ai−Njpj , i ∈ Z

with a finitely supported sequence (ai)i∈Z, called the mask of S.In the study of linear subdivision schemes it is of special importance how

they act on polynomial samples. A subdivision scheme is said to reproduceΠ≤n, the space of polynomials of degree ≤ n, if

SP0p = P1p for all p ∈ Π≤n,

where Pj denotes the sampling operator defined on continuous functions viaPjf = (f(i/N j))i∈Z. It is well known [100, 47] that a linear scheme S hasapproximation order n+ 1 if S reproduces Π≤n. Further, it is well known thatif a linear scheme is convergent, it must reproduce Π0. This property, oftencalled reproduction of constants, is clearly equivalent to∑

j∈Zai−Nj = 1 for all i ∈ Z.

Another important property of linear subdivision schemes related to polynomialsis polynomial generation: A subdivision scheme is said to generate Π≤n if forevery polynomial p ∈ Π≤n there exists another polynomial pS ∈ Π≤n such that

SP0p = P0pS for all p ∈ Π≤n.

There are more precise results regarding the polynomial pS which we shall needlater on (see [100]):

Lemma 2.2.2. If S generates polynomials of degree ≤ n, then for every p ∈Π≤n we have

pS(x) =1

N

∑i∈Z

p(x− iN

)ai. (2.1)


As an example of a linear subdivision scheme we mention the cardinal B-Spline scheme of degree k with dilation factor N which is defined by the gener-ating function of its mask via∑

j∈Zajz

j =(1 + z + · · ·+ zN−1)k+1

(Nz)k.

It is easy to see that S generates polynomials of degree ≤ k + 1. If k = 1 thenS reproduces linear polynomials and thus has approximation order 2.

A useful tool for studying linear schemes are derived schemes. With ∆ beingthe operator that maps a sequence p = (pi)i∈Z to (pi+1−pi)i∈Z, the l-th derivedscheme is defined as the linear subdivision scheme S[l] that satisfies

N l∆l S = S [l] ∆l.

Derived schemes need not exist in general. For instance S [1] exists iff S repro-duces constants. In general the conditions for the existence of derived schemes,called sum rules, are equivalent to polynomial generation.

The derived schemes are intimately related to the smoothness of S. Beforewe can state this more precisely, we need some preparation. For a convergentsubdivision scheme S one can compute the limit function ϕ(x) = S∞δ w.r.t.the initial data sequence δ defined by δ0 = 1 and zero elsewhere. The scheme Sis called stable if there exist constants A1, A2 such that

A1‖p‖∞ ≤ ‖∑i∈Z

piϕ(x− i)‖∞ ≤ A2‖p‖∞ for all p ∈ l∞(Z).

Now we can formulate the following well-known theorem (cf. [48]):

Theorem 2.2.3. Let S be a stable subdivision scheme of Cn smoothness. ThenS generates polynomials of degree ≤ n and derived schemes up to order n + 1exist. Moreover, the derived schemes S [l], l = 1, . . . , n are convergent and satisfy

(S [l])∞∆lp =dl

dxlS∞p.

2.2.2 New results

Below we introduce a new kind of polynomial generation / reproduction prop-erty which will turn out to be convenient for our analysis. More precisely, itcharacterizes the degree of the proximity condition which is satisfied between alinear subdivision scheme S and its geometric analogue.

Definition 2.2.4. A linear subdivision scheme S has polynomial generationdegree (pgd) (d, f) if it generates polynomials of degree < f and for finitely

many polynomials pr, r = 1, . . . , k with deg(∏k

r=1 pr)< f the polynomial

q :=∏kr=1 p

Sr −

(∏kr=1 pr

)Sis of degree < d.

If S reproduces polynomials of degree n, it has pgd (0, n+ 1).The following lemma states a precise characterization of all linear subdivision

schemes which have pgd (d, f) in terms of simple multiplicativity properties ofthe moments of their masks.

2.2. SOME RESULTS ON LINEAR SUBDIVISION 25

Lemma 2.2.5. A linear subdivision scheme S has pgd (d, f) if and only if itgenerates polynomials of degree < f and for all j = 0, . . . , f − d− 1 we have theequality

∑i∈Z

ijaiN

=(∑i∈Z

iaiN

)j. (2.2)

Proof. We shall use the notation p ∼= q for two polynomials if the leading coef-ficients down to degree d of p and q agree. To prove the ”if“–part we need toshow that, provided (2.2) holds,

k∏r=1

pSr∼=( k∏r=1

pr)S.

Let us first examine pS for p =∏kr=1 pr: By (2.1) and Taylor’s formula (writing

p(j) for the j’th derivative of p) we have

( k∏r=1

pr)S

=1

N

∑i∈Z

p(x− iN

)ai ∼=1

N

∑i∈Z

f−d−1∑j=0

1

j!p(j)(

x

N)(−iN

)jai

=1

N

f−d−1∑j=0

(−1

N)j

1

j!

∑i∈Z

ijaip(j)(

x

N)

=1

N

f−d−1∑j=0

(−1

N)j

1

j!

∑i∈Z

ijai∑

l∈Nk0 , |l|1=j

(j

l

) k∏r=1

p(lr)r (

x

N)

=

f−d−1∑j=0

(−1

N)j

1

j!

∑i∈Z

ijaiN

∑l∈Nk0 , |l|1=j

(j

l

) k∏r=1

p(lr)r (

x

N)

=

f−d−1∑j=0

(−1

N)j∑i∈Z

ijaiN

∑l∈Nk0 , |l|1=j

k∏r=1

1

lr!p(lr)r (

x

N).

Here we have used the notation |l|1 :=∑kr=1 |lr|. Furthermore we let

(j

l

):=

j!

l1! · · · · · lk!.


Now we turn to∏kr=1 p

Sr :

k∏r=1

pSr =

k∏r=1

1

N

∑i∈Z

pr(x− iN

)ai ∼=1

Nk

k∏r=1

(∑i∈Z

f−d−1∑j=0

1

j!p(j)r (

x

N)(−1

N)jai

)=

1

Nk

k∏r=1

( f−d−1∑j=0

(−1

N)j

1

j!

∑i∈Z

ijaip(j)r (

x

N))

∼=1

Nk

f−d−1∑j=0

∑l∈Nk0 , |l|1=j

(−1

N)j

k∏r=1

(p(lr)r (

x

N)

1

lr!

∑i∈Z

ilrai)

=1

Nk

f−d−1∑j=0

∑l∈Nk0 , |l|1=j

(−1

N)j

k∏r=1

( 1

lr!p(lr)r (

x

N)N(

∑i∈Z

iaiN

)lr)

=1

Nk

f−d−1∑j=0

(−1

N)j(∑i∈Z

iaiN

)jNk∑

l∈Nk0 , |l|1=j

k∏r=1

1

lr!p(lr)r (

x

N)

=

f−d−1∑j=0

(−1

N)j∑i∈Z

ijaiN

∑l∈Nk0 , |l|1=j

k∏r=1

1

lr!p(lr)r (

x

N) ∼=

( k∏r=1

pr)S.

We have used (2.2) in both the 4th and the last line. This proves the ”if“ part.

For the ”only if“ part we consider the polynomial p(x) := xf−1 =∏f−1r=1 x :=∏k

r=1 pr(x). Along the same lines as the previous computations we see that theterms of degree f − 1− j of pS are given by(

f − 1

j

)(−1

N)j(∑i∈Z

ijaiN

)( xN

)f−1−j.

On the other hand, the terms of the same degree of∏kr=1 p

Sr are given by(

f − 1

j

)(−1

N)j(∑i∈Z

iaiN

)j( xN

)f−1−j.

This proves the statement.

One can draw a number of conclusions from the previous lemma. For the restof our paper we will need the following two:

Corollary 2.2.6. If S has pgd (d, f) it also has pgd (d−k, f−k) for k = 1, . . . , d.

Proof. This follows from the fact that (2.2) only depends on the difference f −d.

Corollary 2.2.7. If S generates polynomials of degree ≤ n, then S has pgd(n− 1, n+ 1).

2.3. PROXIMITY CONDITIONS 27

Proof. This is clear since the condition (2.2) is void for f − d = 2.

It is easy to see that if S reproduces linear polynomials, then S havingpgd (d, f) implies that S generates polynomials of degree < f and reproducesthe coefficients of degree ≥ d: Let p(x) be any polynomial of degree < f and∏kr=1 pr(x) its decomposition into its (complex) linear factors. Then obviously∏r=1 p

Sr = p and thus p−pS is of degree < d. This means that all coefficients of

degree ≥ d are reproduced. This in turn by Corollary 2.2.6 implies that any Swhich reproduces linear polynomials and has pgd (d, f), reproduces polynomialsof degree < f − d. Conversely, if S generates polynomials of degree < f andreproduces polynomials of degree < f − d, it clearly is of pgd (0, f − d). Thisimplies that (2.2) is satisfied for j = 0, . . . , f − d − 1. Hence, by Lemma 2.2.5,S has pgd (d, f).

The previous remark suggests that the pgd quantifies the discrepancy be-tween approximation order on the one hand, and polynomial generation on theother. This is because polynomial reproduction is basically equivalent to ap-proximation order. For us it is mainly of interest since it allows us to characterizethe degree of proximity we can achieve between S and a geometric analogue T ,cf. Section 2.3 below.

2.3 Proximity conditions

2.3.1 Nonlinear subdivision in manifolds

In recent years a number of possible manifold-valued adaptions of linear subdi-vision schemes have been developed and analyzed. We describe a few of them:

Log-exp-analogue Introduced in [46, 59], this nonlinear analogue operatesin any manifold that has some kind of exponential and logarithm mappingavailable to compute difference vectors between points (which lie in the tangentspace) and to add vectors to points. The idea is to locally map points intothe tangent space of one point, to perform linear subdivision in the tangentspace, and to project the resulting tangent vector back onto the manifold. Forexample if G is a Lie group with Lie algebra g, the Log-exp-analogue of a linearsubdivision scheme S with dilation factor 2 is defined as

T p2i+l := pi · exp(∑j∈Z

a2(i−j)+l log(p−1i · pj)

)i ∈ Z, l ∈ 0, 1. (2.3)

In [115, 69] it is shown that for S interpolatory the scheme T is as least assmooth as S. However, for noninterpolatory S, only smoothness equivalenceresults up to C2-smoothness could be obtained in [62]. In [68] it is shown (bycomputational evidence) that indeed there is a breakdown in the smoothness ofT for higher order smoothness. Another drawback of the Log-exp analogue isthat a proof of convergence exists only for very dense initial data p, meaningthat ‖ log(p−1

i · pi)‖ must always be extremely small.Projection analogue The projection analogue of a linear scheme S is de-

fined by T p := P Sp, where P is a retraction mapping onto some submanifoldM of Euclidean space. This definition ensures that one round of subdivisionmaps data in M to data in M. For special cases (P closest point projectiononto M = Sn and S interpolatory) it has been shown in [66] that T has the


same smoothness as S. A substantial generalization of this result has been givenin [114] where it is shown that the projection analogue of any linear scheme S isat least as smooth as S for any submanifold M and any smooth retraction P .One drawback of the projection analogue is that it is only extrinsically defined.Another problem is again that it can only be shown to converge if the initialdata p is very dense.

The g–f analogue The g–f analogue has been introduced in [68]. It isin the spirit of the Log-exp-analogue, but with one important difference: Notethat in (2.3) the point pi is used as ”basepoint“ for the computation of the twopoints T p2i and T p2i+1. Realizing that this does not always yield smooth limitfunctions, Xie and Yu modified this definition to

T pi = pi ·(∑j∈Z

ai−2j log(p−1i · pj)

), p = T ′p, (2.4)

where T ′ is the Log-exp-analogue, in the sense of (2.3), of some linear subdivisionscheme S ′ that has a high enough degree of polynomial reproduction. Withthis modification, it is shown in [68] that the scheme T has indeed the samesmoothness of S. However, there remains the problem that T only for very denseinitial data p there exists a proof of convergence of T . To be more precise, thedensity bounds that the initial data sequence p must satisfy for the proof to gothrough are of the form

‖∆p‖∞ ≤1− µ2C

, (2.5)

where C is a constant depending on the curvature of the manifold and µ < 1is the contraction factor of the linear subdivision scheme S, see [61, 49]. Actu-ally, even this bound is not sufficient to show convergence in general, compareLemma 5.4 in [51]. The actual bound is much smaller and too complicated tobe reproduced here.

Besides, it would be desirable to have a modification at hand that is lessartificial. Another problem is that it is not easy and even infeasible in themultivariate case to preserve certain symmetry properties of the linear scheme S.Let us mention that the g–f analogue has been defined in more generality thanpresented here in that it is not restricted to the log and exp mappings of somemanifold. Rather, the original definition from [68] is based on a vector-bundlesetting and includes all the previous manifold-valued modifications, comparethe geometric analogue below.

Riemannian analogue If a Riemannian structure is available on the man-ifold in which we want to subdivide (this is actually the case for every manifold[42]), then probably the most natural analogue of a linear subdivision schemeis to replace the linear averages

∑j∈Z ai−2jpj with the Riemannian Center of

Mass av((ai−2j)j∈Z,p) which we introduce below in Section 2.4. Hence, theRiemannian analogue of a linear scheme S is simply defined as

T pi = av((ai−2j)j∈Z,p). (2.6)

One immediately sees that in this case it is not necessary to (somewhat artifi-cially) choose a sequence of ”basepoints“ as in the previous examples. Anotherimportant advantage of this definition is that all symmetry properties of S auto-matically carry over to T . This fact is especially important for the multivariatecase which we treat in Section 2.5. Last but not least, in [63] it has been shown


that at least for manifolds with nonpositive curvature and certain schemes S,the Riemannian analogue T converges for all initial data p. We expect thatthis result can even be generalized to handle a much larger class of manifoldsand subdivision schemes, at least we are quite confident that along the samelines as in [63] the bounds of the form (2.5) can be significantly improved.

In addition to these important properties, the present paper shows that theRiemannian analogue satisfies arbitrarily high proximity conditions with S andthus inherits the smoothness properties of S.

Geometric analogue We now describe an analogue which includes allthe previous ones. It is basically the g–f -analogue but with more freedom inchoosing the basepoint sequence. The basic ingredients we need in defining anonlinear scheme which operates in a manifold M are

• a smooth vector bundle Eπ→M,

• a smooth point-vector-addition ⊕ defined locally around the zero sectionwhich maps a point p and a vector v attached to p to another pointq = p⊕ v.

• a smooth difference operation defined locally around the diagonal ofM×M that maps two points p, q to the difference vector v = q p attachedto the point p such that p⊕ (q p) = q for all p, q.

These ingredients allow us to define the geometric analogue with basepoint se-quence p of a subdivision scheme S via

T pi = pi ⊕∑j∈Z

ai−Njpj pi. (2.7)

In order to ensure that T is a subdivision scheme in the sense of Definition2.2.1, we must impose that the basepoint sequence p is also generated by asubdivision scheme T ′ in the sense of Definition 2.2.1, i.e. p = T ′p. However,we do not require that T ′ is the log-exp analogue of some linear scheme withhigh polynomial reproduction.

2.3.2 Notation and statement of the main result

Let us now introduce some notation. In the next subsection we will frequentlyuse the concatenation operator

[v]β := (v, . . . , v︸︷︷︸β-times

), β ∈ N.

In particular for a sequence p the expression [p]β means the β-variate sequence(p, . . . ,p︸︷︷︸β-times

). Sometimes we will encounter expressions of the form ∆d[p]β for some

d = (d1, . . . , dβ) ∈ Nβ . This denotes the β-variate sequence (∆d1p, . . . ,∆dβp).For a vector d = (d1, . . . , dβ) ∈ Nβ and p ≥ 0 we define

|d|p :=

∑βr=1 |dr|p p 6=∞

maxβr=1 |dr| p =∞.


If p is a sequence with values in a finite dimensional vector space with norm‖ · ‖, we let

‖p‖∞ := supi∈Z‖pi‖.

For various asymptotic estimates we shall employ the following notation:

Γr := j ∈ Nr0 :

r∑i=1

iji = r, |j|1 > 1 Γ′r := j ∈ Nr0 :

r∑i=1

iji = r.

Ωr(p) :=∑j∈Γr

r∏i=1

‖∆ip‖ji∞, Ω′r(p) :=∑j∈Γ′r

r∏i=1

‖∆ip‖ji∞.

Since we are primarily interested in local properties such as smoothness, weshall restrict ourselves to charts. In a chart, the role of the maps ⊕ and in the definition of the geometric analogue are played by two maps f : U ⊂Rdim M × Rdim E−dim M → Rdim M and g : V ⊂ Rdim M × Rdim M →Rdim E−dim M, U, V open, with f(p, g(p, p− q)) = q for all p, q (see also [68]):

Definition 2.3.1. For initial data p with values in a compact set K we definethe localized geometric analogue T of a linear subdivision scheme S as

T pi := f(pi,∑j∈Z

ai−Njg(pi, pi − pj)), (2.8)

where pi := T ′pi with an arbitrary (linear or nonlinear) subdivision scheme T ′in the sense of Definition 2.2.1 such that ‖p− p‖∞ = O(‖∆p‖∞).

Now that we have localized the nonlinear scheme T to operate on Euclideandata, we can compare it with S.

Definition 2.3.2. Two subdivision schemes S and T satisfy a proximity con-dition of degree (d, f) if

‖∆d(T − S)p‖∞ = O(Ωf (p)) (2.9)

for all K-valued data p such that Sp and T p are defined.

The goal of the present section is to prove the following theorem:

Theorem 2.3.3. If S has pgd (m,n + 1), m ≤ n − 1 and if T is the localizedgeometric analogue of S with a basepoint sequence satisfying

‖∆rp‖∞ = O(Ω′r(p)) for r = 1, . . . ,m (2.10)

then S and T satisfy a proximity condition of degree (m,n+ 1).

This result has a number of consequences. First of all it implies the resultof [68], where p is chosen as T ′p, and T ′ is the Log-exp-analogue of a linearscheme with polynomial reproduction n. From results in [115] it follows that forthis choice of p the assumption (2.10) is satisfied. However, the assumption thatp = T ′p is much stronger that (2.10), in fact it is one of the key observationsof the present paper that only the rather weak assumption (2.10) is needed.


Theorem 2.3.3 also sheds some light on the fact that for the Log-exp-analogueonly a proximity condition of degree (1, 3) could be shown: The Log-exp-analogue uses the basepoint sequence p with p2i = p2i+1 = pi. With thischoice, p satisfies ‖p− p‖∞ = O(‖∆p‖∞) which implies (2.10) for m = 1, butnot more. If S has pgd (n − 1, n + 1), this implies a proximity condition ofdegree (1, 3), but not more.

Most importantly, in Section 2.4 we are able to apply Theorem 2.3.3 toprove that the Riemannian analogue of a linear subdivision scheme S inheritsthe smoothness properties of S.

2.3.3 Proofs

Lemma 2.3.4. Let T be defined as in Definition 2.3.1 and define for j ∈ Zk, k ∈N the sequence

Aj(i) :=

∏kr=1 ai−Njr − ai−Nj1 , if j1 = · · · = jr∏kr=1 ai−Njr else.

Then, with the exception of terms of order O(‖∆p‖n+1∞ ), the difference

(S − T )pi, i = 0, . . . ,m (2.11)

can be written as a finite linear combination of expressions of the form∑j∈Zα

Aj(i)Ψ([pj1 ]h1 , . . . , [pjα ]hα , [pi]

β), (2.12)

where h1, . . . , hα ≥ 1, n ≥ α ≥ 2 and Ψ is a(∑α

l=1 hl+β)

- multilinear mappingwhose norm only depends on the compact set K.

Proof. We denote by F (k) the symmetric k-multilinear form 1k!d

kf∣∣(p0,0)

, and

by G(l) the symmetric l-multilinear form 1l!d

lg∣∣(p0,0)

with k, l ≤ n. Note that

the operator norms of F (k), G(l) can be bounded uniformly in K. This is whyΨ in (2.11) will be bounded by a constant only depending on K.

The nonlinear scheme T pi can be written as

T pi = f(pi,∑j∈Z

ai−Njg(pi, pi − pj))

=

n∑k=1

F (k)([(pi − p0,

∑j∈Z

ai−Njg(pi, pi − pj))]k)

+R

=∑k=1

∑j∈Zk

k∏r=1

ai−NjrF(k)((pi, g(pi, pi − pj1), . . . , (pi, g(pi, pi − pjk)

)+R

where R is a term of order O(‖(pi− p0,∑j∈Z ai−Njg(pi, pi− pj)‖n+1). We also

write the linear scheme Spi as follows:

Spi =∑j∈Z

ai−Njpj =∑j∈Z

ai−Njf(pi, g(pi, pi − pj))

=

n∑k=1

∑j∈Z

ai−NjF(k)([(pi − p0, g(pi, pi − pj))]k

)+R′,


where R′ is a term of order O(‖(pi − p0, g(pi, pi − pj)‖n+1).We show that we can disregard the terms R,R′ by showing that they are both

of order ‖pi−p0‖n+1: What we need to show is that ‖(pi−p0,∑j∈Z ai−Njg(pi, pi−

pj))‖, ‖(pi − p0, g(pi, pi − pj))‖ = O(‖∆p‖∞). Since∑j∈Z ai−Nj = 1 and (ai)

is of finite support we only need that ‖(pi − p0, g(pi, pi − pj))‖ = O(‖pi − p0‖).Clearly, we only need to obtain a bound on the second factor g(pi, pi − pj).

Since g ∈ Lip 1, there exists a constant C such that

‖g(pi, pi − pj)‖ = ‖g(pi, pi − pj)− g(pi, 0)‖ ≤ C‖pi − pj‖

and this shows that R,R′ = O(‖pi − p0‖n+1). Since ‖p − p‖∞ = O(‖∆p‖∞)this shows that R,R′ = O(‖∆p‖n+1). Hence, with the exception of terms oforder O(‖∆p‖n+1

∞ ) we can write (2.11) as a finite linear combination of termsof the form∑

j∈ZkAj(i)F

(k)((pi − p0, g(pi, pi − pj1)), . . . , (pi − p0, g(pi, pi − pjr ))

). (2.13)

By splitting up the terms (pi−p0, g(pi, pi−pj) = (pi, 0)+(−p0, 0)+(0, g(pi, pi−pj)) and using the multilinearity and symmetry of F (k), we see that we can write(2.11) as a finite linear combination of terms of the form∑j∈Zα′

Aj(i)F(k)((0, g(pi, pi−pj1)), . . . , (0, g(pi, pi−pjα′ )), [(pi, 0)]β

′, [(−p0, 0)]γ

′).

(2.14)By considering the last γ′ terms as constants (which are uniformly bounded bya bound only depending on K), we may write (2.14) as∑

j∈Zα′Aj(i)Ψ

′(g(pi, pi − pj1), . . . , g(pi, pi − pjα′ ), [pi]β′)

(2.15)

with a bounded (α′ + β′)-multilinear form Ψ′.Let us now look at the expressions g(pi, pi − pj): By Taylor expansion we

can write

g(pi, pi − pj) =

n∑l=0

G(l)([(pi − p0, pi − pj)]l

)+R′′,

where R′′ = O(‖(pi − p0, pi − pj)‖n+1). Obviously this implies that R′′ =O(‖∆p‖n+1

∞ ) and thus we may discard it. Inserting into (2.15) we see that(2.11) can be written as linear combination of terms of the form

∑j∈Zα′

Aj(i)Ψ′(G(l1)

([(pi−p0, pi−pj1)]l1

), . . . , G(lα′ )

([(pi−p0, pi−pjα′ )]

lα′), [pi]

β′).

(2.16)Now we expand the terms (pi−p0, pi−pjr ), r = 1, . . . , α′, into (pi, 0)− (p0, 0)+(0, pi)− (0, pjr ), use the multilinearity of Ψ′ and Glr , and arrive at the desiredexpression.

The condition α ≥ 2 follows from the fact that all terms in (2.11), whereα = 0, 1 vanish.


By using the previous lemma we continue to rewrite the difference ∆m(T −S)p0.

Lemma 2.3.5. Let the assumptions and notation be as in Lemma 2.3.4 andm ∈ N0. The difference ∆m(T − S)p0 can be expanded as linear combinationof terms of the form∑

j∈Zα

(∆dAj(i)

)0Ψ

([pj1 ]h1 , . . . , [pjα ]hα ,

(∆e[p]β

)(0,...,0)

)), (2.17)

with d+ |e|1 = m. Again we discard factors of order O(‖∆p‖n+1∞ ).

Proof. For m = 1 by Lemma 2.3.4 we may write ∆(T − S)p0 as a linear com-bination of terms of the form∑j∈Zα

Aj(1)Ψ([pj1 ]h1 , . . . , [pjα ]hα , [p1]β

)−∑j∈Zα

Aj(0)Ψ([pj1 ]h1 , . . . , [pjα ]hα , [p0]β

).

By telescoping we may write the above expression as∑j∈Zα

(∆Aj(i))0Ψ([pj1 ]h1 , . . . , [pjα ]hα , [p1]β

)+

β∑r=1

∑j∈Zα

Aj(0)Ψ([pj1 ]h1 , . . . , [pjα ]hα , [p0]r−1,∆p0, [p1]β−r

).

Iterating this argument gives the desired conclusion.

Our goal is to gain estimates on the norm of (2.17). Regarding the last βvectors in Ψ as constants we may write

(2.17) =∑j∈Zα

(∆dAj(i))0Ψ′([pj1 ]h1 , . . . , [pjα ]hα

)(2.18)

with

Ψ′([pj1 ]h1 , . . . , [pjα ]hα

):= Ψ

([pj1 ]h1 , . . . , [pjα ]hα ,

(∆e[p]β

)(0,...,0)

)).

Obviously

‖Ψ′‖ = O(

β∏r=1

‖∆er p‖∞), (2.19)

the implicit constant only depending on K.

Lemma 2.3.6. Assume that S has pgd (d, f) and let α, h1, . . . , hα be positiveintegers. Then for any γ :=

∑αr=1 hr - multilinearform Φ the expression∑

j∈Zα(∆dAj(i))0Φ

([pj1 ]h1 , . . . , [pjα ]hα

)can be rewritten as a finite linear combination of terms of the form

Φ(∆d[p]γ(j1,...,jγ)

), where |d|1 = f and |d|∞ < f. (2.20)

The coefficients in this linear combination only depend on the mask of S.


Proof. The proof is similar to the arguments used in [115, 114]. We translate thestatement into a combinatorial problem for the multivariate Laurent polynomial

A(x) =∑j∈Zα

(∆dAj(i))0(x1 · · · · · xg1)j1 · · · · · (xγ−hγ+1 · · · · · xγ)jα , x ∈ Rγ .

The assertion that∑

j∈Zα(∆dAj(i))0Φ([pj1 ]h1 , . . . , [pjα ]hα

)can be written as a

linear combination of elements of the form (2.20) is equivalent to the statementthat A(x) is expressible as a linear combination of Laurent polynomials of theform

B(x) =

γ∏r=1

(1− xr)drC(x), |d|1 = f and |d|∞ < f,

with some Laurent polynomial C. This follows if all derivatives of total degree< f and all directional derivatives ( ∂

∂xr)l of A(x) vanish at x = [1]γ . Now let

D = ∂|t|1

∂xt11 ...∂x

tγγ

be a differential operator of degree |t|1. Applying D to A and

evaluating at (1, . . . , 1) gives

DA(1, . . . , 1) =∑j∈Zα

(∆dAj(i))0p1(j1) · · · · · pα(jα)

for polynomials pr with deg(∏r pr) = |t|1. Looking at Lemma 2.3.4 and the

definition of Aj, we note that this expression has an interpretation in terms ofS:

DA(1, . . . , 1) =∑j∈Zα

(∆dAj(i))0p1(j1) · · · · · pα(jα)

= ∆d∑j∈Zα

α∏r=1

ai−Njrpr(jr)−∑j∈Z

ai−Nj

α∏r=1

pr(jr)

= ∆dα∏r=1

(∑j∈Z

ai−Njpr(jr))−∑j∈Z

ai−Nj

α∏r=1

pr(jr)

= ∆d(∏

r

pSr (i)− (∏r

pr)S(i)

).

First, assume that D is a partial differential operator ( ∂∂xr

)l. Then there exists

an r0 with pr0 =∏r pr and therefore the expression

∏r pSr (i) − (

∏r pr)

S(i)vanishes trivially. Now let D be a general differential operator of degree <f . This is where our definition of subdivision schemes of class (d, f) is doingits work: since S is of class (d, f), the expression

∏r pSr (i) − (

∏r pr)

S(i) isa polynomial of degree < d in i and therefore gets annihilated by the ∆d-operator.

Proof. (of Theorem 2.3.3) First let us remark that by translation invariance itis sufficient to estimate ‖∆m(T − S)p0‖, since all constants in our estimatesonly depend on the set K. By Lemma 2.3.6 we may rewrite ∆m(T − S)p0

as a linear combination of terms of the form (2.17) with d + |e|1 = m. ByLemma 2.3.6, Corollary 2.2.6 and (2.19) we can write this expression as a linear

2.4. RIEMANNIAN CENTER OF MASS 35

combination of terms of the form Ψ′(∆d[p]β(j1,...,jγ)

)with |d|1 = d+ n+ 1−m

and |d|∞ < d+ n+ 1−m. We can thus estimate

‖Ψ′(∆d[p]β(j1,...,jγ)

)‖ ≤ ‖Ψ′‖

γ∏s=1

‖∆dγp‖∞ ≤ ‖Ψ′‖Ωd+n+1−m(p).

By (2.19) and (2.10) we can estimate ‖Ψ′‖ = O(Ω′|e|1(p)) = O(Ω′m−d(p)). Sum-marizing we get that

‖Ψ′(∆d[p]β(j1,...,jγ)

)‖ = O(Ωd+n+1−m(p)Ω′m−d(p)) = O(Ωn+1(p)).

The implicit constant only depends on K and therefore we can conclude that

‖∆m(T − S)p‖∞ = O(Ωn+1(p)).

2.4 Applications to manifold-valued subdivisionvia the Riemannian center of mass

2.4.1 The center of mass

Consider the weighted Euclidean average p∗ =∑j αjpj of points p = (pj) with

weights α = (αj) such that∑j αj = 1. This average is characterized as the

unique minimizer of the weighted sum of squared distances given by

fα(x) :=∑j

αjdist (x, pj)2.

Going one step further, the average p∗ is characterized by the infinitesimalcondition

∇fα(p∗) = 0,

or ∑j

αj(pj − p∗) = 0. (2.21)

Interpreting pj − p∗ as difference vector pointing from p∗ to pj , we may rewrite(2.21) as ∑

j

αjpj p∗ = 0. (2.22)

This expression makes sense in any setup where a point-vector addition and adifference vector of two points are available, in particular in our setup for thegeometric analogue. In this framework we may define the geometric averageg-av(α,p) of the points p with weights α as the point p∗ which satisfies (2.22).Of course, such a point need not exist. Also it may not be unique. However foran important special case one can say more: Let M be a Riemannian manifoldwith point-vector addition defined as (p, v) 7→ p⊕ v := exp(p, v) and differenceoperation (p, q) 7→ q p := log(p, q), where exp is the exponential function ofM and log its local inverse. Then the following theorem holds [57]:


Theorem 2.4.1. If ⊕ and are defined as above, then there exists a localconstant C > 0 such that for all p = (pj)

kj=1 and weights α = (αj)

kj=1 with

diam(p) := maxdist(pj1 , pj2) : 1 ≤ j1, j2 ≤ k ≤ C, there exists a uniquep∗ with (2.22) which minimizes the function fα. The constant C depends onthe sectional curvature of M. If the curvature of M is ≤ 0 and M is simplyconnected, we may take C =∞.

The previous theorem allows us to define a Riemannian center of mass for anyRiemannian manifold:

av(α,p) := g-av(α,p).

We define the Riemannian analogue T of a linear subdivision scheme S withmask (ai) via

T pi := av((ai−Nj)j∈Z,p). (2.23)

A few remarks are in order:

First, this definition fits into our framework of the geometric analogue: Be-cause of the equilibrium condition (2.22), T can be regarded as the geometricanalogue of S with basepoint sequence p = T p. This important property willlater allow us to bootstrap from low order proximity conditions to higher orderproximity conditions using the results from Section 2.3.

The Riemannian analogue T can conveniently be computed via fixed-pointiteration [57, 64]: Let p1 be the result of one round of geometric subdivisionin the Riemannian manifold M with some basepoint sequence such that p1

and Sp satisfy a proximity condition (see [68]), and define pj as the result ofone round of geometric subdivision with basepoint sequence pj−1 for j ≥ 2.Then the sequence pj converges to T p as j goes to infinity. One strategy toprove that the Riemannian analogue satisfies a proximity condition with S is toshow inductively that pj satisfies a proximity inequality with Sp if pj−1 does.Actually this can be shown using Theorem 2.3.3 and this was actually the firstapproach that we pursued. However, as j increases, the constant appearing inthe O-term in the proximity inequality goes to infinity. Therefore some otherideas are needed to show that T and S satisfy a proximity condition of arbitraryorder. In the next section we present a bootstrapping argument which does justthat.

2.4.2 Proof of the Proximity condition for the Riemanniananalogue

We are finally ready to prove the main result of this paper, namely that aproximity condition of high order holds between a linear scheme S and its Rie-mannian analogue T :

Theorem 2.4.2. If S is a linear subdivision scheme with pgd (m,n+ 1), m ≤n − 1, then in a chart the Riemannian analogue T of S satisfies the proximitycondition

‖∆m(T − S)p‖∞ = O(Ωn+1(p)), (2.24)

where differences and norms are with respect to a local coordinate chart.

2.4. RIEMANNIAN CENTER OF MASS 37

Proof. We use the equilibrium property of the Riemannian center of mass tobootstrap the desired proximity condition from weaker ones. It is not hard toshow that ‖(T − S)p‖∞ = O(Ω2(p)), see e.g. [63]. If n ≤ 1, this is already thedesired proximity condition. If n ≥ 2, we use the fact that in a chart T can bewritten as localized geometric analogue of S with p = T p. In view of Theorem2.3.3 we thus need to show that

‖∆rp‖∞ = ‖∆rT p‖∞ = O(Ω′r(p)) r = 1, . . . ,m. (2.25)

We regard T as a perturbation of S and note that because of the fact that Sgenerates polynomials of degree ≤ n, it possesses derived schemes up to ordern+1. This implies that ‖∆rSp‖∞ is always bounded by O(Ω′r(p)). It thereforeremains to prove that

‖∆r(T − S)p‖∞ = O(Ω′r(p)) r = 1, . . . ,m. (2.26)

We already showed this for r = 1, 2. Now we perform an induction step. Assumethat

‖∆r(T − S)p‖∞ = O(Ω′r(p)) r = 1, . . . , r0. (2.27)

We would like to show that

‖∆r(T − S)p‖∞ = O(Ω′r(p)) r = 1, . . . , r0 + 1 (2.28)

provided that r0 < m. Note that by Corollary 2.2.6, S is has pgd (r0 + 1, n +1−m+ r0 + 1). Using (2.27) and Theorem 2.3.3 we get that

‖∆r0+1(T − S)p‖∞ = O(Ωn+1−m+r0+1) = O(Ωr0+1).

This implies (2.28) and proves the theorem.

2.4.3 Smooth Splines in CH-Manifolds

Let us now draw some conclusions from Theorem 2.4.2. First, by the results in[60, 68] we have the following theorem:

Theorem 2.4.3. Let S be a stable linear subdivision scheme of Cn-smoothnessand T its Riemannian analogue operating in a Riemannian manifold M. ThenT is of Cn-smoothness for all initial data p such that T converges.

The Riemannian analogue has the remarkable property that it converges for allinitial data p if the manifoldM is a CH-manifold (simply connected, nonpositivesectional curvature) and S is such that the mask is nonnegative, meaning that

ai ≥ 0 for all i ∈ Z,

and such thatN−1maxl=0

∑j∈Z|a[1]l−Nj | ≤ 1,

(a[1]i ) being the mask of the first derived scheme S [1] of S. This has been shown

in [63] (we expect that this result can be generalized). As a corollary to thisand Theorem 2.4.3 we have:


Theorem 2.4.4. Let S be a stable linear Cn subdivision scheme with nonneg-ative mask (ai)i∈Z such that

N−1maxl=0

∑j∈Z|a[1]l−Nj | ≤ 1

and let M be a CH-manifold. Let T be the M-valued Riemannian analogue ofS. Then for all initial data p ∈MZ there exists a Cn limit function T ∞p.

As an example of a CH-Manifold which is of great relevance for practical ap-plications we mention the manifold Posn of symmetric positive definite n × nmatrices. It has constant curvature −1 and its relevance comes from its usagein Diffusion Tensor Imaging [45].

It is now tempting to define smooth splines in CH-manifolds with controlpoints p as limit functions of the Riemannian analogue of the usual cardinalB-spline subdivision schemes. By our results these splines are always as smoothas the linear ones.

As a topic for future research we would like to study further propertiesthese nonlinear splines share with their linear counterpart. For example we areinterested in the following questions:

• is it possible to define an interpolation procedure for nonlinear splines, i.e.is the interpolation problem (at least locally) well-defined?

• is there a way to use the nonlinear splines to obtain quasi-interpolantsthat share the approximation properties of their linear counterparts? Thisquestion has also been asked in [127].

• in case the interpolation procedure is well-defined, does the interpolationminimize an energy analogous to the linear case?

2.5 Generalization to the multivariate case

We describe how to generalize our results to the multivariate case. The reasonsfor our choice to present only the univariate version in full detail and only sketchthe multivariate extension are twofold: first, all relevant ideas and insights arealready contained in the univariate case. And second, the presentation wouldbe harder to follow if we worked in full generality from the start.

We now consider the following situation: Let M ∈ Zs×s be an expandingmatrix, i.e. every eigenvalue is of modulus > 1.

Definition 2.5.1. An m-dimensional subdivision operator (m ∈ Z+) with di-lation matrix M is a mapping T : (Rm)Z

s → (Rm)Zs

which is local and whichobeys

σMy T = T σy,

where σy denotes the right-shift by y ∈ Zs on Zs. Locality means that the value(T p)i ∈ Rm, i ∈ Zs, p ∈ (Rm)Z

s

depends only on a finite number of points.

We call a subdivision scheme with dilation matrix M linear if it is of the form

Spi =∑j∈Zz

ai−Mjpj,

2.5. GENERALIZATION TO THE MULTIVARIATE CASE 39

where (ai)i∈Zs is a finitely supported sequence called the mask. Our first goalis to prove Lemma 2.5.2 below which is a multivariate generalization of Lemma2.2.2. We think that this result is of interest in its own right. Note that thedefinition of polynomial generation for multivariate schemes is the same as forlinear schemes with the exception that s-variate polynomials are considered.

Lemma 2.5.2. If S generates polynomials of degree ≤ n, then for every p ∈Π≤n we have

pS(x) =1

det(M)

∑i∈Zs

p(M−1(x− i)

)ai. (2.29)

In order to prove this we need some preparation. We consider the groupsZs/MZs and Zs/MTZs and choose sets R, R of representatives for these factorgroups. For instance one could set R = M [0, 1)s ∩ Zs and R = MT [0, 1)s ∩ Zs.In particular the groups Zs/MZs and Zs/MTZs have order |det(M)|.For z = (z1, . . . , zs) and i = (i1, . . . , is) we write

zi := zi11 · · · · · ziss .

We defineξd := exp(2πiM−T d), d ∈ R

and note that these are |det(M)| solutions of the equation zMi = 1 for all i ∈ Zs,as shown in [58]. Let

ωdi :=

1

det(M)

∑d∈R

ξi−dd

, i ∈ Zs, d ∈ R.

It follows thatωdi+Mj = ωd

i ,

i.e. the grid function i ∈ Zs 7→ ωdi is M -periodic. For d ∈ R we define the

cosets Ld := d +MZs.

Lemma 2.5.3. ωdi = 1 if i ∈ Ld and zero everywhere else.

Proof. This is a direct consequence of Lemma 1 in [58] which has first beenproven in [43].

Proof. (of Lemma 2.5.2) Define

Ω(i, j) :=∑d∈R

ωdi ω

dj .

From Lemma 2.5.3 it follows that

Ω(i, j) =

1 i− j ∈ L0

0 else.

Using this fact, a straightforward computation yields that

Spi =∑j∈Zs

Ω(i, j)pM−1(i−j)aj.


Note that the index M−1(i − j) is always in Zs for Ω(i, j) 6= 0. We assume(abusing notation) that pi = p(i) for some polynomial p of degree ≤ n. Then

Spi =∑j∈Zs

Ω(i, j)p(M−1(i− j)

)aj =

∑d∈R

ωdi pd(i),

wherepd(x) :=

∑j∈Zs

ωdj p(M−1(x− j)

)aj.

Clearly, Spi = pS(i) for some polynomial if and only if pd = p0 for all d ∈ R.Especially we have

Spi =(∑d∈R

ωdi

)pd0

(i) for all d0 ∈ R.

Let i ∈ Ld0. Since

∑d∈R ω

di = 1 and |R| = |det(M)|, this implies that∑

j∈Zsp(M−1(i− j))aj =

∑j∈Zs

(∑d∈R

ωdi

)p(M−1(i− j))aj

=∑d∈R

pd(i) = |det(M)|pd0(i) = |det(M)|Spi.

This proves the lemma.

We are now able to prove the analogous characterization of the polynomialgeneration degree as in Lemma 2.2.5. The definition of the pgd property carriesover to the multivariate setting unchanged.

Lemma 2.5.4. A linear subdivision scheme S has pgd (d, f) if and only if itgenerates polynomials of degree < f and for all j ∈ Zs with 0 ≤ |j|1 ≤ f − d− 1we have that

det(M)|j|1−1∑i∈Zs

(M−1i)jai =(∑i∈Zs

M−1iai)j

(2.30)

Proof. Using Lemma 2.5.2 and multivariate Taylor expansion, the proof pro-ceeds in the same way as the proof of Lemma 2.2.5.

The important result for us is the following:

Corollary 2.5.5. Corollaries 2.2.6 and 2.2.7 are also valid in the multivariatesetting.

Now we are able to tackle our main goal, namely to prove a result analogousto Theorem 2.3.3 for the multivariate setting. First we define the multivariatedifference operator

∆pi := (∆1pi, . . . ,∆spi)T ∈ Rms,

where we denote by ∆i the univariate difference operator in the i-th directionon Zs. We let

‖∆p‖∞ :=s

maxi=1‖∆ip‖∞.

By replacing the univariate ∆ operator with the multivariate difference ∆ wecan now adapt the notations and definitions of Section 2.3.2. In particular wemake the following definitions:


Definition 2.5.6. For initial data p with values in a compact set K we definethe localized geometric analogue T of a linear subdivision scheme S as

T pi := f(pi,∑j∈Zs

ai−Mjg(pi, pi − pj)), (2.31)

where pi := T ′pi with an arbitrary (linear or nonlinear) subdivision scheme T ′in the sense of Definition 2.2.1 such that ‖p− p‖∞ = O(‖∆p‖∞).

Definition 2.5.7. Two subdivision schemes S and T satisfy a proximity con-dition of degree (d, f) if

‖∆d(T − S)p‖∞ = O(Ωf (p)). (2.32)

The main results are as follows:

Theorem 2.5.8. If S has pgd (m,n + 1), m ≤ n − 1 and if T is the localizedgeometric analogue of S with a basepoint sequence satisfying

‖∆rp‖∞ = O(Ω′r(p)) for r = 1, . . . ,m (2.33)

then S and T satisfy a proximity condition of degree (m,n+ 1).

Proof. The proof goes by adapting the univariate arguments from the proofof Theorem 2.3.3 to the case of multivariate sequences. This works in a ratherstraightforward way, compare the results in [51, 115, 116] for detailed argumentsof this type. We only sketch the proof:

Define for j = (j1, . . . , jr) ∈ Zsk and i ∈ Zs the sequence

Aj(i) :=

∏kr=1 ai−Mjr − ai−Mj1 , if j1 = · · · = jr∏kr=1 ai−Mjr else.

Then, with the exception of terms of the form O(‖∆p‖n+1∞ ), we can write the

difference (S − T )pi, |i|∞ ≤ m as a finite linear combination of terms of theform ∑

j∈ZsαAj(i)Ψ

([pj1 ]h1 , . . . , [pjα ]hα , [pi]

β),

where h1, . . . , hα ≥ 1, α ≥ 2 and Ψ is a(∑α

l=1 hl + β)

- multilinear mappingwhose norm only depends on the compact set K. The proof is the same as theproof of Lemma 2.3.4.

We introduce for l = (l1, . . . , lr) ∈ 1, . . . , sr the notation ∆l := ∆lr · · · ∆l1 . Let us consider one row ∆l(T − S)p0, l ∈ 1, . . . , sm of the expression∆m(T − S)p0. With an analogous telescoping argument as in the proof ofLemma 2.3.5 we obtain that ∆l(T − S)p0 can be expanded as a linear combi-nation of terms of the form∑

j∈Zsα

(∆dAj(i)

)0Ψ

([pj1 ]h1 , . . . , [pjα ]hα ,

(∆e[p]β

)(0,...,0)

)),

with |d|0 + |e|1,0 = m, where e = (e1, . . . , eβ), er ∈ 1, . . . , slr , |e|1,0 :=∑βr=1 |er|0 =

∑βr=1 lr and ∆e[p]β := (∆e1

p, . . . ,∆eβ p). Because S is of pgd


(m,n + 1) and Corollary 2.5.5, this expression can be rewritten as a linearcombination of elements of the form

Ψ

((∆f [p]γ

)(j1,...,jγ)

,(∆e[p]β

)(0,...,0)

)), |f |1,0 = |d|0 + n+ 1−m

and |f |∞,0 < n+ 1,

where γ =∑αr=1 hr, f = (f1, . . . , fγ), fr ∈ 1, . . . , sqr , |f |∞,0 := maxγr=1 |fr|0 =

maxγr=1 qr, compare Lemma 2.3.6 and Lemma 11 in [116]. Now we utilize (2.33),take norms and arrive at the desired inequality.

By replacing scalar indices with multiindices, we can also define the Riemanniananalogue for the multivariate case. Then the following result holds:

Theorem 2.5.9. If S is a linear subdivision scheme with pgd (m,n+ 1), m ≤n − 1, then in a chart the Riemannian analogue T of S satisfies the proximitycondition

‖∆m(T − S)p‖∞ = O(Ωn+1(p)) (2.34)

Proof. The proof is a straightforward generalization of the proof of Theorem2.4.2 (use multiindices instead of univariate indices, replace ∆ with ∆).

As a consequence we obtain the following two theorems which are proven bya direct application of the results in [65]:

Theorem 2.5.10. The multivariate Riemannian analogue of a Cn linear sub-division scheme generating polynomials of degree ≤ n also produces Cn limitfunctions.

If the dilation matrix M is isotropic, meaning that all eigenvalues are equal inmodulus, one can say more. Again, the limit function of the dirac sequencepeaking in 0 ∈ Zs will be denoted by ϕ. A subdivision scheme is called stable ifif there exist constants A1, A2 such that

A1‖p‖∞ ≤ ‖∑i∈Zs

piϕ(x− i)‖∞ ≤ A2‖p‖∞ for all p ∈ l∞(Zs).

The following theorem is well-known (see [56]):

Theorem 2.5.11. Let S be a stable linear subdivision scheme with isotropicdilation matrix M that produces Cn limit functions. Then S generates polyno-mials of degree ≤ n.

By utilizing Theorem 2.5.9 we arrive at the following smoothness equivalenceresult:

Theorem 2.5.12. If T is the Riemannian analogue of a Cn stable linear sub-division scheme with isotropic dilation matrix, then T also produces Cn limitfunctions.

In the case of schemes with non isotropic dilation matrices the situation be-comes more intricate. In particular it turns out that the conventional isotropicsmoothness spaces Cn are not well-suited for the study of such schemes, ratherone characterizes smoothness via anisotropic smoothness spaces which reflect the


anisotropy of M , see [44]. We do not know if it is possible to prove anisotropicsmoothness equivalence properties in the spirit of Theorem 2.5.12 for anisotropicsmoothness spaces with methods similar to the ones developed in this paper.One major obstacle for this is the lack of a nice algebraic characterization ofsmooth subdivision schemes in terms of the mask coefficients such as polynomialgeneration. There is a condition analogous to the polynomial generation prop-erty also for anisotropic schemes, called anisotropic Strang-Fix conditions. In[44] it is shown that if an anisotropic scheme is stable and smooth (measured inan anisotropic smoothness space), then these conditions are satisfied. However,we do not know how to use these conditions to prove proximity inequalities. Thereason why our proximity analysis works is the nice interplay between analyticand algebraic results for linear subdivision schemes with isotropic dilation ma-trix. For anisotropic schemes much less algebraic structure is known. It wouldbe interesting (also in view of being able to systematically search for smoothanisotropic schemes) to find out if this is an inherent lack of structure or if thestructure has simply not been found yet.

Acknowledgments

The author is supported by the Austrian Science Fund (FWF) under grantP-19790.


Bibliography

[41] A.S. Cavaretta, W. Dahmen, and C.A. Micchelli. Stationary Subdivision.American Mathematical Society, 1991.

[42] S.S. Chern, W. Chen, and K. S. Lam. Lectures on differential geometry.World Scientific Publishing Company, 1999.

[43] C.K. Chui and C. Li. A general framework of multivariate wavelets withduals. Applied and Computational Harmonic Analysis, 1(4):368–390, 1993.

[44] A. Cohen, K. Grochenig, and L. F. Villemoes. Regularity of multivariaterefinable functions. Constructive Approximation, 15(2):241–255, 1999.

[45] M. D. Denis Le Bihan, J.F. Mangin, C. Poupon, C.A. Clark, S. Pappata,N. Molko, and H. Chabriat. Diffusion tensor imaging: concepts and appli-cations. Journal of Magnetic Resonance Imaging, 13:534–546, 2001.

[46] D. L. Donoho. Wavelet-type representation of Lie-valued data. talk atthe IMI “Approximation and Computation” meeting, May 12–17, 2001,Charleston, South Carolina, 2001.

[47] S. Dubuc. Interpolation through an iterative scheme. Journal of mathe-matical analysis and applications, 114(1):185–204, 1986.

[48] N. Dyn. Subdivision schemes in computer-aided geometric design. In W. A.Light, editor, Advances in Numerical Analysis Vol. II, pages 36–104. OxfordUniv. Press, 1992.

[49] N. Dyn and D. Levin. Subdivision schemes in geometric modelling. ActaNumer., 11:73–144, 2002.

[50] P. Grohs. Approximation order from stability of nonlinear subdivisionschemes. Geometry Preprint 2008/06, TU Graz, November 2008.

[51] P. Grohs. Smoothness analysis of subdivision schemes on regular grids byproximity. SIAM J. Numerical Analysis, 46:2169–2182, 2008.

[52] P. Grohs. Stability of manifold-valued subdivision schemes and multiscaletransformations. Geometry Preprint 2008/04, TU Graz, October 2008.

[53] P. Grohs. Smoothness equivalence properties of univariate subdivisionschemes and their projection analogues. Num. Math., 2009. to appear.

[54] P. Grohs. Smoothness of multivariate interpolatory subdivision in Liegroups. IMA J. Numer. Anal., 2009. to appear.

45

46 BIBLIOGRAPHY

[55] P. Grohs and J. Wallner. Interpolatory wavelets for manifold-valued data.ACHA, 2009. to appear.

[56] R.Q. Jia. Approximation properties of multivariate wavelets. Mathematicsof Computation, 67(222):647–666, 1998.

[57] H. Karcher. Riemannian center of mass and mollifier smoothing. Commu-nications on Pure and Applied Mathematics, 30(5), 1977.

[58] V. Latour, J. Muller, and W. Nickel. Stationary subdivison for generalscaling matrices. Mathematische Zeitschrift, 227(4):645–661, 1998.

[59] I. Ur Rahman, I. Drori, V. C. Stodden, D. L. Donoho, and P. Schroder.Multiscale representations for manifold-valued data. Multiscale Mod. Sim.,4:1201–1232, 2005.

[60] J. Wallner. Smoothness analysis of subdivision schemes by proximity. Con-str. Approx., 24:289–318, 2006.

[61] J. Wallner and N. Dyn. Convergence and C1 analysis of subdivision schemeson manifolds by proximity. Comput. Aided Geom. Des., 22:593–622, 2005.

[62] J. Wallner, E. Nava Yazdani, and P. Grohs. Smoothness properties of Liegroup subdivision schemes. Multiscale Mod. Sim., 6:493–505, 2007.

[63] J. Wallner, E. Nava Yazdani, and A. Weinmann. Convergence and smooth-ness analysis of subdivision rules in Riemannian and symmetric spaces.Geometry Preprint 2007/05, TU Graz, October 2008.

[64] A. Weinmann. Nonlinear subdivision processes in irregular meshes. Geom-etry Preprint 2008/03, TU Graz, June 2008.

[65] A. Weinmann. Smoothness of nonlinear subdivision schemes with arbitrarydilation matrix. Geometry Preprint 2009/03, TU Graz, April 2009.

[66] G. Xie and T. P.-Y. Yu. Smoothness equivalence properties of manifold-valued data subdivision schemes based on the projection approach. SIAMJ. Numer. Anal., 45:1200–1225, 2007.

[67] G. Xie and T. P.-Y. Yu. Approximation order equivalence properties ofmanifold-valued data subdivision schemes. Manuscript, submitted for pub-lication, April 2008.

[68] G. Xie and T. P.-Y. Yu. Smoothness equivalence properties of generalmanifold-valued data subdivision schemes. Multiscale Modeling and Simu-lation, 2008. to appear.

[69] G. Xie and T. P.-Y. Yu. Smoothness equivalence properties of interpolatoryLie group subdivision schemes. IMA Journal on Numerical Analysis, 2008.to appear.

Chapter 3

[G2]: Interpolatory waveletsfor manifold-valued data

Abstract

Geometric wavelet-like transforms for univariate and multivariate manifold-valued data can be constructed by means of nonlinear stationary subdivisionrules which are intrinsic to the geometry under consideration. We show that inan appropriate vector bundle setting for a general class of interpolatory wavelettransforms, which applies to Riemannian geometry, Lie groups and other ge-ometries, Holder smoothness of functions is characterized by decay rates oftheir wavelet coeffcients.

3.1 Introduction

A great part of work in the analysis of signals, images, and generally real-valued functions concerns the extraction of local information at different levelsof resolution, and the conversion of continuous data to a countable collection ofcoefficients. Wavelet transforms are undoubtedly the most prominent conceptin this area [73].

Topics relevant to wavelet-type transforms include: the computation ofwavelet coefficients; the approximative computation of coefficients from dis-cretely sampled data; re-synthesis of the original continuous data from thecoefficient sequences; the effect which quantizing, thresholding, or otherwiseperturbing coefficients has on the synthesis; and how properties like smoothnesscan be read off the coefficients.

The overwhelming majority of wavelet-type constructions are linear and theirtheory is formulated in terms of topological vector spaces and linear operators.It is a trivial point, which however is important for us, that for linear construc-tions there is often no difference between applying them to real-valued dataand to vector-valued data (at least if one works with so-called scalar subdivi-sion schemes as opposed to vector subdivision schemes, we refer the interestedreader to [87, 89]). Things become different in the analysis of geometric data,

47

48 CHAPTER 3. [G2]: INTERPOLATORY WAVELETS

where the structure of a vector space, even if employed for purposes of coordi-nate representation, is not natural. Functions which take values in surfaces, orRiemannian manifolds, or Lie groups, should be analyzed by intrinsic processes.This basically means invariance under appropriate transformation groups: e.g. itis natural to require that constructions applied to data living in a matrix groupG ≤ GLn be invariant with respect to left translations x 7→ ax where a ∈ G.Likewise, constructions in geometries based on a metric should be invariant un-der isometries. Linear constructions for the purpose of analyzing vector-valueddata occur only as a special case.

Tools common in multiresolution (wavelet) analysis such as spaces spannedby the translates of a refinable function can usually not easily be modifiedso as to apply to data which take values in more general geometries. Withoutfunction spaces, concepts like orthogonality and best approximation are difficultto formulate. The present paper therefore restricts itself to the interpolatingwavelet transforms introduced in [77, 84] which are computable from samplesof a function. We recall their construction and their relation to stationarysubdivision rules in Section 3.1.2. The idea to generalize them to manifold-valued data is not new, but has been proposed some years ago by D. Donoho[78] (see also [91]).

The present paper shows how an interpolating wavelet transform may beconstructed for both univariate and multivariate manifold-valued functions ina way which unifies different kinds of geometries, and that this nonlinear con-struction retains essential properties of the analogous linear construction. Inparticular we show that smoothness of functions directly corresponds to thedecay rate of coefficients.

We mention a few examples of geometries we are thinking of: the Euclideanmotion group SEn (pose data of rigid bodies), the Grassmann manifolds Gn,k

(subspace-valued data), and the symmetric space of positive definite matricesPosn (multivariate data representing diffusion tensor images).

3.1.1 Linear stationary subdivision rules

We here recall properties of linear stationary subdivision rules [71]. Such alinear rule S maps real-valued or vector-valued data p : Zs → Rn (n ≥ 1) todata Sp : Zs → Rn according to

(Sp)α =∑β∈Zs

aα−Nβ · pβ . (3.1)

This definition involves the mask (aα)α∈Zs and the dilation factor N (typically,N = 2). We require a finite mask (#α | aα 6= 0 <∞) and affine invariance ofS:

For all α,∑β∈Zs aα−Nβ = 1. (3.2)

Data p formally defined as a function on the unit grid can be interpreted assamples of a function Fjp on the grid N−jZ. Vice versa, a function f may besampled on a finer grid and converted into data Pjf formally defined on theunit grid. We let

(Fjp)(N−jα) = pα, (Pjf)α = f(N−jα),

3.1. INTRODUCTION 49

so PjFjp = p. A subdivision rule S is interpolatory, if the function F1Spinterpolates the original data (i.e., p = F1Sp|Zs). This is equivalent to a|NZs

being zero except for a0 = 1, and it implies that FiSip = FjSjp|N−iZs wheneveri ≤ j.

The sequence FjSjpj≥0 of functions constructed by subdivision has thelimit S∞p := limj→∞ FjSjp, which is defined in a dense subset of Rs. Con-vergence and Ck smoothness of a subdivision rule S means that for all inputdata p, S∞p is continuous and its unique continuous extension to Rs enjoys Ck

smoothness.We also consider Holder smoothness of subdivision rules: With the notation

(∆hf)(x) = f(x+ h)− f we define the Holder smoothness classes by

f ∈ Lip γ ⇐⇒ ‖f‖Lip γ = ‖f‖∞ + suph∈Rs\0

(h−γ‖∆bγ+1cf‖∞

)<∞ (3.3)

and we say that S has critical Holder regularity r, if all S∞p ∈ Lip γ wheneverγ < r.

A rule S has polynomial reproduction of degree d if for any polynomial f ∈R[x1, . . . , xs] of total degree ≤ d we have Sf |Zs = P1f , i.e., applying S to regularsamples of f produces a denser sampling of the same f . Ck rules have d ≥ k.

Example 3.1.1. Denote by Ltα,...,tβα,...,β the Lagrange interpolation polynomial

which maps each subscript integer to the corresponding superscript. Fix d > 0and let

SpNα = pα, SpNα+β = Lpα−d,...,pα+d+1

α−d,...,α+d+1 (α+ βN ) (β = 1, . . . , N − 1).

Then S is a subdivision rule with dilation factor N and polynomial reproductiondegree 2d+1 [75]. One can show that S has Ck limit functions, with k ≈ d · const.

3.1.2 Linear interpolating wavelet transforms

Introduced by [84, 77] for the univariate case and N = 2, they are based ona “father wavelet” ϕ : R → R with ϕ|Z = δ, where (δα)α∈Z is the Kroneckerdelta sequence (i.e., δ0 = 1 and δα = 0 for α 6= 0). The major example of [77]is that ϕ = S∞δ for some interpolatory subdivision rule S. In the following weconsider the general multivariate case. Interpolatory wavelet-like constructionshave been used in various places, e.g. [88, 85, 86, 74].

The interpolatory wavelet transform associated with an interpolatory subdi-vision rule S maps a function f : Rs → Rn to the coefficient collection (uα)α∈Zs ,(w0

α)α∈Zs , (w1α)α∈Zs , . . . , which is defined by

u = P0f = f |Zs , w0 = P1f − SP0f, w1 = P2f − SP1f, . . .

Smallness of wjβ expresses agreement between (j+1)-st level samples f |N−(j+1)Zsand values (SPjf)β predicted from the j-th level samples f |N−jZs . We recoverf (actually, dense samples of f) by Pjf = wj−1 +Swj−2 + · · ·+Sj−1w0 +Sju.

The following result expresses the fact that smoothness of functions is char-acterized by decay rates of their wavelet coefficients. Both smoothness andcoefficient decay is encoded by finiteness of certain norms. In the present paperwe aim at similar results for the geometric (multivariate and nonlinear) case.


Theorem 3.1.2 ([77], Th. 2.7). Assume that the interpolatory univariate sub-division rule S has polynomial reproduction degree ≥ d, and that ϕ = S∞δhas Holder continuity ≥ r. If r, d > σ > 1

p and 0 < p, q ≤ ∞, the norm

‖(u,w0, w1, . . . )‖ :=∥∥u∥∥

`p(Z)+ ‖ω‖`q(Z+

0 ), where ωj = 2j(σ−1/p)∥∥wj∥∥

`p(Z), on

the interpolating wavelet coefficients of a function f is equivalent to the normof f in the Besov space Bσp,q(R).

3.1.3 Subdivision rules and wavelet transforms in mani-folds

Geometric subdivision rules have been mostly analyzed with regard to smooth-ness (cf. [82, 92, 93, 94, 95, 96, 97] for the univariate case and [81, 83] for themultivariate case), but also with regard to approximation order [79]. Variousdefinitions have been given.

A very general way to define subdivision in a manifold M relies on analoguesof the operation ‘point y minus point x’ and its inverse ‘point x plus vector’ (thevector in question is supposed to lie in an appropriate vector space associatedwith x). We use the notation v = y x and y = x⊕ v for these mappings.

Example 3.1.3. In a Lie group M with Lie algebra g we let yx = log(x−1y),x⊕ v = x exp(v), where v ∈ g and log is the inverse of exp : g→M around e ∈G. In a Riemannian manifold M we let yx = exp−1

x (y), and x⊕ v = expx(v)where v ∈ TxM , and expx is the Riemannian exponential mapping.

Equation (3.2) shows that we can express the subdivision rule S of (3.1) interms of the operations v = y − x, y = x+ v for points x, y and vectors v:

(Sp)Nγ+α =∑β∈Zs

aNγ+α−Nβ · pβ = pγ +∑β∈Zs

aNγ+α−Nβ(pβ − pγ)

for all α, but especially α ∈ 0, . . . , N − 1s. This motivates the followingdefinition:

Definition 3.1.4. Assume that π : E →M is a smooth vector bundle over thebase manifold M (dimE < ∞), and that ⊕ : E → M and : M ×M → Eare smooth and defined locally around M and the diagonal (x, x) ⊂ M ×M ,

respectively. With the notation v ∈ Ex⊕7−→x⊕ v and (x, y)

7−→x y we requirethat y x ∈ Ex, and x ⊕ (y x) = y whenever defined. Then the subdivisionrule T given by

(T p)Nγ+α = pγ ⊕∑β∈Zs

aNγ+α−Nβ(pβ pγ) where α ∈ 0, . . . , N − 1s,

is called the geometric analogue of S. It applies to data p where all instancesof ⊕ and which contribute to T p — terms with aNγ+α−Nβ = 0 do not — aredefined.

In the Lie group case of Ex. 3.1.3, E = M×g and ⊕ is defined globally, whilein the Riemannian case, E = TM and ⊕ is defined globally for complete M . Inboth cases the domain of depends on the respective exponential mappings.E.g. in Cartan-Hadamard manifolds, is globally defined [76].

3.2. RESULTS 51

Example 3.1.5. Consider a surface M ⊂ Rn where P : U → M is a smoothretraction onto M (e.g. a matrix group M is considered as a surface in Rm×mand P is the closest point projection w.r.t. the Frobenius norm). The subdivisionrule T p := P Sp operates on data p : Zs → M and is easily seen to be aninstance of Definition 3.1.4: We let E = M × Rn, y x = y − x ∈ x × Rn,and x⊕ v := P (x+ v).

Having transferred subdivision to the manifold setting, we now define:

Definition 3.1.6. The wavelet transform with respect to the interpolatory subdi-vision rule T for M -valued data maps a function f : Rs →M to the coefficients

u = P0f = f |Zs , w0 = P1f T P0f, w1 = P2f T P1f, . . .

Here we let (pα)α∈Zs (qβ)β∈Zs = (pα qα)α∈Zs . Each of w0, w1, . . . repre-sents bundle-valued data. Note that for topological reasons (worm holes in M)there might be no function f for given u, wjj≥0, not even if all wjβ = 0.

Remark 3.1.7. Example 3.1.3 is due to [78, 91], and Example 3.1.5 is consid-ered also in [97]. In the Lie group case, T ∞p ∈ Lip γ if the Holder regularity ofS exceeds γ and this limit exists (which it does for dense enough input data), asshown in [96, 83]. Analogous results for the univariate retraction case are givenby [82].

3.2 Results

3.2.1 Wavelet coefficient decay and smoothness

The ‘usefulness’ of Definition 3.1.6 is indicated by the fact that like in the linearcase, the smoothness of a function can be read off its wavelet coefficients. Theprecise statements are as follows:

Theorem 3.2.1. Let S be a linear interpolatory subdivision rule of Holdersmoothness r and polynomial reproduction degree d, and let T be its geometricanalogue in the bundle π : E → M . Assume that f : Rs → M is continuous,and that wj : Zs → E are the wavelet coefficients of the function x 7→ f(σx) forsome σ > 0 (whose local existence is guaranteed for small σ).

If f ∈ Lipα and α < d, then ‖wiβ‖ ≤ CN−αi. Conversely, if ‖wiβ‖ ≤ CN−αiand α < r, then f ∈ Lipα. The constant C is understood to be uniform fordata values in a compact set.

Here the symbols ‖wiβ‖ refer to a smooth bundle norm for E (e.g. the Rie-mannian metric in E = TM) the precise choice of which turns out to be irrele-vant.

We break the proof of Theorem 3.2.1 into two steps: (i) Localization of theresult and transfer to a trivial bundle over an open subset of Rm (see below);and (ii) Proof for the simplified setting (see Section 3.3.3).

We start our discussion with the local nature of the result. There is ρ > 0such that the mask coefficient aα = 0 whenever α is outside the ball ρB ofradius ρ. Consequently the wavelet coefficient wjβ of the function x 7→ f(σx) is

determined by f ’s restriction to the ball σN−(j+1)(β + ρB). Smoothness of f(equivalently, smoothness of any f(σ ·)), is a local property. We may therefore,


without loss of generality, restrict the analysis of smoothness of f , and of thewavelet coefficients of f , to arbitrarily small neighbourhoods.

In particular we assume that we work in the domain of a single bundle chartχ from E to the trivial bundle π : E = U × Rn → U , where U is open in someRm. Each E-fiber x × Rn is equipped with the χ-image ‖ · ‖x of the originalbundle metric, which smoothly depends on x. Thus by making U smaller ifnecessary we can achieve that ‖ · ‖x is uniformly equivalent to the standardmetric in Rn.

It is therefore sufficient to show Theorem 3.2.1 for the case that the bundleis U ×Rn, each fiber being equipped with the canonical norm. For convenience,we still use the notation ⊕, , T for the χ-transforms of the respective entities.

3.2.2 Proximity results

The proof of Theorem 3.2.1 relies on the proximity inequality of Theorem 3.2.2which assumes the viewpoint that T is a perturbation of S and which quantifiesthe distance of S from T . Such proximity results are widely employed in theanalysis of nonlinear subdivision schemes.

A result similar to Theorem 3.2.2 is contained in [83], which allows us to keepthe proof short by referring to lemmas also found there. The part which is newin contrast to [83] is that Theorem 3.2.2 applies not only to nonlinear rules Tdefined in matrix groups via the matrix exponential function, but the much moregeneral class of geometric analogues considered here. Nevertheless the algebraicpart of the proof is very similar. Theorem 3.2.2 considers only subdivision rulesin trivial bundles U×Rm, but in view of the previous section this is sufficient forour purposes. We make use of the following notation: Consider data p : Zs → Rnand the canonical basis vectors e1, . . . , es of Rs. Let

(∆ip)β = pβ+ei − pβ , (∆p)β = (∆1pβ , . . . ,∆spβ) ∈ Rns.

Iterating this construction yields data ∆kp : Zs → Rnsk . Further, let ‖p‖ :=supα∈Zs ‖pα‖∞. With these preparations, we formulate:

Theorem 3.2.2. Assume that S is a linear interpolatory rule with polynomialreproduction of degree k, and T is its geometric analogue in the bundle U ×Rm(U open in Rn). For any compact K ⊂ U there is C > 0 such that for K-valueddata p,

‖Sp− T p‖ ≤ C∑

i1, . . . , ik ∈ Z+0

i1 + 2i2 + · · ·+ kik = k + 1

‖∆p‖i1 . . . ‖∆kp‖ik (k > 0). (3.4)

For k = 0 we have the better estimate ‖Sp− T p‖ ≤ C‖∆p‖2.

3.3 Proofs

3.3.1 Proof of the proximity inequality

Recall that the subdivision rule T reads

(T p)Nγ+α = pγ ⊕∑β∈Zs

aNγ+α−Nβ(pβ pγ).

3.3. PROOFS 53

The relation x⊕ (yx) = y implies that the linear rule S of (3.1) is expressibleas (Sp)Nγ+α =

∑β∈Zs aNγ+α−Nβ(pγ⊕(pβpγ)). By introducing some auxiliary

notation we can further rewrite S, T :

Ψp : Rm → U, v 7→ p⊕ v, vβγ := pβ pγ =⇒

T pNγ+α = Ψpγ

( ∑β∈Zs

aNγ+α−Nβvβγ

), SpNγ+α =

∑β∈Zs

aNγ+α−NβΨpγ (vβγ ).

The following lemma concerning the Taylor expansion of Sp − T p is wordedin terms of the r-linear mappings drΨx

∣∣0

which occur in the Taylor expansion

x ⊕ v = x + dΨx

∣∣0(v) + · · · + 1

k!dkΨx

∣∣0(v, . . . , v) + 1

(k+1)!dk+1Ψx

∣∣θv

(v, . . . , v),

where 0 < θ < 1. We also introduce the right inverse Φx : y 7→ y x of thefunction Ψx and consider its expansion y x =

∑kl=1

1l!d

lΦx∣∣x(y − x, . . . , y −

x) +O(‖y − x‖k+1).

Lemma 3.3.1. The difference T p− Sp can be expanded around γ ∈ Zs as

(T p)Nγ+α − (Sp)Nγ+α =

k∑l=0

Bl +O(‖∆p‖k+1), (3.5)

where Bl = 1l!

∑β1,...,βl∈Zs Aβ1,...,βld

lΨpγ

∣∣0(vβ1γ , . . . , v

βlγ ), and the coefficients

A··· are defined as Aβ,...,β := (aNγ+α−Nβ)l−aNγ+α−Nβ , if all indices are equal,and as Aβ1,...,βl := aNγ+α−Nβ1 . . . aNγ+α−Nβl otherwise.

Proof. The proof is the same as for the special case x⊕v = x exp(v) in [83]: Weexpand Sp− T p and estimate the remainder term via vβγ ≈ dΦpγ (pβ − pγ).

By substituting vβγ = Φpγ (pβ) in Bl, we express Bl in terms of the inputdata p:

Bl =1

l!

∑I∈1,...,kl

1

I!

∑β1,...,βl∈Zs

Aβ1,...,βl · F Iβ1,...,βl+O(‖∆p‖k+1),

where I = (i1, . . . , il), I! = i1! · · · il!, and the symbol F Iβ1,...,βlstands for

F Iβ1,...,βl= CI

([pβ1− pγ

]i1 times, . . . ,

[pβl − pγ

]il times), where (3.6)

CI(x1, . . . , x|I|) = dlΨpγ

∣∣0(di1Φpγ (x1, . . . , xi1), . . . , dilΦpγ (. . . , x|I|)). (3.7)

CI : (Rn)|I| → Rn is multilinear. Lemma 3.3.3 below, which gives bounds forBl not in terms of ∆p (which would be easy), but in terms of higher differences,needs

Lemma 3.3.2. Assume that (vτ )τ∈Zs are V -valued data, B : V r → W is amultilinear mapping, and A(v) =

∑τ1,...,τr∈Zs sτ1,...,τrB(vτ1 , . . . , vτr ). With the

notation L(n1, . . . , nr) = (lji ) | 1 ≤ j ≤ r, 1 ≤ i ≤ nj , 1 ≤ lji ≤ s, A(v) isexpressible as∑

τ1, . . . , τr ∈ Zsn1 + · · ·+ nr = k + 1

nj < k + 1

∑l∈L(n1,...,nr)

b(n1,...,nr),lτ1...τr B

(∆l11· · ·∆l1n1

vτ1 , . . . , ∆lr1· · ·∆lrnr

vτr),


if and only if all derivatives of order ≤ k and all partial derivatives ∂k+1

∂xτj0with

j0 ∈ 1, . . . , r, τ ∈ Ns and |τ |1 :=∑sr=1 τr = k + 1 of the Laurent polynomial

fA(x1, . . . ,xr) =∑

τ1,...,τr∈Zssτ1,...,τrx

τ11 . . .xτrr

vanish for (x1, . . . ,xr) = (1, . . . , 1) ∈ Rrs.

The special case that B is matrix multiplication and V = Rn×n is [83,Lemma 1], whose proof carries over unchanged. We use the notation xα =xα1

1 · · ·xαss , α ∈ Zs.

Lemma 3.3.3. If S reproduces polynomials of degree ≤ k then, in the notationof Equation (3.7), there exists a constant C = C(pγ , I, α) > 0, such that∥∥∥ ∑

β1,...,βl∈ZsAβ1,...,βlF

Iβ1,...,βl

∥∥∥ ≤ C ∑n1+···+nk=k+1

‖∆n1p‖ . . . ‖∆nkp‖.

Proof. The left hand sum has the form of the expression “A(v)” in Lemma3.3.2, if we let B = CI and sτ1,...,τ|I| = 0 zero except for sτ1,...,τ|I| = Aβ1,...,βl ,

if (τ1, . . . , τ|I|) = ([β1]i1 times, . . . , [βl]i1 times). Clearly the associated Laurent

polynomial reads

fA =( ∑β1∈Zs

aNγ+α−Nβ1xβ1

1 · · ·xβ1

i1

)· · ·( ∑βl∈Zs

aNγ+α−Nβlxβl|I|−il+1 · · ·x

βl|I|)

−∑β∈Zs

aNγ+α−Nβ xβ1 · · ·xβ|I|.

If D is any differential operator, then DfA(x1, . . . ,x|I|)|(1,...,1) equals

l∏j=1

( ∑β∈Zs

pj(β)aNγ+α−Nβ)−∑β∈Zs

l∏j=1

pj(β)aNγ+α−Nβ ,

where pj are polynomials with deg∏lj=1 pj = deg(D). This expression has an

interpretation in terms of samples and the subdivision rule S such that thepolynomial reproduction property applies: If deg(D) ≤ k, DfA(1, . . . , 1) can beexpressed as∏j

(Spj |Zs)Nγ+α −(S(∏

j

pj |Zs))Nγ+α

=∏j

pj(γ +α

N)− (

∏j

pj)∣∣x=γ+ α

N

= 0.

If D = ∂k+1

∂xτj0, we have

∏j pj = pj0 and we can express DfA(1, . . . , 1) as

∏j

(Spj |Zs)Nγ+α −(S(∏

j

pj |Zs))Nγ+α

=(Spj0 |Zs

)Nγ+α

−(Spj0 |Zs

)Nγ+α

= 0.

By Lemma 3.3.2 we can rewrite A(v) in terms of higher order differences ∆jp.Taking norms yields the desired upper bound.

3.3. PROOFS 55

We now complete the proof of Theorem 3.2.2. First, since we work in acompact set, we can make the constant C(pγ , I, α) in Lemma 3.3.3 independentof pγ . As there are only finitely many indices I and α, it is likewise independentof them. By substituting the upper bounds of Lemma 3.3.3 back into Lemma3.3.1, we obtain

‖Sp− T p‖ ≤ C∑

n1+···+nk=k+1, nj<k+1

‖∆n1p‖ . . . ‖∆nkp‖

for some constant C > 0. Sorting the right hand terms by the exponents niyields the estimate required by Theorem 3.2.2 in the case k ≥ 1. If k = 0,we observe that (3.2) causes the terms of orders 0, 1 in the expansion (3.5) tovanish. Thus, ‖T p− Sp‖ = O(‖vβγ ‖2). Since vβγ = dΦpγ (pβ − pγ) of first order,the result follows.

3.3.2 Wavelet coefficient decay in the linear case

The following theorem, which concerns linear schemes, has the flavor of a knownresult. We were however unable to locate a literature reference and thereforegive a complete proof here. It is interesting that the implication (ii) =⇒ (i) canbe shown using subdivision. Similar results and proofs can be found e.g. in [72].

Theorem 3.3.4. Consider the set Ω =⋃j≥0N

−jZs of N -adic points, a func-

tion f : Ω → Rn, and the wavelet coefficients u, wjj≥0 of f with respect toa fixed linear interpolatory rule S. If S has Lip γ limit functions, the followingare equivalent:

(i) There exists f ′ ∈ Lipα with f ′|Ω = f ;(ii) f is bounded and there is an integer k > α with ‖∆kPjf‖ = O(N−αj);(iii) ‖u‖ <∞ and ‖wj‖ = O(N−jα), provided α < γ.

Proof. Without loss of generality we let n = 1. We first show (i) ⇐⇒ (iii), us-ing approximation methods and discrete interpolation spaces according to [70].Let X consist of the uniformly continuous bounded functions f : Rs → R,and let Y = Lip γ ⊂ X. Define the approximation process Vj by letting

Vjf(x) =∑β∈Zs f( β

Nj )φ(N jx − β), with φ = S∞δ — in other notation,

Vjf(x) = (S∞Pjf)(N jx). It obeys limj→∞ Vjf = f , if f ∈ X. As S is aconvergent rule, the norms ‖Vj‖ w.r.t. ‖ · ‖∞ are bounded independently of j.

It is easy to show the Bernstein-type inequality ‖Vjf‖Y ≤ C(N j)γ‖f‖∞, asφ has compact support and for all λ > 1, ‖f(λ · )‖Lip γ ≤ Cλγ‖f‖Lip γ .

We also show the Jackson inequality ‖Vjf − f‖∞ ≤ CN−jγ‖f‖Y : Any

x ∈ Rs has the form x = h+y with y ∈ N−jZs and ‖h‖ < s12N−j . Let g locally

equal the Taylor polynomial of degree dγe− 1 of f at y, so that |f(x)− g(x)| ≤CN−jγ‖f‖Y , with C independent of x, y. By polynomial reproduction, Vjg = g,and

‖Vjf − f‖∞ ≤ ‖Vjf − Vjg‖∞ + ‖g − f‖∞ ≤ C(‖Vj‖∞ + 1)(N−j)γ‖f‖Y .

Now [70, Th. 3.3.1] implies the norm equivalence [X,Y ]+α,∞,K∼= XJ

α,∞,V forα ∈ (0, γ), in the terminology of [70]. The former space, by interpolation,equals Lipα [90], the latter equals f ∈ X | supj≥0N

jα‖Vjf − Vj−1f‖∞ <


∞. By observing wj = (f − Vj−1f)|N−jZs = (Vjf − Vj−1f)|N−jZs we obtain‖wj‖ ≤ ‖Vjf − Vj−1f‖, i.e., (iii) =⇒ (i). Conversely, Vj−1 = Vj Vj−1 impliesthat ‖Vjf −Vj−1f‖ = ‖Vj(f −Vj−1f)‖ ≤ C‖Vj‖‖wj‖ ≤ C ′‖wj‖, so (iii) impliesf ∈ XJ

α,∞,V = Lipα.The implication (i) =⇒ (ii) follows e.g. from [?, Lemma 2]. For (ii) =⇒ (i),

we employ an auxiliary interpolatory rule S which has a k-th derived schemeS [k] obeying Nk∆kS = S [k]∆k (cf. [80]) and with Ck limit functions (take e.g.tensor products of the rules of Examples 3.1.1). Assuming ‖∆kPjf‖ ≤ CN−αj ,we estimate the interpolatory wavelet coefficients wj of f with respect to S:

‖∆kwj‖ = ‖∆k(SPj − Pj+1)f‖ ≤ ‖∆kSPjf‖+ ‖∆kPj+1f‖

≤ N−k‖S [k]‖‖∆kPjf‖+ ‖∆kPj+1f‖ = O(N−jα).

Now wj |N−j+1Zs = 0 implies that wj itself, not only its k-th differences, is

bounded by O(N−jα). Applying (iii) =⇒ (i) for the rule S completes theproof.

Obviously (i)⇐⇒ (ii) does not have to do anything with subdivision a priori.

3.3.3 Proof of Theorem 3.2.1

Recall that we can restrict ourselves to the bundle U × Rn, with U open inRm, and the Euclidean metric in each fiber x × Rn. We further assume thatwe work on data which take values in a compact set K. By locality this isjustified, as we can simply consider dense enough samples of f . The proofemploys Lipschitz constants C1, C2 > 0 for the function :

C1‖p q‖ ≤ ‖p− q‖ ≤ C2‖p q‖. (3.8)

For the first implication of Theorem 3.2.1, we assume that f ∈ Lipα, α < d andobserve

‖T Pjf − Pj+1f‖ ≤ ‖SPjf − Pj+1f‖+ ‖SPjf − T Pjf‖. (3.9)

Theorem 3.3.4.(iii) bounds the first term with CN−jα. We let k = bαc, so thatk ≤ d. Theorem 3.3.4.(ii) shows that ‖∆lPjf‖ ≤ ClN

−(l−ε)j for l = 1, . . . , kand any ε > 0. The second term in Equation (3.9) is estimated by Theorem3.2.2, as follows:

‖SPjf − T Pjf‖ ≤ C∑

i1+···+kik=k+1

‖∆Pjf‖i1 · · · ‖∆kPjf‖ik

≤ C∑

i1+···+kik=k+1

(N−(1−ε)j)i1 · · · (N−(k−ε)j)ik

= CN−(k+1)j−εj .

These estimates for (3.9) together with (3.8) show

‖wj‖ = ‖T Pjf Pj+1f‖ ≤ C−11 ‖T Pjf − Pj+1f‖ ≤ C(N−αj +N−(k+1−ε)j),

with ε > 0 arbitrary. This proves the desired decay rate as stated by Theo-rem 3.2.1.

3.3. PROOFS 57

For the proof of the converse statement of Theorem 3.2.1, we assume thatwavelet coefficients u, wj are given, samples Pjf for j ≥ 0 are defined, and thatcoefficients decay according to wj ∼ N−jα. Part (i) below makes an additionalcontractivity assumption, which is justified in part (iii).

Part (i): α < 1. For now we assume that T is contractive in the sense that

‖∆T p‖ ≤ µ‖∆p‖ (µ < 1). (3.10)

This allows us to recursively estimate

‖∆Pjf‖ ≤ ‖∆(Pj − T Pj−1)f‖+ ‖∆T Pj−1f‖≤ 2C2‖Pjf T Pj−1f‖+ ‖∆T Pj−1f‖≤ CN−αj + µ‖∆Pj−1f‖ ≤ . . .

which yields

‖∆Pjf‖ ≤ Cj∑l=0

µlN−α(j−l). (3.11)

If µNα < 1, then (3.11), as geometric series, is bounded by C ′N−αj and sam-ples Pjf extend to f ∈ Lipα by Theorem 3.3.4.(ii). If µNα ≥ 1, we chooseν ∈ (µ, 1) — this implies N−α/ν < 1 — and gain an estimate by (3.11)

= Cνj∑jl=0(µ/ν)l(N−α/ν)j−l ≤ Cνj

∑jl=0(µ/ν)l ≤ C νj

1−µ/ν . With ν = N−δ

we have obtained ‖∆Pjf‖ ≤ CN−jδ, showing that samples Pjf extend tof ∈ Lip δ. We increase δ by the following ‘bootstrapping’ argument, whichinvokes Theorem 3.2.2 for k = 0:

‖SPjf − Pj+1f‖ ≤ ‖T Pjf − Pj+1f‖+ ‖T Pjf − SPjf‖≤ C2‖T Pjf Pj+1f‖+ C ′‖∆Pjf‖2

≤ CN−αj + C ′′N−2δj ≤ C ′′′N−min(α,2δ)j .

Thus f ∈ Lip min(α, 2δ). By iteration, we obtain f ∈ Lipα.

Part (ii): α ≥ 1. Here we use induction. If for an integer k > 0 we alreadyknow f ∈ Lip(k − ε) for all ε > 0, we show f ∈ Lip γ for all γ ∈ [k, k + 1),provided γ ≤ α. As part (i) above serves as an induction base (k = 1), thisproves f ∈ Lipα.

We employ as an auxiliary device the wavelet coefficients wj = SPjf−Pj+1fwith respect to the linear rule S. S reproduces polynomials of degree k (becausek ≤ γ ≤ α < r). We invoke Theorem 3.2.2 to estimate the coefficients wj :

‖wj‖ ≤ ‖SPjf − T Pjf‖+ ‖T Pjf − Pj+1f‖ ≤ (′′) + C2‖T Pjf Pj+1f‖

≤ C( ∑i1+2i2+···+kir=k+1

‖∆Pjf‖i1 · · · ‖∆kPjf‖ik +N−αj).

By Theorem 3.3.4, f ∈ Lip(k − ε) implies ‖∆lPjf‖ ≤ ClN−(l−ε)j for l =

1, . . . , k, so

‖wj‖ ≤ C(N−(k+1−ε)j +N−αj

)≤ CN−γj , with C > 0.

Theorem 3.3.4.(iii) shows that f ∈ Lip γ. By induction, f ∈ Lipα.


Part (iii). To complete the proof of Theorem 3.2.1, we have to justify (3.10):It is known (cf. [71]) that for some iterate Sm there is µ′ < 1 with ‖∆Smp‖ ≤µ′‖∆p‖. By [60, Lemma 3], the case k = 1 of Theorem 3.2.2 applies to Sm, T m(since it applies to S, T ). Now [61, Th. 1] says that existence of µ′ implies‖∆T mp‖ ≤ µ‖∆p‖ for some µ < 1, for dense enough input data. Obviouslysamples Pjf are dense enough for j greater than some j0.

We now estimate the wavelet coefficients of f with respect to the subdivisionrule T m, which has dilation factor Nm. Locally T is Lipschitz continuous, sothat ‖T p − T q‖ ≤ D‖p − q‖ (this follows from the construction of T from S).Thus,

‖T mPj Pj+m‖ ≤ C−11 ‖T mPj − Pj+m‖

≤m∑l=0

‖T m−lPj+l − T m−l−1Pj+l+1‖ ≤m∑l=0

Dm−l−1‖T Pj+l − Pj+l+1‖

≤ C2

m∑l=0

Dm−l−1‖T Pj+l Pj+l+1‖ ≤ CN−αj = C(Nm)−jαm

for some C > 0. Part (i) applied to T m yields f ∈ Lip δ, with δ = αm , and so

‖∆Pjf‖ = O(N−δj). From here part (i) goes as above.

3.3.4 Remarks on the reconstruction process

Theorem 3.2.1 assumes that wavelet data u, wjj≥0 come from a continuousfunction f . If we do not know this a priori, we must observe that the bundle-valued sequences wj are not arbitary: The reconstruction procedure Pjf :=T (. . . T (T u ⊕ w0) ⊕ w1 . . .) ⊕ wj is well defined if and only if π wj = T Pjffor j ≥ 0. However, if the fibers Ex are canonically isomorphic to a fixed vectorspace E0 (as in the Lie group and retraction cases), we can view wj as E0-valuedsequences, and the consistency condition is void.

It is clear that the proof of Theorem 3.2.1 applies to data u, wj:

Corollary 3.3.5. In the same setting as Theorem 3.2.1 assume that coefficientsu : Zs → M and wj : Zs → E (j = 0, 1, . . . ) are consistently chosen such thatthe reconstruction procedure is defined. If ‖wjβ‖ ≤ CN−αj with C small enough,and u is dense enough, then the samples Pjf extend to a Lipα function f .

The rather unspecific statements on u being dense enough and C smallenough cannot be avoided. This is because reconstruction of a function withvanishing wavelet coefficients leads to the limit function T ∞u, and there areexamples where that limit does not exist. More specific statements are possibleonly for specific smaller classes of subdivision rules. We leave this problem,which appears to exhibit a big difference between the cases s = 1 and s > 1, asa topic for future research.

Further interesting problems related to our work include analysis of average-interpolating transformations [91], as well as the Lipschitz stability of the re-construction procedure, which is intimately connected with the stability of theunderlying subdivision scheme. Stability is the topic of a forthcoming paper.

3.3. PROOFS 59

Acknowledgments

This work is supported by grant P19780 of the Austrian Science Fund (FWF).We want to thank A. Weinmann for his valuable suggestions.


Bibliography

[70] P. Butzer and K. Scherer. Approximationsprozesse und Interpolationsmeth-oden, volume 826 of BI-Hochschultaschenbucher. Bibliogr. Inst., 1968.

[71] A. S. Cavaretta, W. Dahmen, and C. A. Micchelli. Stationary subdivision.AMS, 1991.

[72] Wolfgang Dahmen. Wavelet and multiscale methods for operator equations.Acta Numerica, 6:55–228, 1997.

[73] I. Daubechies. Ten Lectures on Wavelets. SIAM, 1992.

[74] I. Daubechies, O. Runborg, and W. Sweldens. Normal multiresolutionapproximation of curves. Constr. Approx., 20:399–463, 2004.

[75] G. Deslauriers and S. Dubuc. Symmetric iterative interpolation processes.Constr. Approx., 5:49–68, 1986.

[76] Manfredo P. do Carmo. Riemannian Geometry. Birkhauser, 1992.

[77] D. L. Donoho. Interpolating wavelet transforms. Technical report, StatisticsDep., Stanford, 1992.

[78] D. L. Donoho. Wavelet-type representation of Lie-valued data. talk atthe IMI “Approximation and Computation” meeting, May 12–17, 2001,Charleston, South Carolina, 2001.

[79] N. Dyn, P. Grohs, and J. Wallner. Approximation order of interpolatorynonlinear subdivision schemes. J. Comp. Appl. Math. to appear.

[80] N. Dyn and D. Levin. Subdivision schemes in geometric modelling. ActaNumer., 11:73–144, 2002.

[81] P. Grohs. Smoothness analysis of subdivision schemes on regular grids byproximity. SIAM J. Numer. Anal., 46:2169–2182, 2008.

[82] P. Grohs. Smoothness equivalence properties of univariate subdivisionschemes and their projection analogues. Num. Math., 2009. to appear.

[83] P. Grohs. Smoothness of multivariate interpolatory subdivision in Liegroups. IMA J. Numer. Anal., 2009. to appear.

[84] A. Harten. Multiresolution representation of data: A general framework.SIAM J. Numer. Anal., 33:1205, 1996.

61

62 BIBLIOGRAPHY

[85] A. Harten and S. Osher. Uniformly high-order accurate nonoscillatoryschemes. I. SIAM J. Numer. Anal., pages 279–309, 1987.

[86] M. Holmstrom. Solving hyperbolic PDEs using interpolating wavelets.SIAM J. Sci. Comput., 21(2):405–420, 2000.

[87] R.Q. Jia, S.D. Riemenschneider, and D.X. Zhou. Vector subdivisionschemes and multiple wavelets. Math. Comput., 67(224):1533–1564, 1998.

[88] G. Kutyniok and T. Sauer. Adaptive directional subdivision schemes andshearlet multiresolution analysis. arXiv preprint 0710.2678v1 [math.NA],2007.

[89] C.A. Micchelli and T. Sauer. On vector subdivision. Math. Z., 229(4):621–674, 1998.

[90] H. Triebel. Interpolation theory, function spaces, differential operators.North-Holland, 1978.

[91] I. Ur Rahman, I. Drori, V. C. Stodden, D. L. Donoho, and P. Schroder.Multiscale representations for manifold-valued data. Multiscale Mod. Sim.,4:1201–1232, 2005.

[92] J. Wallner. Smoothness analysis of subdivision schemes by proximity. Con-str. Approx., 24:289–318, 2006.

[93] J. Wallner and N. Dyn. Convergence and C1 analysis of subdivision schemeson manifolds by proximity. Comput. Aided Geom. Des., 22:593–622, 2005.

[94] J. Wallner, E. Nava Yazdani, and P. Grohs. Smoothness properties of Liegroup subdivision schemes. Multiscale Mod. Sim., 6:493–505, 2007.

[95] G. Xie and T. P.-Y. Yu. Smoothness equivalence properties of manifold-valued data subdivision schemes based on the projection approach. SIAMJ. Numer. Anal., 45:1200–1225, 2007.

[96] G. Xie and T. P.-Y. Yu. Smoothness equivalence properties of interpolatoryLie group subdivision schemes. IMA J. Numer. Anal., 2008. to appear.

[97] G. Xie and T. P.-Y. Yu. Smoothness equivalence properties of generalmanifold-valued data subdivision schemes. Multiscale Mod. and Sim.,7:1073–1100, 2009.

Chapter 4

[G3]: Stability ofmanifold-valued subdivisionschemes and multiscaletransformations

Abstract

Linear subdivision schemes can be adapted in various ways so as to operatein nonlinear geometries such as Lie groups or Riemannian manifolds. It is wellknown that along with a linear subdivision scheme a multiscale transformationis defined. Such transformations can also be defined in a nonlinear setting.We show the stability of such nonlinear multiscale transforms. To do this weintroduce a new kind of proximity condition which bounds the difference of thedifferential of a nonlinear subdivision scheme and a linear one. It turns out that –unlike the generic nonlinear case and modulo some minor technical assumptions– in the manifold-valued setting, convergence implies stability of the nonlinearsubdivision scheme and associated nonlinear multiscale transformations.

4.1 Introduction

There exists an ongoing interest in nonlinear subdivision schemes and their ap-plications in data processing. This is because in many situations linear methodsperform poorly and one is forced to employ nonlinear processes in order to ob-tain pleasing results. As examples we mention the presence of non-gaussiannoise where the so-called median interpolating scheme has been introducedin [107], or piecewise smooth data where so-called essentially non-oscillating(ENO) schemes have been developed [118]. Another example is the process-ing of data that does not live in a vector space but in a general manifold. Inthis case linear methods usually cannot even be defined anymore. There hasbeen some effort to also provide approximation tools for this situation using

63

64 CHAPTER 4. [G3]: STABILITY

non linear subdivision schemes which are intrinsically defined in a manifold[126, 112, 124, 103]. These schemes are always defined as nonlinear analogueof some linear scheme. It is certainly of interest to know if these constructionsshare the properties of the corresponding linear ones. Among the propertiesthat have been previously studied are

• convergence and smoothness of the limit function of manifold-valued sub-divison schemes [126, 125, 115, 114, 129, 128]

• approximation order of manifold-valued subdivision schemes [109, 127], or

• smoothness characterization properties of the multiscale decompositionassociated with a manifold valued subdivision scheme [116].

For each one of these properties it has turned out that essentially the nonlinearmanifold-valued constructions do indeed share the same properties with thelinear constructions.

In the spirit of these studies, the present work is concerned with stabilityproperties of manifold valued subdivision schemes.

The main idea of the previous work mentioned above in this direction is toemploy a proximity condition that bounds the difference of a nonlinear subdivi-sion scheme to the linear scheme it is constructed from and to use perturbationresults. Unfortunately, in the case of stability this path is doomed to failure. Inthe course of this paper we will see an example of a nonlinear scheme which isin proximity with a linear one, but which is not stable.

To overcome this difficulty, we introduce a new kind of proximity condition:a differential proximity condition which bounds the difference between the dif-ferential of a nonlinear scheme and a linear scheme. We manage to show thatindeed stability holds for a nonlinear scheme satisfying a differential proximitycondition with a stable linear scheme. We find the surprising result that thetwo notions of proximity – the previously employed notion of proximity and thenew differential proximity condition – are equivalent if the nonlinear scheme issufficiently smooth as an operator on l∞(Z).

The main result of the present paper shows that the manifold-valued ana-logue of a stable subdivision scheme is itself stable and that the associatednonlinear multiscale decomposition is stable, too.

Stability is of course essential for the applicability of any multiscale trans-formation. This is because, for an application in data compression, de-noisingor smoothing, we need to know to what extent a function is affected by thethresholding or perturbing of detail coefficients.

Aside from the practical importance of the stability property, our results alsoprovide strong tools for the theoretical analysis of nonlinear subdivision schemes.As an example we mention the upcoming work [113], where the present resultsare applied to prove some new facts concerning approximation order.

Previous work There are only few general results concerning the stabilityof nonlinear subdivision schemes and their associated multiscale transforms. Avery recent and general work is [117], where necessary and sufficient criteriafor the stability of nonlinear subdivision schemes and multiscale transforms arepresented. These results however cannot be applied to the manifold-valued casesince they assume polynomial generation properties of the underlying subdivi-sion scheme which do not hold in the manifold-valued case. Other results can

4.2. NOTATION AND PRELIMINARIES 65

be found in [101, 99, 121, 103, 98, 102]. All these results rely on the pertur-bation approach of nonlinear subdivision and multiresolution, i.e. they regardthe nonlinear multiresolution as a perturbation of the linear one. This approachdoes not work for manifold-valued schemes either as we will see later. Our workcombines the more analytical methods of [117] with the more algebraic methodsin, say [98], where by algebraic we mean that no differentiation or integrationtakes place in the analysis. As a (rather natural) additional assumption over thisprevious work we shall require the underlying nonlinear subdivision operator tobe a C1-mapping.

Outline After setting up the necessary notation and reviewing some basicfacts on subdivision in Section 4.2, in Section 4.3 we provide an example of anunstable subdivision scheme which is in proximity with a linear stable scheme inthe sense of [126]. Motivated by this result we introduce a new differential prox-imity condition and, using this condition, prove the stability of a large class ofnonlinear subdivision operators. Section 4.4 focuses on the stability of the mul-tiresolution transform associated with a subdivision operator. After discussingsome issues concerning the well-definedness of the reconstruction procedure, weshow that, assuming a differential proximity condition, stability of the multires-olution transform holds. In Section 4.5 we apply the general results of Sections4.3 and 4.4 to some examples of interest: the log-exp analogue in a Lie group,the log-exp analogue in a Riemannian manifold and the projection analogue.

4.2 Notation and preliminaries

The present section is devoted to fixing the notation and reviewing some basicfacts concerning linear subdivision schemes. We shall always use boldface lettersfor sequences p = (pi)i∈Z.

4.2.1 Subdivision schemes

Let us define what we precisely mean by a subdivision scheme:

Definition 4.2.1. An m-dimensional subdivision operator (m ∈ Z+) is a map-ping T : l∞(Z,Rm)→ l∞(Z,Rm) that is local and has dilation factor 2, mean-ing that

σ2 T = T σ, (4.1)

where σ denotes the right-shift on Z. Locality means that the value (T p)i ∈Rm, i ∈ Z, p ∈ l∞(Z,Rm) depends only on a finite number of points.

The dilation factor 2 in this definition for our purposes is completely arbi-trary. It makes no difference if we replace it with any integer > 1 and all ourresults hold for this general case too, with the same proofs.

From the definition of a subdivision scheme it follows that there exists aninterval I ⊂ Z of finite length L such that the value of (T p)2i+k only dependson the values p

∣∣i+I

with i ∈ Z, k ∈ 0, 1. We once and for all fix a norm

‖ · ‖ on Rm, for instance the maximum norm. For an element p ∈ l∞(Z, Rm)we write ‖p‖∞ := supi∈Z ‖pi‖. For functions we use the same symbol ‖ · ‖∞ toindicate the usual sup-norm. For a linear operator l∞(Z,Rm)→ l∞(Z,Rm) weuse the symbol ‖ · ‖ to indicate the usual operator norm induced by ‖ · ‖∞ onl∞(Z,Rm).


The purpose of a subdivision scheme is to be iterated with the interpretationthat the points T npi are approximately the value of some continuous functionT ∞p(t) : R → Rm at t = i/2n. The precise definition of convergence is asfollows:

Definition 4.2.2. A subdivision scheme T is called convergent if for any initialdata p ∈ l∞(Z,Rm) there exists a continuous function T ∞p with

limn→∞

‖T ∞p(t)− (T np)in‖ = 0, whenever limn→∞

in/2n = t.

In this paper we are not concerned with convergence, or smoothness proper-ties of the limit function. If the reader is interested in these topics we refer to[115, 114, 128, 129, 126, 125]. Rather, we are interested in the following type ofquestion:

Question 1: Is the operator T ∞ : l∞(Z,Rm) → C(R,Rm) that maps p toits limit function T ∞p Lipschitz, i.e. does there exist a finite constant C sothat

‖T ∞p(t)− T ∞q(t)‖∞ ≤ C‖p− q‖∞holds (at least locally)?

We will prove that the answer is “yes” for a general class of nonlinear subdivi-sion schemes that includes for example subdivision in manifolds via the log-expanalogy [124].

Our proofs are based on comparing the nonlinear scheme T with a linearscheme S (which is stable) and using perturbation arguments to ensure that T ∞is Lipschitz. First let us recall some well-known properties of linear subdivisionschemes. Comprehensive introductions into the theory of linear subdivisionschemes are provided e.g. by [100, 120, 110]. In the following we always writeS for a linear subdivision scheme and T for a possibly nonlinear one.

Definition 4.2.3. A subdivision scheme S is called linear, if there exists asequence (aj)j∈Z, aj ∈ R with finite support, called the mask of S, such that

(Sp)i =∑j∈Z

ai−2jpj , pj ∈ Rm. (4.2)

It is well known and easy to see from the uniform boundedness principlethat for a convergent linear scheme S the operator norms ‖Sn‖ are uniformlybounded by a constant M ≥ 0 for all n. If S is a convergent linear subdivisionscheme with S∞ 6= 0, it maps constant data to itself:

Sq = q for q = (. . . , q, q, q, . . . ) ∈ l∞(Z,R) with q ∈ Rm. (4.3)

If a (linear or nonlinear) subdivision scheme S satisfies (4.3) we say that Sreproduces constants. For a nonlinear scheme T it might happen that it is notdefined anymore for all sequences. In this case the same definition holds with theadditional (and obvious) restriction that (4.3) needs to hold for every q ∈ Rmsuch that T is defined on q = (. . . , q, q, q, . . . ).

A useful tool in the study of a linear (or nonlinear) subdivision scheme isthe forward difference operator ∆ that maps a sequence p ∈ l∞(Z,Rm) to thesequence (pi+1 − pi)i∈Z. Convergence of a linear subdivision scheme can becharacterized as follows (see e.g. [100]):

4.2. NOTATION AND PRELIMINARIES 67

Theorem 4.2.4. Let S be a linear subdivision scheme. S is convergent if andonly if there exists a constant C ≥ 0 and a constant 0 ≤ µ < 1, called thecontraction factor of S, such that for all initial data p ∈ l∞(Z,Rm)

‖∆Sjp‖∞ ≤ Cµj‖∆p‖∞.

It can be shown that the contraction factor is always greater or equal 1/2unless S∞ maps all data to the zero function.

4.2.2 Multiscale Data Representation using Subdivision

A useful application of subdivision lies in the multiscale representation of data.The principle is as follows: Let f : R → Rm be a continuous function. Wecan analyze f by first sampling it on a grid 2−(j−1)Z to obtain data p =Pj−1f := (f( i

2j−1 ))i∈Z. Then we predict the values on the grid 2−jZ usingthe subdivision scheme T and obtain an approximation T p of the actual val-ues Pjf = (f( i

2j ))i∈Z. From the predicted values T p we extract the “waveletcoefficients” as the differences

λj := λjf := ((λjf)i)i∈Z := ((Pjf)i (T Pj−1f)i) i∈Z

of the predictions and the actual samplings. The symbol serves as an analogueof the conventional difference of two points in Rm and represents a bivariatefunction on Rm×Rm taking values in some vector space (or vector bundle). Weare not concerned with the analysis of a function – this topic is studied in [116]– but with the properties of the reconstruction procedure taking data

(P0, λ1, λ2, . . . ) (4.4)

to the function f : If ⊕ is the inverse operation of in the sense that p⊕(qp) =q for all p, q ∈ Rm, then the values of f on the dyadic rationals can be recon-structed inductively from (4.4) via (Pjf)i = (T Pj−1f)i⊕ (λj)i. If f is continu-ous, then this procedure obviously recovers f from (P0, λ1, . . . ). It is natural toask what effect perturbing the coefficients λj has on the reconstructed function,or in other words:

Question 2: Does there exist a constant C ≥ 0 such that for data P0, P0, λ1, λ1, . . .and functions f, f reconstructed from (P0, λ1, . . . ), (P0, λ1, . . . ) respectively, wehave the inequality

‖f − f‖∞ ≤ C(‖P0 − P0‖∞ +

∞∑i=1

‖λi − λi‖∞) ?

Does such an inequality hold locally?

We provide an affirmative answer to this question which applies to a large classof subdivision schemes and which is true under some restrictions which cannotbe avoided in our setting.

4.2.3 Lipschitz classes

We introduce the so-called Lipschitz function spaces: Let I ⊂ R be an interval.The function spaces Lip γ, where 0 < γ ∈ R \ Z+ are defined as follows forfunctions f : I → Rd and d ∈ Z+: f lies in Lip γ if and only if


‖f‖∞ <∞ and suph∈R\0

‖h−(γ−bγc)(( ddx

)bγcf(x+ h)−

( ddx

)bγcf(x))‖∞ <∞.

It can be shown [104] that this is equivalent to

f ∈ Lip γ ⇔ ‖f‖∞ <∞ and |f |Lip γ := supj∈Z+

2γj‖∆bγc+1Pjf‖∞ <∞. (4.5)

For γ ∈ Z+ the definition of Lipschitz spaces can be obtained by interpolationmethods [123]. With this definition (4.5) remains valid also for integers γ.

During the course of our analysis we will occasionally encounter multivariateLipschitz function spaces containing functions f : G → Rd for some domainG ⊂ Rs, s ∈ Z+. In this case the definition is similar, with the differentialoperator d

dx replaced by the total differential, see [123] for more information.The characterization (4.5) still holds in the multivariate case with the ∆ operatorreplaced by a multivariate divided difference operator, see e.g. [116].

4.3 Stability from Proximity

This section is devoted to the answer to Question 1 above. We first discuss theconditions that we impose on the nonlinear scheme T . Then we show that theseconditions suffice to prove stability.

One technical issue that arises in the study of many nonlinear subdivisionschemes is the fact the they may not be defined for all data p ∈ l∞(Z,Rm)anymore. We therefore introduce a family of subsets of l∞(Z,Rm) convenientfor our analysis:

Definition 4.3.1. The set PM,δ denotes all M -valued sequences such that

‖∆p‖∞ < δ.

For simplicity we assume that M ⊆ Rm is open and convex.

The requirement for M to be open and convex is no real restriction. Infact, if M is open, we can make M smaller to get a convex set. The convexityassumption will be needed later in some proofs in order to apply the mean valuetheorem for real-valued multivariate functions restricted to a line connecting twopoints. It is however no essential assumption, with a little more technical effortin the proofs it can just as well be replaced by path-connectedness. The typicalsets M that we will encounter in the examples in Section 4.5 are domains ofdefinition of charts on a differentiable manifold.

Our analysis strongly relies on so-called proximity conditions which are sat-isfied between a nonlinear and a linear subdivision scheme. They have firstbeen used in [103] for the analysis of normal multiresolution of curves. The spe-cific conditions that we introduce below have first been used in [126] to proveconvergence and smoothness properties of nonlinear, manifold-valued schemes:

Definition 4.3.2. S and T satisfy a proximity condition with exponent α > 1for the set PM,δ if there exists a constant C ≥ 0 such that for all p ∈ PM,δ

‖Sp− T p‖∞ ≤ C‖∆p‖α∞.

4.3. STABILITY FROM PROXIMITY 69

It is perhaps not very surprising that such an inequality is useful in thestudy of convergence properties, and indeed, using a perturbation result, it ispossible to deduce convergence and smoothness properties of T from analogousproperties of S.

This kind of proximity condition does not seem well-suited for the study ofstability properties. With perturbation arguments similar to [126] it seems veryhard to prove stability of T from stability of S. As a matter of fact it turns outthat, without requiring anything else, the implication proximity ⇒ stability isfalse:

Example 4.3.3. Define a subdivision scheme T by

(T p)2j+k := (Sp)2j+k + ‖∆pj+5‖2|f(pj)|, k ∈ 0, 1,

where S is the linear B-spline scheme defined via

(Sp)2j = pj and (Sp)2j+1 =1

2pj +

1

2pj+1,

and f is some bounded function with f(0) = 0 and f /∈ Lip 1 at 0. Clearly, Tis a subdivision scheme which is in proximity with S with α = 2.

Lemma 4.3.4. There exists initial data p,qn, n ∈ Z+ such that the limitfunctions T ∞p, T ∞qn exist and such that

lim supn→∞

‖T ∞p− T ∞qn‖∞‖p− qn‖∞

=∞.

In particular, T is not stable.

Proof. First we note that by the general results in [126], the limit function T ∞pexists for all initial data p with ‖∆p‖∞ sufficiently small. Now set p5 = δ andpj = 0 otherwise with δ small enough for T ∞p to exist. By the assumptionthat f is not in Lip 1 at 0 it follows that there exists a sequence (εn)n∈Z+ suchthat limn→∞ εn = 0, εn ≤ δ for all n ∈ Z+ and lim supn→∞ |f(εn)|/|εn| = ∞.Now define qn with qi = pi for i 6= 0 and q0 = εn. Again by the results in [126]it can be shown that T ∞qn exists for all n. Since

|(T iqn)0| ≥ ‖(∆qn)5‖2|f(εn)| = δ2|f(εn)| for all i ≥ 1,

it follows that

‖T ∞p− T ∞qn‖∞/‖p− qn‖∞ ≥ δ2|f(εn)|/|εn| → ∞.

This proves the statement.

The previous example shows that a proximity condition as defined in Defini-tion 4.3.2 is not the right one to study stability properties of nonlinear subdivi-sion schemes. It is clear that in our example the obstruction to stability comesfrom the fact that f is not in Lip 1, or to put it differently, that the subdivisionoperator T is not a Lipschitz-continuous operator.

We therefore restrict our attention to subdivision schemes T such that Tis a C1-mapping and introduce a more natural proximity condition for prov-ing stability properties (which obviously are related to the differentials of thesubdivision operators).


Note that the requirement on T to be C1 is a very small restriction for ourpurposes. In most of the applications which we have in mind the scheme Tturns out to be even analytic, see Section 4.5.

In the following we are going to use the differential dT . At first sight it mightbe necessary to first adapt a proper definition of differentiating a subdivisionoperator T acting on an infinite dimensional Banach space. However, since byassumption T is a local operator we are not confronted with such issues andthe differential dT of T in fact consists only of the differentials of two functionsas can easily be verified from the defining assumptions of a subdivision scheme.Keeping this in mind the following definition makes perfect sense.

Definition 4.3.5. A nonlinear subdivision scheme T ∈ C1 and a linear schemeS satisfy a differential proximity condition with exponent β > 0 for the set PM,δ

if there exists a constant C ≥ 0 such that for all p ∈ PM,δ

‖dT∣∣p− S‖ ≤ C‖∆p‖β∞. (4.6)

Intuitively, this definition should work better than Definition 4.3.2, becauseif ‖∆p‖∞ becomes small (which it does for a convergent scheme T ), then the dif-ference between the differentials of T and S becomes small, too. These heuristicarguments provide some hope that, using a differential proximity condition, itshould be possible to show uniform boundedness of the differentials dT n, andconsequently stability of T . Theorem 4.3.8 below turns this heuristic into arigorous proof.

First we need the following perturbation result which will be useful in severalplaces:

Lemma 4.3.6. Let Ai, i ∈ Z+ be operators on a normed space with norm‖ · ‖ with supk∈Z+ ‖

∏kl=1Al‖ ≤ M for a constant M ≥ 0. If Ui, i ∈ Z+ are

operators such that ‖Ui‖ ≤ Cµi for some C ≥ 0, µ < 1 (‖Ui‖ denoting the

induced operator norm), then ‖∏kl=1(Al + Ul)‖ is bounded by a constant only

depending on C, µ and M .

Proof. Define the norm ‖x‖′ := supk∈Z+ ‖∏kl=1Alx‖. Clearly, we have

‖x‖ ≤ ‖x‖′ ≤M‖x‖.

The operators Al have norm ≤ 1 with respect to ‖ · ‖′. Therefore

‖k∏l=1

(Al + Ul)x‖ ≤ ‖k∏l=1

(Al + Ul)x‖′ ≤k∏l=1

‖Al + Ul‖′‖x‖′

≤∞∏i=1

(1 +MCµi)‖x‖′ ≤M∞∏i=1

(1 +MCµi)‖x‖.

This implies our statement.

In order to be able to speak of stability we need T to be convergent whichis usually related to the notion of contractivity :

Definition 4.3.7. A subdivision scheme T is contractive for data in PM,δ ifthere exists a constant C ≥ 0 and µ < 1 such that for all data p ∈ PM,δ

‖∆T jp‖∞ ≤ Cµj‖∆p‖∞ j ∈ Z+. (4.7)


Now we can formulate and prove our main result regarding stability of non-linear subdivision schemes

Theorem 4.3.8. Assume that T is a C1-subdivision scheme satisfying a dif-ferential proximity condition with exponent β > 0 and data in PM,δ with aconvergent linear scheme S. Assume further that T is contractive on PM,δ.Then there exists a constant C ≥ 0 such that for all p, q ∈ PM,δ we have

‖T ∞p(t)− T ∞q(t)‖∞ ≤ C‖p− q‖∞.

Proof. We show that the operator norms ‖dT n∣∣p‖ are uniformly bounded in

n by a constant C for all p ∈ PM,δ. To do this, we apply the chain rule toT n = T T n−1 and obtain the following expansion, with certain operators Uj :

dT n∣∣p

= dT∣∣T n−1p

· dT n−1∣∣T n−2p

· . . . · dT∣∣p

=: (S + Un)(S + Un−1) · . . . · (S + U1).

By assumption, the schemes S and T satisfy the differential proximity condition(4.6). Hence, thanks to the contractivity assumption (4.7),

‖Ui‖ ≤ C‖∆T i−1p‖β∞ ≤ C ′(µβ)i‖∆p‖β∞, i = 1, . . . , n.

Now it remains to apply Lemma 4.3.6 with Ai := S to conclude that the norms‖dT n

∣∣p‖ are indeed bounded.

In order to obtain a bound for ‖T ∞p − T ∞q‖ we employ the definition ofconvergence and see that for t ∈ I,

T ∞p(t) = limn→∞

(T np)in ,

with integers in chosen such that in/2n → t. An analogous formula holds for q.

Now let c(s) := p + s(q−p). The uniform boundedness of ‖dT n‖ shown aboveimplies that

‖(T np− T nq)in‖ ≤ supt∈(0,1)

‖dT n∣∣c(t)‖‖p− q‖∞ ≤ C‖p− q‖∞.

We let n→∞ and arrive at

‖T ∞p(t)− T ∞q(t)‖∞ ≤ C‖p− q‖∞

for all t.

Using this general result we go on to prove that proximity in the sense of4.3.2 does imply stability if we impose a mild regularity assumption on thesubdivision operator T .

The proof is based on the remarkable fact that if we require the subdivisionscheme T : l∞(Z,Rm) → l∞(Z,Rm) to be smooth, then Definition 4.3.2 andDefinition 4.3.5 almost coincide. Recall that T ∈ Lip α for 1 < α < 2 at pmeans that there exists a constant C ≥ 0 such that for all q locally around p

‖dT∣∣p− dT

∣∣q‖ ≤ C‖p− q‖α−1

∞ .


Lemma 4.3.9. Let T be a subdivision scheme with T ∈ Lip α, 1 < α < 2 atconstant sequences and S a linear scheme. Then the following are equivalent:

(i) S and T satisfy a proximity condition with exponent α for PM,δ,

(ii) T reproduces constants and S, T satisfy a differential proximity conditionwith exponent α− 1 for p ∈ PM,δ,

(iii) T reproduces constants and for all constant data q = (. . . , q, q, q, . . . ) withq ∈M ,

S = dT∣∣q. (4.8)

Proof. In the following we denote by Tk,l, Sk,l, k ∈ Z, l ∈ 1, . . . ,m the real-valued functions that map initial data p to the l-th coordinate of T pk, Spk,respectively. Note that every function Tk,l, Sk,l depends only on p

∣∣i+I

withk = 2i or k = 2i+ 1 and I a fixed interval.

(ii) ⇒ (iii) is clear, since ∆ maps constant data to zero.(iii) ⇒ (ii): Since T ∈ Lip α at constant sequences q = (. . . , q, q, q, . . . ) for

all q ∈ Rm we have by (iii)

‖dTk,l∣∣p− Sk,l‖ = ‖dTk,l

∣∣p− dTk,l

∣∣q‖ ≤ C‖p

∣∣i+I− q

∣∣i+I‖α−1∞ . (4.9)

By choosing the value q ∈ Rm appropriately, we get

‖p∣∣i+I− q

∣∣i+I‖∞ ≤ C ′‖∆p‖∞

with a uniform constant C ′. In summary we arrive at

‖dT∣∣p− S‖ ≤ C ′′‖∆p‖α−1

∞

with another constant.(iii)⇒ (i): For any constant sequence q = (. . . , q, q, q, . . . ), q = (q1, . . . , qm) ∈

Rm we let c : [0, 1]→ l∞(Z,Rm) be the straight line with c(0) = q and c(1) = p.Since T ∈ Lip α,

Tk,lp = Tk,lq + dTk,l∣∣c(θ)

(p∣∣i+I− q

∣∣i+I

)

= ql + (dTk,l∣∣q

+ U)(p∣∣i+I− q

∣∣i+I

) = Sk,lp + U(p∣∣i+I− q

∣∣i+I

)

with 0 < θ < 1 and ‖U‖ ≤ C‖p∣∣i+I− q

∣∣i+I‖α−1∞ . By choosing q appropriately

we can conclude that

|Tk,lp− Sk,lp| ≤ ‖U‖‖p∣∣i+I− q

∣∣i+I‖∞

≤ C‖p∣∣i+I− q

∣∣i+I‖α−1∞ ‖p

∣∣i+I− q

∣∣i+I‖∞

≤ C ′‖∆p‖α∞

with another uniform constant C ′. This implies (i).(i) ⇒ (iii): We let e be the sequence with ej = er and zero elsewhere,

er ∈ Rm is the r-th canonical basis vector and r ∈ 1, . . . ,m, j ∈ Z. For t > 0and q = (. . . , q, q, q, . . . ), q = (q1, . . . , qm) ∈ Rm we define p := q + te. Then∆p = t∆e. We have

Tk,lp = Tk,lq + dTk,l∣∣q+t′e

(te) = ql + dTk,l∣∣q(te) + (dTk,l

∣∣q+t′e

− dTk,l∣∣q)(te)


with 0 < t′ < t. We let U := (dTk,l∣∣q+t′e

− dTk,l∣∣q) and note that

‖U‖ ≤ Ct′α−1‖e‖α−1∞ < Ctα−1‖e‖α−1

∞ (4.10)

with a uniform constant C.

Since Sp = q + Ste by (4.3),

Tk,lp− Sk,lp− Ute = (dTk,l∣∣q− Sk,l)te.

By (i) we have that |Tk,lp − Sk,lp| ≤ C ′‖∆p‖α∞ ≤ C ′tα‖∆e‖α∞ with someconstant C ′. Since by (4.10) also |Ute| ≤ Ctα‖e‖∞, the triangle inequalityimplies that

C ′′tα ≥ t|(dTk,l∣∣q− Sk,l)e|

with some uniform constant C ′′. Dividing by t and letting t→ 0 yields

(dT∣∣q− S)e = 0.

This implies (iii).

The equivalence of the two notions of proximity (Definitions 4.3.2 and 4.3.5)in particular implies that the schemes S and T are in proximity if and only ifthey agree up to first order on constant data. This insight might open up roomfor new ideas and we plan to study it in more depth in the future.

As already mentioned above, a proximity condition already implies conver-gence and contractivity of T . The following theorem has been proven in [112]

Theorem 4.3.10. Assume that T and S satisfy a proximity condition withexponent α > 1 for data p ∈ PM,δ. If S is a convergent linear scheme, thenthere exists M ′ ⊆M, δ′ > 0, 0 ≤ µ < 1 such that for all p ∈ PM ′,δ′

‖∆T jp‖∞ ≤ Cµj‖∆p‖∞ (4.11)

and T jp ∈ PM,δ.

With the help of these results we are now able to prove that any smoothT is stable, provided it satisfies a proximity condition with a convergent linearscheme.

Theorem 4.3.11. Assume that T ∈ C1 is a subdivision scheme, Lip α atconstant sequences, satisfying a proximity condition with exponent α > 1 anddata in PM,δ with a convergent linear scheme S. Then there exists a constantC ≥ 0 such that for all p, q ∈ PM ′,δ′ and M ′, δ′ from Theorem 4.3.10:

‖T ∞p(t)− T ∞q(t)‖∞ ≤ C‖p− q‖∞.

Proof. This is a direct consequence of Lemma 4.3.9 and Theorems 4.3.8 and4.3.10.


4.4 Stability of the Multiscale Decompositionassociated with a Subdivision operator

4.4.1 Definition of the multiscale decomposition associ-ated with a nonlinear subdivision scheme

We proceed by answering Question 2: Does there exist a constant C ≥ 0 suchthat for data P0, P0, λ1, λ1, . . . we have the inequality

‖f − f‖∞ ≤ C(‖P0 − P0‖∞ +

∞∑i=1

‖λi − λi‖∞)

for the functions f, f reconstructed from data (P0, λ1, . . . ), (P0, λ1, . . . ) respec-tively? Does such an inequality hold locally?

Recall that we have defined Pj to be the sampling operator that mapsa bounded, continuous function f : R → Rm to its samples (f(i/2j))i∈Z ∈l∞(Z,Rm). For S the linear B-spline scheme, Faber [111] introduced a mul-tiscale decomposition built from the samples of a function f ∈ Lip γ alreadyin 1908. The idea is that one stores not the sampling data Pjf , but insteadthe “prediction error” that one makes in predicting the values of Pjf from thecoarser values Pj−1f via a subdivision scheme. Hence, a function f gets mappedto its initial samples P0f together with the prediction errors

λj = SPj−1 − Pj . (4.12)

The intuitive reason for doing this is that for a continuous function f and fora convergent scheme S, the values λj should decrease rather quickly. In fact,it is possible to characterize the smoothness order of a continuous function fby the decay rate of the coefficient sequences λj [106, 116]. One of the aims ofour approach is to make such decompositions available for manifold-valued dataand to prove that the properties of the linear decomposition carry over to themanifold setting. The key property a manifold-valued transformation shouldsatisfy is to be intrinsically defined. While most quantitative results can beshown just by considering a chart and making all computations in a Euclideansetting, we still need an intrinsic way to define our multiscale transformation.We come back to this issue later in Section 4.5. For now we simply remark thatthe definition of λj via (4.12) has no intrinsic meaning if applied to local coor-dinate representations of a manifold. We therefore need a little more generalityin our definition. Our setup is that we have an m-dimensional image space Rmfor the functions f and a k-dimensional vector space Rk of wavelet coefficients.

Definition 4.4.1. The multiscale transformation maps a function f ∈ Lip γwith γ > 0 to a coarse sample P0f and a countable collection of wavelet coeffi-cients λj ∈ l∞(Z,Rk) according to

λji := (λjf)i := (Pjf)i (T Pj−1f)i, j ∈ Z+, i ∈ Z.

Here, the operation is a nonlinear analogue of the operation “ minus” andrepresents a bivariate function φ : Rm × Rm → Rk mapping two points totheir difference vector. In order to be able to reconstruct our function f from

4.4. STABILITY OF THE MULTISCALE DECOMPOSITION 75

data (P0, λ1, . . . ) we need an inverse operation ⊕ serving as an analogue of theeuclidean point-vector addition +. It must satisfy the relation

p⊕ (q p) = q for all p, q ∈ Rm.

We write ψ(p, v) for p ⊕ v and p ∈ Rm, v ∈ Rk for some k ∈ Z+. For techni-cal reasons we need the following not too restrictive properties to hold for thefunction ψ:

• ψ together with its inverse φ : (p, q) 7→ p q are C1 mappings,

• d1ψ ∈ Lip ν for some ν > 0 where d1 means differentiation with respectto the variable p,

• ψ(p, 0) = p⊕ 0 = p for all p ∈ Rm.

We now define the reconstruction procedure.

Definition 4.4.2. The reconstruction procedure consists of the functions

Pn : l∞(Z,Rm)× l∞(Z,Rk)n → l∞(Z,Rm),

which are inductively defined by

Pn(P0, λ1, . . . , λn) := Ψ(T Pn−1(P0, λ1, . . . , λn−1), λn) P0 := P0,

and Ψ(T Pn−1, λn)i := ψ((T Pn−1)i, (λn)i), i ∈ Z.

Again, by locality, although Pn is a function acting on an infinite dimensionalspace, it makes sense to speak of the derivatives d

dλlPn and d

dP0Pn.Although we abuse notation by calling the reconstruction functions Pn it

will be possible to distinguish them from the sampling operator Pn which actson continuous functions.

So far we have not looked at the question for which initial data (P0, λ1, . . . )the reconstruction procedure is well-defined. This means that there exists acontinuous function f with

Pnf = Pn(P 0, . . . , λn) for all n ∈ Z+.

Conditions for well-definedness are studied in the next section.

4.4.2 Well-definedness of the reconstruction procedure

We denote by B the open unit ball in Rm and use the notation U + V for theMinkowski sum between two subsets U, V ⊂ Rm.

Lemma 4.4.3. Let T be in proximity with a linear convergent scheme S forall initial data p ∈ PM,δ and assume that S has contraction factor µ0. Assumethat 0 < µ0 < µ < 1 and let M ′ ⊂M be any subset of M that stays away fromthe boundary of M , i.e. there exists ρ > 0 with

M ′ + ρ ·B ⊂M (4.13)

Then with ν := − log2 µ and R ∈ Z there exist constants δ′, C such that alldata

(P0, λ1, λ2, . . . )


satisfying

P0 ∈ PM ′,δ′ and ‖λj‖∞ ≤ Cµj (4.14)

belong to a function f ∈ Lip ν with |f |Lip ν depending only on δ′, C.

Proof. We first note that we can without loss of generality assume that thefunction is given by ordinary subtraction (p, q) 7→ p − q. The general casecan be easily obtained by noting that, because is a C1 function, locally thereexist Lipschitz constants C1, C2 with

C1‖p q‖ ≤ ‖p− q‖ ≤ C2‖p q‖.

By assumption there exists a constant CS with

‖∆Sp‖∞ ≤ CSµj0‖∆p‖∞ for all p ∈ l∞(Z,Rm), j ∈ Z+. (4.15)

Since S and T are in proximity, there exists a constant CP such that

‖Sp− T p‖∞ ≤ CP ‖∆p‖α∞ for all p ∈ PM,δ.

Further it is well-known [100] that there exists a constant CL with

supi∈Z

infk∈Z‖(Sp)i − pk‖ ≤ CL‖∆p‖∞ for all p ∈ l∞(Z,Rm).

Without loss of generality, we may assume that CS , CL ≥ 1. Write τ := µ0/µ <1 and find constants C, δ′ > 0 such that

CP

(8CS1− τ

)αCα−1µ−1 < 1, (4.16)

δ′ ≤ inf

((Cµ

CP

( 2CS1− τ

− 1))1/α

,4C

1− τ

), (4.17)

4CCS1− τ

+ CSδ′ < δ, (4.18)

andC

1− µ+ CP

(8CCS1− τ

)α1

1− µα+ CL

8CCS1− τ

1

1− µ< ρ. (4.19)

The above inequalities assure that the following estimates go through. Afterour choice of C and δ′, we now show inductively that for data satisfying (4.14),the reconstructed data Pj lies in the class Pδ,M and

‖∆Pj‖∞ ≤4CCS1− τ

µj + CSµj0‖∆P0‖∞. (4.20)

for all j ∈ Z. Let us start with the case j = 1: We have that

‖∆P1‖∞ ≤ ‖∆(P1 − T P0)‖∞ + ‖∆(T P0 − SP0)‖∞ + ‖∆SP0‖∞≤ 2Cµ+ 2CP ‖∆P0‖α∞ + CSµ0‖∆P0‖∞

≤ (2C + 2CP δ′α/µ)µ+ CSµ0‖∆P0‖∞ ≤

4CCS1− τ

µ+ CSµ0‖∆P0‖∞


because of (4.17). In particular, because of (4.18), we have ‖∆P1‖∞ ≤ δ. Weshow that P1 takes values in M :

supi∈Z

infx∈M ′

‖(P1)i − x‖ ≤ supi∈Z

infk∈Z‖(P1)i − (P0)k‖

≤ supi∈Z

infk∈Z

(‖(P1)i − (T P0)i‖

+‖(T P0)i − (SP0)i‖+ ‖(SP0)i − (P0)k‖)

≤ Cµ+ CP (δ′)α + CSδ′ < ρ

because of (4.17), (4.19) and the assumption that CS , CL ≥ 1. With (4.13) wecan conclude that P1 takes values in M and hence P1 ∈ PM,δ. We go on toprove the induction step j − 1 7→ j and estimate

‖∆Pj‖∞ ≤j∑

k=1

(‖∆(Sj−kPk − Sj−kT Pk−1)‖∞

+‖∆(Sj−kT Pk−1 − Sj−kSPk−1)‖∞)

+ ‖∆SjP0‖∞

≤j∑

k=1

(CSµ

j−k0 ‖∆(Pk − T Pk−1)‖∞ + CSµ

j−k0 ‖∆(T Pk−1 − SPk−1)‖∞

)+CSµ

j0‖∆P0‖∞

≤ 2CSC

j∑k=1

µj−k0 µk + 2CSCP

j∑k=1

µj−k0 ‖∆Pk−1‖α∞

+CSµj0‖∆P0‖∞

≤ 2CSC

1− τµj + 2CSCP

j∑k=1

µj−k0

(4CCS1− τ

µk−1 + CSµk−10 ‖∆P0‖∞)α

+CSµj0‖∆P0‖∞

≤ 2CSC

1− τµj + 2CSCP

j∑k=1

µj−k0

(8CCS1− τ

µk−1)α

+ CSµj0‖∆P0‖∞

≤ 2CSC

1− τµj(

1 + CP( 8CS

1− τ)αCα−1µ−1

)+ CSµ

j0‖∆P0‖∞

≤ 4CCS1− τ

µj + CSµj0‖∆P0‖∞.

We have used (4.17) and (4.16). In particular, because of (4.18), we get that‖∆Pj‖∞ < δ. Now all that needs to be shown is that Pj still takes values inM :

supi∈Z

infx∈M ′

‖(Pj)i − x‖ ≤ supi∈Z

infk∈Z‖(Pj)i − (P0)k‖

≤j∑l=1

supi∈Z

infk∈Z‖(P l)i − (P l−1)k‖

≤j∑l=1

supi∈Z

infk∈Z

(‖(P l)i − (T P l−1)i‖+


‖(T P l−1)i − (SP l−1)i‖+ ‖(SP l−1)i − (P l−1)k‖)

≤ C

j∑l=1

µj + CP

j∑l=1

‖∆P l−1‖α∞ + CL‖∆P l−1‖∞

≤ C

1− µ+ CP

j∑l=1

(4CCS1− τ

µl−1 + δ′µl−10 )α

+CL

j∑l=1

(4CCS1− τ

µl−1 + δ′µl−10 )

≤ C

1− µ′+ CP

j∑l=1

(8CCS1− τ

µl−1)α + CL

j∑l=1

8CCS1− τ

µl−1

≤ C

1− µ+ CP

(8CCS1− τ

)α1

1− µα+ CL

8CCS1− τ

1

1− µ< ρ.

With (4.13) we can conclude that Pj takes values in M . We have now shownthat for data satisfying (4.14) the reconstruction procedure is well-defined andthe reconstructed data satisfies (4.20). Now all assertions follow from (4.5).

Remark 4.4.4. It is well-known that the contractivity factor of a convergentlinear scheme S must satisfy 1

2 ≤ µ0 as shown e.g. in [125].

4.4.3 Stability of the reconstruction procedure

Similar to the proof of the stability of the subdivision scheme, the key to stabilityis to show that the differentials of the reconstruction procedure are uniformlybounded. This is done by the next lemma.

Lemma 4.4.5. Let f ∈ Lip γ, 1 > γ > 0 with wavelet decomposition(P0f, λ1f, . . . ) with respect to the subdivision scheme T ∈ Lip α, α > 1, and theoperations , ⊕. Assume that T satisfies a proximity condition with a linear,convergent subdivision scheme S with exponent α for data in PM,δ. Then, ifPnf ∈ PM,δ for all n ∈ Z+, the operator norms

∥∥ d

dλlPn∣∣(P0f,λ1f,... )

∥∥ and∥∥ d

dP0Pn∣∣(P0f,λ1f,... )

∥∥are bounded by a constant C ≥ 0 independent of l and n and depending only on‖∆P0f‖∞ and the seminorm |f |Lip γ .

Proof. We break up the proof into several parts. In order to keep the notationsimple, we shall denote several different constants by C and contraction factorswhich are greater than 0 and less than 1 by µ. Recall that we write ψ(p, v) forp ⊕ v and φ(p, q) for p q, p, q ∈ Rm, v ∈ Rk. We require that ψ and φ arecontinuously differentiable mappings with d1ψ ∈ Lip ν for some ν > 0.

(i): We first show that for f ∈ Lip γ the wavelet coefficients decay geomet-rically: Note that for points p, q ∈ Rm and some 0 < θ < 1

‖(p q)l‖ = ‖φ(q, p)l‖ = ‖φ(q, q)l + d2φ∣∣c(θ)

(p− q)l‖ ≤ ‖d2φ‖‖p− q‖


with c(t) := u+ t(v − u), d2 meaning differentiation with respect to the secondvariable and the subscript l the l-th coordinate in Rk, l ∈ 1, . . . , k. Weestimate the wavelet coefficients as follows:

‖λjf‖∞ = ‖(T Pj−1f Pjf‖∞≤ sup

0<t<1‖d2Φ

∣∣c(t)‖‖T Pj−1f − Pjf‖∞

≤ sup0<t<1

‖d2Φ∣∣c(t)‖(‖T Pj−1f − SPj−1f‖∞

+‖SPj−1f − Pjf‖∞).

Since S, T satisfy a proximity condition, we can estimate the first term in thebrackets by a constant times ‖∆Pjf‖α∞. Since f ∈ Lip γ, there exist C ≥0, µ < 1 with ‖∆Pjf‖∞ ≤ Cµj with C = |f |Lip γ and µ = 2−γ . Therefore thefirst term is bounded by a constant times µαj . Now to the second term: Recallthat for i ∈ Z, k ∈ 0, 1 the value of Sp2i+k depends only on p restricted toi + I where I is a fixed interval ⊂ Z of length L. By (4.3) we have for everyq ∈ Rm, q = (. . . , q, q, q, . . . ),

‖(SPj−1f)2i+k − (Pjf)2i+k‖ = ‖S(Pj−1f − q)2i+k − ((Pjf)2i+k − q)‖≤ ‖S‖ max

l∈i+I‖(Pj−1f)l − q‖+ ‖(Pjf)2i+k − q‖

By choosing q appropriately, we obtain

‖SPj−1f − Pjf‖∞ ≤ C‖∆Pjf‖∞ ≤ C ′µj

for another constant C ′. Combining these two estimates we arrive at

‖λjf‖∞ ≤ C2−jγ

for yet another uniform constant C.(ii): Now we want to obtain bounds on the differentials of the reconstruction

procedure. In what follows we write λj := λjf . We begin by computing forl < n

d

dλlPn =

d

dλlΨ(T Pn−1, λn) = d1Ψ

∣∣(T Pn−1,λn)

dT∣∣Pn−1

d

dλlPn−1

= (d1Ψ∣∣(T Pn−1,0)

+ Un)dT∣∣Pn−1

d

dλlPn−1

= (I + Un)(S + Un)d

dλlPn−1, (4.21)

with I the identity mapping. Let us first obtain a bound for ‖Un‖: Sinced1Ψ ∈ Lip ν, we obtain

‖Un‖ = ‖d1Ψ∣∣(T Pn−1,0)

− d1Ψ∣∣(T Pn−1,λn)

‖ ≤ C‖λn‖ν∞ ≤ C ′(µν)n,

because by (i) the wavelet coefficients decay geometrically. The Matrix Unis bounded by a constant times ‖∆Pn−1‖α−1

∞ since we assumed that a prox-imity condition with exponent α > 1 holds. By Lemma 4.3.9 (i) ⇒ (ii),this implies that a differential proximity condition holds and further, ‖Un‖ ≤


C‖∆Pn−1‖α−1∞ . By the above estimate and the fact that f ∈ Lip γ we finally

get‖Un‖, ‖Un‖ ≤ Cµn

for another constant C ≥ 0 and some 0 < µ < 1. Since also ‖S‖ is uniformlybounded, this implies

d

dλlPn = (S + Vn)

d

dλlPn−1, (4.22)

where ‖Vn‖ ≤ Cµn with a uniform constant C ≥ 0.

For l > n we have ddλlPn = 0. If l = n, then d

dλlPn = d2Ψ

∣∣(T Pn−1,λn)

. Now we

apply (4.22) inductively and obtain

d

dλlPn =

n∏j=l+1

(S + Vj)d

dλlP l =

n∏j=l+1

(S + Vj)d2Ψ∣∣(T Pl−1,λl)

with ‖Vj‖ ≤ Cµj . Applying Lemma 4.3.6 yields a uniform bound for ‖ ddλlPn‖.

We still need to estimate ‖ ddP0Pn‖. Similar to the derivation of (4.21),

(4.22), we obtain

d

dP0Pn =

n∏j=1

(S +Wj)

with matrices ‖Wj‖ ≤ Cµj . Using Lemma 4.3.6 again, we arrive at bounded-ness of ‖ d

dP0Pn‖. We have now proven that the norms ‖ ddλlPn‖, ‖ d

dP0Pn‖ arebounded by a constant C ≥ 0 independent of l, n and depending only on γ andC.

We state and prove our main stability theorem for Euclidean data.

Theorem 4.4.6. With the assumptions of Lemma 4.4.5, let f ∈ Lip γ with0 < γ < − log2 µ0 ≤ 1 (µ0 being the contraction factor of S) and Pnf ∈ PM ′′,δfor all n ∈ Z+ where M ′′ is a set that stays away from the boundary of M ′ ofLemma 4.4.3. Then there exist δ0, ε0 > 0 and a constant C ≥ 0 depending only(in a monotonely decreasing way) on ‖∆P0‖ and |f |Lip γ such that for all initialdata

(P0, λ1, λ2, . . . ) (4.23)

with

‖P0 − P0f‖∞ ≤ δ0 and ‖λj − λjf‖∞ ≤ ε0/2γj , j ∈ Z+ (4.24)

the reconstruction from (4.23) yields a well-defined continuous function f with

‖Pn(f − f)‖∞ ≤ C(‖P0(f − f)‖∞ +

n∑j=1

‖λj − λj‖∞).

Proof. From the first part of the proof of Lemma 4.4.5 we know that, since f ∈Lip γ, the norms of the wavelet coefficients decay according to ‖λj‖∞ = O(µj)with the hidden constant depending only on γ and |f |Lip γ and µ = 2−γ . It


is also clear by definition that ‖∆Pjf‖∞ ≤ |f |Lip γ2−γj . Hence there exists j0such that ‖∆Pj0‖∞ < δ′ and ‖λj0+j‖ < C ′/2γj for all j ≥ 0 with δ′, C ′ fromLemma 4.4.3. Denoting Pj := Pj(P0, λ1, . . . ), we can now pick ε0, δ0 > 0 suchthat for all data satisfying (4.24) we have ‖∆Pj0‖ < δ′, ‖λj0+j‖ < C ′/2γj , andPj0 assumes values in M ′. Any such data, by Lemma 4.4.3, defines a continuousfunction g ∈ Lip γ with |g|Lip γ only depending on δ′, C ′. For all data whichobey (4.24), by Lemma 4.4.5, the norms ‖ d

dλj0+kPj0+j‖, ‖ ddPj0 P

j0+j‖, k, j ≥ 0are uniformly bounded by a constant C1 > 0. Define

C2 := maxj,k=1,...,j0

sup(P0,λ1,...,λj0 ) with (4.24)

‖ d

dλkPj‖ <∞.

We now show that the theorem holds with C := max(C1, C1 · C2, C2):For data (P0, λ1, . . . ) with (4.24), the estimate

‖Pn(P0, λ1, . . . , λn) − Pn(P0, λ1, . . . , λn)‖∞

≤ ‖Pn(P0, λ1, . . . , λn)− Pn(P0, λ1, . . . , λn)‖∞+

‖Pn(P0, λ1, . . . , λn)− Pn(P0, λ1, λ2, . . . , λn)‖∞

+ · · ·+ ‖Pn(P0, λ1, . . . , λn−1, λn)−

Pn(P0, λ1, . . . , λn−1, λn)‖∞

≤ supt∈(0,1)

‖ d

dP0Pn∣∣(tP0+(1−t)P0,λ1,...,λn)

‖‖P0 − P0‖∞+

supt∈(0,1)

‖ d

dλ1Pn∣∣(P0,tλ1+(1−t)λ1,...,λn)

‖‖λ1 − λ1‖∞

+ · · ·+ supt∈(0,1)

‖ d

dλnPn∣∣(P0,λ1,...,λn−1,tλn+(1−t)λn)

‖‖λn − λn‖∞

holds. Now we distinguish three cases to show that the norms ‖ ddλiP

j‖, i, j ∈ Z+

are uniformly bounded by C:Case 1: i, j > j0. For this case, boundedness has been shown with the constantC1.Case 2: i, j ≤ j0. In this case the expression is bounded by C2.Case 3: i > j0, j ≤ j0. Here we apply the chain rule to obtain

‖ d

dλiPj‖ ≤ ‖ d

dP0Pj‖‖ d

dλjPj0‖ ≤ C1 · C2.

The proof is complete.

Comparison with the results in [117] We would like to say a few wordson how our results compare to the recent work [117]. The idea of both ourwork as well as [117] is to obtain global estimates on the differentials of theiterates of the subdivision and reconstruction operators. However, other thanthat, the approaches are quite different. The fundamental assumption in [117] isthat the nonlinear subdivision scheme admits some algebraic factorization prop-erties which allow for the construction of so-called derived schemes which are


intertwining maps between a difference operator and the subdivision operator.Based on the differentials of these derived schemes a spectral quantity is intro-duced which turns out to accurately describe the stability properties. It alsoturns out that convergence of the nonlinear scheme, stability of the nonlinearscheme and stability of the reconstruction procedure are in general independentof each other.

The algebraic factorization properties assumed in [117] do not hold for theapplications we have in mind, namely manifold-valued subdivision, so the resultsof [117] cannot be applied at all. The nonlinearity of manifold-valued schemeshas no algebraic structure; it stems from the rather arbitrary curvature proper-ties of some manifold – this is the reason why algebraic factorization propertiesdo not hold in this case. We overcome this difficulty by using a variant of theperturbation approach which has been successfully used e.g. in [126, 103] to-gether with the above mentioned idea to bound the differentials of the iteratesof the nonlinear subdivision operator.

A remarkable difference between our stability results and the results in [117]is that with our assumptions (i.e. a proximity condition), stability of the sub-division scheme and the reconstruction procedure essentially follow from con-vergence of the scheme. Another difference lies in the assumptions made on theinitial data sequences for the reconstruction procedure: we assume exponentialdecay of the detail coefficients while in [117] no such assumption is made. Inour framework this assumption is natural and poses almost no restriction: ascan be seen from the proofs in this section, the detail coefficients coming fromany reasonable function satisfy it.

4.5 Examples

In the previous sections we have shown some general results concerning thestability of nonlinear subdivision schemes. In that we have always required ourdata to lie in some Euclidean space. In the present section we discuss someexamples of nonlinear multiscale decompositions which operate on manifold-valued data. While again quantitative results can easily be obtained by using achart, we still need to state an analogue of Theorem 4.4.6 in an intrinsic setting.In particular, it is desirable to have an intrinsic way of comparing two waveletcoefficients which may be, for example, two tangent vectors at different points,i.e., two elements of different vector spaces. If a metric is available for ourmanifold, then perhaps the most natural way of comparing two such vectors isto parallel translate the first vector into the tangent space of the second onealong a geodesic curve. We discuss this below, but first we treat the simplercase of a Lie group, where the tangent bundle is trivial.

4.5.1 The log-exp analogue in Lie groups

The log-exp analogue in Lie groups and the corresponding multi-scale decompo-sition has first been studied in [124]. Given a Lie group (G, ·) with Lie-algebra g,the log-exp analogue T of a linear subdivision scheme S is defined on G-valuedsequences p = (pi)i∈Z via

(T p)2i+k := pi · exp(∑j∈Z

a2i+k−2j log(p−1i · pj)) =: pi ⊕

(∑j∈Z

a2i+k−jpj pi).

4.5. EXAMPLES 83

Here exp is the exponential mapping of G that diffeomorphically maps a neigh-borhood of 0 ∈ g to a neighborhood of the unit element in G, log is its inverse,and (ai) is the mask of S. The detail coefficients of a continuous G-valuedfunction f are consequently defined via

(λj)i := log((Pjf)−1

i · (T Pj−1)i

)= (T Pj−1)i (Pjf)i.

It is convenient that in the case of a Lie group all detail coefficients lie in thesame vector space g. This makes it possible to directly apply our stability resultsfrom the previous sections using local charts. We fix a norm ‖ · ‖ in g and definefor two points p, q ∈ G with p−1 · q small enough for q p to be defined

d(p, q) := ‖q p‖.

For G-valued sequences p,q we write d(p,q)∞ := supi∈Z d(pi, qi). Similarly, fora sequence λ = (λi)i∈Z we write ‖λ‖∞ := supi∈Z ‖λi‖. It is easy to see and hasbeen shown e.g. in [112] that locally a proximity condition of order α = 2 issatisfied between S and T . Now we can state our main stability theorem forthe log-exp analogue in Lie groups.

Theorem 4.5.1. For all G-valued functions f ∈ Lip γ there exist δ0, ε0 > 0and a constant C ≥ 0 such that for all initial data

(P0, λ1, λ2, . . . )

with

d(P0,P0f)∞ ≤ δ0 and ‖λj − λjf‖∞ ≤ ε0/2γj , j ∈ Z+ (4.25)


d(Pnf,Pnf)∞ ≤ C

(d(P0f, P0)∞ +

n∑j=1

‖λjf − λj‖∞).

The constants δ0, ε0, C are uniform for data values in a compact set.

Proof. We only sketch the argument. The first simple remark is that the re-quirement 0 < γ < − log2 µ0 ≤ 1, where µ0 is the contraction factor of S posesno loss of generality since all our assumptions hold also for any 0 < γ′ < γ ifthey hold for γ > 0. The proof is then just a simple application of Theorem4.4.6 after pulling back the subdivision and reconstruction procedure to Eu-clidean space using a local chart such that the domain of definition of the chartis open and convex. For each chart we get different constants δ0, ε0, C, thereforeuniform constants can in general only be achieved for data values in a compactset.

Remark 4.5.2. Actually, more can be said regarding the smoothness of thereconstructed function f . The decay rate of wavelet coefficients of f is clearlythe same as the decay rate of f . This implies by the results in [116] that, providedthe underlying linear scheme S is interpolatory and sufficiently smooth, f is assmooth as f . The same is true if S is not interpolatory if we define T ina different way [128]. This remark also applies to the log-exp analogue in aRiemannian manifold which is studied below.


4.5.2 The log-exp analogue in Riemannian manifolds

Before we proceed further, we introduce some basic facts of Riemannian geome-try. For more information on this topic, the interested reader is referred to [105].

Some basics of Riemannian geometry An m-dimensional Riemannianmanifold is by definition an m-dimensional smooth differentiable manifold Mtogether with a smooth (0, 2)-tensor field

p ∈M 7→ 〈·, ·〉p (4.26)

such that for each p ∈M, the bilinear form (u, v) 7→ 〈u, v〉p ∈ TpM is a positivedefinite inner product on TpM. With the help of the inner product we definethe length of a smooth curve c : I →M, where I is some interval ⊆ R, by

l(c) :=

∫I

‖c(t)‖dt :=

∫I

〈c(t), c(t)〉1/2c(t)dt.

A vector field V along a curve c : I →M is a mapping that smoothly assignsto each parameter value t ∈ I a tangent vector V (t) ∈ Tc(t)M. A Rieman-nian manifold possesses a canonical way of differentiating a vector field alonga curve via the Riemannian connection. This derivative is called the covariantderivative. With respect to a chart ϕ : U ⊂ Rm →M it can be written as

D

dtV (t) :=

(dvldt

+ vkxjΓljk)∂l,

where V (t) is a vector field along γ(t). This means that V (t) ∈ Tc(t)M with

coordinate representation vj∂j . The coefficients xk represent the coefficientsof the tangent vector field c, i.e. c = xk∂k and ∂k are the basis vector fieldsinduced by the parametrization (ϕ, dϕ). The quantities Γljk are the Christoffelsymbols of the first kind. Note that we have used the Einstein sum convention.

The vector field V is called parallel along c, if DdtV ≡ 0. By the linearity of

the equation DVdt = 0, for curves c with c(0) = p and v ∈ TpM, there exists

a unique vector field V along c with V (0) = v and V is parallel along c. Ifc(1) = q, then the vector V (1) is called the result of parallel transport of v fromp to q along c. We denote the linear mapping that maps the vector v to thevector V (1) by Pt1

0(c) : TpM→ TqM.Obviously, the tangent vector field t 7→ c(t) is a vector field along c. The

curve c is called a geodesic, if c is parallel along c. It is well known that geodesicslocally minimize arc length. Moreover, locally any two points can be joined bya unique shortest curve which then is a geodesic. Conversely, given a pointp ∈ M and a vector v ∈ TpM of sufficiently small norm, then there exists aunique geodesic c : [−2, 2] → M with c(0) = p and c(0) = v. The point c(1)is called expp(v). This mapping (p, v) 7→ expp(v) is a smooth map in bothvariables and it also possesses an inverse logp with logp(expp(v)) = v which isalso smooth in both variables. As the notation already suggests, the functionexp is called exponential mapping. The notion of arc length naturally induces ametric on M: For p, q ∈M we define

d(p, q) := infl(c) : c(0) = p and c(1) = q.

4.5. EXAMPLES 85

The multiscale decomposition in Riemannian manifolds We now ex-plain the multiscale transformation associated with the so-called log-exp ana-logue of a linear subdivision scheme S: It is defined by

(T p)2i+k := exppi(∑j∈Z

a2i+k−2j logpi(pj)) =: pi ⊕(∑j∈Z

a2i+k−jpj pi),

where (ai) is the mask of S and i ∈ Z, k ∈ 0, 1. The multiscale transformationis set up as follows:

(λj)i := log(Pjf)i((T Pj−1)i) = (T Pj−1)i (Pjf)i.

The reconstruction of a continuous function from data (P0, λ1, λ2, . . . ) is definedinductively by Pj = T Pj−1⊕λj . Observe that now each wavelet coefficient liesin a different vector space, namely (λj)i ∈ T(T Pj−1)iM. The question now ishow to compare two wavelet coefficients which lie in different tangent spaces. If asubdivision scheme T operates inM, we can by the locality of the reconstructionprocedure and the fact that our data becomes arbitrarily dense, pull the databack into Euclidean space via a chart ϕ. This chart induces a chart of TM via(ϕ, dϕ) and we can compare two wavelet coefficients in the Euclidean setting.The problem in using Theorem 4.4.6 directly with respect to this chart is thatthe quantities ‖λi − λi‖ in Theorem 4.4.6 have no intrinsic meaning.

Therefore we need to find an intrinsic way of comparing wavelet coefficientsliving in different tangent spaces. Probably the most natural way is the follow-ing:

Definition 4.5.3. Assume that p, q ∈M can be connected by a unique geodesicc with c(0) = p and c(1) = q. The distance d(v, w) between two tangent vectorsv ∈ TpM and w ∈ TqM is defined as ‖Pt10(c)v − w‖q, where ‖ · ‖q is the norminduced by the Riemannian metric in TqM.

Remark 4.5.4. The distance d is symmetric with respect to v and w. Thisfollows from the fact that the parallel transport is an isometry.

Now we can state our main stability theorem for the Riemannian log-expanalogue:

Theorem 4.5.5. For all M-valued functions f ∈ Lip γ there exist δ0, ε0 > 0and a constant C ≥ 0 such that for all initial data

(P0, λ1, λ2, . . . )

with

d(P0,P0f)∞ ≤ δ0 and d(λj , λjf

)∞ ≤ ε0/2

γj , j ∈ Z+ (4.27)


d(Pnf,Pnf)∞ ≤ C

(d(P0f, P0)∞ +

n∑j=1

d(λjf, λj

)∞

). (4.28)

The constants δ0, ε0, C are uniform for data values in a compact set.


Proof. Unlike the Lie group case, here we cannot simply invoke Theorem 4.4.6to establish our result. This is because, by pulling back with respect to a chartand using 4.4.6 we lose the geometric meaning of the distance d. The solutionto this problem is as follows: Instead of assuming that all wavelet coefficients liein different tangent spaces we regard the wavelet coefficients (λj)i as elementsof T(T Pj−1f)iM instead of elements of T(T Pj−1)i

M, the symbol Pj as usual

denoting the j-th reconstruction step from the data (P0, λ1, λ2, . . . ). Accordingto this interpretation we need to redefine the functions

Pj(P0, λ1, . . . , λj

)i

:= (Pj−1)i ⊕ (λj)i,

where i ∈ Z andp ⊕ v := p⊕ Ptpq(v) for v ∈ TqM.

Here Ptpq denotes the parallel transport along the unique shortest geodesic con-necting q and p. Using this interpretation of the multiscale transformation thedetail coefficients live in fixed vector spaces determined by the function f . Thismakes it possible to compare two vectors (λj)i and (λj)i (which now live inthe same vector space) using the Riemannian metric in (T Pj−1)i. This differ-ence is clearly the same as d((λj)i, (λ

j)i) where now (λj)i is interpreted as theusual detail coefficient living in T(T Pj−1)i

M. We can thus directly apply ourresults from the previous section and repeat the proof of Theorem 4.5.1 to proveTheorem 4.5.5.

4.5.3 The projection analogue

We also want to say something about the so-called projection analogue [114]. Itis defined by first applying a linear subdivision scheme S to data living in Rmand then applying a retraction mapping P , i.e. the projection analogue of S isdefined via

T p := P Sp.

We consider the case where a (hyper-) surfaceM in Rm is given, and P is somesmooth projection onto M defined locally around M, for example the closestpoint-projection. The results of the previous sections cannot be directly appliedto this particular nonlinear scheme, since T is only defined on M-valued data.In Definition 4.3.1 we have explicitly asked for an open and convex domain ofdefinition for T . Neither of these requirements is satisfied by the projectionanalogue of a linear scheme. The proof that T is stable is however still almostthe same: We can just repeat the arguments in Section 4.3 and replace everyoccurrence of a line c(t) connecting two points by the unique geodesic connectingthe points. Since the norms of tangent vectors of a geodesic remain constantalong the geodesic, all the arguments still go through. This implies that thesubdivision operator T = P S is stable.

Another problem exists: The multiscale transformation associated with T isdefined via

(λjf)i = (Pjf)i − (T Pj−1f)i. (4.29)

Observe that the values for (λj)i cannot be chosen arbitrarily, but must bechosen such that (T Pj−1f)i + (λj)i is in M. The solution for this problem iseasy: We replace (4.29) with

(λjf)i = log(T Pj−1)i((Pjf)i). (4.30)

4.6. CONCLUDING REMARKS 87

It is well-known and easy to see that the two definitions are equivalent in that thenorms of both definitions are proportional. Now the reconstruction procedurefits into the framework of the log-exp analogue for Riemannian manifolds andwe can apply Theorem 4.5.5.

4.6 Concluding remarks

We conclude our work with a few remarks. First, we would like to mention thatall our results remain valid in the following more general framework: Instead ofdefining Pnf := (f(i/2j))i∈Z, we can define

Pnf := U Pnf,

where U is a local smoothing operator and

(λj)i := (Pjf)i (T Pj−1f)i.

The purpose of defining such a decomposition is to reduce aliasing effects andto make the methods better suited for applications in noise-removal. As anexample we mention [124], where U is a nonlinear average imputing operator.For the linear case compare also [108, 122].

As already mentioned in the introduction, our results also remain valid forany dilation factor > 1 with the same proofs. Also the extension of our resultsto the multivariate case is straightforward using the methods developed in [112].A very natural question for future research is if the regularity conditions on Tare really necessary. Example 4.3.3 shows that T ∈ Lip 1 is definitely necessary.Is this also sufficient? Does proximity and T ∈ Lip 1 imply stability? We donot know the answer.

As another possible direction for future research we would like to mentionthe recent work [119], where directionality is encompassed into the (linear) mul-tiscale decomposition. It seems that these constructions can also be defined formanifold-valued data. It will be an interesting task to study the properties ofthese nonlinear decompositions.

Acknowledgments

The author gratefully acknowledges support by the Austrian Science Fund(FWF) under grant number P19780.


Bibliography

[98] S. Amat, K. Dadourian, and J. Liandrat. Analysis of a class of non linearsubdivision schemes and associated multi-resolution transforms. Arxivpreprint arXiv:0810.1146, 2008.

[99] S. Amat and J. Liandrat. On the stability of PPH nonlinear multiresolu-tion. Appl. Comp. Harm. Anal., 18(2):198–206, 2005.

[100] A. S. Cavaretta, W. Dahmen, and C. A. Micchelli. Stationary Subdivision.American Mathematical Society, 1991.

[101] A. Cohen, N. Dyn, and B. Matei. Quasilinear subdivision schemes withapplications to ENO interpolation. Appl. Comp. Harm. Anal., 15:89–116,2001.

[102] K. Dadourian. Schemas de subdivision, analyses multiresoltuions non-lineaires. applications. PhD thesis, Universite de Provence, 2008.

[103] I. Daubechies, O. Runborg, and W. Sweldens. Normal multiresolution ap-proximation of curves. Constructive Approximation, 20(3):399–463, 2004.

[104] Z. Ditzian. Moduli of smoothness using discrete data. J. Approx. Th.,49(2):115–129, 1987.

[105] M. P. do Carmo. Riemannian Geometry. Birkhauser, 1992.

[106] D. L. Donoho. Interpolating wavelet transforms. Technical report, 1992.available from http://citeseer.ist.psu.edu/donoho92interpolating.html.

[107] D. L. Donoho and T. P.-Y. Yu. Nonlinear pyramid transforms based onmedian-interpolation. SIAM Journal of Math. Anal., 31(5):1030–1061,2000.

[108] David L. Donoho. Smooth wavelet decompositions with blocky coeffi-cient kernels. In In Recent Advances in Wavelet Analysis, pages 259–308.Academic Press, 1993.

[109] N. Dyn, P. Grohs, and J. Wallner. Approximation order of interpolatorynonlinear subdivision schemes. J. Comp. Appl. Math., 2009. to appear.

[110] N. Dyn and D. Levin. Subdivision schemes in geometric modelling. ActaNumerica, 11:73–144, 2003.

[111] G. Faber. Uber stetige Funktionen. Mathematische Annalen, 66(1):81–94,1908.

89

90 BIBLIOGRAPHY

[112] P. Grohs. Smoothness analysis of subdivision schemes on regular grids byproximity. SIAM J. Numerical Analysis, 46:2169–2182, 2008.

[113] P. Grohs. Approximation order from stability of nonlinear subdivisionschemes. Technical report, TU Graz, November 2009. Geometry preprint2008/06.

[114] P. Grohs. Smoothness equivalence properties of univariate subdivi-sion schemes and their projection analogues. Numerische Mathematik,113(3):163–180, 2009.

[115] P. Grohs. Smoothness of interpolatory multivariate subdivision in Liegroups. IMA J. Numer. Math., 2009. to appear.

[116] P. Grohs and J. Wallner. Interpolatory wavelets for manifold-valued data.Appl. Comp. Harm. Anal., 2009. to appear.

[117] S. Harizanov and P. Oswald. Stability of nonlinear subdivision and mul-tiscale transforms. Constructive Approximation, 2009. submitted.

[118] A. Harten, S. Osher, B. Engquist, and S. R. Chakravarthy. Some resultson uniformly high-order accurate essentially nonoscillatory schemes. Appl.Numer. Math., 2(3-5):347–378, 1986.

[119] G. Kutyniok and T. Sauer. Adaptive directional subdivision schemes andshearlet multiresolution analysis. SIAM Journal of Math. Anal., 2009. toappear.

[120] C. A. Micchelli. Mathematical Aspects of Geometric Modeling. Society forIndustrial Mathematics, 1995.

[121] O. Runborg. Introduction to Normal Multiresolution Approximation.In Classification, Clustering, and Data Mining Applications: Proceedingsof the Meeting of the International Federation of Classification Societies(IFCS), Illinois Institute of Technology, Chicago, 15-18 July 2004, page205. Springer, 2004.

[122] W. Sweldens. The Lifting Scheme: A Construction of Second GenerationWavelets. SIAM J. Math. Anal., 29:511–546, 1998.


[124] I. Ur Rahman, I. Drori, V. C. Stodden, D.L. Donoho, and P. Schroder.Multiscale representations for manifold-valued data. Multiscale Modelingand Simulation, 4(4):1201–1232, 2005.

[125] J. Wallner. Smoothness Analysis of Subdivision Schemes by Proximity.Constructive Approximation, 24(3):289–318, 2006.

[126] J. Wallner and N. Dyn. Convergence and C1 analysis of subdivisionschemes on manifolds by proximity. Computer Aided Geometric Design,22(7):593–622, 2005.

BIBLIOGRAPHY 91

[127] G. Xie and T. P.-Y. Yu. Approximation order equivalence propertiesof manifold-valued data subdivision schemes. Manuscript, submitted forpublication, April 2008.

[128] G. Xie and T. P.-Y. Yu. Smoothness equivalence properties of generalmanifold-valued data subdivision schemes. Multiscale Modeling and Sim-ulation, 7(3):1073–1100, 2008.

[129] G. Xie and T. P.-Y. Yu. Smoothness equivalence properties of interpola-tory Lie group subdivision schemes. IMA Journal on Numerical Analysis,2008. to appear.

92 BIBLIOGRAPHY

Chapter 5

[G4]: Approximation orderfrom stability of nonlinearsubdivision schemes

Abstract

This paper proves approximation order properties of various nonlinear subdi-vision schemes. Building on some recent results on the stability of nonlinearmultiscale transformations, we are able to give very short and concise proofs.In particular we point out an interesting connection between stability propertiesand approximation order of nonlinear subdivision schemes.

5.1 Introduction

For various reasons many different nonlinear subdivision schemes have beenintroduced in the last decades. Among those reasons are the presence of non-Gaussian noise, Gibbs-like phenomena which are inherently present in manylinear procedures, or the need to process data which lie in a nonlinear geometry,e.g. a manifold. It turns out that most of these subdivision schemes can beanalyzed by viewing them as a perturbation of a linear subdivision scheme[134, 146].

The present paper is devoted to approximation order of nonlinear schemes,i.e., to answering the following question: If we take a dense sampling of a smoothfunction and compute the limit function with respect to our subdivision scheme,how well does this limit function approximate the initial function in relation tothe sampling density? By revealing an intimate connection between stabilityof a subdivision operator and approximation order properties, we are able toanswer the above question for a large class of nonlinear subdivision schemes.Our approach allows for particularly short and natural proofs, once stabilityhas been established. The results of this paper may roughly be subsumed bythe statement that once stability is known, it is usually easy to understandapproximation order.

93

94 CHAPTER 5. [G4]: APPROXIMATION ORDER

There has been recent progress in proving stability properties for nonlinearsubdivision schemes [141, 139]. By making use of these results we are able toprove optimal approximation order properties for a number of examples likethe median interpolating scheme, the power-p scheme and the log-exp-analoguewhich operates in manifolds.

Other work related to nonlinear multiscale decompositions and their prop-erties can be found e.g. in [142, 144, 130, 133]. Most of this work is howeverconcerned with convergence and stabililty properties, not with approximationorder.

The outline of the paper is as follows: We first introduce notation and basicdefinitions in Section 5.2. Then, in Section 5.3, we prove our main result, namelythat essentially stability of a MRA associated with the nonlinear subdivisionoperator T (as defined is Section 5.2) implies approximation order if a directtheorem (as defined in Section 5.2) is valid for T . We also prove that if T isa suitable perturbation of a linear subdivision scheme, then a direct theoremalways holds. In Section 5.4 we show that in most cases stability of the MRAassociated with T can be replaced with the weaker assumption that T is stable.To show this we use the framework developed in [141]. Finally, in Section 5.5we apply our results to some examples.

5.2 Notation and Definitions

Let us describe our setup.

Definition 5.2.1. An m-dimensional subdivision operator (m ∈ N) is a map-ping T : l∞(Z,Rm) → l∞(Z,Rm) that is local and has dilation factor N > 1,meaning that

σN T = T σ,where σ denotes the right-shift on Z. Locality means that the value (T p)i ∈Rm, i ∈ Z, p ∈ l∞(Z,Rm) depends only on a finite number of points.

Denote by Fl(T lp) the piecewise linear function that interpolates the dataT lp on the grid 1

N lZ. Then T is called convergent if the functions Fl(T lp) uni-

formly converge to a nontrivial continuous limit function T ∞p for all nontrivialinitial data p 6= (. . . , 0, 0, 0, . . . ).

In the present paper we are interested in the approximation order of subdi-vision schemes. For α ∈ R+ let Lip α be the space of all bounded functionssuch that with γ := dαe − 1 we have f ∈ Cγ and ( ddt )

γf ∈ Lip (α − γ).For 0 < γ ≤ 1 the space Lip γ consists of all bounded functions f with

suph>0 ‖f(·+h)−f(·)|h|γ ‖C(R) <∞.

Remark 5.2.2. The definition of Lipschitz classes given here is nonstandardin case of α ∈ N, where usually higher order divided differences have to be takeninto account. This would yield the so-called Zygmund classes [145] and ensurethat these spaces are all containted in a Besov scale. For our purposes, however,the above definition turns out to be the most convenient.

Definition 5.2.3. A convergent subdivision scheme T provides approximationorder α > 0 if for any f ∈ Lip α there exists a constant C > 0 such that

‖D1/hT ∞f∣∣hZ − f‖C(R) ≤ C|h|α for all h > 0

5.2. NOTATION AND DEFINITIONS 95

with D1/h being the dilation operator f(·) 7→ f(·/h).

Definition 5.2.4. With the sampling operators

Pj : L∞(R,Rm)→ l∞(Z,Rm), f 7→ Pjf := (f(i

N j))i∈Z (5.1)

the MRA associated with T of a function f ∈ L∞(R,Rm) is defined as the data

(P0, λ1, λ2, . . . ), (5.2)

where P0 := P0f and λjf = Pjf − T Pj−1f . Given data (5.2), the reconstruc-tion procedure is defined by the functions

Pj(P0, λ1, λ2, . . . ) := T Pj−1(P0, λ1, λ2, . . . ) + λj . (5.3)

If there exists a continuous function f with Pjf = Pj(P0, λ1, λ2, . . . ), we writeP∞(P0, λ1, λ2, . . . ) := f .

In the linear case such an MRA is also known as interpolating wavelet trans-form [135]. A nonlinear version of these transforms has been analyzed in [140].The first paper that uses the idea of building an MRA from subdivision is, tothe best of the author’s knowledge, Faber’s 1908 paper [138].

Definition 5.2.5. We say that the MRA associated with T is stable if there ex-ists a constant C ≥ 0 such that for initial data (P0, λ1, λ2, . . . ) and (P0, λ1, λ2, . . . )we have

‖Pj(P0, λ1, λ2, . . . )−Pj(P0, λ1, λ2, . . . )‖∞ ≤ C(‖P0−P0‖∞+

j∑i=1

‖λi− λi‖∞).

In approximation theory one often tries to relate the convergence order of anapproximation process to the membership of a function in some function space.In this spirit we make the following definition:

Definition 5.2.6. The subdivision scheme T admits a direct theorem of orderα > 0 if for every f ∈ Lip α there exists a constant Cf such that

‖λjf‖∞ ≤ CfN−αj .

For technical reasons we require that the constant CDτf continuously dependson τ ∈ [1/N, 1].

Of special interest are linear subdivision schemes which we shall alwaysdenote by S. They can be written as

(Sp)Ni+l =∑j∈Z

aNi+l−Njpj , i ∈ Z, l = 0, . . . , N − 1

with a finitely supported sequence (ai)i∈Z, called the mask of S. A subdivisionscheme is said to reproduce Πk, the space of polynomials of degree ≤ k, if

SP0p = P1p for all p ∈ Πk.


It is well known [131, 137] that a linear scheme S has approximation orderk + 1 if S reproduces Πk. Further, it is well known that if a linear scheme isconvergent, it must reproduce Π0. This property, often called reproduction ofconstants, is clearly equivalent to∑

j∈Zai−Nj = 1 for all i ∈ Z.

As an example of a linear subdivision scheme we mention the linear splinescheme with dilation factor N which is defined by the generating function of itsmask via ∑

j∈Zajz

j =(1 + z + · · ·+ zN−1)2

Nz.

It is easy to verify that this subdivision scheme reproduces Π1, i.e. has approx-imation order 2.

A useful tool for studying linear schemes are derived schemes. With ∆ beingthe operator that maps a sequence p = (pi)i∈Z to (pi+1−pi)i∈Z, the l-th derivedscheme is defined as the linear subdivision scheme that satisfies

∆l S = S [l] ∆l.

Derived schemes need not exist in general. For instance S [1] exists iff S repro-duces constants. In general the conditions for the existence of derived schemes,called sum rules, are purely algebraic. We remark that if S reproduces Πk, thenthe derived scheme S [k+1] exists.

5.3 Main theorems

Theorem 5.3.1. Assume that the subdivision scheme T admits a direct theo-rem of order α > 0. If the MRA associated with T is stable, then T providesapproximation order α.

Proof. Let f ∈ Lip α with decomposition (P0, λ1, . . . ). We first consider valuesh = 1/N j . In this case, the expression

‖D1/hT ∞f∣∣hZ − f‖C(R)

assumes the form

‖P∞(P0, λ1, . . . , λj ,0,0, . . . )− P∞(P0, λ1, . . . , λj , λj+1, λj+2, . . . )‖∞,

where 0 denotes the constant zero sequence. By stability and the fact that Tadmits a direct theorem, this expression can be bounded by

C∑k>j

‖λk‖∞ ≤ C ′∑k>j

N−αk = C ′1

1−N−αN−αj .

Now we let h = τ/N j with τ ∈ [1/N, 1]. Then

‖D1/hT ∞f∣∣hZ − f(·)‖C(R) = ‖DNjT ∞(Dτf)

∣∣N−jZ −Dτf‖C(R).

5.4. WEAKENING THE ASSUMPTIONS 97

This expression is bounded by CτN−αj = Cττ

−αhα ≤ NαCτhα with a constant

Cτ depending continuously on τ ∈ [1/N, 1]. Since [1/N, 1] is a compact set, weconclude that

‖D1/hT ∞f∣∣hZ − f‖C(R) ≤ Chα

for some constant C ≥ 0 and h > 0.

The following theorem settles a conjecture in [147] under a stability assump-tion on T . As we shall see in Section 5.5, it can be applied to a wide class ofnonlinear subdivision schemes.

Theorem 5.3.2. Let T be a subdivision scheme such that the associated MRAis stable and let S be a linear scheme reproducing Πk. Assume that there existsa constant C ≥ 0 such that

‖Sp− T p‖∞ ≤ CΩk(p) for all p, (5.4)

where

Ωk(p) :=∑γ∈Γk

k+1∏i=1

‖∆ip‖γi∞ and

Γk :=γ = (γ1, . . . , γk+1) : γi ∈ Z+,

k+1∑i=1

iγi = k + 1.

Then T has approximation order k + 1.

Proof. We need to show that T admits a direct theorem of order k + 1. Notethat for every f ∈ Lip (k + 1), we have that

‖∆lPjf‖∞ ≤ ClN−lj l = 1, . . . , k + 1

with nonnegative constants C1, . . . , Ck+1 depending continously on τ 7→ Dτf .With ∆hf := f(·+h)−f(·) we may for instance take Ci := suph>0 ‖h−i∆i

hf‖∞,i = 1, . . . , k + 1. Putting these estimates into (5.4), we arrive at

Ωk(Pjf) ≤(

maxi=1,...,k+1

Ci)k+1

N−(k+1)j . (5.5)

Now we estimate the wavelet coefficients

‖λjf‖∞ = ‖T Pj−1f − Pjf‖∞ ≤ ‖SPj−1f − Pjf‖∞ + ‖(S − T )Pj−1f‖∞.

In [135] it is shown that if S is linear and reproduces Πk, then S satisfies adirect theorem of order k + 1. Hence, the first term is bounded by a constanttimes N−(k+1)j . Because of (5.4) and (5.5) the second term is bounded by ananaloguous expression and we arrive at the result.

5.4 Weakening the assumptions

In the present section we show how to weaken our assumption that the MRAassociated with T is stable to the simpler property that T is stable, i.e. thereexists C ≥ 0 such that

‖T ∞p− T ∞q‖∞ ≤ C‖p− q‖∞.


From [141] we know that stability of T does not necessarily imply stability ofthe MRA associated with T . The dyadic median interpolating scheme providesa counterexample. However, within the framework of [141] it is possible to provethat stability of T (almost) implies that for an iterate T n the associated MRAis stable. A careful examination of the proof of Theorem 5.3.1 shows that thisis already enough to show approximation order, provided that T n satisfies adirect theorem (Note that if T has dilation N , then T n has dilation Nn). Inwhat follows we shall only consider subdivision schemes T ∈ C1(l∞, l∞). Wecan thus define the differential DT in an obvious way. By locality of T theoperator DT behaves like a finite dimensional operator. We say that T ∈ Ok iffor l = 1, . . . , k there exists another subdivision operator T [l] with

∆l T = T [l] ∆.

We also define the spectral radius associated with the MRA as

ρMRA(T , k) := lim infj→∞

sup(pj−1,...,p0)∈lj∞

‖DT [k]∣∣pj−1DT [k]

∣∣pj−2 . . . DT [k]

∣∣p0‖1/j

and the spectral radius of T via

ρS(T , k) := lim infj→∞

supp∈l∞

‖DT [k]∣∣(T [k])j−1p

DT [k]∣∣(T [k])j−2p

. . . DT [k]∣∣p‖1/j .

The main result of [141] reads as follows:

Theorem 5.4.1. T is stable if ρS(T , k) < 1 for some k ∈ Z+. If ρS(T , k) > 1,then T is not stable. The MRA associated with T is stable if ρMRA(T , k) < 1for some k ∈ Z+. If ρMRA(T , k) > 1, then the MRA is not stable.

Lemma 5.4.2. Assume that ρS(T , k) < 1. Then there exists an n ∈ Z+ suchthat ρMRA(T n, k) < 1.

Proof. The assumption ρS(T , k) < 1 implies that there exists n ∈ Z+ such thatif we write T := T n the following holds:

‖DT [k]∣∣p‖ = ‖DT [k]

∣∣(T [k])n−1p

DT [k]∣∣(T [k])n−2p

. . . DT [k]∣∣p‖ =: ρ < 1

for any p ∈ l∞. It follows that ρMRA(T , k) ≤ ρ < 1.

Lemma 5.4.3. Assume that S is a linear scheme such that the k+ 1-st derivedscheme S [k+1] exists. Then, if (5.4) holds between T and S, there exist constantsCn such that

‖Snp− T np‖∞ ≤ CnΩk(p) for all n ∈ Z+.

Proof. First note that there exists a constant C :=(

maxi=1,...,k+1 ‖S [i]‖)k+1

such that

Ωk(Sp) =∑γ∈Γk

k+1∏i=1

‖∆iSp‖γi∞ =∑γ∈Γk

k+1∏i=1

‖S[i]∆ip‖γi∞ ≤ CΩk(p).

Further, since T ∈ C1, there exists a constant D such that

‖T p− T q‖∞ ≤ D‖p− q‖∞.

5.5. APPLICATIONS 99

Thus,

‖T np− Snp‖∞ ≤n−1∑i=0

‖T n−iSip− T n−i−1Si+1p‖∞

≤n−1∑i=0

Dn−i−1‖(T − S)Sip‖∞ ≤n−1∑i=0

Dn−i−1Ωk(Sip)

≤n−1∑i=0

Dn−i−1CiΩk(p) =: CnΩk(p).

Now we are able to prove approximation order only using stability of T :

Theorem 5.4.4. Assume that T ∈ Ol for some l ∈ Z+ satisfies (5.4) witha linear subdivision scheme S reproducing Πk. If ρS(T , l) < 1, then T hasapproximation order k + 1.

Proof. The proof follows from Lemmas 5.4.2, 5.4.3 and from Theorem 5.3.1.

5.5 Applications

We make use of recent stability results [139, 141] and the results of the precedingsection to prove approximation order for some nonlinear schemes. It turns outthat our point of view allows for very short and natural proofs, once stabilityhas been established.

5.5.1 Manifold-Valued Subdivision

We first recover the recent result of [147] using our methods. It is possible toadapt a linear subdivision scheme S so as to operate on nonlinear (manifold-valued) data. The idea is as follows: Since any convergent linear scheme S withmask (ai)i∈Z reproduces constants, we can write

(Sp)Ni+l = mi,l +∑j∈Z

aNi+l−Nj(pj −mi,l), i ∈ Z, l = 0, . . . , N − 1 (5.6)

where mi,l is some sequence that is not too far away from the point pi: Forinstance, mi,l = pi will do for our purposes. We can interpret (5.6) as addingthe vector

∑j∈Z aNi+l−Nj(pj mi,l) to the point mi,l, where p q denotes the

vector pointing from q to p. We write mi,l ⊕∑j∈Z aNi+l−Nj(pj mi,l) and let

⊕ denote the point-vector addition which clearly satisfies

q ⊕ (p q) = p. (5.7)

With this geometric interpretation in mind we can now easily define an analogueT of S which operates in a manifold. All that we need is a notion of point-vectoraddition and difference vector pq of two points p, q such that (5.7) is satisfied.


Example 5.5.1. The standard example is when the manifold is a Lie group(G, ). In this case we let vectors live in the Lie algebra g of G and we definep q := log(q−1 p) and p⊕ v := p exp(v) for all p, q ∈ G and v ∈ g such thatthe expressions make sense. Now we can define the G valued Log-exp analogueof S by

(T p)Ni+l := mi,l exp(∑j∈Z

aNi+l−Nj log(m−1i,l pj)

),

where exp and log denote the exponential function of G and its (locally defined)inverse. For general manifolds an analogous construction exists, but the vectorsnow live in the tangent bundle (which is in general not trivial). The point-vectoraddition and point-point difference can be realized by choosing a Riemannianmetric on the manifold and then using the exponential mapping with respect tothis Riemannian metric (See e.g. [132] for more information on Lie groups andmanifolds).

By restricting ourselves to a chart, which is possible by the local natureof subdivision, we can always assume that T operates on data living in someRm. It is shown in [140] that for any two smooth functions and ⊕ satisfying(5.7) and any linear subdivison scheme S reproducing Πk, k ≥ 0 there exists aconstant C such that

‖Sp− T p‖ ≤ C∑

γ1+2γ2+···+kγk=k+1

‖∆p‖γ1 . . . ‖∆kp‖γk (5.8)

and‖Sp− T p‖ ≤ C‖∆p‖2 (5.9)

where we define the geometric analogue T of S via

(T p)Ni+l := mi,l ⊕∑j∈Z

aNi+l−Nj(pj mi,l).

Equation (5.9) together with the smoothness of the functions ⊕, by theresults of [139] implies that T is stable. Further, we observe that (5.8) is justan instance of (5.4) where the term ‖∆k+1p‖ does not occur. Since for linearschemes reproduction of Πk is equivalent to approximation order k + 1, we canapply Theorem 5.3.2 and obtain

Theorem 5.5.2. The approximation order of the geometric analogue of a con-vergent linear subdivision scheme S equals at least the approximation order ofS.

This result has recently been obtained in [147] using different methods. Weremark that the cited result is slightly stronger than ours, since it only requires(5.9) and (5.8) to hold. We additionally assumed that T is smooth as a map-ping l∞ → l∞. For the applications on manifold-valued data this assumptionis however no restriction. Actually, smoothness of T is required to establishinequalities (5.9), (5.8) in the first place, see [140, 148].

5.5.2 Triadic Median Interpolation

The median interpolating scheme has been introduced in [136] for the use innoise-removal applications where the noise is non-Gaussian. It is defined as

5.5. APPLICATIONS 101

follows: Let

med(f ; I) := supγ : µ(x : f(x) < γ) ≤ 1

2µ(I)

,

be the median of a continuous function f on an interval I where µ denotesthe Lebesque measure. For initial data p ∈ l∞(Z,R) let qi(x) be the uniquequadratic polynomial satisfying

med(qi; [i− l, i− l + 1]) = pi−l, l = −1, 0, 1.

The median interpolating scheme T is defined via

(T p)3i+l = med(qi;[3i+ l

3,

3i+ l + 1

3

]), l = 0, 1, 2.

Clearly, T has dilation factor 3. Moreover it can be shown [143] that if S is thelinear spline rule with dilation factor 3, there exists a constant C such that

‖T p− Sp‖∞ ≤ C‖∆2p‖∞ for all p.

This inequality is of the form (5.4) for k = 1. In [141] it is shown that theMRA associated with T is stable. Further, it is obvious that S reproduces Π1.Applying Theorem 5.3.2 yields

Theorem 5.5.3. The triadic median interpolating scheme has approximationorder 2.

5.5.3 Power-p Scheme

Let 1 ≤ p <∞. Then the power-p subdivision scheme T is defined via

(T p)2i = pi, (T p)2i+1 =pi + pi+1

2− 1

8Hp

((∆2p)i−1, (∆

2p)i),

where

Hp(x, y) :=

x+y2

(1−

∣∣x−yx+y

∣∣p), xy > 0,

0 xy ≤ 0.

Lemma 3.6 in [141] implies that Hp is Lipschitz continuous, and thus we havethat

‖T p− Sp‖ ≤ C‖∆2p‖

for some constant C ≥ 0, S being the linear spline scheme with dilation factor2. Clearly, S reproduces Π1. Further, for 1 ≤ p < 8/3 it is shown in [141] thatthe MRA associated with T is stable. Theorem 5.3.2 implies

Theorem 5.5.4. For 1 ≤ p < 8/3 the power-p scheme has approximation order2.

5.5.4 Dyadic Median Interpolation

The dyadic median interpolating scheme T is defined as

(T p)2i+l = med(qi;[2i+ l

2,

2i+ l + 1

2

]), l = 0, 1,


where the polynomial qi is defined as for the triadic median interpolating scheme.The dyadic median interpolating scheme is an example where T is stable butthe associated MRA is unstable. In [141] it is shown that ρS(T , 1) < 1 butρMRA(T , 1) > 1. As for the triadic median interpolating scheme, T satisfies‖T p−Sp‖ ≤ C‖∆2p‖, where S is the linear spline scheme with dilation factor2 (see [143]). With Theorem 5.4.4 we can conclude that

Theorem 5.5.5. The dyadic median interpolating scheme has approximationorder 2.

5.6 Conclusion

We have shown a number of general results which state that stability propertiesof a nonlinear subdivision scheme imply approximation order. The analysis ofstability is a subject of current research and we expect that new results willappear in the near future. Since most nonlinear schemes which are studiedsatisfy (5.4), our results imply that if a new stability result is proven, thenapproximation order comes for free.

Acknowledgements

The author gratefully acknowledges support from the Austrian Science Fund(FWF) under grant number P19780.

Bibliography

[130] S. Amat, K. Dadourian, and J. Liandrat. Analysis of a class of non linearsubdivision schemes and associated multi-resolution transforms. Arxivpreprint arXiv:0810.1146, 2008.

[131] A.S. Cavaretta, W. Dahmen, and C.A. Micchelli. Stationary Subdivision.American Mathematical Society, 1991.

[132] S.S. Chern, W. Chen, and KS Lam. Lectures on Differential Geometry.World Scientific Publishing Company, 1999.

[133] K. Dadourian. Schemas de subdivision, analyses multiresoltuions non-lineaires. applications., 2008.

[134] I. Daubechies, O. Runborg, and W. Sweldens. Normal MultiresolutionApproximation of Curves. Constructive Approximation, 20(3):399–463,2004.

[135] D. L. Donoho. Interpolating wavelet transforms. Technical report, 1992.

[136] D. L. Donoho and T. P.-Y. Yu. Nonlinear pyramid transforms based onmedian-interpolation. SIAM Journal of Math. Anal., 31(5):1030–1061,2000.

[137] S. Dubuc. Interpolation through an iterative scheme. Journal of mathe-matical analysis and applications, 114(1):185–204, 1986.

[138] G. Faber. Ueber stetige Funktionen. Mathematische Annalen, 66(1):81–94, 1908.

[139] P. Grohs. Stability of manifold-valued subdivision. Manuscript, submittedfor publication, September 2008.

[140] P. Grohs and J. Wallner. Interpolatory wavelets for manifold-valued data.Manuscript, submitted for publication, July 2008.

[141] S. Harizanov and P. Oswald. Stability of nonlinear subdivision and mul-tiscale transforms. Manuscript, submitted for publication, July 2008.

[142] A. Harten, S. Osher, B. Engquist, and S. R. Chakravarthy. Some resultson uniformly high-order accurate essentially nonoscillatory schemes. Appl.Numer. Math., 2(3-5):347–378, 1986.

[143] P. Oswald. Smoothness of Nonlinear Median-Interpolation Subdivision.Advances in Computational Mathematics, 20(4):401–423, 2004.

103

104 BIBLIOGRAPHY

[144] O. Runborg. Introduction to Normal Multiresolution Approximation.In Classification, Clustering, and Data Mining Applications: Proceedingsof the Meeting of the International Federation of Classification Societies(IFCS), Illinois Institute of Technology, Chicago, 15-18 July 2004, page205. Springer, 2004.


[146] J. Wallner and N. Dyn. Convergence and C 1 analysis of subdivisionschemes on manifolds by proximity. Computer Aided Geometric Design,22(7):593–622, 2005.

[147] G. Xie and T. P.-Y. Yu. Approximation order equivalence propertiesof manifold-valued data subdivision schemes. Manuscript, submitted forpublication, April 2008.

[148] G. Xie and T. P.-Y. Yu. Smoothness equivalence properties of generalmanifold-valued data subdivision schemes. Multiscale Modeling and Sim-ulation, 2008.

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