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Nonlinear multi-scale transforms: Lp theory
Journal of the Franklin Institute 349 (2012) 1619–1636 www.elsevier.com/locate/jfranklin
Nonlinear multi-scale transforms: Lp theory Peter Oswald Jacobs University Bremen, Bremen, Germany Received 27 December 2009; received in revised form 3 April 2011; accepted 4 June 2011 Available online 15 June 2011
Abstract We treat the Lp theory ð1rpo1Þ for univariate nonlinear subdivision schemes and multi-scale transforms based on the concept of offset invariance and nonlinear derived subdivision operators. The paper covers convergence, smoothness, and stability issues, and complements the recent survey [8], where results for Lp have been included without proof. & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction Nonlinear subdivision schemes and multi-scale transforms have recently attracted some attention, as means to battle certain deficiencies of linear methods, and to deal with nonlinearly constrained data sets (e.g., manifold-valued functions). Although most applications require multivariate parameterizations, the theory of multi-scale transforms has so far been developed predominantly in the univariate case, and for the uniform metric. The reason is that, on the one hand, most of the univariate results have straightforward extensions to shift-invariant multivariate grid topologies, a case that thus does not represent a serious mathematical challenge, and, on the other hand, existing nonlinear multi-scale algorithms, e.g., for the extraction of directional features in two- and three-dimensional data sets, require completely new ideas, and a new theoretical framework for their analysis. Concerning the concentration on the uniform metric, it is justified in areas such as geometry processing where local deviations and singularities can easily be spotted in a visualization and thus need to be controlled. Some other applications of multi-scale algorithms, such as modeling of densities with integral balances, image processing using BV type norms, or multi-level schemes for numerically solving elliptic Tel.:þ494212003179; fax:þ494212003103.
E-mail address: [email protected] 0016-0032/$32.00 & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2011.06.006
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problems in Sobolev spaces, necessitate results for the Lp setting ð1rpo1Þ, especially with p ¼ 1 and 2. The present paper contributes to the latter topic in the functional setting, i.e., when the multi-scale transform applies to scalar data, and convergence and smoothness are treated using standard parameterizations of discrete real-valued data sequences by piecewise linear functions on uniform grids. It complements Section 2 of the survey [8], where most of the results below are formulated without proof, and a more detailed discussion of other approaches, the case p ¼ 1, and illustrating examples are given. In contrast to previous research [6,13], where Lp theory is developed using quasi-uniform subdivision operators, we start with the notion of offset invariance for Pk originally introduced for k ¼ 1 in [16], and extended to arbitrary kZ1 in [11]. This notion allows us to define nonlinear derived subdivision operators S ½m satisfying the commutator property Dm S ¼ S½m Dm , m ¼ 1,y,k, and systematically work with them, similar to the linear case. We establish close-to-final convergence and smoothness results in the Besov scale in terms of spectral radii of the S ½m , and also show Lp stability following the recent paper [11] covering the case p ¼ 1. In the concluding section, the results are applied to the family of nonlinear power-q schemes. 2. Offset invariance and derived subdivision operators Throughout the paper, we consider local, r-shift invariant, stationary multi-scale transforms, recursively acting on data sequences from ‘p ðZÞ ð1rpr1Þ according to v j1 ¼ Rv j ,
d j ¼ Dðv j Sv j1 Þ,
v j ¼ Sv j1 þ Pd j , jZ1,
ð1Þ
with bounded but generally nonlinear operators P,D,R,S : ‘p ðZÞ-‘p ðZÞ. For consistency in Eq. (1), the relation ðIdPDÞðIdSRÞ ¼ 0 needs to hold ðId is the identity operator). Abusing a bit conventions, throughout this paper we call a (not necessarily linear) operator T : X -Y between two Banach spaces X and Y bounded if there is a constant C0 such that JTxJY rC0 JxJX for all x 2 X , and Lipschitz continuous if there exists a constant C1 such that JTxTx0 JY rC1 JxyJX for all x,y 2 X . The infimum of all such constants C0 is denoted by JTJX -Y . Since the operators in Eq. (1) are independent of the scale index jZ1, the scheme is stationary, while its r-shift invariance is equivalent to requiring ST 1 ¼ Tr S,
RT r ¼ T1 R,
PT 1 ¼ T1 P,
DT 1 ¼ T1 D,
where Tm, m 2 Z, is the shift operator given by ðTm vÞi ¼ viþm , i 2 Z. Finally, locality of an r-shift invariant transform means that we can write ðSvÞriþs ¼ fs ðviL1 , . . . ,viþL2 Þ,
s ¼ 0, . . . ,r1, i 2 Z,
ð2Þ
with functions fs : R -R, where L ¼ L1 þ L2 þ 1 is the support length of the subdivision part of the transform, and the integers L1 ,L2 are fixed. Similarly, L
ðRvÞi ¼ fR ðvriL3 , . . . ,vriþL4 Þ,
i 2 Z,
ð3Þ
and ðDvÞi ¼ fD ðviL5 , . . . ,viþL6 Þ,
ðPvÞi ¼ fP ðviL7 , . . . ,viþL8 Þ, i 2 Z,
for some functions fR , fD , fP , and fixed integers Ls, s ¼ 3, . . . ,8. It is easy to see that due to locality and r-shift invariance, boundedness (Lipschitz continuity, C1 property, etc.) of S on ‘p ðZÞ spaces is equivalent to the boundedness (Lipschitz continuity, C1 property, etc.) of the coordinate functions fs representing S independently of p, similarly for R, D, and P.
P. Oswald / Journal of the Franklin Institute 349 (2012) 1619–1636
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Moreover, the definition of these local operators automatically extends to the space ‘ðZÞ of all real-valued sequences. We will always silently assume that S0 ¼ R0 ¼ D0 ¼ P0 ¼ 0, where 0 is the zero sequence given by 0i ¼ 0, i 2 Z. Sometimes, especially if nonlinear schemes are considered as perturbations of associated linear schemes, the alternative representation L1 X
ðSvÞriþs ¼
arlþs ðviL1 , . . . ,viþL2 Þvil ,
s ¼ 0, . . . ,r1,
l ¼ L2
or, equivalently, X ðSvÞj ¼ ajri ðvjI½j=r Þvi ,
j2Z
ð4Þ
i2Z
is chosen. To shorten the notation, by vjIi we have denoted the restriction of v to the finite index set Ii :¼ fiL1 , . . . ,i þ L2 g, i 2 Z. Coefficient functions with index s= 2frL1 , . . . , rðL2 þ 1Þ1g vanish for all arguments: as ðÞ 0. Based on Eq. (4) we can now formally write the action of S as an infinite matrix–vector product Sv ¼ Sv v,
ð5Þ
where Sv is a bi-infinite, data-dependent matrix operator with entries identified from Eq. (4): ðSv Þj,i :¼ ajri ðvjI½j=r Þ,
j,i 2 Z:
The assumed boundedness of S follows if we assume boundedness of all coefficient functions as. Note that for a linear S, the matrix operator Sv does not depend on v, and is given by the finitely supported sequence a :¼ fal gl2Z called mask of the subdivision operator. The representation (4)–(5) was introduced in [6], and was the departure point for a systematic theory of data-dependent, so-called quasi-linear, subdivision schemes and multi-scale transforms developed in [6,13]. The transition from Eq. (2) to Eqs. (4)–(5) is (due to the assumed boundedness of S) always possible but not unique. The concept of polynomial reproduction for subdivision operators is fundamental in the study of multi-scale transforms, therefore we start the exposition with it. For nonlinear S, there are two slightly different extensions of the familiar definition for linear subdivision operators. The first extension proposed in [6] uses the representation (4)–(5), and requires that each Sv reproduces polynomials of order k, we will not pursue it here. Alternatively, following [16,11] we can introduce the notion of offset invariance for Pk , where Pk denotes the set of algebraic polynomials of degree ok or, equivalently, of order rk. Let 1 be the constant sequence given by 1i :¼ 1, i 2 Z. Definition 1. A local r-shift invariant subdivision operator S is offset invariant for Pk , kZ1, if for each v 2 ‘ðZÞ, and any polynomial p of degree m, 0rmok, there exists a polynomial q of degree om such that Sðv þ pjZ Þ ¼ Sv þ ðp þ qÞjr1 Z : In particular, S is offset invariant for constants (i.e., the set P1 Þ if Sðv þ a1Þ ¼ Sv þ a1,
8a 2 R, 8v 2 ‘ðZÞ:
Note that the formulation automatically ensures that offset invariance for Pk , k41, implies offset invariance for Pm for all 1rmok. The proof of Theorem 2 given below
1622
P. Oswald / Journal of the Franklin Institute 349 (2012) 1619–1636
shows that Definition 1 is equivalent to a recursive one: S ¼ S½0 is offset invariant for kZ2 if it is offset invariant for Pk1 , and the scaled version of its associated (k1) derived ½k1 operator S~ ¼ rk1 S ½k1 is offset invariant for constants (which in turn guarantees the existence of the kth derived subdivision operator S ½k Þ. For linear S, Definition 1 coincides with the usual definition of polynomial reproduction of order k, where the above property is required for v ¼ 0. As many examples [8] indicate, offset invariance for Pk of a nonlinear scheme usually holds with k ¼ 1 or k ¼ 2, but no nonlinear examples of practical relevance are known for larger k. However, it is the right concept for the extension of the notion of derived subdivision operators to the nonlinear setting. Theorem 2. Let S be a local, r-shift invariant subdivision operator. If S is offset invariant for Pk for some integer kZ1 then there exist local, r-shift invariant derived subdivision operators S½m such that Dm Sv ¼ S½m Dm v,
8v 2 ‘p ðZÞ
ð6Þ
for m ¼ 1,2, . . . ,k. Moreover, if S is written in the form (2) then its derived subdivision operators S½m , m¼ 1,y,k, inherit such a representation with the same (or smaller) L1 ,L2 , and with functions f½m that are obtained from the fs by superpositions involving only linear transformations. In s particular, if S and thus the functions fs ðÞ are bounded (continuous, Lipschitz continuous, C1, etc.) then so are S ½m and the functions f½m s .f In particular, if S is bounded then JS ½m wJ‘p ðZÞ rrmþ1=p JwJ‘p ðZÞ þ CJDwJ‘p ðZÞ ,
ð7Þ
and if S is Lipschitz continuous then JS ½m wS ½m w0 J‘p ðZÞ rrmþ1=p Jww0 J‘p ðZÞ þ CJDðww0 ÞJ‘p ðZÞ ,
ð8Þ
m ¼ 0,1, . . . ,k1, with constants C independent of w,w0 2 ‘p ðZÞ. Proof. The proof extends the standard argument for linear S, see [3,9]. For k ¼ 1 it was first given in [16, Theorem 2.5], see also [11, Lemma 2.12]. The case k41 was suggested in [11, Section 2.1] and can be obtained by induction from k ¼ 1. For convenience we give the complete argument. Let first k ¼ 1. For any fixed i 2 Z, write 8 l1 X > > > ðDvÞiþm , l40, > > > > > 1 X > > > > ðDvÞiþm , lo0: > : m¼l
Now apply the definition of offset invariance for k ¼ 1, and Eq. (2): ðSvÞriþs ¼ ðSðw þ vi 1ÞÞriþs ¼ vi þ fs ðwiL1 , . . . ,wiþL2 Þ,
s ¼ 0, . . . ,r1,
ð9Þ
note that w depends on the arbitrarily fixed i 2 Z. From this, we see that ðDSvÞriþs ¼ fsþ1 ðwiL1 , . . . ,wiþL2 Þfs ðwiL1 , . . . ,wiþL2 Þ ¼: fs½1 ððDvÞiL1 , . . . ,ðDvÞiþL2 1 Þ
P. Oswald / Journal of the Franklin Institute 349 (2012) 1619–1636
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for s ¼ 0, . . . ,r2, and ðDSvÞriþr1 ¼ ðDvÞi þ f0
1 X m ¼ L1
ðDvÞiþ1þm , . . . ,
L 2 1 X
! ðDvÞiþ1þm
m¼0
fr1 ðwiL1 , . . . ,wiþL2 Þ ¼: f½1 r1 ððDvÞiL1 , . . . ,ðDvÞiþL2 Þ, which yields the existence of S½1 (uniqueness is obvious). The argument also shows that boundedness and differentiability properties of the functions fs defining S via Eq. (2) ½1 ½0 automatically carry over to the functions f½1 and Eq. s defining S . Moreover, by S ¼ S (9) the inequalities (7) and (8) automatically hold for m ¼ 0 if S resp. the functions fs are bounded resp. Lipschitz continuous. This establishes Theorem 2 for k ¼ 1. Now, suppose that S is a offset invariant for Pk for some k41 (thus also offset invariant for ½k1 Pk1 Þ, and that the existence of S½k1 is already established. Observe that S~ ¼: rk1 S ½k1 must be offset invariant for constants. Indeed, fix i 2 Z. For given w, find v such that Dk1 viþl ¼ wiþl , l ¼ L1 , . . . ,L2 , where L1 ,L2 are as in the representation (2) of S resp. S½k1 . Also, set pðxÞ ¼ xk1 =ðk1Þ! 2 Pk , and note that Dk1 ðpjZ Þ ¼ rðk1Þ Dk1 ðpjr1 Z Þ ¼ 1. Then ðS~
½k1
ðw þ a1ÞÞriþs ¼ rk1 ðS ½k1 Dk1 ðv þ apjZ ÞÞriþs ¼ rk1 ðDk1 Sðv þ apjZ ÞÞriþs ¼ rk1 ðDk1 ðSv þ ðap þ qÞjr1 Z ÞÞriþs ¼ rk1 ðDk1 SvÞriþs þ ark1 Dk1 ðpjr1 Z Þriþs ½k1 ¼ ðS~ w þ a1Þjriþs
holds for all s ¼ 0, . . . ,k1 (note that Dk1 ðqjr1 Z Þ ¼ 0 since q 2 Pk1 Þ. This proves the offset ½k1 ½k1 invariance of S~ for P1 . Applying the already proved result for m¼ 0 to S~ yields the ½k1 desired S½k :¼ rðk1Þ ðS~ Þ½1 satisfying Eq. (6): ½k1 ½1 ½k1 k1 S½k Dk ¼ rðk1Þ ððS~ Þ DÞDk1 ¼ rk1 DðS~ D Þ ¼ DðS ½k1 Dk1 Þ ¼ Dk S:
By repeatedly using the corresponding result for k¼ 1, properties of S, such as boundedness, Lipschitz continuity, etc., are inherited by S ½k . Similarly, the inequalities (7) and (8) also follow from the case k¼ 1. Indeed, by the above observation for any fixed m ¼ 0, . . . ,k1 the ½m subdivision operator S~ ¼ rm S ½m is offset invariant for constants. Thus, by applying the result for k¼ 1 to it, we get from ½m JS½m wJ‘p ðZÞ ¼ rm JS~ wJ‘p ðZÞ rrm ðr1=p JwJ‘p ðZÞ þ CJDwJ‘p ðZÞ Þ ½m the desired (7) if S (and thus S½m and S~ Þ are bounded while ½m ½m JS½m wS ½m w0 J‘p ðZÞ ¼ rm JS~ wS~ w0 J‘p ðZÞ rrm ðr1=p Jww0 J‘p ðZÞ þ CJDðww0 ÞJ‘p ðZÞ Þ
yields Eq. (8) under the assumption of Lipschitz continuity for S. & 3. Convergence and smoothness In the univariate case, Lp convergence of the reconstruction part v j ¼ Sv j1 þ Pd j ,
jZ1,
ð10Þ
P. Oswald / Journal of the Franklin Institute 349 (2012) 1619–1636
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of a multi-scale transform, resp., the subdivision scheme v j ¼ Sv j1 ,
jZ1,
ð11Þ
associated with it to a limit function f 1 2 Lp ðRÞ, and the Lp smoothness of the latter, is studied by associating with v j its linear spline interpolant f j on the grid G j ¼ rj Z: f j ðxÞ ¼ ði þ 1r j xÞv ji þ ðr j xiÞv jiþ1 , r j x 2 ½i,ði þ 1ÞÞ, i 2 Z: ð12Þ P Alternatively, we can write f j ¼ i v ji B2 ðr j iÞ using linear B-spline series with the hat function B2 ðxÞ ¼ ð1jxjÞþ , or think of f j as the limit of a linear interpolating subdivision process governed by the subdivision operator ðS2 vÞriþs ¼ r1 ððrsÞvi þ sviþ1 Þ,
s ¼ 0, . . . ,r1, i 2 Z:
Definition 3. The multi-scale reconstruction algorithm (10) is called Lp convergent if, for any v0 2 ‘p ðZÞ and detail sequences d j 2 ‘p ðZÞ satisfying X rj=p Jd j J‘p ðZÞ o1, ð13Þ jZ1
the corresponding sequence of linear spline interpolants f j converges in Lp ðRÞ to a limit function f 1 2 Lp ðRÞ. Similarly, the subdivision (11) associated with S is called Lp convergent if f j -f 1 c0 in Lp ðRÞ for any 0av0 2 ‘p ðZÞ.
In applications to multi-scale solvers for operator equations [7,5] and geometric modeling [9] the smoothness characteristics of the limits f 1 matter. Smoothness of functions that are limits of approximation processes (in our case the recursively constructed sequences ff j g of linear splines) is conveniently measured in the scale of Besov spaces Bsp,q ðRÞ, see [13] for various equivalent definitions including the standard one based on moduli of smoothness: f 2 Lp ðRÞ belongs to Bsp,q ðRÞ for some 0osok, and 1rp,qr1 if jf jBsp,q ðRÞ :¼ Jr sj ok ðrj ,f ÞLp ðRÞ J‘q ðZþ Þ o1, where ok ðt,f ÞLp ðRÞ ¼ sup JDkh f JLp ðRÞ ,
t40:
0ohot
The choice of the integer k4s is not critical, the resulting norms are equivalent for any two such integers. We will deal only with the case 1rp ¼ qo1, and give an alternative definition using an approximation-theoretic characterization of Besov spaces that is very convenient for our setup: For small values 0osr1 of the smoothness parameter a function f 2 Lp ðRÞ belongs to Bsp ðRÞ : ¼ Bsp,p ðRÞ if and only if there exists at least one Lp convergent series representation f¼
1 X j¼0
h j,
P. Oswald / Journal of the Franklin Institute 349 (2012) 1619–1636
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where the functions h j are linearPsplines on the grid G j ¼ rj Z (and thus possess a B-spline representation of the form h j ¼ i c ji B2 ðr j iÞÞ satisfying the constraint 1 X rspj Jh j JpLp ðRÞ o1: j¼0
Moreover, we can define an equivalent norm in Bsp ðRÞ by setting !1=p 1 X spj j p r Jh JLp ðRÞ , Jf JBsp ðRÞ :¼ inf
ð14Þ
j¼0
where the infimum is taken with respect to all such representations. For larger values mosrm þ 1, where mZ1 is an integer, we have by induction f 2 Bsp ðRÞ iff f ðm1Þ is absolutely continuous, and f ðmÞ 2 Bsm ðRÞ. In this case, an equivalent norm is given by p Jf JBsp ðRÞ :¼ Jf JLp ðRÞ þ Jf ðmÞ JBsm ðRÞ : p
ð15Þ
Proofs for these statements based on Jackson–Bernstein inequalities for linear splines and further references can be found in, e.g., [4,14,7]. Finally, note that for the important subcase p¼ 2, the scale Bsp ðRÞ, s40, coincides with the scale of Sobolev spaces H s ðRÞ ¼ W2s ðRÞ. Definition 4. The subdivision scheme (11) associated with S possesses Lp smoothness s40 if it is Lp convergent, and limit functions satisfy f 1 2 Bsp ðRÞ for all v0 2 ‘p ðZÞ. The supremum of all such s40 is called Lp smoothness exponent of S, and denoted by sp(S). Theorem 5. Let S be a local, r-shift invariant, bounded subdivision operator. Assume that S is offset invariant for Pk for some integer kZ1, and 1rpo1. (a) If 1=j
rp,k ðSÞ ¼ rp ðS½k Þ :¼ lim supJðS½k Þ j J‘p ðZÞ-‘p ðZÞ or1=p
ð16Þ
j-1
then S is Lp convergent, and sp ðSÞZminðk,logr ðr1=p rp,k ðSÞÞÞ40:
ð17Þ
(b) If, in addition, S is Lipschitz continuous, and P bounded, then (16) also implies the Lp convergence of the multi-scale reconstruction (10). Moreover, if s is non-integer, satisfies 0osominðk,logr ðr1=p rp,k ðSÞÞÞ, and 0
j
Jfv ,d gjZ1 Jp,s;r :¼
Jv0 Jp‘p ðZÞ
þ
1 X
!1=p r
jðsp1Þ
Jd j Jp‘p ðZÞ
o1,
j¼1
then the limit function f 1 belongs to Bsp ðRÞ, and Jf JBsp ðRÞ rCJfv0 ,d j gjZ1 Jp,s;r
ð18Þ
We conjecture that the restriction on non-integer s in part (b) can be removed (see the remark at the end of this section). For a discussion of the properties of the introduced spectral radii, and their connection to similar quantities defined in [6,13] and to the smoothness equivalence conjecture [16,15], we refer to [8]. We just mention some useful
P. Oswald / Journal of the Franklin Institute 349 (2012) 1619–1636
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properties. Firstly, the characterization rp,k ðSÞ ¼ inf JðS ½k Þ j J‘p ðZÞ-‘p ðZÞ ¼ inffr : JDk S j vJ‘p ðZÞ rCr r j JDk vJ‘p ðZÞ g jZ1
ð19Þ
highlights the associated geometric decay property of the quantities JDk S j vJ‘p ðZÞ that is crucially used both in the proofs, and for obtaining numerical upper bounds for rp,k ðSÞ. Secondly, we have rp,m ðSÞrmaxðrmþ1=p ,rp,k ðSÞÞ,
m ¼ 1, . . . ,k1:
ð20Þ
Finally, we note that computing exact values for rp,k ðSÞ (and similar spectral radii) is a hard problem in general, as it is equivalent to evaluating ‘p joint spectral radii of finite families of nonlinear maps acting on RL . Proof. Theorem 5 is established following the same strategy as for proving similar statements for linear S (in the nonlinear setting see [13] and for p ¼ 1 also [1]). We give the argument for part (b), the simple changes for part (a) are discussed at the end. Due to Eq. (20), the assumption on rp,k ðSÞ implies rp,1 ðSÞor1=p . Thus, we can assume k ¼ 1. By definition, P proving Lp convergence of Eq. (10) is the same as proving Lp convergence of the series jZ0 h j composed of linear splines h0 ¼ f 0 , h j ¼ f j f j1 , jZ1, where f j are the linear interpolants to the sequences v j produced by Eq. (10). Recall that the B-spline subdivision operator S2 is interpolating, and produces f j as the limit function if recursively applied to v j. Moreover, the shifts of the B-spline B2 are Lp stable. This implies Jh0 JLp ðRÞ ^Jv0 J‘p ðZÞ , and Jh j JLp ðRÞ rCrj=p JSv j1 þ Pd j S2 v j1 J‘p ðZÞ rCrj=p ðJDvJ‘p ðZÞ þ Jd j J‘p ðZÞ Þ,
ð21Þ
jZ1. The first estimation step in Eq. (21) follows since the nodal values of the linear spline f j1 on the refined grid rj Z are given by the sequence S2 v j1 , the second one from the triangle inequality combined with the boundedness of P and with JSvS2 vJ‘p ðZÞ rCJDvJ‘p ðZÞ ,
v 2 ‘p ðZÞ:
ð22Þ
The latter inequality in turn follows since both S and S2 are offset invariant for constants, use the representation (9). Next we prove geometric decay of the quantities Aj :¼ rj=p Jw j J‘p ðZÞ , where wj :¼ Dv j . Note that by Eq. (10) w j ¼ DðSv j1 þ Pd j Þ ¼ S½1 w j1 þ DPd j ¼ ðS½1 Þ2 w j2 þ S ½1 ðS½1 w j2 þ DPd j1 ÞS½1 ðS ½1 w j2 Þ þ DPd j n1 X ¼ ðS½1 Þn w jn þ ðS ½1 Þl ðS½1 w jl1 þ DPd jl ÞðS½1 Þl ðS ½1 w jl1 Þ þ DPd j : l¼1
Thus, using the boundedness of P and the Lipschitz continuity of S ½1 , JS ½1 w0 S ½1 wJ‘p ðZÞ rC1 Jw0 wJ‘p ðZÞ , we get Jw j J‘p ðZÞ rJðS½1 Þn w jn J‘p ðZÞ þ
n1 X l¼1
ðC1 Þl JDPd jl J‘p ðZÞ þ JDPd j J‘p ðZÞ rJðS ½1 Þn w jn J‘p ðZÞ þ C
n1 X l¼0
Jd jl J‘p ðZÞ ,
P. Oswald / Journal of the Franklin Institute 349 (2012) 1619–1636
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where the constant C depends on n. According to the definition of the spectral radius rp,1 ðSÞ, for any fixed r such that rp,1 ðSÞoror1=p , there is an n ¼ nr such that JðS ½1 Þn wJ‘p ðZÞ rrn JwJ‘p ðZÞ : Substituting into the previous inequality, we get Jw j J‘p ðZÞ rrn Jwjn J‘p ðZÞ þ C
n1 X
rl Jd jl J‘p ðZÞ ,
l¼0
and, after multiplying by r Aj rdn Ajn þ C
n1 X
j=p
and setting d :¼ rr1=p o1,
dl rðjlÞ=p Jd jl J‘p ðZÞ ,
l¼0
with a constant C depending on r. By recursion, if j is given by j ¼ nm þ j 0 , where 0rj 0 on, then ! j nm1 X X nm l ðjlÞ=p ji i=p jl j 0 i Aj rd Aj0 þ C dr Jd J‘p ðZÞ rC d Jv J‘p ðZÞ þ d r Jd J‘p ðZÞ , i¼1
l¼0
ð23Þ where the last step in the estimate follows from the boundedness of S and P: ! j0 X j0 0 j0i i=p j0=p j0 i Aj 0 r2r Jv J‘p ðZÞ rC d Jv J‘p ðZÞ þ d r Jd J‘p ðZÞ , i¼1 0
for j ¼ 0,1, . . . ,n1, with another constant depending on r. Thus, substituting into Eq. (21), we arrive at 1 X
Jh JLp ðRÞ rC Jv J‘p ðZÞ þ j
0
j¼0
1 X
! r
j=p
j
j
Jw J‘p ðZÞ þ Jd J‘p ðZÞ
j¼1
rC Jv J‘p ðZÞ þ 0
1 X
r
j=p
j
Jd J‘p ðZÞ þ
j¼1
rC Jv J‘p ðZÞ þ 0
1 X
1 X
rC Jv J‘p ðZÞ þ
1 X
0
ðd Jv J‘p ðZÞ þ
j¼1
r
j=p
j
Jd J‘p ðZÞ þ
j¼1
0
j
1 X i¼1
j X
! d
ji i=p
r
i
Jd J‘p ðZÞ Þ
i¼1
i i=p
d r
i
Jd J‘p ðZÞ
1 X
! d
j
j¼i
! r
j=p
j
Jd J‘p ðZÞ Þ :
j¼1
P j This proves Lp convergence of the series 1 j ¼ 0 h and consequently of the multi-scale reconstruction (10) under the stated assumption (13). The remaining statements about the Besov smoothness of the limit function and the estimate (18) are proved by induction in k. Let first k ¼ 1, and 0osomin ð1,logr ðr1=p rp,1 ðSÞÞÞ. By our Eq. (14) of the norm in a Besov space, the limit function f belongs to Bsp ðRÞ and Eq. (18) holds if we can show that the above decomposition
P. Oswald / Journal of the Franklin Institute 349 (2012) 1619–1636
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f1 ¼
P1
j¼0
1 X
h j of f 1 into linear splines satisfies
rjsp Jh j JpLp ðRÞ rCJfv0 ,d j gjZ1 Jpp,s,r ,
j¼0
provided that the right-hand side of this inequality is finite. But this follows almost line-byline from the proof of Lp convergence if we replace the quantities Aj by the quantities ^ j :¼ rjðs1=pÞ JDv j J‘ ðZÞ , choose e40 such that A p r1=p 4r1=ps 4r^ :¼ r1=pse 4maxðrp,1 ðSÞ,r1=p1 Þ ^ s1=p ¼ re o1. (that this is possible follows from the assumption on s), and set d^ :¼ rr Then, Eq. (23) translates into ! j X ji iðs1=pÞ j 0 i d^ r Jd J‘p ðZÞ , jZ0: A^ j rC d^ Jv J‘p ðZÞ þ i¼1
Applying the inequality !p j j X X ji ji d^ ai rCr,p d^ ap
ðai Z0Þ,
i
i¼0
i¼1
we have p A^ i rC
d^ j Jv0 Jp‘p ðZÞ
þ
j X
^ ji iðsp1Þ r
d
! Jd i Jp‘p ðZÞ
jZ0:
,
i¼1
Substitution of this bound yields the desired estimate: 1 X
rjsp Jh j JpLp ðRÞ rC
Jv0 Jp‘p ðZÞ
þ
j¼0
1 X
p ðA^ j1
þ
!
rjðsp1Þ Jd j Jp‘p ðZÞ Þ
j¼1
rC Jv0 Jp‘p ðZÞ þ
1 X
rjðsp1Þ Jd j Jp‘p ðZÞ
j¼1
þ
1 X
^j
d
Jv0 Jp‘p ðZÞ
j¼0
rC
Jv0 Jp‘p ðZÞ
þ
j X
^ ji i=p
d
r
!! Jd i Jp‘p ðZÞ
i¼1
þ
1 X
rjðsp1Þ Jd j Jp‘p ðZÞ
j¼1 p 0 j rCJfv ,d gjZ1 Jp,s,r o1:
þ
1 X i¼1
^ i iðsp1Þ
d r
Jd i J‘p ðZÞ
1 X
! d^ j
j¼i
This proves the result for k ¼ 1 for 1rpo1. The induction step for the case k41 is shown for k ¼ 2, it can be recursively repeated without difficulty. We can concentrate on the case rp,2 ðSÞor1þ1=p , and assume 1rsominð2,,logr ðr1=p rp,2 ðSÞÞÞ. Indeed, if rp,2 ðSÞZr1þ1=p then rp,1 ðSÞ ¼ rp,2 ðSÞZ r1þ1=p by Eq. (20), and the already established result for k ¼ 1 applies. Similarly for the case rp,2 ðSÞor1þ1=p and so1, since then rp,1 ðSÞrr1þ1=p orsþ1=p , and again the result ½1 for k ¼ 1 is applicable. Consider S~ :¼ rS½1 . Obviously, this operator satisfies the ½1 conditions of the theorem with k ¼ 1 and s replaced by s~ ¼ s1 since rp,1 ðS~ Þ ¼ rrp,2 ðSÞ
P. Oswald / Journal of the Franklin Institute 349 (2012) 1619–1636
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and thus ½1 0o~s ominð2,logr ðr1=p rp,2 ðSÞÞÞ1 ¼ minð1,logr ðr1=p rp,2 ðS~ ÞÞÞ:
Thus, if we consider the reconstruction algorithm ½1
w j ¼ S~ w j1 þ rDPd j ,
w0 ¼ Dv0 ,
then its limit function g belongs to Bs1 p ðRÞ, and rCJfDv0 ,rd j gjZ1 Jpp,s1,r rCJfv0 ,d j gjZ1 Jpp,s,r : JgJBs1 p ðRÞ ½1 Now, from the definition of S~ and the recursion for wj it is obvious that w j ¼ r j Dv j . Thus, P the piecewise constant functions g j :¼ i wji B1 ðr j iÞ, where B1 is the characteristic function of [0,1), represent the derivatives of the piecewise linear interpolants f j associated with the sequence v j. From this and the already established Lp convergence f j -f , it is not difficult to see that g ¼ f 0 in Lp ðRÞ. The desired result now follows since for s41
Jf JBsp ðRÞ rCðJf JLp ðRÞ þ Jf 0 JBs1 Þ: p ðRÞ The changes for part (a) are minor. The proof can be repeated by setting d j ¼ 0. This implies w j ¼ ðS ½1 Þn wjn without any reference to the Lipschitz stability of S, and convergence and smoothness estimates follow for k ¼ 1. The induction argument for k41 is also the same (note that for the proof of Eq. (17) the consideration of integer s is not necessary due to the monotonicity of the Besov scale with respect to s). This finishes the proof of Theorem 5. &
We end this section with the following remark. The restriction to non-integer s in part (b) of Theorem 5 is due to the method of proof (as a matter of fact, the restriction neither mentioned in [8], nor does it appear in [13] although the proof given there also does not cover integer values of s). The technical hurdle is the crude estimate (22) which suffices for the case 0osok ¼ 1 but not for s ¼ 1 and k ¼ 2. What would be needed is a replacement of JDvJ‘p ðZÞ by JD2 vJ‘p ðZÞ in the right-hand side of the inequality in Eq. (22). Such a replacement is obvious if S is interpolatory, or if the construction of f j by linear B-spline series interpolating v j on rj Z, as done in this paper, is modified by introducing an appropriate shift (this trick works for linear S). We believe that this gap in the statement of Theorem 5 can also be closed in the nonlinear case.
4. Stability Stability of multi-scale transforms, i.e., the robustness with respect to small perturbations, is not a major issue for linear schemes since convergence of a linear subdivision scheme implies stability. However, stability is by no means obvious for nonlinear schemes, and deserves consideration. In this section, we consider only Lipschitz stability in ‘p ðZÞ of the simplified version (1) of a nonlinear multi-scale transform.
P. Oswald / Journal of the Franklin Institute 349 (2012) 1619–1636
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Definition 6. The reconstruction algorithm (10) is called Lp stable if there is a constant CR such that ! J X J=p J J 0 0 j=p j ~j r Jv ~v J‘ ðZÞ rCR Jv ~v J‘ ðZÞ þ r Jd d J‘ ðZÞ p
p
p
j¼1
holds for all v0 , v~ 0 2 ‘p ðZÞ, and JZ1. The subdivision algorithm (11) is called Lp stable if there is a constant CS such that rJ=p JvJ ~v J J‘p ðZÞ rCS Jv0 ~v 0 J‘p ðZÞ holds for all v0 , v~ 0 2 ‘p ðZÞ, and JZ1. For all these definitions it is assumed that the associations vJ 2fv0 ,d 1 , . . . ,d J g, 1
J
v~ J 2f~v 0 , d~ , . . . , d~ g are given by the corresponding recursions in Eq. (1), where in the subdivision case detail sequences are set to 0. For a brief discussion of the stability of the decomposition part of the multi-scale transform (1), see [8]. We stick to the above finite-dimensional versions of Lp stability since in this form they are valuable for realistic algorithms, e.g., for compression based on detail thresholding. We also note that the inclusion of the fore-factors rj=p is dictated by the interpretation of the sequences v j as coarse-scale representations of an Lp limit function. Indeed, assuming Lp convergence, the stability of Eq. (10) implies ! 1 X 1 ~1 0 0 j=p j ~j Jf f JL ðRÞ rCU Jv ~v J‘ ðZÞ þ r Jd d J‘ ðZÞ p
p
p
j¼1
for the Lp limits of the associated sequences ff j gjZ0 and ff~ j gjZ0 . General results on the Lp stability of the multi-scale reconstruction (10) and of the subdivision algorithm (11) have been developed in [6,13] for 1rpr1 (note that these papers consider the limit case J-1, and also the Besov space setting), and more recently for p ¼ 1 in [11,1] (see also [12,2] for earlier stability results). We extend the result from [11] to the range 1rpo1. We state it for k ¼ 1. Theorem 7. Let S be an r-shift invariant, local, offset invariant for P1 , and Lipschitz continuous subdivision operator, and P be Lipschitz continuous. Then the existence of a 0oro1 and some integer nZ1 such that rn=p JDðvn ~v n ÞJ‘p ðZÞ rrJDðv0 ~v 0 ÞJ‘p ðZÞ þ C
n X
l rl=p Jd l d~ J‘p ðZÞ
ð24Þ
l¼1
for any two sets fv0 ,d j g, f~v 0 , d~ j g of multi-scale data implies that the multi-scale reconstruction (10) is Lp stable. If Eq. (24) holds in the special case when v0 , v~ 0 2 ‘p ðZÞ are arbitrary but d j ¼ d~ j ¼ 0, j¼ 1,y,n, then the subdivision scheme (11) is Lp stable.
P. Oswald / Journal of the Franklin Institute 349 (2012) 1619–1636
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Proof. From the assumptions and Eq. (8) with m ¼ 0 we have J
rrJ=p ðJSvJ1 Sv~ J1 J‘p ðZÞ þ JPd J Pd~ J‘p ðZÞ
rJ=p JvJ ~v J J‘p ðZÞ
J rrJ1=p JvJ1 ~v J1 J‘p ðZÞ þ CrJ=p ðJDðvJ1 ~v J1 ÞJ‘p ðZÞ þ Jd J d~ J‘p ðZÞ Þ J J 1 X X rJv0 ~v 0 J‘p ðZÞ þ C rj=p Jd j d~ j J‘p ðZÞ þ C rl=p JDðvl ~v l ÞJ‘p ðZÞ : j¼1
j¼0
To see that the last sum is majorized by the remaining terms in this upper bound, we now explore Eq. (24). The reasoning is analogous to our estimates for the Lp convergence proof for Theorem 5. Write j ¼ sn þ j 0 with j 0 ¼ 0, . . . ,r1, and denote r~ :¼ r1=n . Then rj=p JDðv j ~v j ÞJ‘p ðZÞ
j X
rrrðjnÞ=p ðJDðvjn ~v jn ÞJ‘p ðZÞ þ C
l rl=p Jd l d~ J‘p ðZÞ
l ¼ jnþ1 j X
rr~ n rðjnÞ=p ðJDðvjn ~v jn ÞJ‘p ðZÞ þ C
l
r~ jl rl=p Jd l d~ J‘p ðZÞ
l ¼ jnþ1
rr~ sn JDðvj0 ~v j0 ÞJ‘p ðZÞ þ C
j X
l r~ jl rl=p Jd l d~ J‘p ðZÞ
l ¼ j 0 þ1
rC r~ Jv ~v J‘p ðZÞ þ C j
0
0
j X
r~
jl l=p
r
l
!
Jd d~ J‘p ðZÞ : l
l¼1
Here, the last step comes from recursively estimating finitely many expressions JDðvj0 ~v j0 ÞJ‘p ðZÞ , j 0 ¼ 0, . . . ,n1, using the Lipschitz continuity of S and P, clearly, the constant C depends on the fixed values of ro1 and n. With this at hand, it remains to substitute and change order of summation: J 1 X
rl=p JDðvl ~v l ÞJ‘p ðZÞ
j¼0
rC
J 1 X
r~ j Jv0 ~v 0 J‘p ðZÞ þ C
j¼0 0
J 1 X
j
r~ þ
j¼0
rC Jv ~v J‘p ðZÞ þ 0
l
!
r~ jl rl=p Jd l d~ J‘p ðZÞ
l¼1
rC Jv ~v J‘p ðZÞ 0
j X
0
J 1 X
J 1 X
l l=p
r~ r
l¼1
r
l=p
l
l Jd d~ J‘p ðZÞ l
!
J 1 X
! r~
j
j¼l
Jd d~ J‘p ðZÞ : l
l¼1
This shows the desired estimate, and the proof of the Lp stability of Eq. (10) is complete. The Lp stability of Eq. (11) follows under the stated conditions if one repeats the above argument with d j ¼ d~ j ¼ 0. & The statement of this theorem carries over to k41 if an estimate of the form JSvSwJ‘p ðZÞ rr1=p JvwJ‘p ðZÞ þ CJDk ðvwÞJ‘p ðZÞ
ð25Þ
can be established, and Eq. (24) holds with D replaced by Dk , see [11]. Moreover, in [11] the condition (24) has been replaced by a spectral radius estimate on the derivatives of derived
P. Oswald / Journal of the Franklin Institute 349 (2012) 1619–1636
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subdivision operators. For the sake of simplicity, we assume that S and thus all S½m , mrk, are C1, i.e., all functions fs in Eq. (2) of S possess uniformly bounded, continuous partial derivatives, and refer to [11] for the exact conditions of piecewise continuous differentiability under which the statement can be proved. Denote by Dv S½k the Frechet derivative of S ½k at v 2 ‘p ðZÞ. Due to our simplifying assumptions, the linear operator family Dv S ½k : ‘p ðZÞ-‘p ðZÞ depends continuously on v. Now define the spectral radii stab ½k ½k ~ stab ~ stab rstab p,k ðSÞ ¼ r p ðS Þ as follows: p,k ðSÞ ¼ rp ðS Þ and r 1=j
½k ½k ½k ½k rstab p ðS Þ :¼ lim sup sup JDS ðS ½k Þj1 w DS ðS ½k Þj2 w . . . DS w J‘p ðZÞ-‘p ðZÞ j-1
ð26Þ
w2‘p ðZÞ
and ½k r~ stab p ðS Þ :¼ lim sup j-1
sup w0 ,w1 ,...,wj1 2‘p ðZÞ
1=j
JDS ½k DS ½k . . . DS ½k J : wj1 wj2 w0 ‘p ðZÞ-‘p ðZÞ
ð27Þ
Note that in Eq. (27) the supremum is taken with respect to an arbitrary collection of wl, l ¼ 0, . . . ,j1, while in Eq. (26) it is taken with respect to a single w. Set wl ¼ ðS½k Þl w, l ¼ 0, . . . ,j1, to see that ~ stab rstab p,k ðSÞrr p,k ðSÞ: As demonstrated in [11] for the dyadic median interpolating scheme, this inequality can be strict. On the other hand, in [10, Lemma 4.2] it was observed that lim r~ stab ððS ½k Þn Þ1=n n-1 p
¼ rstab p,k ðSÞ,
a property that is useful for establishing approximation results, see [10,8]. The proof of the following theorem, and its generalization to certain classes of piecewise-differentiable S, is given in [11, Section 2.3] in the case p ¼ 1, it straightforwardly carries over to 1rpo1. & Theorem 8. Let S be an r-shift invariant, local, C1 continuous subdivision operator, and P be bounded and Lipschitz continuous. In addition, assume that S is offset invariant for Pk , and 1=p that Eq. (25) holds. The multi-scale reconstruction (10) is Lp stable if r~ stab , and the p,k ðSÞor stab 1=p subdivision algorithm (11) is Lp stable if rp,k ðSÞor . We note that under mild conditions Lp stability fails to hold when the corresponding inequalities are reversed. The C1 assumption on S can be replaced by the weaker kdifferentiability, as done in [11] for p ¼ 1. This extension is necessary to deal with the example considered in the next section. 5. Example For future reference, we want to summarize the currently known results on Lp convergence, Besov smoothness, and stability for the family of nonlinear interpolating power-q subdivision operators. They represent a simple test case for the application of general theories. Despite the simplicity of power-q schemes, there are still some open questions about their Lp smoothness and stability if q42.
P. Oswald / Journal of the Franklin Institute 349 (2012) 1619–1636
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The power-q subdivision operator is defined by ðSvÞ2i ¼ vi ,
ðSvÞ2iþ1 ¼
vi þ viþ1 1 Hq ðD2 vi1 ,D2 vi Þ, i 2 Z, 8 2
where the so-called limiter Hq is defined by 8 q > < x þ y 1 xy , xy40, x þ y 2 Hq ðx,yÞ ¼ > : 0, xyr0:
ð28Þ
ð29Þ
The parameter q 2 ½1, þ 1Þ is fixed. References and basic facts can be found in [8]. The most studied case is q ¼ 2. Obviously, S realizes a nonlinear, data-dependent interpolation between the subdivision rule S2 which results if ðD2 vi1 ÞðD2 vi Þr0 and the standard Deslauriers-Dubuc 4-point scheme the definition of which coincides with the case D2 vi1 ¼ D2 vi in Eq. (28). The S of the power-q family are Lipschitz continuous for all q 2 ½1,1Þ, are 2-shift invariant, and offset invariant for P2 . Thus, both S ½1 and S ½2 are well-defined, and our theorems can be used with either k ¼ 1 or 2: ðS ½1 wÞ2i ¼
wi 1 Hq ðDwi1 ,Dwi Þ, 2 8
ðS ½1 wÞ2iþ1 ¼ ðS ½2 wÞ2i ¼
wi 1 þ Hq ðDwi1 ,Dwi Þ, 2 8
1 Hq ðwi1 ,wi Þ, 4
ðS ½2 wÞ2iþ1 ¼
i 2 Z,
wi 1 ðHq ðwi1 ,wi Þ þ Hq ðwi ,wiþ1 ÞÞ: 2 8
ð30Þ
By working with S½1 , the Lp convergence of the power-q multi-scale reconstruction algorithm with P ¼ Id (and thus also the Lp convergence of the associated subdivision scheme) can easily be established. By Theorem 5 it is sufficient to show that rp,1 ðSÞo21=p which evidently follows from the stronger statement JS½1 J‘p ðZÞ-‘p ðZÞ o21=p ,
1rpo1:
Indeed, since 1 1 8 jHq ðDwi1 ,Dwi Þjr16 jDwi1
1 þ Dwi jr16 ðjwi1 j þ jwiþ1 jÞ,
we have 1 JS½1 wJp‘p ðZÞ r2J12 jwi j þ 16 ðjwi1 j þ jwiþ1 jÞJp‘p ðZÞ r2ð58JwJ‘p ðZÞ Þp ,
independently of q 2 ½1,1Þ. It was conjectured in [8] that a much sharper result holds, namely, that for all q 2 ½1,1Þ rp,2 ðSÞ ¼
1 1 ¼) sp ðSÞ ¼ 1 þ , 2 p
1rpr1:
ð31Þ
The upper bound sp ðSÞr1 þ 1=p is obvious since the limit function S1 d for the d-sequence is the linear B-spline B2, independently of q, and o2 ðt,B2 ÞLp ðRÞ ^t1þ1=p . The lower bound in
P. Oswald / Journal of the Franklin Institute 349 (2012) 1619–1636
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(31) follows from the stronger inequality JS ½2 J‘p ðZÞ-‘p ðZÞ ¼ 12,
1rpo1,
ð32Þ
for the range 1rqr2 (for p ¼ 1, this equality holds for all q, see [8]). Indeed, for this range of values the q-power scheme is convexity preserving: If a finite segment of v is convex, i.e., for w ¼ D2 v we have wj1 ,wj 0 þ1 r0owi , i ¼ j, . . . ,j 0 , for some jrj 0 , then Sv remains convex there, more precisely ðS ½2 wÞ2j ¼ ðS ½2 wÞ2j0 þ2 ¼ 0,
ðS ½2 wÞi 40, i ¼ 2j þ 1, . . . ,2j 0 þ 1:
Similarly for concave segments, with all inequality signs reversed. This follows from the elementary inequality 0oHq ðx,yÞ ¼ Hq ðx,yÞoqminðx,yÞ, x,y40. Since any finitely supported v decomposes into finitely many convex/concave segments, in order to prove Eq. (32) it is enough to prove the corresponding ‘p inequality for a finite convex segment, i.e., to show that j X wi 0
i¼j
2
xi1 xi
p
0
þ
j 1 X
j p X wi 0
ð2xi Þp r
i¼j
i¼j
2
,
where the notation 0oxi :¼ 18 Hq ðwi ,wiþ1 Þo14minðwi ,wiþ1 Þ,
i ¼ j, . . . ,j 0 1,
is used ðxj1 ¼ xj 0 ¼ 0Þ. But this follows by induction in the length j 0 j of the underlying index segment ½j,j 0 . The case of a length j 0 j ¼ 0 segment is trivial. Suppose j 0 j40. Since xj o2xj ,
1 1 2 wj xj o2 wj ,
0oxj ,
1 1 2 wjþ1 xjþ1 xj o2wjþ1 xjþ1 ,
the convexity of fðtÞ ¼ jtjp implies ð12 wj xj Þp þ ð2xj Þp rð12wj Þp þ xpj and xpj þ ð12 wjþ1 xjþ1 xj Þp rð12wjþ1 xjþ1 Þp : Substitution yields ð12 wj xj Þp þ ð2xj Þp þ ð12 wjþ1 xjþ1 xj Þp rð12 wj Þp þ ð12wjþ1 xjþ1 Þp : This reduces the proof of the above ‘p estimate for the index segment ½j,j 0 to the case of the shorter index segment ½j þ 1,j 0 , and the induction is complete. For q42, it is easy to see that Eq. (32) is not necessarily true anymore (e.g., consider p¼ 1, the finitely supported sequence v given by v0 ¼ ð1 þ eÞ, v1 ¼ v1 ¼ 1, vi ¼ 0, jijZ2, and choose e40 small enough. Thus, Eq. (31) is formally established for 1rqr2, and still open for 2oqo1. Concerning stability, since we know that S is 2-differentiable (see [11, Section 3.2]), and satisfies Eq. (25), Theorem 8 is applicable with k ¼ 2. By crude estimates, we can show that JDv S½2 J‘p ðZÞ-‘p ðZÞ o21=p ,
1rpo1
ð33Þ
holds, independently of v, for a certain range q 2 ½1,qp Þ, where 2oqp o4. This obviously 1=p implies r~ stab . For p ¼ 1, the latter inequality was established for q 2 ½1,8=3Þ in p,2 ðSÞo2 [11], there it was also shown that r~ stab 1,2 ðSÞ41 if q44.
P. Oswald / Journal of the Franklin Institute 349 (2012) 1619–1636
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To prove Eq. (33), we assume q 2 ½1,4, and set a :¼ q=4 2 ½1=4,1. From [11, Section 3.2] we quote the following properties of the matrix representation of Dv S ½2 : jðDv S ½2 uÞ2i jr2ai jui1 j þ 2a0i jui j, jðDv S ½2 uÞ2iþ1 jrai jui1 j þ ð1=2a0i aiþ1 Þjui j þ a0iþ1 juiþ1 j, where the data-dependent coefficients ai ,a0i satisfy 0oai oai þ a0i oa=2. For even indices, we get X X jðDv S ½2 uÞ2i jp rap1 2ai jui1 jp þ 2a0i jui jp r2ap JuJp‘p ðZÞ , i2Z
i2Z
and for odd indices X X jðDv S ½2 uÞ2iþ1 jp rð12 þ aÞp1 ai jui1 jp þ ð12a0i aiþ1 Þjui jp þ a0iþ1 juiþ1 jp i2Z
i2Z
r12 ð12 þ aÞp1 JuJp‘p ðZÞ : Together this gives JDv S ½2 uJp‘p ðZÞ rð2ap þ 12 ð12 þ aÞp1 ÞJuJp‘p ðZÞ , and Eq. (33) follows as long as 2ap þ 12 ð12 þ aÞp1 o2. If ar1=2 it is trivially satisfied for all p. Since the left-hand side as a function of a 2 ð0,1 is monotone and continuous for each fixed p, and exceeds 2 if a ¼ 1, there is an 1=2oap o1 such that Eq. (33) holds iff 0oaoap whichpffiffiffiffiffi is equivalent to q 2 ½1,qp Þ with some 2oqp o4. In particular, q1 ¼ 3, q2 ¼ ð 571Þ=2 3:275, and qp -2 if p-1. To summarize, for the subfamily of convexity-preserving power-q schemes ð1rqr2Þ Lp convergence, smoothness exponents, and stability are fully established while for q42 only partial results are known. References [1] S. Amat, K. Dadourian, J. Liandrat, Analysis of a class of subdivision schemes and associated non-linear multiresolution transforms, Adv. Comput. Math. 34 (2011) 253–277. [2] S. Amat, J. Liandrat, On the stability of the PPH nonlinear multiresolution, Appl. Comput. Harmon. Anal. 18 (2005) 198–206. [3] A.S. Cavaretta, W. Dahmen, C.A. Micchelli, Stationary Subdivision, Memoirs American Mathemetical Society, vol. 93, AMS, Providence, 1991. [4] Z. Ciesielski, Constructive function theory and spline systems, Studia Math. 58 (1975) 277–302. [5] A. Cohen, Numerical Analysis of Wavelet Methods, Elsevier, 2003. [6] A. Cohen, N. Dyn, B. Matei, Quasilinear subdivision schemes with application to ENO interpolation, Appl. Comput. Harmon. Anal. 15 (2003) 89–116. [7] W. Dahmen, Wavelet and multiscale methods for operator equations, Acta Numer. 6 (1997) 55–228. [8] N. Dyn, P. Oswald, Univariate subdivision and multiscale transforms: the nonlinear case, in: R.A. DeVore, A. Kunoth (Eds.), Multiscale, Nonlinear, and Adaptive Approximation, Springer, Berlin, 2009, pp. 203–247. [9] N. Dyn, D. Levin, Subdivision schemes in geometric modelling, Acta Numer. 11 (2002) 73–144. [10] P. Grohs, Approximation order from stability of nonlinear subdivision schemes, J. Approx. Theory 162 (2010) 1085–1094. [11] S. Harizanov, P. Oswald, Stability of nonlinear subdivision and multiscale transforms, Constr. Approx. 31 (2009) 359–393. [12] F. Kuijt, R. van Damme, Stability of subdivision schemes, TW Memorandum 1469, Faculity of Applied Mathematics, University of Twente, 1998.
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[13] B. Matei, Smoothness characterization and stability in nonlinear multiscale framework: theoretical results, Asymptotic Anal. 41 (2005) 277–309. [14] P. Oswald, Multilevel Finite Element Approximation: Theory & Applications, Teubner, Leipzig, 1994. [15] G. Xie, T.P.-Y. Yu, On a linearization principle for nonlinear p-mean subdivision schemes, in: M. Neamtu, E.B. Saff (Eds.), Advances in Constructive Approximation, Nashboro Press, 2004, pp. 519–533. [16] G. Xie, T.P.-Y. Yu, Smoothness analysis of nonlinear subdivision schemes of homogeneous and affine invariant type, Constr. Approx. 22 (2005) 219–254.
Nonlinear multi-scale transforms: Lp theory Peter Oswald Jacobs University Bremen, Bremen, Germany Received 27 December 2009; received in revised form 3 April 2011; accepted 4 June 2011 Available online 15 June 2011
Abstract We treat the Lp theory ð1rpo1Þ for univariate nonlinear subdivision schemes and multi-scale transforms based on the concept of offset invariance and nonlinear derived subdivision operators. The paper covers convergence, smoothness, and stability issues, and complements the recent survey [8], where results for Lp have been included without proof. & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction Nonlinear subdivision schemes and multi-scale transforms have recently attracted some attention, as means to battle certain deficiencies of linear methods, and to deal with nonlinearly constrained data sets (e.g., manifold-valued functions). Although most applications require multivariate parameterizations, the theory of multi-scale transforms has so far been developed predominantly in the univariate case, and for the uniform metric. The reason is that, on the one hand, most of the univariate results have straightforward extensions to shift-invariant multivariate grid topologies, a case that thus does not represent a serious mathematical challenge, and, on the other hand, existing nonlinear multi-scale algorithms, e.g., for the extraction of directional features in two- and three-dimensional data sets, require completely new ideas, and a new theoretical framework for their analysis. Concerning the concentration on the uniform metric, it is justified in areas such as geometry processing where local deviations and singularities can easily be spotted in a visualization and thus need to be controlled. Some other applications of multi-scale algorithms, such as modeling of densities with integral balances, image processing using BV type norms, or multi-level schemes for numerically solving elliptic Tel.:þ494212003179; fax:þ494212003103.
E-mail address: [email protected] 0016-0032/$32.00 & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2011.06.006
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problems in Sobolev spaces, necessitate results for the Lp setting ð1rpo1Þ, especially with p ¼ 1 and 2. The present paper contributes to the latter topic in the functional setting, i.e., when the multi-scale transform applies to scalar data, and convergence and smoothness are treated using standard parameterizations of discrete real-valued data sequences by piecewise linear functions on uniform grids. It complements Section 2 of the survey [8], where most of the results below are formulated without proof, and a more detailed discussion of other approaches, the case p ¼ 1, and illustrating examples are given. In contrast to previous research [6,13], where Lp theory is developed using quasi-uniform subdivision operators, we start with the notion of offset invariance for Pk originally introduced for k ¼ 1 in [16], and extended to arbitrary kZ1 in [11]. This notion allows us to define nonlinear derived subdivision operators S ½m satisfying the commutator property Dm S ¼ S½m Dm , m ¼ 1,y,k, and systematically work with them, similar to the linear case. We establish close-to-final convergence and smoothness results in the Besov scale in terms of spectral radii of the S ½m , and also show Lp stability following the recent paper [11] covering the case p ¼ 1. In the concluding section, the results are applied to the family of nonlinear power-q schemes. 2. Offset invariance and derived subdivision operators Throughout the paper, we consider local, r-shift invariant, stationary multi-scale transforms, recursively acting on data sequences from ‘p ðZÞ ð1rpr1Þ according to v j1 ¼ Rv j ,
d j ¼ Dðv j Sv j1 Þ,
v j ¼ Sv j1 þ Pd j , jZ1,
ð1Þ
with bounded but generally nonlinear operators P,D,R,S : ‘p ðZÞ-‘p ðZÞ. For consistency in Eq. (1), the relation ðIdPDÞðIdSRÞ ¼ 0 needs to hold ðId is the identity operator). Abusing a bit conventions, throughout this paper we call a (not necessarily linear) operator T : X -Y between two Banach spaces X and Y bounded if there is a constant C0 such that JTxJY rC0 JxJX for all x 2 X , and Lipschitz continuous if there exists a constant C1 such that JTxTx0 JY rC1 JxyJX for all x,y 2 X . The infimum of all such constants C0 is denoted by JTJX -Y . Since the operators in Eq. (1) are independent of the scale index jZ1, the scheme is stationary, while its r-shift invariance is equivalent to requiring ST 1 ¼ Tr S,
RT r ¼ T1 R,
PT 1 ¼ T1 P,
DT 1 ¼ T1 D,
where Tm, m 2 Z, is the shift operator given by ðTm vÞi ¼ viþm , i 2 Z. Finally, locality of an r-shift invariant transform means that we can write ðSvÞriþs ¼ fs ðviL1 , . . . ,viþL2 Þ,
s ¼ 0, . . . ,r1, i 2 Z,
ð2Þ
with functions fs : R -R, where L ¼ L1 þ L2 þ 1 is the support length of the subdivision part of the transform, and the integers L1 ,L2 are fixed. Similarly, L
ðRvÞi ¼ fR ðvriL3 , . . . ,vriþL4 Þ,
i 2 Z,
ð3Þ
and ðDvÞi ¼ fD ðviL5 , . . . ,viþL6 Þ,
ðPvÞi ¼ fP ðviL7 , . . . ,viþL8 Þ, i 2 Z,
for some functions fR , fD , fP , and fixed integers Ls, s ¼ 3, . . . ,8. It is easy to see that due to locality and r-shift invariance, boundedness (Lipschitz continuity, C1 property, etc.) of S on ‘p ðZÞ spaces is equivalent to the boundedness (Lipschitz continuity, C1 property, etc.) of the coordinate functions fs representing S independently of p, similarly for R, D, and P.
P. Oswald / Journal of the Franklin Institute 349 (2012) 1619–1636
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Moreover, the definition of these local operators automatically extends to the space ‘ðZÞ of all real-valued sequences. We will always silently assume that S0 ¼ R0 ¼ D0 ¼ P0 ¼ 0, where 0 is the zero sequence given by 0i ¼ 0, i 2 Z. Sometimes, especially if nonlinear schemes are considered as perturbations of associated linear schemes, the alternative representation L1 X
ðSvÞriþs ¼
arlþs ðviL1 , . . . ,viþL2 Þvil ,
s ¼ 0, . . . ,r1,
l ¼ L2
or, equivalently, X ðSvÞj ¼ ajri ðvjI½j=r Þvi ,
j2Z
ð4Þ
i2Z
is chosen. To shorten the notation, by vjIi we have denoted the restriction of v to the finite index set Ii :¼ fiL1 , . . . ,i þ L2 g, i 2 Z. Coefficient functions with index s= 2frL1 , . . . , rðL2 þ 1Þ1g vanish for all arguments: as ðÞ 0. Based on Eq. (4) we can now formally write the action of S as an infinite matrix–vector product Sv ¼ Sv v,
ð5Þ
where Sv is a bi-infinite, data-dependent matrix operator with entries identified from Eq. (4): ðSv Þj,i :¼ ajri ðvjI½j=r Þ,
j,i 2 Z:
The assumed boundedness of S follows if we assume boundedness of all coefficient functions as. Note that for a linear S, the matrix operator Sv does not depend on v, and is given by the finitely supported sequence a :¼ fal gl2Z called mask of the subdivision operator. The representation (4)–(5) was introduced in [6], and was the departure point for a systematic theory of data-dependent, so-called quasi-linear, subdivision schemes and multi-scale transforms developed in [6,13]. The transition from Eq. (2) to Eqs. (4)–(5) is (due to the assumed boundedness of S) always possible but not unique. The concept of polynomial reproduction for subdivision operators is fundamental in the study of multi-scale transforms, therefore we start the exposition with it. For nonlinear S, there are two slightly different extensions of the familiar definition for linear subdivision operators. The first extension proposed in [6] uses the representation (4)–(5), and requires that each Sv reproduces polynomials of order k, we will not pursue it here. Alternatively, following [16,11] we can introduce the notion of offset invariance for Pk , where Pk denotes the set of algebraic polynomials of degree ok or, equivalently, of order rk. Let 1 be the constant sequence given by 1i :¼ 1, i 2 Z. Definition 1. A local r-shift invariant subdivision operator S is offset invariant for Pk , kZ1, if for each v 2 ‘ðZÞ, and any polynomial p of degree m, 0rmok, there exists a polynomial q of degree om such that Sðv þ pjZ Þ ¼ Sv þ ðp þ qÞjr1 Z : In particular, S is offset invariant for constants (i.e., the set P1 Þ if Sðv þ a1Þ ¼ Sv þ a1,
8a 2 R, 8v 2 ‘ðZÞ:
Note that the formulation automatically ensures that offset invariance for Pk , k41, implies offset invariance for Pm for all 1rmok. The proof of Theorem 2 given below
1622
P. Oswald / Journal of the Franklin Institute 349 (2012) 1619–1636
shows that Definition 1 is equivalent to a recursive one: S ¼ S½0 is offset invariant for kZ2 if it is offset invariant for Pk1 , and the scaled version of its associated (k1) derived ½k1 operator S~ ¼ rk1 S ½k1 is offset invariant for constants (which in turn guarantees the existence of the kth derived subdivision operator S ½k Þ. For linear S, Definition 1 coincides with the usual definition of polynomial reproduction of order k, where the above property is required for v ¼ 0. As many examples [8] indicate, offset invariance for Pk of a nonlinear scheme usually holds with k ¼ 1 or k ¼ 2, but no nonlinear examples of practical relevance are known for larger k. However, it is the right concept for the extension of the notion of derived subdivision operators to the nonlinear setting. Theorem 2. Let S be a local, r-shift invariant subdivision operator. If S is offset invariant for Pk for some integer kZ1 then there exist local, r-shift invariant derived subdivision operators S½m such that Dm Sv ¼ S½m Dm v,
8v 2 ‘p ðZÞ
ð6Þ
for m ¼ 1,2, . . . ,k. Moreover, if S is written in the form (2) then its derived subdivision operators S½m , m¼ 1,y,k, inherit such a representation with the same (or smaller) L1 ,L2 , and with functions f½m that are obtained from the fs by superpositions involving only linear transformations. In s particular, if S and thus the functions fs ðÞ are bounded (continuous, Lipschitz continuous, C1, etc.) then so are S ½m and the functions f½m s .f In particular, if S is bounded then JS ½m wJ‘p ðZÞ rrmþ1=p JwJ‘p ðZÞ þ CJDwJ‘p ðZÞ ,
ð7Þ
and if S is Lipschitz continuous then JS ½m wS ½m w0 J‘p ðZÞ rrmþ1=p Jww0 J‘p ðZÞ þ CJDðww0 ÞJ‘p ðZÞ ,
ð8Þ
m ¼ 0,1, . . . ,k1, with constants C independent of w,w0 2 ‘p ðZÞ. Proof. The proof extends the standard argument for linear S, see [3,9]. For k ¼ 1 it was first given in [16, Theorem 2.5], see also [11, Lemma 2.12]. The case k41 was suggested in [11, Section 2.1] and can be obtained by induction from k ¼ 1. For convenience we give the complete argument. Let first k ¼ 1. For any fixed i 2 Z, write 8 l1 X > > > ðDvÞiþm , l40, > > > >
Now apply the definition of offset invariance for k ¼ 1, and Eq. (2): ðSvÞriþs ¼ ðSðw þ vi 1ÞÞriþs ¼ vi þ fs ðwiL1 , . . . ,wiþL2 Þ,
s ¼ 0, . . . ,r1,
ð9Þ
note that w depends on the arbitrarily fixed i 2 Z. From this, we see that ðDSvÞriþs ¼ fsþ1 ðwiL1 , . . . ,wiþL2 Þfs ðwiL1 , . . . ,wiþL2 Þ ¼: fs½1 ððDvÞiL1 , . . . ,ðDvÞiþL2 1 Þ
P. Oswald / Journal of the Franklin Institute 349 (2012) 1619–1636
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for s ¼ 0, . . . ,r2, and ðDSvÞriþr1 ¼ ðDvÞi þ f0
1 X m ¼ L1
ðDvÞiþ1þm , . . . ,
L 2 1 X
! ðDvÞiþ1þm
m¼0
fr1 ðwiL1 , . . . ,wiþL2 Þ ¼: f½1 r1 ððDvÞiL1 , . . . ,ðDvÞiþL2 Þ, which yields the existence of S½1 (uniqueness is obvious). The argument also shows that boundedness and differentiability properties of the functions fs defining S via Eq. (2) ½1 ½0 automatically carry over to the functions f½1 and Eq. s defining S . Moreover, by S ¼ S (9) the inequalities (7) and (8) automatically hold for m ¼ 0 if S resp. the functions fs are bounded resp. Lipschitz continuous. This establishes Theorem 2 for k ¼ 1. Now, suppose that S is a offset invariant for Pk for some k41 (thus also offset invariant for ½k1 Pk1 Þ, and that the existence of S½k1 is already established. Observe that S~ ¼: rk1 S ½k1 must be offset invariant for constants. Indeed, fix i 2 Z. For given w, find v such that Dk1 viþl ¼ wiþl , l ¼ L1 , . . . ,L2 , where L1 ,L2 are as in the representation (2) of S resp. S½k1 . Also, set pðxÞ ¼ xk1 =ðk1Þ! 2 Pk , and note that Dk1 ðpjZ Þ ¼ rðk1Þ Dk1 ðpjr1 Z Þ ¼ 1. Then ðS~
½k1
ðw þ a1ÞÞriþs ¼ rk1 ðS ½k1 Dk1 ðv þ apjZ ÞÞriþs ¼ rk1 ðDk1 Sðv þ apjZ ÞÞriþs ¼ rk1 ðDk1 ðSv þ ðap þ qÞjr1 Z ÞÞriþs ¼ rk1 ðDk1 SvÞriþs þ ark1 Dk1 ðpjr1 Z Þriþs ½k1 ¼ ðS~ w þ a1Þjriþs
holds for all s ¼ 0, . . . ,k1 (note that Dk1 ðqjr1 Z Þ ¼ 0 since q 2 Pk1 Þ. This proves the offset ½k1 ½k1 invariance of S~ for P1 . Applying the already proved result for m¼ 0 to S~ yields the ½k1 desired S½k :¼ rðk1Þ ðS~ Þ½1 satisfying Eq. (6): ½k1 ½1 ½k1 k1 S½k Dk ¼ rðk1Þ ððS~ Þ DÞDk1 ¼ rk1 DðS~ D Þ ¼ DðS ½k1 Dk1 Þ ¼ Dk S:
By repeatedly using the corresponding result for k¼ 1, properties of S, such as boundedness, Lipschitz continuity, etc., are inherited by S ½k . Similarly, the inequalities (7) and (8) also follow from the case k¼ 1. Indeed, by the above observation for any fixed m ¼ 0, . . . ,k1 the ½m subdivision operator S~ ¼ rm S ½m is offset invariant for constants. Thus, by applying the result for k¼ 1 to it, we get from ½m JS½m wJ‘p ðZÞ ¼ rm JS~ wJ‘p ðZÞ rrm ðr1=p JwJ‘p ðZÞ þ CJDwJ‘p ðZÞ Þ ½m the desired (7) if S (and thus S½m and S~ Þ are bounded while ½m ½m JS½m wS ½m w0 J‘p ðZÞ ¼ rm JS~ wS~ w0 J‘p ðZÞ rrm ðr1=p Jww0 J‘p ðZÞ þ CJDðww0 ÞJ‘p ðZÞ Þ
yields Eq. (8) under the assumption of Lipschitz continuity for S. & 3. Convergence and smoothness In the univariate case, Lp convergence of the reconstruction part v j ¼ Sv j1 þ Pd j ,
jZ1,
ð10Þ
P. Oswald / Journal of the Franklin Institute 349 (2012) 1619–1636
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of a multi-scale transform, resp., the subdivision scheme v j ¼ Sv j1 ,
jZ1,
ð11Þ
associated with it to a limit function f 1 2 Lp ðRÞ, and the Lp smoothness of the latter, is studied by associating with v j its linear spline interpolant f j on the grid G j ¼ rj Z: f j ðxÞ ¼ ði þ 1r j xÞv ji þ ðr j xiÞv jiþ1 , r j x 2 ½i,ði þ 1ÞÞ, i 2 Z: ð12Þ P Alternatively, we can write f j ¼ i v ji B2 ðr j iÞ using linear B-spline series with the hat function B2 ðxÞ ¼ ð1jxjÞþ , or think of f j as the limit of a linear interpolating subdivision process governed by the subdivision operator ðS2 vÞriþs ¼ r1 ððrsÞvi þ sviþ1 Þ,
s ¼ 0, . . . ,r1, i 2 Z:
Definition 3. The multi-scale reconstruction algorithm (10) is called Lp convergent if, for any v0 2 ‘p ðZÞ and detail sequences d j 2 ‘p ðZÞ satisfying X rj=p Jd j J‘p ðZÞ o1, ð13Þ jZ1
the corresponding sequence of linear spline interpolants f j converges in Lp ðRÞ to a limit function f 1 2 Lp ðRÞ. Similarly, the subdivision (11) associated with S is called Lp convergent if f j -f 1 c0 in Lp ðRÞ for any 0av0 2 ‘p ðZÞ.
In applications to multi-scale solvers for operator equations [7,5] and geometric modeling [9] the smoothness characteristics of the limits f 1 matter. Smoothness of functions that are limits of approximation processes (in our case the recursively constructed sequences ff j g of linear splines) is conveniently measured in the scale of Besov spaces Bsp,q ðRÞ, see [13] for various equivalent definitions including the standard one based on moduli of smoothness: f 2 Lp ðRÞ belongs to Bsp,q ðRÞ for some 0osok, and 1rp,qr1 if jf jBsp,q ðRÞ :¼ Jr sj ok ðrj ,f ÞLp ðRÞ J‘q ðZþ Þ o1, where ok ðt,f ÞLp ðRÞ ¼ sup JDkh f JLp ðRÞ ,
t40:
0ohot
The choice of the integer k4s is not critical, the resulting norms are equivalent for any two such integers. We will deal only with the case 1rp ¼ qo1, and give an alternative definition using an approximation-theoretic characterization of Besov spaces that is very convenient for our setup: For small values 0osr1 of the smoothness parameter a function f 2 Lp ðRÞ belongs to Bsp ðRÞ : ¼ Bsp,p ðRÞ if and only if there exists at least one Lp convergent series representation f¼
1 X j¼0
h j,
P. Oswald / Journal of the Franklin Institute 349 (2012) 1619–1636
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where the functions h j are linearPsplines on the grid G j ¼ rj Z (and thus possess a B-spline representation of the form h j ¼ i c ji B2 ðr j iÞÞ satisfying the constraint 1 X rspj Jh j JpLp ðRÞ o1: j¼0
Moreover, we can define an equivalent norm in Bsp ðRÞ by setting !1=p 1 X spj j p r Jh JLp ðRÞ , Jf JBsp ðRÞ :¼ inf
ð14Þ
j¼0
where the infimum is taken with respect to all such representations. For larger values mosrm þ 1, where mZ1 is an integer, we have by induction f 2 Bsp ðRÞ iff f ðm1Þ is absolutely continuous, and f ðmÞ 2 Bsm ðRÞ. In this case, an equivalent norm is given by p Jf JBsp ðRÞ :¼ Jf JLp ðRÞ þ Jf ðmÞ JBsm ðRÞ : p
ð15Þ
Proofs for these statements based on Jackson–Bernstein inequalities for linear splines and further references can be found in, e.g., [4,14,7]. Finally, note that for the important subcase p¼ 2, the scale Bsp ðRÞ, s40, coincides with the scale of Sobolev spaces H s ðRÞ ¼ W2s ðRÞ. Definition 4. The subdivision scheme (11) associated with S possesses Lp smoothness s40 if it is Lp convergent, and limit functions satisfy f 1 2 Bsp ðRÞ for all v0 2 ‘p ðZÞ. The supremum of all such s40 is called Lp smoothness exponent of S, and denoted by sp(S). Theorem 5. Let S be a local, r-shift invariant, bounded subdivision operator. Assume that S is offset invariant for Pk for some integer kZ1, and 1rpo1. (a) If 1=j
rp,k ðSÞ ¼ rp ðS½k Þ :¼ lim supJðS½k Þ j J‘p ðZÞ-‘p ðZÞ or1=p
ð16Þ
j-1
then S is Lp convergent, and sp ðSÞZminðk,logr ðr1=p rp,k ðSÞÞÞ40:
ð17Þ
(b) If, in addition, S is Lipschitz continuous, and P bounded, then (16) also implies the Lp convergence of the multi-scale reconstruction (10). Moreover, if s is non-integer, satisfies 0osominðk,logr ðr1=p rp,k ðSÞÞÞ, and 0
j
Jfv ,d gjZ1 Jp,s;r :¼
Jv0 Jp‘p ðZÞ
þ
1 X
!1=p r
jðsp1Þ
Jd j Jp‘p ðZÞ
o1,
j¼1
then the limit function f 1 belongs to Bsp ðRÞ, and Jf JBsp ðRÞ rCJfv0 ,d j gjZ1 Jp,s;r
ð18Þ
We conjecture that the restriction on non-integer s in part (b) can be removed (see the remark at the end of this section). For a discussion of the properties of the introduced spectral radii, and their connection to similar quantities defined in [6,13] and to the smoothness equivalence conjecture [16,15], we refer to [8]. We just mention some useful
P. Oswald / Journal of the Franklin Institute 349 (2012) 1619–1636
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properties. Firstly, the characterization rp,k ðSÞ ¼ inf JðS ½k Þ j J‘p ðZÞ-‘p ðZÞ ¼ inffr : JDk S j vJ‘p ðZÞ rCr r j JDk vJ‘p ðZÞ g jZ1
ð19Þ
highlights the associated geometric decay property of the quantities JDk S j vJ‘p ðZÞ that is crucially used both in the proofs, and for obtaining numerical upper bounds for rp,k ðSÞ. Secondly, we have rp,m ðSÞrmaxðrmþ1=p ,rp,k ðSÞÞ,
m ¼ 1, . . . ,k1:
ð20Þ
Finally, we note that computing exact values for rp,k ðSÞ (and similar spectral radii) is a hard problem in general, as it is equivalent to evaluating ‘p joint spectral radii of finite families of nonlinear maps acting on RL . Proof. Theorem 5 is established following the same strategy as for proving similar statements for linear S (in the nonlinear setting see [13] and for p ¼ 1 also [1]). We give the argument for part (b), the simple changes for part (a) are discussed at the end. Due to Eq. (20), the assumption on rp,k ðSÞ implies rp,1 ðSÞor1=p . Thus, we can assume k ¼ 1. By definition, P proving Lp convergence of Eq. (10) is the same as proving Lp convergence of the series jZ0 h j composed of linear splines h0 ¼ f 0 , h j ¼ f j f j1 , jZ1, where f j are the linear interpolants to the sequences v j produced by Eq. (10). Recall that the B-spline subdivision operator S2 is interpolating, and produces f j as the limit function if recursively applied to v j. Moreover, the shifts of the B-spline B2 are Lp stable. This implies Jh0 JLp ðRÞ ^Jv0 J‘p ðZÞ , and Jh j JLp ðRÞ rCrj=p JSv j1 þ Pd j S2 v j1 J‘p ðZÞ rCrj=p ðJDvJ‘p ðZÞ þ Jd j J‘p ðZÞ Þ,
ð21Þ
jZ1. The first estimation step in Eq. (21) follows since the nodal values of the linear spline f j1 on the refined grid rj Z are given by the sequence S2 v j1 , the second one from the triangle inequality combined with the boundedness of P and with JSvS2 vJ‘p ðZÞ rCJDvJ‘p ðZÞ ,
v 2 ‘p ðZÞ:
ð22Þ
The latter inequality in turn follows since both S and S2 are offset invariant for constants, use the representation (9). Next we prove geometric decay of the quantities Aj :¼ rj=p Jw j J‘p ðZÞ , where wj :¼ Dv j . Note that by Eq. (10) w j ¼ DðSv j1 þ Pd j Þ ¼ S½1 w j1 þ DPd j ¼ ðS½1 Þ2 w j2 þ S ½1 ðS½1 w j2 þ DPd j1 ÞS½1 ðS ½1 w j2 Þ þ DPd j n1 X ¼ ðS½1 Þn w jn þ ðS ½1 Þl ðS½1 w jl1 þ DPd jl ÞðS½1 Þl ðS ½1 w jl1 Þ þ DPd j : l¼1
Thus, using the boundedness of P and the Lipschitz continuity of S ½1 , JS ½1 w0 S ½1 wJ‘p ðZÞ rC1 Jw0 wJ‘p ðZÞ , we get Jw j J‘p ðZÞ rJðS½1 Þn w jn J‘p ðZÞ þ
n1 X l¼1
ðC1 Þl JDPd jl J‘p ðZÞ þ JDPd j J‘p ðZÞ rJðS ½1 Þn w jn J‘p ðZÞ þ C
n1 X l¼0
Jd jl J‘p ðZÞ ,
P. Oswald / Journal of the Franklin Institute 349 (2012) 1619–1636
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where the constant C depends on n. According to the definition of the spectral radius rp,1 ðSÞ, for any fixed r such that rp,1 ðSÞoror1=p , there is an n ¼ nr such that JðS ½1 Þn wJ‘p ðZÞ rrn JwJ‘p ðZÞ : Substituting into the previous inequality, we get Jw j J‘p ðZÞ rrn Jwjn J‘p ðZÞ þ C
n1 X
rl Jd jl J‘p ðZÞ ,
l¼0
and, after multiplying by r Aj rdn Ajn þ C
n1 X
j=p
and setting d :¼ rr1=p o1,
dl rðjlÞ=p Jd jl J‘p ðZÞ ,
l¼0
with a constant C depending on r. By recursion, if j is given by j ¼ nm þ j 0 , where 0rj 0 on, then ! j nm1 X X nm l ðjlÞ=p ji i=p jl j 0 i Aj rd Aj0 þ C dr Jd J‘p ðZÞ rC d Jv J‘p ðZÞ þ d r Jd J‘p ðZÞ , i¼1
l¼0
ð23Þ where the last step in the estimate follows from the boundedness of S and P: ! j0 X j0 0 j0i i=p j0=p j0 i Aj 0 r2r Jv J‘p ðZÞ rC d Jv J‘p ðZÞ þ d r Jd J‘p ðZÞ , i¼1 0
for j ¼ 0,1, . . . ,n1, with another constant depending on r. Thus, substituting into Eq. (21), we arrive at 1 X
Jh JLp ðRÞ rC Jv J‘p ðZÞ þ j
0
j¼0
1 X
! r
j=p
j
j
Jw J‘p ðZÞ þ Jd J‘p ðZÞ
j¼1
rC Jv J‘p ðZÞ þ 0
1 X
r
j=p
j
Jd J‘p ðZÞ þ
j¼1
rC Jv J‘p ðZÞ þ 0
1 X
1 X
rC Jv J‘p ðZÞ þ
1 X
0
ðd Jv J‘p ðZÞ þ
j¼1
r
j=p
j
Jd J‘p ðZÞ þ
j¼1
0
j
1 X i¼1
j X
! d
ji i=p
r
i
Jd J‘p ðZÞ Þ
i¼1
i i=p
d r
i
Jd J‘p ðZÞ
1 X
! d
j
j¼i
! r
j=p
j
Jd J‘p ðZÞ Þ :
j¼1
P j This proves Lp convergence of the series 1 j ¼ 0 h and consequently of the multi-scale reconstruction (10) under the stated assumption (13). The remaining statements about the Besov smoothness of the limit function and the estimate (18) are proved by induction in k. Let first k ¼ 1, and 0osomin ð1,logr ðr1=p rp,1 ðSÞÞÞ. By our Eq. (14) of the norm in a Besov space, the limit function f belongs to Bsp ðRÞ and Eq. (18) holds if we can show that the above decomposition
P. Oswald / Journal of the Franklin Institute 349 (2012) 1619–1636
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f1 ¼
P1
j¼0
1 X
h j of f 1 into linear splines satisfies
rjsp Jh j JpLp ðRÞ rCJfv0 ,d j gjZ1 Jpp,s,r ,
j¼0
provided that the right-hand side of this inequality is finite. But this follows almost line-byline from the proof of Lp convergence if we replace the quantities Aj by the quantities ^ j :¼ rjðs1=pÞ JDv j J‘ ðZÞ , choose e40 such that A p r1=p 4r1=ps 4r^ :¼ r1=pse 4maxðrp,1 ðSÞ,r1=p1 Þ ^ s1=p ¼ re o1. (that this is possible follows from the assumption on s), and set d^ :¼ rr Then, Eq. (23) translates into ! j X ji iðs1=pÞ j 0 i d^ r Jd J‘p ðZÞ , jZ0: A^ j rC d^ Jv J‘p ðZÞ þ i¼1
Applying the inequality !p j j X X ji ji d^ ai rCr,p d^ ap
ðai Z0Þ,
i
i¼0
i¼1
we have p A^ i rC
d^ j Jv0 Jp‘p ðZÞ
þ
j X
^ ji iðsp1Þ r
d
! Jd i Jp‘p ðZÞ
jZ0:
,
i¼1
Substitution of this bound yields the desired estimate: 1 X
rjsp Jh j JpLp ðRÞ rC
Jv0 Jp‘p ðZÞ
þ
j¼0
1 X
p ðA^ j1
þ
!
rjðsp1Þ Jd j Jp‘p ðZÞ Þ
j¼1
rC Jv0 Jp‘p ðZÞ þ
1 X
rjðsp1Þ Jd j Jp‘p ðZÞ
j¼1
þ
1 X
^j
d
Jv0 Jp‘p ðZÞ
j¼0
rC
Jv0 Jp‘p ðZÞ
þ
j X
^ ji i=p
d
r
!! Jd i Jp‘p ðZÞ
i¼1
þ
1 X
rjðsp1Þ Jd j Jp‘p ðZÞ
j¼1 p 0 j rCJfv ,d gjZ1 Jp,s,r o1:
þ
1 X i¼1
^ i iðsp1Þ
d r
Jd i J‘p ðZÞ
1 X
! d^ j
j¼i
This proves the result for k ¼ 1 for 1rpo1. The induction step for the case k41 is shown for k ¼ 2, it can be recursively repeated without difficulty. We can concentrate on the case rp,2 ðSÞor1þ1=p , and assume 1rsominð2,,logr ðr1=p rp,2 ðSÞÞÞ. Indeed, if rp,2 ðSÞZr1þ1=p then rp,1 ðSÞ ¼ rp,2 ðSÞZ r1þ1=p by Eq. (20), and the already established result for k ¼ 1 applies. Similarly for the case rp,2 ðSÞor1þ1=p and so1, since then rp,1 ðSÞrr1þ1=p orsþ1=p , and again the result ½1 for k ¼ 1 is applicable. Consider S~ :¼ rS½1 . Obviously, this operator satisfies the ½1 conditions of the theorem with k ¼ 1 and s replaced by s~ ¼ s1 since rp,1 ðS~ Þ ¼ rrp,2 ðSÞ
P. Oswald / Journal of the Franklin Institute 349 (2012) 1619–1636
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and thus ½1 0o~s ominð2,logr ðr1=p rp,2 ðSÞÞÞ1 ¼ minð1,logr ðr1=p rp,2 ðS~ ÞÞÞ:
Thus, if we consider the reconstruction algorithm ½1
w j ¼ S~ w j1 þ rDPd j ,
w0 ¼ Dv0 ,
then its limit function g belongs to Bs1 p ðRÞ, and rCJfDv0 ,rd j gjZ1 Jpp,s1,r rCJfv0 ,d j gjZ1 Jpp,s,r : JgJBs1 p ðRÞ ½1 Now, from the definition of S~ and the recursion for wj it is obvious that w j ¼ r j Dv j . Thus, P the piecewise constant functions g j :¼ i wji B1 ðr j iÞ, where B1 is the characteristic function of [0,1), represent the derivatives of the piecewise linear interpolants f j associated with the sequence v j. From this and the already established Lp convergence f j -f , it is not difficult to see that g ¼ f 0 in Lp ðRÞ. The desired result now follows since for s41
Jf JBsp ðRÞ rCðJf JLp ðRÞ þ Jf 0 JBs1 Þ: p ðRÞ The changes for part (a) are minor. The proof can be repeated by setting d j ¼ 0. This implies w j ¼ ðS ½1 Þn wjn without any reference to the Lipschitz stability of S, and convergence and smoothness estimates follow for k ¼ 1. The induction argument for k41 is also the same (note that for the proof of Eq. (17) the consideration of integer s is not necessary due to the monotonicity of the Besov scale with respect to s). This finishes the proof of Theorem 5. &
We end this section with the following remark. The restriction to non-integer s in part (b) of Theorem 5 is due to the method of proof (as a matter of fact, the restriction neither mentioned in [8], nor does it appear in [13] although the proof given there also does not cover integer values of s). The technical hurdle is the crude estimate (22) which suffices for the case 0osok ¼ 1 but not for s ¼ 1 and k ¼ 2. What would be needed is a replacement of JDvJ‘p ðZÞ by JD2 vJ‘p ðZÞ in the right-hand side of the inequality in Eq. (22). Such a replacement is obvious if S is interpolatory, or if the construction of f j by linear B-spline series interpolating v j on rj Z, as done in this paper, is modified by introducing an appropriate shift (this trick works for linear S). We believe that this gap in the statement of Theorem 5 can also be closed in the nonlinear case.
4. Stability Stability of multi-scale transforms, i.e., the robustness with respect to small perturbations, is not a major issue for linear schemes since convergence of a linear subdivision scheme implies stability. However, stability is by no means obvious for nonlinear schemes, and deserves consideration. In this section, we consider only Lipschitz stability in ‘p ðZÞ of the simplified version (1) of a nonlinear multi-scale transform.
P. Oswald / Journal of the Franklin Institute 349 (2012) 1619–1636
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Definition 6. The reconstruction algorithm (10) is called Lp stable if there is a constant CR such that ! J X J=p J J 0 0 j=p j ~j r Jv ~v J‘ ðZÞ rCR Jv ~v J‘ ðZÞ þ r Jd d J‘ ðZÞ p
p
p
j¼1
holds for all v0 , v~ 0 2 ‘p ðZÞ, and JZ1. The subdivision algorithm (11) is called Lp stable if there is a constant CS such that rJ=p JvJ ~v J J‘p ðZÞ rCS Jv0 ~v 0 J‘p ðZÞ holds for all v0 , v~ 0 2 ‘p ðZÞ, and JZ1. For all these definitions it is assumed that the associations vJ 2fv0 ,d 1 , . . . ,d J g, 1
J
v~ J 2f~v 0 , d~ , . . . , d~ g are given by the corresponding recursions in Eq. (1), where in the subdivision case detail sequences are set to 0. For a brief discussion of the stability of the decomposition part of the multi-scale transform (1), see [8]. We stick to the above finite-dimensional versions of Lp stability since in this form they are valuable for realistic algorithms, e.g., for compression based on detail thresholding. We also note that the inclusion of the fore-factors rj=p is dictated by the interpretation of the sequences v j as coarse-scale representations of an Lp limit function. Indeed, assuming Lp convergence, the stability of Eq. (10) implies ! 1 X 1 ~1 0 0 j=p j ~j Jf f JL ðRÞ rCU Jv ~v J‘ ðZÞ þ r Jd d J‘ ðZÞ p
p
p
j¼1
for the Lp limits of the associated sequences ff j gjZ0 and ff~ j gjZ0 . General results on the Lp stability of the multi-scale reconstruction (10) and of the subdivision algorithm (11) have been developed in [6,13] for 1rpr1 (note that these papers consider the limit case J-1, and also the Besov space setting), and more recently for p ¼ 1 in [11,1] (see also [12,2] for earlier stability results). We extend the result from [11] to the range 1rpo1. We state it for k ¼ 1. Theorem 7. Let S be an r-shift invariant, local, offset invariant for P1 , and Lipschitz continuous subdivision operator, and P be Lipschitz continuous. Then the existence of a 0oro1 and some integer nZ1 such that rn=p JDðvn ~v n ÞJ‘p ðZÞ rrJDðv0 ~v 0 ÞJ‘p ðZÞ þ C
n X
l rl=p Jd l d~ J‘p ðZÞ
ð24Þ
l¼1
for any two sets fv0 ,d j g, f~v 0 , d~ j g of multi-scale data implies that the multi-scale reconstruction (10) is Lp stable. If Eq. (24) holds in the special case when v0 , v~ 0 2 ‘p ðZÞ are arbitrary but d j ¼ d~ j ¼ 0, j¼ 1,y,n, then the subdivision scheme (11) is Lp stable.
P. Oswald / Journal of the Franklin Institute 349 (2012) 1619–1636
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Proof. From the assumptions and Eq. (8) with m ¼ 0 we have J
rrJ=p ðJSvJ1 Sv~ J1 J‘p ðZÞ þ JPd J Pd~ J‘p ðZÞ
rJ=p JvJ ~v J J‘p ðZÞ
J rrJ1=p JvJ1 ~v J1 J‘p ðZÞ þ CrJ=p ðJDðvJ1 ~v J1 ÞJ‘p ðZÞ þ Jd J d~ J‘p ðZÞ Þ J J 1 X X rJv0 ~v 0 J‘p ðZÞ þ C rj=p Jd j d~ j J‘p ðZÞ þ C rl=p JDðvl ~v l ÞJ‘p ðZÞ : j¼1
j¼0
To see that the last sum is majorized by the remaining terms in this upper bound, we now explore Eq. (24). The reasoning is analogous to our estimates for the Lp convergence proof for Theorem 5. Write j ¼ sn þ j 0 with j 0 ¼ 0, . . . ,r1, and denote r~ :¼ r1=n . Then rj=p JDðv j ~v j ÞJ‘p ðZÞ
j X
rrrðjnÞ=p ðJDðvjn ~v jn ÞJ‘p ðZÞ þ C
l rl=p Jd l d~ J‘p ðZÞ
l ¼ jnþ1 j X
rr~ n rðjnÞ=p ðJDðvjn ~v jn ÞJ‘p ðZÞ þ C
l
r~ jl rl=p Jd l d~ J‘p ðZÞ
l ¼ jnþ1
rr~ sn JDðvj0 ~v j0 ÞJ‘p ðZÞ þ C
j X
l r~ jl rl=p Jd l d~ J‘p ðZÞ
l ¼ j 0 þ1
rC r~ Jv ~v J‘p ðZÞ þ C j
0
0
j X
r~
jl l=p
r
l
!
Jd d~ J‘p ðZÞ : l
l¼1
Here, the last step comes from recursively estimating finitely many expressions JDðvj0 ~v j0 ÞJ‘p ðZÞ , j 0 ¼ 0, . . . ,n1, using the Lipschitz continuity of S and P, clearly, the constant C depends on the fixed values of ro1 and n. With this at hand, it remains to substitute and change order of summation: J 1 X
rl=p JDðvl ~v l ÞJ‘p ðZÞ
j¼0
rC
J 1 X
r~ j Jv0 ~v 0 J‘p ðZÞ þ C
j¼0 0
J 1 X
j
r~ þ
j¼0
rC Jv ~v J‘p ðZÞ þ 0
l
!
r~ jl rl=p Jd l d~ J‘p ðZÞ
l¼1
rC Jv ~v J‘p ðZÞ 0
j X
0
J 1 X
J 1 X
l l=p
r~ r
l¼1
r
l=p
l
l Jd d~ J‘p ðZÞ l
!
J 1 X
! r~
j
j¼l
Jd d~ J‘p ðZÞ : l
l¼1
This shows the desired estimate, and the proof of the Lp stability of Eq. (10) is complete. The Lp stability of Eq. (11) follows under the stated conditions if one repeats the above argument with d j ¼ d~ j ¼ 0. & The statement of this theorem carries over to k41 if an estimate of the form JSvSwJ‘p ðZÞ rr1=p JvwJ‘p ðZÞ þ CJDk ðvwÞJ‘p ðZÞ
ð25Þ
can be established, and Eq. (24) holds with D replaced by Dk , see [11]. Moreover, in [11] the condition (24) has been replaced by a spectral radius estimate on the derivatives of derived
P. Oswald / Journal of the Franklin Institute 349 (2012) 1619–1636
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subdivision operators. For the sake of simplicity, we assume that S and thus all S½m , mrk, are C1, i.e., all functions fs in Eq. (2) of S possess uniformly bounded, continuous partial derivatives, and refer to [11] for the exact conditions of piecewise continuous differentiability under which the statement can be proved. Denote by Dv S½k the Frechet derivative of S ½k at v 2 ‘p ðZÞ. Due to our simplifying assumptions, the linear operator family Dv S ½k : ‘p ðZÞ-‘p ðZÞ depends continuously on v. Now define the spectral radii stab ½k ½k ~ stab ~ stab rstab p,k ðSÞ ¼ r p ðS Þ as follows: p,k ðSÞ ¼ rp ðS Þ and r 1=j
½k ½k ½k ½k rstab p ðS Þ :¼ lim sup sup JDS ðS ½k Þj1 w DS ðS ½k Þj2 w . . . DS w J‘p ðZÞ-‘p ðZÞ j-1
ð26Þ
w2‘p ðZÞ
and ½k r~ stab p ðS Þ :¼ lim sup j-1
sup w0 ,w1 ,...,wj1 2‘p ðZÞ
1=j
JDS ½k DS ½k . . . DS ½k J : wj1 wj2 w0 ‘p ðZÞ-‘p ðZÞ
ð27Þ
Note that in Eq. (27) the supremum is taken with respect to an arbitrary collection of wl, l ¼ 0, . . . ,j1, while in Eq. (26) it is taken with respect to a single w. Set wl ¼ ðS½k Þl w, l ¼ 0, . . . ,j1, to see that ~ stab rstab p,k ðSÞrr p,k ðSÞ: As demonstrated in [11] for the dyadic median interpolating scheme, this inequality can be strict. On the other hand, in [10, Lemma 4.2] it was observed that lim r~ stab ððS ½k Þn Þ1=n n-1 p
¼ rstab p,k ðSÞ,
a property that is useful for establishing approximation results, see [10,8]. The proof of the following theorem, and its generalization to certain classes of piecewise-differentiable S, is given in [11, Section 2.3] in the case p ¼ 1, it straightforwardly carries over to 1rpo1. & Theorem 8. Let S be an r-shift invariant, local, C1 continuous subdivision operator, and P be bounded and Lipschitz continuous. In addition, assume that S is offset invariant for Pk , and 1=p that Eq. (25) holds. The multi-scale reconstruction (10) is Lp stable if r~ stab , and the p,k ðSÞor stab 1=p subdivision algorithm (11) is Lp stable if rp,k ðSÞor . We note that under mild conditions Lp stability fails to hold when the corresponding inequalities are reversed. The C1 assumption on S can be replaced by the weaker kdifferentiability, as done in [11] for p ¼ 1. This extension is necessary to deal with the example considered in the next section. 5. Example For future reference, we want to summarize the currently known results on Lp convergence, Besov smoothness, and stability for the family of nonlinear interpolating power-q subdivision operators. They represent a simple test case for the application of general theories. Despite the simplicity of power-q schemes, there are still some open questions about their Lp smoothness and stability if q42.
P. Oswald / Journal of the Franklin Institute 349 (2012) 1619–1636
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The power-q subdivision operator is defined by ðSvÞ2i ¼ vi ,
ðSvÞ2iþ1 ¼
vi þ viþ1 1 Hq ðD2 vi1 ,D2 vi Þ, i 2 Z, 8 2
where the so-called limiter Hq is defined by 8 q > < x þ y 1 xy , xy40, x þ y 2 Hq ðx,yÞ ¼ > : 0, xyr0:
ð28Þ
ð29Þ
The parameter q 2 ½1, þ 1Þ is fixed. References and basic facts can be found in [8]. The most studied case is q ¼ 2. Obviously, S realizes a nonlinear, data-dependent interpolation between the subdivision rule S2 which results if ðD2 vi1 ÞðD2 vi Þr0 and the standard Deslauriers-Dubuc 4-point scheme the definition of which coincides with the case D2 vi1 ¼ D2 vi in Eq. (28). The S of the power-q family are Lipschitz continuous for all q 2 ½1,1Þ, are 2-shift invariant, and offset invariant for P2 . Thus, both S ½1 and S ½2 are well-defined, and our theorems can be used with either k ¼ 1 or 2: ðS ½1 wÞ2i ¼
wi 1 Hq ðDwi1 ,Dwi Þ, 2 8
ðS ½1 wÞ2iþ1 ¼ ðS ½2 wÞ2i ¼
wi 1 þ Hq ðDwi1 ,Dwi Þ, 2 8
1 Hq ðwi1 ,wi Þ, 4
ðS ½2 wÞ2iþ1 ¼
i 2 Z,
wi 1 ðHq ðwi1 ,wi Þ þ Hq ðwi ,wiþ1 ÞÞ: 2 8
ð30Þ
By working with S½1 , the Lp convergence of the power-q multi-scale reconstruction algorithm with P ¼ Id (and thus also the Lp convergence of the associated subdivision scheme) can easily be established. By Theorem 5 it is sufficient to show that rp,1 ðSÞo21=p which evidently follows from the stronger statement JS½1 J‘p ðZÞ-‘p ðZÞ o21=p ,
1rpo1:
Indeed, since 1 1 8 jHq ðDwi1 ,Dwi Þjr16 jDwi1
1 þ Dwi jr16 ðjwi1 j þ jwiþ1 jÞ,
we have 1 JS½1 wJp‘p ðZÞ r2J12 jwi j þ 16 ðjwi1 j þ jwiþ1 jÞJp‘p ðZÞ r2ð58JwJ‘p ðZÞ Þp ,
independently of q 2 ½1,1Þ. It was conjectured in [8] that a much sharper result holds, namely, that for all q 2 ½1,1Þ rp,2 ðSÞ ¼
1 1 ¼) sp ðSÞ ¼ 1 þ , 2 p
1rpr1:
ð31Þ
The upper bound sp ðSÞr1 þ 1=p is obvious since the limit function S1 d for the d-sequence is the linear B-spline B2, independently of q, and o2 ðt,B2 ÞLp ðRÞ ^t1þ1=p . The lower bound in
P. Oswald / Journal of the Franklin Institute 349 (2012) 1619–1636
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(31) follows from the stronger inequality JS ½2 J‘p ðZÞ-‘p ðZÞ ¼ 12,
1rpo1,
ð32Þ
for the range 1rqr2 (for p ¼ 1, this equality holds for all q, see [8]). Indeed, for this range of values the q-power scheme is convexity preserving: If a finite segment of v is convex, i.e., for w ¼ D2 v we have wj1 ,wj 0 þ1 r0owi , i ¼ j, . . . ,j 0 , for some jrj 0 , then Sv remains convex there, more precisely ðS ½2 wÞ2j ¼ ðS ½2 wÞ2j0 þ2 ¼ 0,
ðS ½2 wÞi 40, i ¼ 2j þ 1, . . . ,2j 0 þ 1:
Similarly for concave segments, with all inequality signs reversed. This follows from the elementary inequality 0oHq ðx,yÞ ¼ Hq ðx,yÞoqminðx,yÞ, x,y40. Since any finitely supported v decomposes into finitely many convex/concave segments, in order to prove Eq. (32) it is enough to prove the corresponding ‘p inequality for a finite convex segment, i.e., to show that j X wi 0
i¼j
2
xi1 xi
p
0
þ
j 1 X
j p X wi 0
ð2xi Þp r
i¼j
i¼j
2
,
where the notation 0oxi :¼ 18 Hq ðwi ,wiþ1 Þo14minðwi ,wiþ1 Þ,
i ¼ j, . . . ,j 0 1,
is used ðxj1 ¼ xj 0 ¼ 0Þ. But this follows by induction in the length j 0 j of the underlying index segment ½j,j 0 . The case of a length j 0 j ¼ 0 segment is trivial. Suppose j 0 j40. Since xj o2xj ,
1 1 2 wj xj o2 wj ,
0oxj ,
1 1 2 wjþ1 xjþ1 xj o2wjþ1 xjþ1 ,
the convexity of fðtÞ ¼ jtjp implies ð12 wj xj Þp þ ð2xj Þp rð12wj Þp þ xpj and xpj þ ð12 wjþ1 xjþ1 xj Þp rð12wjþ1 xjþ1 Þp : Substitution yields ð12 wj xj Þp þ ð2xj Þp þ ð12 wjþ1 xjþ1 xj Þp rð12 wj Þp þ ð12wjþ1 xjþ1 Þp : This reduces the proof of the above ‘p estimate for the index segment ½j,j 0 to the case of the shorter index segment ½j þ 1,j 0 , and the induction is complete. For q42, it is easy to see that Eq. (32) is not necessarily true anymore (e.g., consider p¼ 1, the finitely supported sequence v given by v0 ¼ ð1 þ eÞ, v1 ¼ v1 ¼ 1, vi ¼ 0, jijZ2, and choose e40 small enough. Thus, Eq. (31) is formally established for 1rqr2, and still open for 2oqo1. Concerning stability, since we know that S is 2-differentiable (see [11, Section 3.2]), and satisfies Eq. (25), Theorem 8 is applicable with k ¼ 2. By crude estimates, we can show that JDv S½2 J‘p ðZÞ-‘p ðZÞ o21=p ,
1rpo1
ð33Þ
holds, independently of v, for a certain range q 2 ½1,qp Þ, where 2oqp o4. This obviously 1=p implies r~ stab . For p ¼ 1, the latter inequality was established for q 2 ½1,8=3Þ in p,2 ðSÞo2 [11], there it was also shown that r~ stab 1,2 ðSÞ41 if q44.
P. Oswald / Journal of the Franklin Institute 349 (2012) 1619–1636
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To prove Eq. (33), we assume q 2 ½1,4, and set a :¼ q=4 2 ½1=4,1. From [11, Section 3.2] we quote the following properties of the matrix representation of Dv S ½2 : jðDv S ½2 uÞ2i jr2ai jui1 j þ 2a0i jui j, jðDv S ½2 uÞ2iþ1 jrai jui1 j þ ð1=2a0i aiþ1 Þjui j þ a0iþ1 juiþ1 j, where the data-dependent coefficients ai ,a0i satisfy 0oai oai þ a0i oa=2. For even indices, we get X X jðDv S ½2 uÞ2i jp rap1 2ai jui1 jp þ 2a0i jui jp r2ap JuJp‘p ðZÞ , i2Z
i2Z
and for odd indices X X jðDv S ½2 uÞ2iþ1 jp rð12 þ aÞp1 ai jui1 jp þ ð12a0i aiþ1 Þjui jp þ a0iþ1 juiþ1 jp i2Z
i2Z
r12 ð12 þ aÞp1 JuJp‘p ðZÞ : Together this gives JDv S ½2 uJp‘p ðZÞ rð2ap þ 12 ð12 þ aÞp1 ÞJuJp‘p ðZÞ , and Eq. (33) follows as long as 2ap þ 12 ð12 þ aÞp1 o2. If ar1=2 it is trivially satisfied for all p. Since the left-hand side as a function of a 2 ð0,1 is monotone and continuous for each fixed p, and exceeds 2 if a ¼ 1, there is an 1=2oap o1 such that Eq. (33) holds iff 0oaoap whichpffiffiffiffiffi is equivalent to q 2 ½1,qp Þ with some 2oqp o4. In particular, q1 ¼ 3, q2 ¼ ð 571Þ=2 3:275, and qp -2 if p-1. To summarize, for the subfamily of convexity-preserving power-q schemes ð1rqr2Þ Lp convergence, smoothness exponents, and stability are fully established while for q42 only partial results are known. References [1] S. Amat, K. Dadourian, J. Liandrat, Analysis of a class of subdivision schemes and associated non-linear multiresolution transforms, Adv. Comput. Math. 34 (2011) 253–277. [2] S. Amat, J. Liandrat, On the stability of the PPH nonlinear multiresolution, Appl. Comput. Harmon. Anal. 18 (2005) 198–206. [3] A.S. Cavaretta, W. Dahmen, C.A. Micchelli, Stationary Subdivision, Memoirs American Mathemetical Society, vol. 93, AMS, Providence, 1991. [4] Z. Ciesielski, Constructive function theory and spline systems, Studia Math. 58 (1975) 277–302. [5] A. Cohen, Numerical Analysis of Wavelet Methods, Elsevier, 2003. [6] A. Cohen, N. Dyn, B. Matei, Quasilinear subdivision schemes with application to ENO interpolation, Appl. Comput. Harmon. Anal. 15 (2003) 89–116. [7] W. Dahmen, Wavelet and multiscale methods for operator equations, Acta Numer. 6 (1997) 55–228. [8] N. Dyn, P. Oswald, Univariate subdivision and multiscale transforms: the nonlinear case, in: R.A. DeVore, A. Kunoth (Eds.), Multiscale, Nonlinear, and Adaptive Approximation, Springer, Berlin, 2009, pp. 203–247. [9] N. Dyn, D. Levin, Subdivision schemes in geometric modelling, Acta Numer. 11 (2002) 73–144. [10] P. Grohs, Approximation order from stability of nonlinear subdivision schemes, J. Approx. Theory 162 (2010) 1085–1094. [11] S. Harizanov, P. Oswald, Stability of nonlinear subdivision and multiscale transforms, Constr. Approx. 31 (2009) 359–393. [12] F. Kuijt, R. van Damme, Stability of subdivision schemes, TW Memorandum 1469, Faculity of Applied Mathematics, University of Twente, 1998.
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[13] B. Matei, Smoothness characterization and stability in nonlinear multiscale framework: theoretical results, Asymptotic Anal. 41 (2005) 277–309. [14] P. Oswald, Multilevel Finite Element Approximation: Theory & Applications, Teubner, Leipzig, 1994. [15] G. Xie, T.P.-Y. Yu, On a linearization principle for nonlinear p-mean subdivision schemes, in: M. Neamtu, E.B. Saff (Eds.), Advances in Constructive Approximation, Nashboro Press, 2004, pp. 519–533. [16] G. Xie, T.P.-Y. Yu, Smoothness analysis of nonlinear subdivision schemes of homogeneous and affine invariant type, Constr. Approx. 22 (2005) 219–254.