Stability of manifold-valued subdivision schemesand multiscale transformations

Philipp Grohs

October 21, 2008

Abstract

Linear subdivision schemes can be adapted in various ways so as tooperate in nonlinear geometries such as Lie groups or Riemannian man-ifolds. It is well known that along with a linear subdivision scheme amultiscale transformation is defined. Such transformations can also bedefined in a nonlinear setting. We show the stability of such nonlinearmultiscale transforms. To do this we introduce a new kind of proximitycondition which bounds the difference of the differential of a nonlinearsubdivision scheme and a linear one. Surprisingly, it turns out that thisnew proximity condition is almost equivalent to the previously studiedproximity condition introduced in [26].

Keywords: lipschitz stability, interpolatory wavelet, manifold valued data,nonlinear subdivision.

2000 Mathematics Subject Classification. Primary 41AXX, Secondary41A25, 53B, 22E.

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 introduced in[8], or piecewise smooth data where so-called essentially non-oscillating (ENO)schemes have been developed [19]. Another example is the processing of datathat does not live in a vector space but in a general manifold. In this caselinear methods usually cannot even be defined anymore. There has been someeffort to also provide approximation tools for this situation using non linearsubdivision schemes which are intrinsically defined in a manifold [26, 15, 24, 4].

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These schemes are always defined as nonlinear analogue of some linear scheme.It is certainly of interest to know if these constructions share the properties ofthe corresponding linear ones. Among the properties that have been previouslystudied are

• convergence and smoothness of the limit function of manifold-valued sub-divison schemes [26, 25, 16, 14, 30, 29]

• approximation order of manifold-valued subdivision schemes [10, 28], or

• smoothness characterization properties of the multiscale decompositionassociated with a manifold valued subdivision scheme [17].

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 [13], where the present resultsare applied to prove some new facts concerning approximation order.

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1.1 Previous work

There are only few general results concerning the stability of nonlinear subdi-vision schemes and their associated multiscale transforms. A very recent andgeneral work is [18], where necessary and sufficient criteria for the stability ofnonlinear subdivision schemes and multiscale transforms are presented. Theseresults however cannot be applied to the manifold-valued case. Other resultscan be found in [3, 1].

1.2 Outline

After setting up the necessary notation and reviewing some basic facts on subdi-vision in Section 2, in Section 3 we first introduce the new differential proximitycondition and, using this condition we prove the stability of nonlinear subdivi-sion operators. Section 4 focuses on the stability of the multiresolution transformassociated with a subdivision operator. After discussing some issues concerningthe well-definedness of the reconstruction procedure, we show that, assuming adifferential proximity condition, stability of the multiresolution transform holds.In Section 5 we apply the general results of Sections 3 and 4 to some examples ofinterest: the log-exp analogue in a Lie group, the log-exp analogue in a Rieman-nian manifold and the projection analogue. Finally, in Section 6 we demonstratethe necessity of the differential proximity condition by constructing an exampleof an unstable subdivision scheme which is in proximity with a linear stablescheme in the sense of [26].

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.

2.1 Subdivision schemes

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

Definition 2.1. An m-dimensional subdivision operator (m ∈ Z+) is a mappingT : l∞(Z,Rm)→ l∞(Z,Rm) that is local and has dilation factor 2, meaning that

σ2 ◦ T = T ◦ σ, (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.

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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 depends onthe 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 ‖ · ‖∞ to indicatethe usual sup-norm. For a linear operator l∞(Z,Rm) → l∞(Z,Rm) we use thesymbol ‖ · ‖ to indicate the usual operator norm induced by ‖ · ‖∞ on l∞(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 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[16, 14, 29, 30, 26, 25]. Rather, we are interested in the following type of question:

Question 1: Is the operator T ∞ : l∞(Z,Rm) → C(R,Rm) that maps p toits limit function T ∞p a bounded operator, i.e. does there exist a finite con-stant C so that

‖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 [24].

Our proofs are based on comparing the nonlinear scheme T with a lin-ear scheme S (which is bounded) and using perturbation arguments to ensureboundedness of T ∞. First let us recall some well-known properties of linearsubdivision schemes. Comprehensive introductions into the theory of linearsubdivision schemes are provided e.g. by [2, 21, 11]. In the following we alwayswrite S for a linear subdivision scheme and T for a possibly nonlinear one.

Definition 2.3. A subdivision scheme S is called linear, if there exists a se-quence (aj)j∈Z, aj ∈ R with finite support, called the mask of S, such that

(Sp)i =∑j∈Z

ai−2jpj, pj ∈ Rm. (2)

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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. (3)

If a (linear or nonlinear) subdivision scheme S satisfies (3) we say that S repro-duces constants.

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. [2]):

Theorem 2.4. Let S be a linear subdivision scheme. S is convergent if and onlyif there exists a constant C ≥ 0 and a constant 0 ≤ µ < 1, called the contractionfactor 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.

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 := (λjfi)i∈Z := (Pjfi T Pj−1fi) 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 [17]– but with the properties of the reconstruction procedure taking data

(P0, λ1, λ2, . . . ) (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) via (Pjf)i = (T Pj−1f)i⊕ (λj)i. If f is continuous,

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then this procedure obviously recovers f from (P0, λ1, . . . ). It is natural to askwhat effect perturbing the coefficients λj has on the reconstructed function, orin 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.

2.3 Lipschitz classes

We introduce the so-called Lipschitz function spaces : Let G ⊂ RR be a domainwith R ∈ Z+. The function spaces Lip γ, where 0 < γ ∈ R \ Z+ are defined asfollows for functions f : G→ RS and S ∈ Z+:

f ∈ Lip γ ⇔ ‖f‖∞ <∞ and suph∈RR\{0}

‖h−γ+bγc(Dbγcf(x+h)−Dbγcf(x))‖∞ <∞,

where bγc denotes the greatest integer ≤ γ and D is the total differential op-erator. For R = 1 we assume that G is an interval and the definition reducesto

‖f‖∞ <∞ and suph∈R\{0}

‖h−(γ−bγc)(( ∂∂x

)bγcf(x+ h)−

( ∂∂x

)bγcf(x))‖∞ <∞.

It can be shown [5] that this is equivalent to

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

2γj‖∆bγc+1Pjf‖∞ <∞. (5)

For γ ∈ Z+ the definition of Lipschitz spaces can be obtained by interpolationmethods [23]. In the univariate case, (5) remains valid also for integers γ.

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.

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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 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. In fact,if M is open, we can make M smaller to get a convex set. If M is not open,but open with respect to the subset topology induced by a linear subspace V ofRm, we can again make M convex by reducing its size. In this case (which isnot explicitly treated in the following considerations) all the statements madein this paper remain true with only trivial modifications in the proofs.

Our analysis strongly relies on so-called proximity conditions which are sat-isfied between a nonlinear and a linear subdivision scheme. They have first beenused in [4] for the analysis of normal multiresolution of curves. The specific con-ditions that we introduce below have first been used in [26] to prove convergenceand smoothness properties of nonlinear, manifold-valued schemes:

Definition 3.2. S and T satisfy a proximity condition with exponent α > 1 forthe set PM,δ if there exists a constant C ≥ 0 such that for all p ∈ PM,δ

‖Sp− T p‖∞ ≤ C‖∆p‖α∞

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 analoguousproperties of S.

This kind of proximity condition does not seem well-suited for the study ofstability properties. With perturbation arguments similar to [26] it seems veryhard to prove stability of T from stability of S. In fact, we will see in Section 6that, without requiring anything else, the implication proximity ⇒ stability isfalse.

We therefore introduce a more natural condition for proving stability proper-ties (which obviously are related to the differentials of the subdivision operators).

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.

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Definition 3.3. A nonlinear subdivision scheme T and a linear scheme S sat-isfiy a differential proximity condition with exponent α > 0 for the set PM,δ ifthere exists a constant C ≥ 0 such that for all p ∈ PM,δ

‖dT∣∣p− S‖ ≤ C‖∆p‖α∞. (6)

Intuitively, this definition should work better than Definition 3.2, because if‖∆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 3.6 turns this heuristic into a rigorousproof.

Much more can be said: It is remarkable that if we require the subdivisionscheme T : l∞(Z,Rm) → l∞(Z,Rm) to be smooth, then Definition 3.2 andDefinition 3.3 almost coincide. Recall that T ∈ Lip α for 1 < α < 2 at p meansthat there exists a constant C ≥ 0 such that for all q locally around p

‖dT∣∣p− dT

∣∣q‖ ≤ C‖p− q‖α−1

∞ .

Lemma 3.4. 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. (7)

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∞ . (8)

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

∞

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

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

Tk,lp = Tk,lq + dTk,l∣∣γ(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

∞ (9)

with a uniform constant C.Since Sp = q + Ste by (3),

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

By (i), |Tk,lp− Sk,lp| ≤ C ′‖∆p‖α∞ ≤ C ′tα‖∆e‖α∞ with some constant C ′. Sinceby (9) also |Ute| ≤ Ctα‖e‖∞, the triangle inequality implies 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 3.2 and 3.3) inparticular implies that the schemes S and T are in proximity if and only if theyagree up to first order on constant data. This insight might open up room fornew ideas and we plan to study it in more depth in the future.

We continue with our goal of proving stability. The next perturbation resultwill be useful:

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Lemma 3.5. Let Ai, i ∈ Z+ be operators on a normed space with norm ‖·‖ withsupk∈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 operatornorm), then ‖

∏kl=1(Al + Ul)‖ is bounded by a constant only depending on C, µ

and M .

Proof. Define the norm ‖x‖′ := supk∈Z+ ‖∏k

l=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.

As already mentioned above, a proximity condition implies convergence andcontractivity of T . The following theorem has been proven in [15]

Theorem 3.6. Assume that T and S satisfy a proximity condition with exponentα > 1 for data p ∈ PM,δ. If S is a convergent linear scheme, then there existsM ′ ⊆M, δ′ > 0, 0 ≤ µ < 1 such that for all p ∈ PM ′,δ′

‖∆T jp‖∞ ≤ Cµj‖∆p‖∞ (10)

and T jp ∈ PM,δ.

With the help of these results we are now able to prove that T is Lipschitzstable.

Theorem 3.7. Assume that T is a subdivision scheme satisfying a proximitycondition with exponent α > 1 and data in PM,δ with a convergent linear schemeS (such that T ∈ Lip α). Then there exists a constant C ≥ 0 such that for allp, q ∈ PM ′,δ′ and M ′, δ′ from Theorem 3.6:

‖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).

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By Lemma 3.4 (i)⇒ (ii), the schemes S and T satisfy the differential proximitycondition (6). Hence, thanks to Theorem 3.6,

‖Ui‖ ≤ C‖∆T i−1p‖α−1∞ ≤ C ′(µα−1)i‖∆p‖α−1

∞ , i = 1, . . . , n.

Now it remains to apply Lemma 3.5 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 analoguous formula holds for

q. Now let γ(s) := p + s(q − p). The uniform boundedness of ‖dT n‖ shownabove implies that

‖(T np− T nq)in‖ ≤ supt∈(0,1)

‖dT n∣∣γ(t)‖‖p− q‖∞ ≤ C‖p− q‖∞.

We let n→∞ and arrive at

‖T ∞p(t)− T ∞q(t)‖∞ ≤ C‖p− q‖∞

for all t.

4 Stability of the Multiscale Decomposition as-

sociated with a Subdivision operator

4.1 Definition of the multiscale decomposition associatedwith 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 linear schemes S, Faber [12] introduced a multiscale decompo-sition built from the samples of a function f ∈ Lip γ already in 1908. The ideais that one stores not the sampling data Pjf , but instead the “prediction error”

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that one makes in predicting the values of Pjf from the coarser values Pj−1fvia a subdivision scheme. Hence, a function f gets mapped to its initial samplesP0f together with the prediction errors

λj = SPj−1 − Pj. (11)

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 [7, 17]. 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 5. For now we simply remark that thedefinition of λj via (11) has no intrinsic meaning if applied to local coordinaterepresentations of a manifold. We therefore need a little more generality in ourdefinition. Our setup is that we have an m-dimensional image space Rm for thefunctions f and a k-dimensional vector space Rk of wavelet coefficients.

Definition 4.1. The multiscale transformation maps a function f ∈ Lip γ withγ > 0 to a coarse sample P0f and a countable collection of wavelet coefficientsλj ∈ 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 to theirdifference vector. In order to be able to reconstruct our function f from data(P0, λ1, . . . ) we need an inverse operation ⊕ serving as an analogue of the eu-clidean 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 technicalreasons we need the following weak properties to hold for the function 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 respect tothe variable p,

• ψ(p, 0) = p⊕ 0 = p for all p ∈ Rm.

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We now define the reconstruction procedure.

Definition 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−1i , λni ), i ∈ Z.

Remark 4.3. Again, by locality, although Pn is a function acting on an infinitedimensional space, it makes sense to speak of the derivatives d

dλlPn and ddP0Pn.

Remark 4.4. Although we abuse notation by calling the reconstruction functionsPn it will be possible to distinguish them from the sampling operator Pn whichacts on continuous functions.

Remark 4.5. Our results remain valid in the following more general framework:Instead of defining Pnf := (f(i/2j))i∈Z, we can define

Pnf := U ◦ Pnf,

where U is a local smoothing operator and

λji := Pjfi T Pj−1fi.

The purpose of defining such a decomposition is to reduce anti-aliasing effectsand to make the methods better suited for applications in noise-removal. As anexample we mention [24], where U is a nonlinear average imputing operator.For the linear case compare also [9, 22].

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.

4.2 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.

13

Lemma 4.6. Let T be in proximity with a linear convergent scheme S for allinitial data p ∈ PM,δ and assume that S has contraction factor µ0. Assume that0 < µ0 < µ < 1 and let M ′ ⊂ M be any subset of M that stays away from theboundary of M , i.e. there exists ρ > 0 with

M ′ + ρ ·B ⊂M (12)

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

(P0, λ1, λ2, . . . )

satisfyingP0 ∈ PM ′,δ′ and ‖λj‖∞ ≤ Cµj (13)

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+. (14)

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 [2] 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 < 1and find constants C, δ′ > 0 such that

CP

(8CS

1− τ

)αCα−1µ−1 < 1, (15)

δ′ ≤ inf

((Cµ

CP

( 2CS1− τ

− 1))1/α

,4C

1− τ

), (16)

4CCS1− τ

+ CSδ′ < δ, (17)

14

andC

1− µ+ CP

(8CCS1− τ

)α1

1− µα+ CL

8CCS1− τ

1

1− µ< ρ. (18)

The above inequalities assure that the following estimates go through. After ourchoice of C and δ′, we now show inductively that for data satisfying (13), thereconstructed data Pj lies in the class Pδ,M and

‖∆Pj‖∞ ≤4CCS1− τ

µj + CSµj0‖∆P0‖∞. (19)

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 (16). In particular, because of (17), we have ‖∆P1‖∞ ≤ δ. We showthat 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 (16), (18) and the assumption that CS, CL ≥ 1. With (12) we canconclude that P1 takes values in M and hence P1 ∈ PM,δ. We go on to provethe 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)‖∞

+CSµj−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+

15

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 (16) and (15). In particular, because of (17), we get that‖∆Pj‖∞ < δ. Now all that needs to be shown is that Pj still takes values in M :

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 (12) we can conclude that Pj takes values in M . We have now shownthat for data satisfying (13) the reconstruction procedure is well-defined and thereconstructed data satisfies (19). Now all assertions follow from (5).

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.

16

Lemma 4.7. Let f ∈ Lip γ, γ > 0 with wavelet decomposition (P0f, λ1f, . . . )with respect to the subdivision scheme T ∈ Lip α, α > 1, and the operations, ⊕. Assume that T satisfies a proximity condition with a linear, convergentsubdivision scheme S with exponent α for data in PM,δ. Then, if Pnf ∈ 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∣∣γ(θ)

(p− q)l‖ ≤ ‖d2φ‖‖p− q‖

with γ(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

∣∣γ(t)‖‖T Pj−1f − Pjf‖∞

≤ sup0<t<1

‖d2Φ∣∣γ(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 the first termis bounded by a constant times µαj. Now to the second term: Recall that fori ∈ Z, k ∈ {0, 1} the value of Sp2i+k depends only on p restricted to i + Iwhere I is a fixed interval ⊂ Z of length L. By (3) we have for every q ∈ Rm,q = (. . . , q, q, q, . . . ),

‖SPj−1f2i+k − Pjf2i+k‖ = ‖S(Pj−1f − q)2i+k − (Pjf2i+k − q)‖≤ ‖S‖max

j∈i+I‖Pj−1fj − q‖+ ‖Pjf2i+k − q‖

By choosing q appropriately, we obtain

‖SPj−1f − Pjf‖∞ ≤ C‖∆Pjf‖∞ ≤ C ′µj

17

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, (20)

with I the identity mapping. Let us first obtain a bound for ‖Un‖: Since d1Ψ ∈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 Un isbounded by a constant times ‖∆Pn−1‖α−1

∞ since we assumed that a proximitycondition with exponent α > 1 holds. By Lemma 3.4 (i)⇒ (ii), this implies thata differential proximity condition holds and further, ‖Un‖ ≤ C‖∆Pn−1‖α−1

∞ . Bythe 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, (21)

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 (21) 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 3.5 yields a uniform bound for ‖ ddλlPn‖.

We still need to estimate ‖ ddP0Pn‖. Similar to the derivation of (20), (21),

we obtaind

dP0Pn =

n∏j=1

(S +Wj)

18

with matrices ‖Wj‖ ≤ Cµj. Using Lemma 3.5 again, we arrive at boundedness of‖ ddP0Pn‖. We have now proven that the norms ‖ d

dλlPn‖, ‖ ddP0Pn‖ are bounded

by a constant C ≥ 0 independent of l, n and depending only on γ and C.

We state and prove our main stability theorem for Euclidean data.

Theorem 4.8. With the assumptions of Lemma 4.7, let f ∈ Lip γ with Pnf ∈PM ′′,δ for all n ∈ Z+ where M ′′ is a set that stays away from the boundary ofM ′ of Lemma 4.6. Then there exist δ0, ε0 > 0 and a constant C ≥ 0 dependingonly (in a monotonely decreasing way) on ‖∆P0‖ and |f |Lip γ such that for allinitial data

(P0, λ1, λ2, . . . ) (22)

with‖P0 − P0f‖∞ ≤ δ0 and ‖λj − λjf‖∞ ≤ ε0/2

γj, j ∈ Z+ (23)

the reconstruction from (22) yields a well-defined continuous function f with

‖Pn(f − f)‖∞ ≤ C(‖P0(f − f)‖∞ +

n∑j=1

‖λj − λj‖∞).

Proof. Without loss of generality we can assume that 1 > 2−γ > µ0, where µ0

is the contraction factor of S. From the first part of the proof of Lemma 4.7 weknow that, since f ∈ Lip γ, the norms of the wavelet coefficients decay accordingto ‖λj‖∞ ≤ Cµj with C depending only on γ and |f |Lip γ and µ = 2−γ. It is alsoclear by definition that ‖∆Pjf‖∞ ≤ |f |Lip γ2

−γj. Hence there exists j0 such that‖∆Pj0‖∞ < δ′ and ‖λj0+j‖ < C ′/2γj for all j ≥ 0 with δ′, C ′ from Lemma 4.6.Denoting Pj := Pj(P0, λ1, . . . ), we can now pick ε0, δ0 > 0 such that for all datasatisfying (23) we have ‖∆Pj0‖ < δ′, ‖λj0+j‖ < C ′/2γj, and Pj0 assumes valuesin M ′. Any such data, by Lemma 4.6, defines a continuous function g ∈ Lip γwith |g|Lip γ only depending on δ′, C ′. For all data which obey (23), by Lemma4.7, the norms ‖ d

dλj0+kPj0+j‖, ‖ ddPj0Pj0+j‖, k, j ≥ 0 are uniformly bounded by

a constant C > 0. Define

C1 := maxj,k=1,...,j0

sup(P0,λ1,...,λj0 ) with (23)

‖ d

dλkPj‖ <∞.

We now show that the theorem holds with C2 := max(C,C · C1, C1):For data (P0, λ1, . . . ) with (23), 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)‖∞ + · · ·+

19

‖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 ‖ ddλiPj‖, i, j ∈ Z+

are uniformly bounded by C2:Case 1: i, j > j0. For this case, boundedness has been shown with the constantC.Case 2: i, j ≤ j0. In this case the expression is bounded by C1.Case 3: i > j0, j ≤ j0. Here we apply the chain rule to obtain

‖ ddλiPj‖ ≤ ‖ d

dP0Pj‖‖ d

dλjPj0‖ ≤ C · C1.

The proof is complete.

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 4.8 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.

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 [24]. 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).

20

Here exp is the exponential mapping of G that diffeomorphically maps a neigh-bourhood of 0 ∈ g to a neighbourhood of the unit element in G, log is its inverse,and (ai) is the mask of S. The detail coefficients of a continuous G-valued func-tion f are consequently defined via

λji := log((Pjf)−1

i · (T Pj−1)i)

= T Pj−1i Pjfi.

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). Similarily, fora sequence λ = (λi)i∈Z we write ‖λ‖∞ := supi∈Z ‖λi‖. It is easy to see and hasbeen shown e.g. in [15] that locally a proximity condition of order α = 2 issatisfied between S and T . Now we can state our main stability theorem for thelog-exp analogue in Lie groups.

Theorem 5.1. For all G-valued functions f ∈ Lip γ there exist δ0, ε0 > 0 anda constant C ≥ 0 such that for all initial data

(P0, λ1, λ2, . . . )

withd(P0,P0f)∞ ≤ δ0 and ‖λj − λjf‖∞ ≤ ε0/2

γj, j ∈ Z+ (24)

the reconstruction yields a well-defined continuous function f with

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 proof is just a simple application ofTheorem 4.8 after pulling back the subdivision and reconstruction procedure toEuclidean space using a local chart. For each chart we get different constantsδ0, ε0, C, therefore uniform constants can in general only be achieved for datavalues in a compact set.

Remark 5.2. Actually, more can be said regarding the smoothness of the re-constructed function f . The decay rate of wavelet coefficients of f is clearly thesame as the decay rate of f . This implies by the results in [17] 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 in a differ-ent way [29]. This remark also applies to the log-exp analogue in a Riemannianmanifold which is studied below.

21

5.2 The log-exp analogue in Riemannian manifolds

Before we proceed further, we introduce some basic facts of Riemannian geom-etry. For more information on this topic, the interested reader is referred to [6].

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 (25)

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 γ : I →M, where I is some interval ⊆ R, by

l(γ) :=

∫I

‖γ(t)‖dt :=

∫I

〈γ(t), γ(t)〉1/2γ(t)dt.

A vector field V along a curve γ : I →M is a mapping that smoothly assignsto each parameter value t ∈ I a tangent vector V (t) ∈ Tγ(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) ∈ Tγ(t)M withcoordinate representation vj∂j. The coefficients xk represent the coefficients ofthe tangent vector field γ, i.e. γ = xk∂k and ∂k are the basis vector fields inducedby the parametrization (ϕ, dϕ). The quantities Γljk are the Christoffel symbolsof the first kind. Note that we have used the Einstein sum convention.

The vector field V is called parallel along γ, if DdtV ≡ 0. By the linearity of

the equation DVdt

= 0, for curves γ with γ(0) = p and v ∈ TpM, there existsa unique vector field V along γ with V (0) = v and V is parallel along γ. Ifγ(1) = q, then the vector V (1) is called the result of parallel transport of v fromp to q along γ. We denote the linear mapping that maps the vector v to thevector V (1) by Pt1

0(γ) : TpM→ TqM.Obviously, the tangent vector field t 7→ γ(t) is a vector field along γ. The

curve γ is called a geodesic, if γ is parallel along γ. It is well known thatgeodesics locally minimize arc length. Moreover, locally any two points can bejoined by a unique shortest curve which then is a geodesic. Conversely, given apoint p ∈M and a vector v ∈ TpM of sufficiently small norm, then there existsa unique geodesic γ : [−2, 2] → M with γ(0) = p and γ(0) = v. The pointγ(1) is called expp(v). This mapping (p, v) 7→ expp(v) is a smooth map in both

22

variables 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) := inf{l(γ) : γ(0) = p and γ(1) = q}.

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:

λji := logPjfi(T Pj−1

i ) = T Pj−1i Pjfi.

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 λji ∈ TT Pj−1

iM. The question now is how

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 4.8 directly with respect to this chart is that thequantities ‖λi − λi‖ in Theorem 4.8 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 following:

Definition 5.3. Assume that p, q ∈M can be connected by a unique geodesic γwith γ(0) = p and γ(1) = q. The distance d(v, w) between two tangent vectorsv ∈ TpM and w ∈ TqM is defined as ‖Pt10(γ)v − w‖q, where ‖ · ‖q is the norminduced by the Riemannian metric in TqM.

Remark 5.4. The distance d is symmetric with respect to v and w. This followsfrom the fact that the parallel transport is an isometry.

Now we can state our main stability theorem for the Riemannian log-expanalogue:

Theorem 5.5. For all M-valued functions f ∈ Lip γ there exist δ0, ε0 > 0 anda constant C ≥ 0 such that for all initial data

(P0, λ1, λ2, . . . )

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withd(P0,P0f)∞ ≤ δ0 and d

(λj, λjf

)∞ ≤ ε0/2

γj, j ∈ Z+ (26)

the reconstruction yields a well-defined continuous function f with

d(Pnf,Pnf)∞ ≤ C

(d(P0f, P0)∞ +

n∑j=1

d(λjf, λj

)∞

). (27)

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 4.8 toestablish our result. This is because, by pulling back with respect to a chartand using 4.8 we lose the geometric meaning of the distance d. The solution tothis problem is as follows: Instead of assuming that all wavelet coefficients liein different tangent spaces we regard the wavelet coefficients λji as elements ofTT Pj−1fi

M instead of elements of TT Pj−1iM, the symbol Pj as usual denoting

the j-th reconstruction step from the data (P0, λ1, λ2, . . . ). According to thisinterpretation we need to redefine the functions

Pj(P0, λ1, . . . , λj

)i

:= (Pj−1)i ⊕ λji ,

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 λji and λji (which now live in the samevector space) using the Riemannian metric in T Pj−1

i . This difference is clearlythe same as d(λji , λ

ji ) where now λji is interpreted as the usual detail coefficient

living in TT Pj−1iM. We can thus directly apply our results from the previous

section and repeat the proof of Theorem 5.1 to prove Theorem 5.5.

5.3 The projection analogue

We also want to say something about the so-called projection analogue [14]. Itis defined by first applying a linear subdivision scheme S to data living in Rm

and 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.

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In definition 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 3 and replace everyoccurrence of a line γ(t) connecting two points by the unique geodesic connectingthe points. Since that 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

λjfi = Pjfi − T Pj−1fi. (28)

Observe that the values for λji cannot be chosen arbitrarily, but must be chosensuch that T Pj−1fi + λji is in M. The solution for this problem is easy: Wereplace (28) with

λjfi = logT Pj−1i

(Pjfi). (29)

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 5.5.

6 On the sharpness of our results

In the previous sections we have shown that for a nonlinear subdivision schemeT with smooth subdivision rules a proximity condition of order α > 1 witha linear, stable scheme S implies stability of T and the associated multiscaletransform. It is interesting to ask if the smoothness of the functions T is reallynecessary. We cannot fully answer this question, but we are able to show thatat least T ∈ Lip 1 is necessary. In what follows we will construct an exampleof an unstable nonlinear scheme T with T /∈ Lip 1 and such that T satisfies aproximity condition with the midpoint insertion scheme S. The construction isinspired by an idea due to A. Weinmann [27]. Let S be defined as follows:

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

2pj−1 +

1

2pj.

Clearly, this is a convergent linear subdivision scheme which is stable (As amatter of fact, any convergent interpolatory linear scheme S would do). Let usdefine

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

where f denotes any function with f(0) = 0 and f /∈ Lip 1 at 0. Clearly, T islocal and has dilation factor 2. Moreover it satisfies a proximity condition with Sof order 2. Now set p5 = 1 and pj = 0 otherwise. Then T ip0 = 0 for all i ∈ Z+.

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Since f /∈ Lip 1 at 0, there exists a sequence εn → 0 such that |f(εn)|/|εn| → ∞.If we choose initial data qn with qi = pi for i 6= 0 and q0 = εn, then

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

It follows that

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

This shows that T is not stable.

7 Conclusion

We conclude our work with a few remarks. First, as already mentioned in theintroduction, our results remain valid for any dilation factor > 1 with the sameproofs. Also the extension of our results to the multivariate case is straight-forward using the methods developed in [15]. As possible direction for futureresearch we would like to mention the recent work [20], where directionality isencompassed into the (linear) multiscale decomposition. It seems that these con-structions can also be defined for manifold-valued data. It will be an interestingtask to study the properties of these nonlinear decompositions.

Acknowledgments

The author gratefully acknowledges support by the Austrian Science Fund (FWF)under grant number P19780.

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