an analysis of wavelet frame based scattered data reconstruction · 2019. 1. 28. · an analysis of...

An analysis of wavelet frame based scattered

data reconstruction

Jianbin Yang ∗ ab, Dominik Stahl †b, and Zuowei Shen ‡b

aDepartment of Mathematics, Hohai University, ChinabDepartment of Mathematics, National University of Singapore, Singapore

Abstract

In real world applications many signals contain singularities, like edges in images.Recent wavelet frame based approaches were successfully applied to reconstruct scat-tered data from such functions while preserving these features. In this paper wepresent a recent approach which determines the approximant from shift invariantsubspaces by minimizing an `1-regularized least squares problem which makes ad-ditional use of the wavelet frame transform in order to preserve sharp edges. Wegive a detailed analysis of this approach, i.e., how the approximation error behavesdependent on data density and noise level. Moreover, a link to wavelet frame basedimage restoration models is established and the convergence of these models is an-alyzed. In the end, we present some numerical examples, for instance how to applythis approach to handle coarse-grained models in molecular dynamics.

Keywords: framelet; scattered data reconstruction; `1-regularized least squares;

image restoration; asymptotic approximation analysis

1 Introduction

The task of scattered data reconstruction is to determine a function that approximatesa given set of unorganized points. It finds applications in various fields, for instance,terrain modeling, surface reconstruction and the numerical solution of partial differentialequations, see e.g., [39]. Moreover, it can be used to approximate sparse range data [24],and it can be even applied to fit coarse-grained force functions in structural biology[34, 31], as we will learn below.

Let f : Rd → R be a function, which usually is only known on some scattered datasites Ξ = {ξ1, ξ2, . . . , ξn} ⊂ Rd and additionally is disturbed by some noise, i.e., the given∗[email protected], corresponding author†[email protected]‡[email protected]

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data is y(ξi) = f(ξi) + �i for i = 1, 2, . . . , n. The task of scattered data reconstruction isnow to determine a function f∗ from some function space V that approximates the noisydata {ξi, y(ξi)}ni=1. Most approaches determine the function f∗ by solving a regularizedleast squares problem of the form

ming∈V

n∑i=1

(g(ξi)− y(ξi))2 + Γ(g) , (1.1)

where the first term measures the fitting error while the regularization term Γ(g) givespreferences to properties of the approximant f∗. It can for instance be chosen such thatthe roughness of f∗ is penalized or such that f∗ comes close to a piecewise continuousfunction, as we will present in this paper.

There are also several choices for the function space V in (1.1), often consideredspaces are the Beppo-Levi space Hm, C2 or as we will use in this paper, a shift invariantspace

Sh(φ) := closure{∑α∈Zd

u(α)φ(·h− α) : u(α) ∈ R with u(α) = 0 for almost all α ∈ Zd},

which is spanned by integer translates of just one compactly supported function φ thatin turn is scaled by a h > 0. Besides its structural simplicity shift invariant spaces havethe beneficial property that for special choices of φ they provide good approximationorders to sufficiently smooth functions, see [25, 26, 29]. The compact support of φ alsoresults in sparse system matrices which is of computational interest [29, 40]. Such scalingfunctions φ are for instance B-splines, which in turn give rise to associated wavelet framesystems, as we discuss below.

Inspired by some recent wavelet frame based image restoration methodologies [7, 36,24], we determine the approximating function f∗ ∈ Sh(φ) by minimizing the functional

Eh(u) :=∑ξ∈Ξ

(∑α∈I

u(α)φ(ξ

h− α)− y(ξ))2 + ν‖diag(λ)Wu‖`1(Zd) , (1.2)

where u ∈ `1(Zd), I is some properly chosen index set, ν is a positive parameter,W is thediscrete framelet transform and diag(λ) is a diagonal matrix based on the vector λ whichscales the different wavelet channels. The basic idea behind the regularization term in(1.2) is to make use of the interaction between the framelet transform and the `1-norm.It is a known fact that the wavelet coefficient sequence of a signal, which is sampledfrom a piecewise smooth function, is sparse. Furthermore, because of the `1-norm,the regularization term ‖diag(λ)Wu‖`1 gives preference to a solution u whose waveletcoefficient sequence is sparse, and to keep the singularities of functions. This propertymakes the approach (1.2) predestined for applications like range data approximation, asit is demonstrated in [24]. However in [24] the location of the scattered data sites Ξ isonly considered on regular grids, i.e., scaled subsets of Z2 and no error analysis of themethod is given.

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The main focus of this paper is to extend this to bounded subsets Ω ⊂ Rd and to givea proper approximation analysis to the approach connected to (1.2), i.e., let f ∈ W k1 ,given some scattered data y(ξi) = f(ξi)+�i with ξi ∈ Ξ ⊂ Ω and some noise �i, how does‖f∗ − f‖Lp(Ω) behave dependent on the data density, the noise level and the dilation h,where f∗ :=

∑α u∗(α)φ(·/h−α) with u∗ = arg minuEh(u) being the minimizer of (1.2).

Similar has been considered in [29] for the model

ming∈Sh(φ)

∑ξ∈Ξ

(g(ξ)− y(ξ))2 + ν|g|2Hm ,

whose minimizer can be seen as an approximation to the thin plate smoothing spline.Hence gives a smooth approximation to the scattered data, meaning that discontinuitiesare not displayed very well. Additionally in numerical considerations the regularizationterm |g|2Hm has to be discretized, but no approximation result is given for this discretecase. This is another advantage of the approach (1.2), because no discretization of theregularization term is needed as it is already in discrete form and so the approximationresults which we present stay valid. Since in real world applications many signals containsingularities, the approach (1.2) finds various applications.

For instance almost every image contains sharp edges. In [9] a similar model to (1.2)is applied to reconstruct images from uniformly sampled pixels, which is referred to asan image inpainting problem. An error estimation in terms of probability is presented in[9] but no asymptotic approximation estimation is given with regard to the resolution ofimages, which we will present in this paper. Furthermore, a connection between approach(1.2) and wavelet frame based image restoration models [8, 15, 36, 7] is established,additionally convergence of these models is analyzed.

The model (1.2) is also used in [24] to approximate range data which is knownto contain discontinuities. Range data is usually obtained by lasers or photometricapproaches using for instance structured light, where the main concept behind theseapproaches is triangulation, see [21] for a tutorial. Photogrammetric technology evolveddrastically in the last two decades, because of the need in the manufacturing industries,especially in casting, to quickly inspect whether a produced part is matching the originalCAD model within the tolerances.

Molecular dynamics is a field in computational physics which simulates the interac-tions of atoms based on Newton’s equations of motion [19]. To model molecular inter-actions in big systems it can be necessary to simplify the system by combining severalatoms to one group which is then treated as one single interaction site. The resultingmodel is then called coarse-grained model, which can also be used to investigate the longtime behavior of a molecular system. In this paper we show that the model (1.2) is agood approach to determine the force functions between several interaction sites in acoarse-grained model.

The paper is organized as follows. Below we set up the notation and review somebasic properties of B-splines and framelets. In Section 2.2 we establish a connectionbetween the regularity of functions in Sh(φ) and the discrete wavelet transform of their

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coefficients, see Proposition 2.1. Moreover, we give an approximation result dependenton the scattered data sites. Finally, we present an asymptotic approximation analysisof the model Eh(u) and its minimizer. In Section 2.3 we provide some lemmas andtechnical details to give the proofs to the Propositions from Section 2.2. In Section 3we apply our approach to wavelet frame based image restoration models and investigatethe convergence of their solutions. In the last section we explain how to treat theminimization problem numerically and present some examples.

2 Analysis of the model

2.1 Notation and Preliminaries

We use multi-index notation µ = (µ1, . . . , µd), ν = (ν1, . . . , νd) ∈ Nd, where µ ≥ νmeans µj ≥ νj for all j = 1, . . . , d. If µ ≥ ν and µ 6= ν, we write µ > ν. Moreover,|µ| := µ1 + · · ·+ µd and Dµ equals Dµ11 · · ·D

µdd , where Dj denotes the partial derivative

with respect to the j-th coordinate.

We use X ∼ Y to denote that two variables X and Y are equivalent, that is, cY ≤X ≤ CY for some positive constants c and C.

Let φ be a real valued continuous and compactly supported function, and the shiftinvariant space S(φ) generated by φ can be defined by

S(φ) := closure{φ ∗′ a : a ∈ `0(Zd)},

whereφ ∗′ a :=

∑α∈Zd

a(α)φ(· − α)

is the semi-discrete convolution and `0(Zd) denotes the set of all finitely supportedsequences in Zd. The shift invariant space S(φ) can be refined by a dilation h > 0

Sh(φ) := {f( ·h

) : f ∈ S(φ)}.

Let φ ∈ Lp(Rd) be a compactly supported function. We say that the shifts of φare `p-stable (1 ≤ p ≤ ∞) if there exist positive constants cp and Cp such that for allsequences a ∈ `p(Zd),

cp‖a‖`p ≤ ‖∑α∈Zd

a(α)φ(· − α)‖Lp ≤ Cp‖a‖`p .

It is known from [27] that if the shifts of φ are `p0-stable for some 1 ≤ p0 ≤ ∞, thenthey are `p-stable for all 1 ≤ p ≤ ∞.

The Fourier transform of a function f ∈ L1(Rd) is defined by

f̂(ξ) :=

∫Rdf(x)e−ix·ξdx, ξ ∈ Rd,

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where x · ξ is the inner product of two vectors x and ξ in Rd. The domain of the Fouriertransform can be naturally extended to functions in L2(Rd) and tempered distributions.Similarly, if c ∈ `1(Zd), its Fourier series is defined by

ĉ(ξ) :=∑α∈Zd

c[α]e−iα·ξ, ξ ∈ Rd.

We denote the Sobolev space that consists of all distributions f such that Dµf ∈L1(Rd) for all |µ| ≤ k byW k1 (Rd) and the Sobolev semi-norm by |f |Wk1 :=

∑|µ|=k ‖Dµf‖L1 .

We say that f satisfies the Strang-Fix conditions [38] of order k if

f̂(0) 6= 0, and Dµf̂(2πα) = 0, ∀α ∈ Zd\{0}, |µ| < k.

The Fourier series of the first order difference filter ∇1 := [1,−1] is 2ie−iξ2 sin ( ξ2). In

general, the symmetric `-th order difference filter ∇` can be defined by its Fourier series

2ìè−ij`ξ2 sin` (

ξ

2), (2.1)

where

j` =

{1, ` is odd,

0, ` is even.(2.2)

When ` is odd, the difference filter ∇` is symmetric around 12 ; when ` is even, ∇` is

symmetric around the origin.

Similarly, the symmetric `-th order average filter 4` can be defined by its Fourierseries

2è−ij`ξ2 cos` (

ξ

2),

where 41 := [1, 1].Let µ = (µ1, µ2, . . . , µd) ∈ Nd, the difference filter ∇µ on `1(Zd) is defined by the

tensor product∇µ := ∇µ1e1∇

µ2e2 . . .∇

µded.

Let Bm be the centered B-spline of order m, which is defined in the Fourier domainby

B̂m(ξ) = e−ijm ξ2

sinm ( ξ2)

( ξ2)m

,

with jm as in (2.2). By the well-known recurrence formula of B-splines, the derivativeof the centered B-splines can be computed in terms of lower order splines as follows, seealso [11, 37].

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When m is odd,

d

dxBm = Bm−1(·)−Bm−1(· − 1)

= Bm−1 ∗′ ∇1.

For 1 ≤ ` < m− 1,d`

dx`Bm = Bm−` ∗′ ∇`. (2.3)

When m is even, for 1 ≤ ` < m− 1 and ` is odd,

d`

dx`Bm = Bm−`(·+ 1) ∗′ ∇`. (2.4)

For 1 < ` < m− 1 and ` is even,

d`

dx`Bm = Bm−` ∗′ ∇`.

It is known that Bm is refinable with refinement mask

ĥ0(ξ) = e−ijm ξ2 cosm (

ξ

2). (2.5)

By Bm and h0, we define m framelet masks as

ĥ`(ξ) := −i`e−ijmξ2

√(m

`

)sin`(

ξ

2) cosm−`(

ξ

2), ` = 1, 2, . . . ,m, (2.6)

with jm as in (2.2). See [35, 12] for more details on framelet masks. By the m + 1filters defined above, we can define the discrete wavelet frame transform in the multi-dimensional case by tensor product.

Let the index i = (i1, i2, . . . , id) with 0 ≤ i1, i2, . . . , id ≤ m. The wavelet frame filters(hi[k])k∈Zd ∈ `1(Zd) are defined by

hi[k] := hi1 [k1]hi2 [k2] . . . hid [kd], (2.7)

where ir denotes the ir-th vanishing moment of hir corresponding to the r-th variableand k = (k1, k2, . . . , kd) ∈ Zd. For the (` + 1)-th level of wavelet frame transform, thefilters are defined by h`,i := h̃`,i ∗ h̃`−1,i ∗ . . . ∗ h̃0,i, where

h̃`,i[k] =

{hi[2

−`k], k ∈ 2`Zd,0, k /∈ 2`Zd.

Let u ∈ `1(Zd), the discrete framelet transform are defined byW`,iu := h`,i[−·]∗u. Ingeneral, we denote the undecimated framelet transform (see, e.g., [15, 7]) with L levelsof decomposition as

Wu = {W`,iu : 0 ≤ ` ≤ L− 1, 0 ≤ i1, i2, . . . , id ≤ m}.

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For simplicity, we choose L = 1, while our analysis can be extended to the general caseswith L > 1. In this case we define

Wiu :=W0,iu = hi[−·] ∗ u for i = (i1, i2, . . . , id). (2.8)

Let B(r) := {|x| < r, x ∈ Rd}. The scattered data sites Ξ are assumed to lie in abounded subset Ω ⊂ Rd. Suppose Ω has a Lipschitz boundary and the cone property,that is there exist positive constants ρΩ, rΩ such that for all x ∈ Ω, there exists y ∈ Ωsuch that |x− y| = ρΩ and

x+ t(y − x+ rΩB(1)) ⊂ Ω, ∀ t ∈ [0, 1].

The separation distance in Ξ is defined by

sep(Ξ) := inf{|ξ − ξ′| : ξ, ξ′ ∈ Ξ, ξ 6= ξ′}.

The density level of Ξ in Ω is defined by

δ := δ(Ξ,Ω) := supx∈Ω

infξ∈Ξ|x− ξ|.

And the accumulation of Ξ in Ω is defined by

γ := γ(Ξ,Ω, δ) := maxx∈Ω

#{ξ ∈ Ξ : |x− ξ| ≤ δ}.

2.2 Asymptotic approximation analysis

In this section we determine an asymptotic approximation analysis to the wavelet framebased approach (1.2) to approximate noisy scattered data {y(ξi)}ni=1 by a function f∗ ona bounded subset Ω ⊂ Rd, where the scattered data is sampled at the sites ξi ∈ Ξ ⊂ Ωfrom an usually unknown function f , i.e., y(ξi) = f(ξi) + �i with added noise satisfying‖y − f‖`2(Ξ) ≤ �. More precisely, let f ∈ W k1 (Rd) with k ≥ d, we investigate how‖f∗−f‖Lp(Ω) behaves dependent on the data density, the scaling parameter and the noiselevel, where f∗ :=

∑α∈I u

∗(α)φ( ·h − α) and u∗ being the minimizer of the functional

Eh(u). Recall, this functional is defined as

Eh(u) :=

n∑i=1

(∑α∈I

u(α)φ(ξih− α)− y(ξi))2 + ν‖diag(λ)Wu‖`1(Zd) , (2.9)

where u ∈ `1(Zd), φ ∈ W k1 (Rd) with stable shifts, W is the discrete framelet transformand diag(λ) is a diagonal matrix based on the vector λ which scales the different waveletchannels. This means that we select the approximant f∗ from the shift invariant subspaceSh(φ,Ω) spanned by the integer translates of φ(·/h), i.e.,

Sh(φ,Ω) = {∑α∈I

u(α)φ(·h− α) : u(α) ∈ R},

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withI = {α ∈ Zd : suppφ( ·

h− α) ∩ Ω 6= ∅}.

Before we present the asymptotic analysis to the minimizer of Eh(u), which givesrise to the approximation f∗, we need to establish a connection between the regularityof functions

g =∑α∈I

u(α)φ(·/h− α)

and the discrete framelet transform of their coefficients. This link is provided in the nextProposition.

Proposition 2.1. (I) Let φ ∈W k1 (Rd) be a compactly supported function which satisfiesthe Strang-Fix conditions of order k, and the shifts of φ are stable. Let

g(x) =∑α∈Zd

u(α)φ(x

h− α),

where u ∈ `1(Zd). Then for any β ∈ Nd and 1 ≤ |β| ≤ k,

‖Dβg‖L1 ∼ hd−|β|∑i≥β‖Wiu‖`1

and|g|Wk1 ∼ h

d−k∑|i|≥k

‖Wiu‖`1 .

(II) If f ∈ W k1 (Rd) with k ≥ d and h > 0 is sufficiently small, then for any β ∈ Ndand 1 ≤ |β| ≤ k,

‖Dβf‖L1 ∼ hd−|β|∑i≥β‖Wif(hα)‖`1

and|f |Wk1 ∼ h

d−k∑|i|≥k

‖Wif(hα)‖`1 ,

where (f(hα))α∈Zd is the discrete function value sequence.

Proof. See Section 2.3.

Here, φ is assumed to satisfy the Strang-Fix conditions of order k, this ensures thatf ∈ W k1 (Rd) can be approximated by functions in Sh(φ) with high order. An explicitexample of such a generator is the tensor product of B-splines,

φ(x) = Bm(x1)Bm(x2) · · ·Bm(xd), with x = (x1, x2, . . . , xd) ∈ Rd.

Since Bm ∈ Wm−11 (R) satisfies the m-th order Strang-Fix conditions, it holds thatφ ∈W k1 (Rd) and satisfies the k-th order Strang-Fix conditions if m ≥ k + 1.

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This assumption on φ ensures that there exits b ∈ `0(Zd) such that ϕ := φ∗′b satisfiesthe Strang-Fix conditions of order k and the conditions ϕ∗′ q :=

∑α∈Zd ϕ(·−α)q(α) = q

for all q ∈ πdk−1 [26], where πdk−1 denotes the set of all polynomials in d variables withdegree ≤ k−1. For example, if φ is given by tensor product of B-splines, by [30, Chapter4.2], we can explicitly construct a finitely supported sequence b such that b̂(0) = 1 andϕ = φ∗′b satisfies the conditions ϕ∗′q = q for all q ∈ πdk−1. The main idea of constructionthose ϕ is to make ϕ̂ very flat at the origin. Other examples of such ϕ are pseudo-splines[12, 14].

Let f̃ ∈ Sh(φ) be defined by

f̃ : =∑j∈Zd

f(hj)ϕ(·h− j)

=∑j∈Zd

(f(h·) ∗ b)(j)φ( ·h− j).

Proposition 2.2. There exists δ0 > 0 (depending only on ρΩ and rΩ) such that if Ξ ⊂ Ωsatisfies δ := δ(Ξ,Ω) ≤ δ0, then for f̃ defined above and 0 < h ≤ 1, it holds that

‖f̃ − f‖`2(Ξ) ≤ Cφ,kh(k−

d2

)δ−d2√γ|f |Wk1 , ∀f ∈W

k1 ,

where Cφ,k is a positive constant dependent on φ and k.

Proof. See Section 2.3.

With Proposition 2.1 and Proposition 2.2 in hand, we are now ready to give anasymptotic approximation analysis to the minimizer of Eh(u).

Theorem 2.3. Suppose f ∈ W k1 (Rd) with k ≥ d. Given noisy data {ξi, y(ξi)}ni=1 ⊂Rd ×R, where y(ξi) = f(ξi) + �i. Suppose u∗ ∈ `1(Zd) minimizes Eh(u) defined by (2.9)with diag(λ) ∼ diag(hd−k) and Wu given by those Wiu for which |i| ≥ k. Let

f∗ =∑α∈I

u∗(α)φ(·h− α) ∈ Sh(φ,Ω) ,

1 ≤ p ≤ ∞ and τ := max{p, 2}. Then, if h and δ are sufficiently small,

‖f∗ − f‖Lp(Ω) ≤ Cφ,p,k,Ω

(δk−d+d/p(ν−1h(2k−d)δ−dγ|f |2

Wk1+ ν−1�2 + |f |Wk1 )

+ δd/τ (�+ h(k−d2

)δ−d2√γ|f |Wk1 +

√ν|f |

12

Wk1)),

where Cφ,p,k,Ω is a constant dependent on φ, p, k and Ω.

Proof. By Duchon’s inequality (see [16, 1], or Lemma 2.9), we have

‖f∗ − f‖Lp(Ω) ≤ Cp,k,Ω1

(δk−d+d/p|f∗ − f |Wk1 (Ω) + δ

d/τ‖f∗ − f‖`2(Ξ))

≤ Cp,k,Ω1(δk−d+d/p(|f∗|Wk1 (Ω) + |f |Wk1 (Ω)) + δ

d/τ (‖f∗ − y‖`2(Ξ) + ‖y − f‖`2(Ξ)))

≤ Cp,k,Ω1(δk−d+d/p(|f∗|Wk1 (Ω) + |f |Wk1 (Ω)) + δ

d/τ (‖f∗ − y‖`2(Ξ) + �)).

(2.10)

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In the following, we estimate |f∗|Wk1 (Ω) and ‖f∗ − y‖`2(Ξ).

Since u∗ is a minimizer of Eh(u), we have

‖f∗ − y‖2`2(Ξ) + ν‖diag(λ)Wu∗‖`1(Zd)

= Eh(u∗) ≤ Eh(f(h·) ∗ b)= ‖f̃ − y‖2`2(Ξ) + ν‖diag(λ)W(f(h·) ∗ b)‖`1(Zd)≤ ‖f̃ − y‖2`2(Ξ) + C

φ2 ν‖diag(λ)W(f(h·))‖`1(Zd)

≤ 2(‖f̃ − f‖2`2(Ξ) + ‖f − y‖2`2(Ξ)

) + Cφ2 ν‖diag(λ)W(f(h·))‖`1(Zd).

By Proposition 2.1 (II),∑|i|≥k

‖diag(λ)Wi(f(h·))‖`1(Zd) ≤ C3 |f |Wk1 .

This together with Proposition 2.2 implies

‖f∗ − y‖2`2(Ξ) + ν‖diag(λ)Wu∗‖`1(Zd) = E

h(u∗) ≤ Eh(f(h·) ∗ b)

≤ 2‖f̃ − f‖2`2(Ξ) + 2‖f − y‖2`2(Ξ)

+ Cφ2C3 ν|f |Wk1≤ Cφ,k4 h

(2k−d)δ−dγ|f |2Wk1

+ 2�2 + Cφ2C3 ν|f |Wk1 .

Furthermore, by Proposition 2.1 (I),

|f∗|Wk1 (Ω) ≤

∣∣∣∣∣∣∑α∈Zd

u∗(α)φ(·h− α)

∣∣∣∣∣∣Wk1

≤ Cφ5∑|i|≥k

‖diag(λ)Wiu∗‖`1(Zd).

Hence,

‖f∗ − y‖2`2(Ξ) + ν|f∗|Wk1 (Ω) ≤ C

φ,k6

(h(2k−d)δ−dγ|f |2

Wk1+ �2 + ν|f |Wk1

). (2.11)

Therefore, combining (2.10) and (2.11), we conclude that

‖f∗ − f‖Lp(Ω) ≤ Cφ,p,k,Ω7

(δk−d+d/p(ν−1h(2k−d)δ−dγ|f |2

Wk1+ ν−1�2 + |f |Wk1 )

+ δd/τ (�+ h(k−d2

)δ−d2√γ|f |Wk1 +

√ν|f |

12

Wk1)).

By Theorem 2.3, it is obvious that if the dilation h of the shift invariant space ischosen equivalent to the data density δ, we have the following asymptotic approximationresult.

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Corollary 2.4. With the same conditions as in Theorem 2.3 it holds that

‖f∗ − f‖Lp(Ω) ≤ Cφ,p,k,Ω

(δk−d+d/p(ν−1δ(2k−2d)γ|f |2

Wk1+ ν−1�2 + |f |Wk1 )

+ δd/τ (�+ δk−d√γ|f |Wk1 +

√ν|f |

12

Wk1)),

if h ∼ δ, for 1 ≤ p ≤ ∞ and τ := max{p, 2}.

The generalized wavelet frame based scattered data reconstruction model can begiven by minimizing the functional

Gh(u) :=∑ξ∈Ξ|∑α∈I

u(α)φ(ξ

h− α)− y(ξ)|m + ν‖diag(λ)Wu‖`q(Zd) .

In Theorem 2.3 we chose m = 2 and q = 1 to keep the singularities of the approximationfunctions. In fact, if the underlying functions are smooth enough, q = 2 is recommended.An approximation analysis of Gh(u) for 1 ≤ m, q ≤ ∞ can be done similarly. In [29]the model (1.1) is investigated for m = 2 and q = 2 with the regularized term Γ(g) =|g|2

Wk2. There no approximation result for the discrete case is given, additionally the

regularization term has to be discretized.

In Theorem 2.3 we assumed that f ∈ W k1 (Rd). Furthermore, the approximationresult was restrict to Ω ⊂ Rd, so no boundary conditions were considered. If we assumethat f is defined on Ω, but on information about f outside Ω is known, the problem getsmore difficult. In that case boundary wavelets should be considered.

2.3 Proof of Proposition 2.1 and 2.2

Here, we give the proofs of Proposition 2.1 and Proposition 2.2. We start by providingsome technical details and lemmas.

Lemma 2.5. Let φ(x) := Bm(x1)Bm(x2) · · ·Bm(xd), x = (x1, x2, . . . , xd) ∈ Rd,and

g(x) =∑α∈Zd

u(α)φ(x

h− α),

where u ∈ `1(Zd), h > 0 is a dilation. For any β = (β1, β2, . . . , βd) ∈ Nd, with 1 ≤ |β| =β1 + β2 + . . .+ βd < m− 1, it holds that

‖Dβg‖L1 ∼ hd−|β|‖∇β ∗ u‖`1 .

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Proof. First, we consider the case that m is odd. By equation (2.3), the following applies

Dβg(x) = h−|β|∑

α=(α1,α2,...,αd)∈Zdu(α)(Dβφ)(

x

h− α)

= h−|β|∑α∈Zd

u(α)((B(m−β1) ∗

′ ∇β1)(x1h− α1) · . . . (B(m−βd) ∗

′ ∇βd)(xdh− αd)

)= h−|β|

∑α∈Zd

(∇β1e1 · · · ∇βded∗ u)(α)B(m−β1)(

x1h− α1) · . . . B(m−βd)(

xdh− αd).

Let θ(x) := Bm−β1(x1)Bm−β2(x2) · · ·Bm−βd(xd), then

Dβg(x) = h−|β|∑α∈Zd

(∇β ∗ u)(α)θ(xh− α).

Thus, ∫Rd|Dβg(x)|dx =

∫Rd|h−|β|

∑α∈Zd

(∇β ∗ u)(α)θ(xh− α)|dx

= hd−|β|∫Rd|∑α∈Zd

(∇β ∗ u)(α)θ(x− α)|dx.

Since the shifts of θ(x) are stable, there exist two positive constants Cφ1 and Cφ2 inde-

pendent of u such that

Cφ1‖∇β ∗ u‖`1 ≤

∫Rd|∑α∈Zd

(∇β ∗ u)(α)θ(x− α)|dx ≤ Cφ2‖∇β ∗ u‖`1 .

Therefore,Cφ1h

d−|β|‖∇β ∗ u‖`1 ≤ ‖Dβg‖L1 ≤ Cφ2h

d−|β|‖∇β ∗ u‖`1 .

In case that m is even the proof works analogously.

Lemma 2.6. Let the framelet masks h` be given by (2.5) and (2.6) for 0 ≤ ` ≤ m. Letk ∈ N and 0 ≤ k ≤ m. For every sequence u ∈ `1(Z),

‖∇k ∗ u‖`1 ∼m∑`=k

‖h`[−·] ∗ u‖`1 .

Proof. By the unitary extension principle (UEP) [35], it holds that

m∑`=0

ĥ`(ξ)ĥ`(ξ) = 1.

Hence,

2kike−ijkξ2 sink (

ξ

2)û(ξ) =

m∑`=0

ĥ`(ξ)ĥ`(ξ)2kike−ijk

ξ2 sink (

ξ

2)û(ξ).

12

For simplicity, we use the notation ‖ĉ(ξ)‖`1 := ‖c‖`1 for c ∈ `1(Z) in the followingcontext. The `1-norm ‖ĉ(ξ)‖`1 is defined on the time domain essentially, and we usethis notation for applying the propositions of Fourier transform easily. By (2.1) and thediscrete Young’s inequality (see e.g. [3]), we conclude that

‖∇k ∗ u‖`1 = ‖m∑`=0

ĥ`(ξ)ĥ`(ξ)2kike−ijk

ξ2 sink (

ξ

2)û(ξ)‖`1

≤ C1m∑`=0

‖ĥ`(ξ)2kike−ijkξ2 sink (

ξ

2)û(ξ)‖`1

≤ C2m∑`=0

‖ − (−i)`eijmξ2

√(m

`

)sin`+k(

ξ

2) cosm−(`+k)(

ξ

2) cosk(

ξ

2)û(ξ)‖`1

≤ C3( m∑`=k


√(m

`

)sin`(

ξ

2) cosm−`(

ξ

2)û(ξ)‖`1 + ‖ĥm(ξ)û(ξ)‖`1

)≤ C4

m∑`=k

‖h`[−·] ∗ u‖`1 .

On the other hand, applying the discrete Young’s inequality again, we have

m∑`=k

‖h`[−·] ∗ u‖`1 =m∑`=k

‖ĥ`(ξ)û(ξ)‖`1

=

m∑`=k


√(m

`

)sin`(

ξ

2) cosm−`(

ξ

2)û(ξ)‖`1

≤ C5m∑`=k

‖2kike−ijkξ2 sink (

ξ

2)û(ξ)‖`1

≤ C6‖∇k ∗ u‖`1 .

Here, the positive constants Ci, i = 1, . . . , 6 are independent of the sequence u.

Note, by the discrete Young’s inequality, there exists a postive constant C such thatfor all sequences u ∈ `1(Z),

‖h`[−·] ∗ u‖`1 ≤ C‖∇k ∗ u‖`1 ,

when ` ≥ k. But there does not exist a positive constant C such that the inverseinequality ‖∇k ∗ u‖`1 ≤ C‖h`[−·] ∗ u‖`1 holds for all u ∈ `1(Z).

Lemma 2.7. Let hi be defined by (2.7) and the framelet transform Wi be defined by(2.8) for i = (i1, i2, . . . , id) ∈ Nd with each ir ≤ m. Then, for β = (β1, β2, . . . , βd) ∈ Ndwith each βr ≤ m,

‖∇β ∗ u‖`1 ∼∑i≥β‖Wiu‖`1 ,

13

for every sequence u ∈ `1(Zd).

Proof. By Lemma 2.6, the commutativity and associativity of convolution, the followingapplies

‖∇β ∗ u‖`1 = ‖∇β1e1 ∗ (∇β2e2 . . . ∗ ∇

βded∗ u)‖`1

∼∑i1≥β1

‖hi1 [−·] ∗ (∇β2e2 · · · ∗ ∇βded∗ u)‖`1

∼∑

i1≥β1,i2≥β2

‖hi1 [−·] ∗ hi2 [−·] ∗ (∇β3e3 · · · ∗ ∇βded∗ u)‖`1

∼∑

i1≥β1,i2≥β2,··· ,id≥βd

‖hi1 [−·] ∗ hi2 [−·] ∗ · · · ∗ hid [−·] ∗ u‖`1

=∑i≥β‖hi[−·] ∗ u‖`1

=∑i≥β‖Wiu‖`1 .

By Lemma 2.5 and Lemma 2.7, the following proposition holds.

Proposition 2.8. Let φ(x) := Bm(x1)Bm(x2) · · ·Bm(xd), x = (x1, x2, . . . , xd) ∈ Rd.Let

g(x) :=∑α∈Zd

u(α)φ(x

h− α),

where u ∈ `1(Zd). Then‖Dβg‖L1 ∼ hd−|β|

∑i≥β‖Wiu‖`1

and|g|Wk1 ∼

∑|i|≥k

hd−k‖Wiu‖`1

for β ∈ Nd and 1 ≤ |β| < m− 1.

In Proposition 2.8, under the assumption that φ is given by the tensor productof splines, the regularity of functions in Sh(φ) can be estimated by the norm of theirframelet coefficients. In fact, for a general function φ, similar results hold, see Proposition2.1.

Proof of Proposition 2.1. (I) Let ω(x) = (Dβφ)(x). Then

ω̂(ξ) = (iξ1)β1(iξ2)

β2 . . . (iξd)βd φ̂(ξ), for ξ = (ξ1, ξ2, . . . , ξd).

14

Since φ satisfies the Strang-Fix conditions of order k, by Leibniz’s rule, for every µ < β,it holds that Dµω̂(2πα) = 0 for all α ∈ Zd. It follows from [22, Lemma 3.1], there existsa function q having polynomial decay such that ω = q ∗′ ∇β.

Since the shifts of φ are stable, φ̂ has no 2π-periodic zero points. Hence by theConvolution Theorem,

q̂(ξ) =ω̂(ξ)∏d

`=1(2βìβè−ijβ`

ξ`2 sinβ` ( ξ`2 ))

=φ̂(ξ)

∏d`=1(iξ`)

β`∏d`=1(2

βìβè−ijβ`ξ`2 sinβ` ( ξ`2 ))

∼ φ̂(ξ)d∏`=1

ξβ``sinβ` ( ξ`2 )

.

It can be verified that q̂ also has no 2π-periodic zero points. This implies that the shiftsof q are stable (see e.g. [27]).

Note that

Dβg(x) =∑α∈Zd

h−|β|u(α)(Dβφ)(x

h− α)

=∑α∈Zd

h−|β|u(α)ω(x

h− α)

=∑α∈Zd

h−|β|u(α)(q ∗′ ∇β)(xh− α)

=∑α∈Zd

h−|β|(∇β ∗ u)(α)q(xh− α).

Then by using the same method as in Lemma 2.5, we can prove that

‖Dβg‖L1 ∼ hd−|β|‖∇β ∗ u‖`1 .

Thus, by Lemma 2.7, we have

‖Dβg‖L1 ∼ hd−|β|∑i≥β‖Wiu‖`1

and|g|Wk1 ∼ h

d−k∑|i|≥k

‖Wiu‖`1 .

(II) First we consider the case that each βi ≥ 1 and βi is odd.

15

Let η(x) := Bβ1(x1)Bβ2(x2) . . . Bβd(xd), x = (x1, x2, . . . , xd) ∈ Rd. By (2.3), thefollowing holds

〈Dβf, η( ·h− α)〉

= (−1)|β|−d〈D(1,1,...,1)f,D(β1−1,β2−1,...,βd−1)η( ·h− α)〉

= (−1)|β|−dhd−|β|〈D(1,1,...,1)f, (D(β1−1)Bβ1)(x1h− α1) . . . (D(βd−1)Bβd)(

x1h− αd)〉

= (−1)|β|−dhd−|β|〈D(1,1,...,1)f, (B1 ∗′ ∇(β1−1))(x1h− α1) . . . (B1 ∗′ ∇(βd−1))(

xdh− αd)〉,

for all α = (α1, α2, . . . , αd) ∈ Zd.Let the sequence c[α] := f(hα), α ∈ Zd. The Sobolev embedding theorem implies

that f is continuous. Moreover, by a standard density argument, we can find a series ofsmooth functions to approximate B1, and their derivatives approximate δ0 − δ1. Hencethe following applies

〈Dβf, η( ·h− α)〉

= (−1)|β|−dhd−|β|〈f, (δ1 − δ0h

) ∗′ ∇(β1−1)(x1h− α1) . . . (

δ1 − δ0h

) ∗′ ∇(βd−1)(xdh− αd)〉

= (−1)|β|−dhd−|β|((∇β−(1,...,1)[−·] ∗ c)[α+ (1, . . . , 1)]− (∇β−(1,...,1)[−·] ∗ c)[α]

).

Furthermore, since ∇β−(1,...,1) is symmetric around the origin, we have

〈Dβf, η( ·h− α)〉 = (−1)|β|−dhd−|β|(∇β ∗ c)[α+ (1, . . . , 1)].

Thus, by [25, Theorem 3.1], it holds that

‖Dβf −∑α∈Zd〈Dβf, h−dη( ·

h− α)〉φ( ·

h− α)‖L1(Rd)

= ‖Dβf −∑α∈Zd

(−1)|β|−dhd−|β|(∇β ∗ c)[α+ (1, . . . , 1)]h−dφ( ·h− α)‖L1(Rd)

−→ 0 as h −→ 0. (2.12)

Since the shifts of φ(x) are stable, we get

‖∑α∈Zd

(−1)|β|−dhd−|β|(∇β ∗ c)[α+ (1, . . . , 1)]h−dφ( ·h− α)‖L1(Rd)

∼ hd−|β|∑α∈Zd

|(∇β ∗ c)(α)|. (2.13)

Combining (2.12) and (2.13) together yields if h is sufficiently small,

‖Dβf‖L1 ∼ hd−|β|∑α∈Zd

|∇β ∗ f(hα)|. (2.14)

16

If there exits some even βi, by (2.4), D(βi−1)Bβi = B1(· + 1) ∗′ ∇(βi−1) and ∇(βi−1) is

symmetric around 12 , then [α+(1, . . . , 1)] in (2.13) should be substituted by α plus someother integer point, (2.14) also holds.

In case some component βi = 0, for example, β1 = 0, the proof works analogouslyby choosing η(x) := B1(x1)Bβ2(x2) . . . Bβd(xd). Since f is continuous,

‖D(0,β2,...,βd)f‖L1 ∼ hd−1−|β|∑

α=(0,α2,...,αd)∈{0}×Zd−1|∫R∇(0,β2,...,βd) ∗ f(x1, hα2, . . . , hαd)dx1|

∼ hd−|β|∑α∈Zd

|∇β ∗ f(hα)|.

Proof of Proposition 2.2. The idea of the proof follows [28]. Let

C := [−12,1

2)d, N := {j ∈ Zd : suppϕ(· − j)

⋂C 6= ∅},

and r be a positive number such that N ⊂ B(r) and C ⊂ B(r).Let fh := f(h·) and f̃h := f̃(h·) = ϕ ∗′ f(h·). Since f ∈ W k1 , by the generalized

Poincaré inequality (see e.g. [17, Theorem 3.2]), for every α ∈ Zd, there exists qα ∈ πdk−1such that

‖fh − qα‖L∞(B(r)+α) ≤ Cr,k1 |fh|Wk1 (B(r)+α).

Note that ϕ ∗′ q = q for all q ∈ πdk−1. It follows that for any given α ∈ Zd,

‖f̃h − fh‖L∞(C+α) = ‖f̃h − qα − (fh − qα)‖L∞(C+α)≤ ‖

∑j∈N+α

(fh(j)− qα(j)

)ϕ(· − j)‖L∞(C+α) + ‖fh − qα‖L∞(C+α)

≤ ‖∑j∈N

(fh(α+ j)− qα(α+ j)

)ϕ(· − j − α)‖L∞(C+α) + ‖fh − qα‖L∞(C+α)

≤ (1 + #(N )‖ϕ‖L∞)‖fh − qα‖L∞(B(r)+α)≤ Cφ2‖fh − qα‖L∞(B(r)+α)≤ Cr,k1 C

φ2 |fh|Wk1 (B(r)+α).

Noticing that |fh|Wk1 = hk−d|f |Wk1 , for any bounded set of points E ⊂ R

d, if µ :=

sep(E) > 0, it holds that

‖f̃h − fh‖2`2(E) ≤ Cd3 µ−d∑α∈Zd

‖f̃h − fh‖2L∞(C+α)

≤ Cφ,k4 µ−d∑α∈Zd

|fh|2Wk1 (B(r)+α)

≤ Cφ,k5 µ−dh2(k−d)|f |2

Wk1.

17

By virtue of [29, Lemma 2.1], it is possible to partition Ξ as Ξ =⋃Ni=1 Ξi such that

N ≤ Cd6γ and sep(Ξi) ≥ δ, i = 1, 2, . . . , n. Therefore, with Ξ̃i := h−1Ξi,

‖f̃ − f‖2`2(Ξ) = ‖f̃h − fh‖2`2(h−1Ξ)

=N∑i=1

‖f̃h − fh‖2`2(Ξ̃i)

≤N∑i=1

Cφ,k5 sep(Ξ̃i)−dh2(k−d)|f |2

Wk1

≤ Cφ,k5 Cd6(h−1δ)−dh2(k−d)γ|f |2

Wk1

≤ Cφ,k7 h(2k−d)δ−dγ|f |2

Wk1.

It follows that

‖f̃ − f‖`2(Ξ) ≤√Cφ,k7 h

(k− d2

)δ−d2√γ|f |Wk1 .

The following Duchon’s inequality plays an important role in the proof of Theorem2.3. It was first proved in [16], and the generalized form can be found in [1, Theorem4.1].

Lemma 2.9. (Duchon’s inequality) Let 1 ≤ p, q,m ≤ ∞. Suppose k ≥ d if q = 1;k > d/q if 1 < q < ∞ or k ∈ N if q = ∞. Let τ := max{p, q,m}. Then there existsδ0 > 0 (dependent on Ω, d, k) such that if Ξ ⊂ Ω satisfies δ := δ(Ξ,Ω) ≤ δ0, it holdsthat

‖g‖Lp(Ω) ≤ Cp,k,Ω,q,m

(δk−d/q+d/p|g|Wkq (Ω) + δ

d/τ‖g‖`m(Ξ)), ∀g ∈W kq (Ω),

where Cp,k,Ω,q,m is a positive constant dependent on p, k, Ω, q, and m.

3 Applications to image restoration models

3.1 Image inpainting

In this section, we link the image restoration problems to the scattered data reconstruc-tion. Ideas of section 2.2 are applied to analyze the asymptotic behavior of the waveletframe models in image restoration.

Let f be a function on Ω := [0, 1] × [0, 1] which represents an image. Given partialobservations

v[α] =

{f(2−nα) + �[α], α ∈ Λn,unknown, α ∈ In\Λn,

(3.1)

with In := {α = (α1, α2) ∈ Z2, 0 ≤ α1, α2 ≤ 2n − 1}, Λn ⊂ In, which are disturbed bysome noise �. The image inpainting problem is to recover f on Ω, or its discrete versionon In.

18

Image inpainting is a fundamental problem in image processing and has been widelyinvestigated during the last decade (see e.g. [2, 6, 8, 15, 36, 9, 7]). It can be seen as aspecial case of the scattered data reconstruction problem if we consider

Ξ = {2−nα, α ∈ Λn} ⊂ Ω

as our scattered data sites and the pixel values as

y(2−nα) = v[α] = f(2−nα) + �[α].

Hence, similar to the scattered data reconstruction, the underlying image function fcan be approximated from Sh(φ,Ω). Additionally, if φ is chosen to be an interpolatoryfunction, that is φ(0) = 1 and φ(α) = 0 for all α ∈ Z2\{0}, the matrix Aξ,α := φ(2nξ−α)with ξ, α ∈ In is the identity matrix. Thus, the wavelet frame based inpainting model[8, 36, 9]

minu‖u− v‖2`2(Λn) + ‖diag(λ)Wu‖`1

= minu

∑ξ∈Ξ

(∑α∈In

u(α)φ(ξ

h− α)− y(ξ))2 + ‖diag(λ)Wu‖`1 (3.2)

matches our approach with h = 2−n.

Let u∗n be a minimizer of (3.2). Then

g∗n :=∑α∈In

u∗n(α)φ(·h− α)

approximates the image function f . In case Λn is uniformly sampled from the imagedomain In, an error analysis of ‖u∗n − f(2−n·)‖`2 in terms of probability is given in [9]based on the uniform law of large numbers. In this section, we consider the observationsof image as scattered data sites and image restoration as a scattered data reconstructionproblem. Under some mild conditions for f , the convergence of g∗n to f is given in section3.3.

3.2 Image denoising

In general the pixel values are considered as local weighted averages of the analog imagefunction f . For example, let φ be the B-spline function and denote the scaled functionsby φn,α := 2

nφ(2n · −α). Let f ∈ L2(Ω). Then each pixel value p is obtained by adiscrete operator Tφ,n (see [33, 10, 9, 7]),

p[α] := Tφ,nf [α] := 2n〈f, φn,α〉, α = (α1, α2), 0 ≤ α1, α2 ≤ 2n − 1. (3.3)

The consistency between pointwise sampling (3.1) and local weighted averages sam-pling (3.3) is proved in [13, Lemma 6.1]. We assume that ‖p‖`∞ ≤ M holds for all

19

n, where M is a given constant. In image denoising, an usually unknown image f isreconstructed or denoised, by observations v that are disturbed by additive noise �

v[α] = p[α] + �[α],

see [33, 7]. Let �n := ‖�‖`2(In) denote the noise level. In [8, 15, 36, 7] a wavelet framebased approach has been used for image denoising, where the denoised image

g∗n :=∑α∈In

u∗n(α)φ(2n · −α)

is obtained by a minimizer u∗n of the model

En(u) := ‖u− v‖2`2(In) + ‖diag(λ)Wu‖`1(In).

Let hi be 2-D masks given by (2.7) with d = 2 and i := (i1, i2). Then the corre-sponding 2-D refinable function and framelets are defined by

ψi(x, y) = ψi1(x)ψi2(y), 0 ≤ i1, i2 ≤ m; (x, y) ∈ R2,

with ψi := 22∑

k∈Z2 hi[k]φ(2 · −k) and ψ0 := φ = Bm(x1)Bm(x2). We denote byMn ⊂ In the index set which contains all α such that ψi,n,α := 2nψi(2n ·−α) is completelysupported in Ω for all i.

For our analysis below we use two assumptions, one on the smoothness of f in theinterior of Ω, which can be characterized by the decay of the wavelet frame coefficients,see [23, 5]. Moreover, we assume the noise level to be relatively small. Hence, we assume

A1: There exits s > 0 such that∑

α∈Mn 22sn|〈f, ψi,n,α〉|2 is uniformly bounded for

all |i| ≥ 1.A2: limn→∞ 2

−n�n = 0.As most images possess sharp edges, they are usually modeled by piecewise smoothfunctions to preserve these discontinuities. Therefore, small s is desirable in order toreflect the low regularity of the underlying image. We assume 0 < s < 1.

Let 0 < s0 < s and A1 be satisfied, then by the Cauchy-Schwarz inequality, for|i| ≥ 1,

limn→∞

2(s−s0)n(2−2n

∑α∈Mn

|Wip[α]|)

= limn→∞

∑α∈Mn

2(s−s0−1)n2−n|WiTφ,nf [α]|

= limn→∞

2(s−s0)n(2−2n

∑α∈Mn

|Wi〈f(2−n·), φ(· − α)〉|)

= limn→∞

∑α∈Mn

2(s−s0−1)n|〈f, ψi,n−1,α〉|

≤ limn→∞

2−s0n( ∑α∈Mn

22sn|〈f, ψi,n,α〉|2) 1

2

= 0. (3.4)

20

The decay of the wavelet coefficients links to the regularity of the underlying imagefunction where the pixel values p derived from. It is reasonable to assume that for 22n

pixel values, the average values of the discrete frame coefficients in Mn decay to zerowhen n tends to infinity, as (3.4) indicates.

Proposition 3.1. Let u∗n be a minimizer of En with λ0,0 = 0 and λi ≤ 2(s−s0)n for some0 < s0 < s and all |i| ≥ 1, then

limn→∞

‖g∗n − f‖L2(Ω) = 0

if (A1) and (A2) are satisfied.

Proof. Applying the triangle inequality, we get

‖∑α∈In

u∗n(α)φ(2n · −α)− f‖L2(Ω)

≤ ‖∑α∈In

〈f(2−n·), φ(· − α)〉φ(2n · −α)− f‖L2(Ω)

+ ‖∑α∈In

(u∗n(α)− 〈f(2−n·), φ(· − α)〉)φ(2n · −α)‖L2(Ω).

By [7, Lemma 4.1], we obtain

limn→∞

‖∑α∈Mn

〈f(2−n·), φ(· − α)〉φ(2n · −α)− f‖L2(Ω) = 0.

Since #{α : α ∈ In\Mn} ≤ C12n and the Lebesgue measure of {x ∈ R2 : x ∈ supp φ(2n ·−α)} is less than C22−2n, where C1 and C2 depend only on the support of ψi and φ, itholds

limn→∞

‖∑

α∈In\Mn

〈f(2−n·), φ(· − α)〉φ(2n · −α)‖L2(Ω)

≤ limn→∞

M‖∑

α∈In\Mn

|φ(2n · −α)|‖L2(Ω)

≤ limn→∞

M‖φ‖L∞C12nC22−2n

= 0.

Therefore,

limn→∞

‖∑α∈In

〈f(2−n·), φ(· − α)〉φ(2n · −α)− f‖L2(Ω) = 0.

21

Furthermore, since the shifts of φ are stable, we have

‖∑α∈In

(u∗n(α)− 〈f(2−n·), φ(· − α)〉)φ(2n · −α)‖L2(Ω)

≤ ‖∑α∈In

(u∗n(α)− 〈f(2−n·), φ(· − α)〉)φ(2n · −α)‖L2(R2)

≤ C32−n‖u∗n − p‖`2(In)≤ C32−n(‖u∗n − v‖`2(In) + ‖p− v‖`2(In))≤ C32−n(‖u∗n − v‖`2(In) + �n).

Thus, to complete the proof it is left to show that

limn→∞

2−n‖u∗n − v‖`2(In) = 0.

Since u∗n is a minimizer of En, the following applies

2−2n‖u∗n − v‖2`2(In) ≤ 2−2nEn(u

∗n) ≤ 2−2nEn(p)

= 2−2n‖p− v‖2`2(In) + 2−2n‖diag(λ)Wp‖`1(In)

≤ 2−2n�2n + 2(s−s0)n(2−2n

∑|i|≥1

∑α∈In

|Wip[α]|). (3.5)

By assumption A2, we obtain that

limn→∞

2−2n�2n = 0. (3.6)

For the second term of (3.5), we first consider the terms with α ∈ Mn. By assumptionA1, (3.4) holds, i.e. for |i| ≥ 1,

limn→∞

2(s−s0)n(2−2n

∑α∈Mn

|Wip[α]|)

= 0. (3.7)

Finally, we consider the terms with α ∈ In\Mn in (3.5). Note that p is bounded, thus

limn→∞

2(s−s0)n(2−2n

∑α∈In\Mn

|Wip[α]|)≤ lim

n→∞C12

(s−s0)n2−2n2nM = 0. (3.8)

Combining (3.6), (3.7) and (3.8), we conclude

limn→∞

2−n‖u∗n − v‖`2(In) = 0.

This completes the proof of the Proposition.

Proposition 3.1 shows that if the noise level is relatively small, λi can be chosensmaller than 2(s−s0)n. If the noise level is high, for example limn→∞2

−n�n > 0, meaningthat the average noise of each pixel does not converge to 0 as n tends to∞. In this case,

22

in order to fit the image function well and suppress noise, we need a larger weight forthe regularization term. In the following, we show that for different parameters in En,g∗n has a different asymptotic approximation behavior.

Following the line of [7], let f be an analog (noisy) image function, v := Tφ,nf be theobserved digital image and u?n be a minimizer of En with λ0,0 = 0 and λi =

12ci

2n for|i| ≥ 1, where ci (see [7, P1054]) are positive constants which are only dependent on ψi.Let

g?n :=∑α∈In

u?n(α)φ̃(2n · −α),

where φ̃ is the dual of φ.

By [7, Propsition 3.1], g?n is a minimizer of

Fn(g) := ‖Tφ,ng − Tφ,nf‖2`2 + ‖diag(λ)WTφ,ng‖`1 ,

with g ∈ W 11 (Ω). Furthermore, by [7, Theorem 3.2], 2−2nFn Γ-converges to the totalvariational model

F (g) := ‖g − f‖2L2(Ω) + |g|W 11 (Ω).

Thus, g?n approximates the minimizer of F (g) in case n is sufficiently large [7, Corollary3.1].

The essential difference of the asymptotic state between g∗n and g?n is due to the

different choice of parameters. Compared to g∗n, in the model corresponding to g?n,

we introduced a larger weight for the regularization term. This makes sense when thenoise level increases proportional to the number of samples. But when the noise levelis bounded, or even if the samples are from a clean analog image function, g?n may notconverge to f . As a result, the wavelet frame based approach has a relaxed asymptoticstate behavior, and it can approximate various models by choosing a proper waveletframe transform and parameters [7].

3.3 Image inpainting (Continued)

Let f ∈ L2(Ω) be an analog image function and p := Tφ,nf be the pixel values. Supposethe noisy observations are given by

v[α] =

{p[α] + �[α], α ∈ Λn,unknown, α ∈ In\Λn,

(3.9)

with Λn ⊂ In and �[α] represents the noise.Besides the smoothness assumption (A1), we need assumptions on the density of

known pixels and on the noise level, under which we can prove that f can be recoveredby observations v[α]. Let �n := ‖�‖`2(Λn). We assume

A3: The density of known pixels δ := δ(Λn,Ω) ≤ (2(s13n−1) − 12)2

−n for some 0 <s1 < s.

23

A4: limn→∞ 2(s1−2)n�2n = 0.

By assumption A3, for every sub-square matrix with length 2δ, that is 22s13n total pixel

positions, there exists at least one known pixel value. As f gets smoother, s becomeslarger and thus δ can be larger. Note that assumption A4 is different from A2. Here,noise � is defined on Λn and we expect a lower bound.

Inpainting can also be done by a wavelet frame based approach by minimizing

Qn(u) := ‖u− v‖2`2(Λn) + ‖diag(λ)Wu‖`1(In).

Proposition 3.2. Let the assumptions (A1), (A3) and (A4) be satisfied. Moreover, letu∗n be a minimizer of Qn where λ is chosen such that λ0,0 = 0 and 0 < λi ≤ 2(s−s0)nfor some s1 < s0 < s, of the same order and satisfying limn→∞

1λi

2(s1−2)n�2n = 0 for all|i| ≥ 1. Let

g∗n :=∑α∈In

u∗n(α)φ(2n · −α),

thenlimn→∞

‖g∗n − f‖L2(Ω) = 0.

Proof. Similar to Proposition 3.1, we only need to prove that

limn→∞

2−n‖u∗n − p‖`2(In) = 0.

The proof is organized as follows, first we prove

limn→∞

2−n‖u∗n − p‖`2(Λn) = 0,

thenlimn→∞

2−n‖u∗n − p‖`2(In\Λn) = 0.

Since u∗n is a minimizer of Qn, it holds that

2(s1−2)nQn(u∗n)

≤ 2(s1−2)nQn(p)≤ 2(s1−2)n(�2n + ‖diag(λ)Wp‖`1(In))

≤ 2(s1−2)n(�2n + 2

(s−s0)n∑|i|≥1

(‖Wip‖`1(Mn) + ‖Wip‖`1(In\Mn)))

≤ 2(s1−2)n�2n +∑|i|≥1

2(s1−s0)n2sn2−n∑α∈Mn

|〈f, ψi,n−1,α〉|+ 2(s+s1−s0−2)nC12nM

≤ 2(s1−2)n�2n +∑|i|≥1

2(s1−s0)n(∑α∈Mn

22sn|〈f, ψi,n−1,α〉|2)12 + 2(s+s1−s0−1)nC1M.

24

For each sub-square matrix with length < 4δ, there exist at most 22s13n+2 total pixel

positions. It follows that∑α∈In\Λn

|u∗n(α)− p(α)|2

≤ 2s13n2

2s13n48( ∑α∈In

(|∇e1u∗n(α)|2 + |∇e2u∗n(α)|2) +∑α∈In

(|∇e1p(α)|2 + |∇e2p(α)|2)

+∑α̃∈Λn

|u∗n(α̃)− p(α̃)|2)

≤ 2s1n48M( ∑α∈In

(|∇e1u∗n(α)|+ |∇e2u∗n(α)|) +∑α∈In

(|∇e1p(α)|+ |∇e2p(α)|)

+ ‖u∗n − p‖2`2(Λn)).

This together with Lemma 2.7 implies∑α∈In\Λn

|u∗n(α)− p(α)|2 ≤ 2s1n48M( ∑|i|≥1

(‖Wiu∗n‖`1(In) + ‖Wip‖`1(In)) + ‖u∗n − p‖2`2(Λn)

).

Therefore, by (3.11) we conclude

limn→∞

2−2n‖u∗n − p‖2`2(In\Λn)

≤ limn→∞

2s1n2−2n48M( ∑|i|≥1

(‖Wiu∗n‖`1(In) + ‖Wip‖`1(In)) + ‖u∗n − p‖2`2(Λn)

)≤ lim

n→∞2(s1−2)n48M(C + 1)

(λ−1i �

2n +

∑|i|≥1

‖Wip‖`1(In) + ‖u∗n − p‖2`2(Λn)

)≤ lim

n→∞48M(C + 1)

(2(s1−2)nλ−1i �

2n +

∑|i|≥1

2(s1−2)n‖Wip‖`1(In) + 2(s1−2)n‖u∗n − p‖2`2(Λn)

)≤ lim

n→∞48M(C + 1)

(2(s1−2)nλ−1i �

2n +

∑|i|≥1

2(s1−2)n‖Wip‖`1(Mn) + 2(s1−2)n‖Wip‖`1(In\Mn)

+ 2(s1−2)n‖u∗n − p‖2`2(Λn))

= 0,

where the last equality follows from (3.4) and (3.10).

Remark: The parameters λi in the model Qn(u) are chosen of the same order tomake all of the wavelet channels ‖Wiu∗n‖`1(Mn) decay for i 6= (0, 0). They should beneither too large in order to fit the observations, nor be too small in order to permeatethe missing region. If f is continuous and the observed values v in (3.9) are defined byv[α] = f(2−nα) + �[α], α ∈ Λn, an asymptotic analysis of the inpainting model can bedone similarly.

26

4 Algorithm and Experiments

In this section we explain how to numerically solve the minimization problem

minu

n∑i=1

(∑α∈I

u(α)φ(ξih− α)− y(ξi))2 + ‖diag(λ)Wu‖`1 (4.1)

and further present some numerical experiments. The problem (4.1) can be written inmatrix vector form as

minu∈Rm

‖Au− y‖2`2 + ‖diag(λ)Wu‖`1 , (4.2)

with y = [y(ξ1), y(ξ2), . . . , y(ξn)]T , Aij = φ(ξi/h − kj), Ξ = {ξ1, . . . , ξn} and I =

{k1, . . . , km}. So, (4.2) is an ordinary least squares problem with an `1-regularizationterm. It means that this problem cannot be solved straight forward by solving only onesystem of equations. But there are iterative solvers like the split Bregman algorithmwhich “splits” the `2-problem from the `1-problem and then uses the Bregman iteration,see, e.g., [20, 8]. The iteration step i→ i+ 1 of the split-Bregman algorithm in terms ofproblem (4.2) reads

ui+1 = arg minu‖Au− y‖2`2 +

µ

2‖Wu− di + bi‖2`2 (4.3)

di+1 = Tλ/µ(Wui+1 + bi) (4.4)bi+1 = bi +Wui+1 − di+1 (4.5)

with initial u0 = 0, d0 = 0 and b0 = 0. For µ > 0, Tλ/µ is the soft-threshold operator

Tλ/µ(x) := [tλ/µ(x1), tλ/µ(x2), . . . , tλ/µ(xM )] ,

with tλ/µ(xi) := sgn(xi) max{0, |xi| − λµ}. The stoping criteria of the iteration is

‖di −Wui‖`2 ≤ �

for some positive constant �. The solution to (4.3) can be determined by solving thesystem of equations

(2ATA+ µWTW)u = 2ATy + µWT (di − bi)

which, because of WTW = I, can be simplified to

(2ATA+ µI)u = 2ATy + µWT (di − bi) . (4.6)

Since (2ATA + µI) is symmetric positive definite, the system of equations (4.6) can beefficiently solved by applying a conjugate gradient method. Further note, that WT (di−bi) in (4.6) is determined by performing the inverse framelet transform rather than byusing its matrix representation, similar in the iterations (4.4) and (4.5).

27

Wavelet frame based image restoration has found various applications, for instanceimage denoising, inpainting and scene reconstruction, see [6, 8, 24, 36, 7, 13] and thereferences therein. Below, we also present two numerical examples. First we reconstructa piecewise continuous function from some scattered data samples; second we fit theinteraction force function between water and water molecules in terms of their distance,which is a simple example in molecular dynamics simulations [32].

4.1 Piecewise continuous function

In the first experiment we show the advantage of the approach (1.2) by reconstructinga piecewise continuous function on Ω = [0, 1]2 from some noisy samples. We constructa testfunction by using the well-known Franke function, which is the weighted sum offour exponential functions

franke(x, y) =3

4e−((9x−2)

2+(9y−2)2)/4 +3

4e−((9x+1)

2)/49−(9y+1)/10

+1

2e−((9x−7)

2+(9y−3)2)/4 − 15e−(9x−4)

2−(9y−7)2 .

The function was introduced in [18] and is widely used in scattered data approximationbenchmarks. To obtain our piecewise continuous test function f we substract 0.2 fromthe Franke function on the domain [0.25, 0.75]2, i.e.,

f(x, y) =

{franke(x, y)− 0.2 if (x, y) ∈ [0.25, 0.75]2

franke(x, y) else. (4.7)

We randomly sample sites {ξi}4000i=1 from Ω and choose the scattered data samples asy(ξi) = f(ξi) + �i with �i drawn from a normal distribution with 0 mean and standarddeviation 0.02. As scaling parameter we choose h = 1/180. We choose the cubic B-spline

B4(x) =

x3/6 if 0 ≤ x < 1(−3x3 + 12x2 − 12x+ 4)/6 if 1 ≤ x < 2(3x3 − 24x2 + 60x− 44)/6 if 2 ≤ x < 3(4− x)3/6 if 3 ≤ x < 40 else

as generator for the shift invariant subspace and its associated tight wavelet frame systemwith the corresponding masks h1 = [1/16,−1/4, 3/8,−1/4, 1/16], h2 = [−1/8, 1/4, 0,− 1/4, 1/8], h3 = [

√6/16, 0,−

√6/8, 0,

√6/16] and h4 = [−1/8,−1/4, 0, 1/4, 1/8].

In figure 1a the approximation which is obtained by minimizing (2.9) is depicted,whereas in figure 1b the approximation with a Laplacian regularization is depicted,i.e., ‖Au − y‖2`2 + νu

∗Lu with L the discrete Laplacian is minimized to determine theapproximant. It can be well seen that the Laplacian regularization which punishes the

28

roughness is not able to preserve the discontinuities, whereas the approach (2.9) is ableto do so.

(a) Approximation by minimizing (2.9) (b) Approximation using Laplacian regu-larization

Figure 1: Approximation of 4000 samples from testfunction (4.7)

4.2 Application to CG models in Molecular dynamics

Molecular dynamics (MD) numerically simulates the interactions of atoms. This hascontributed key insight into structural biology, one of the most vibrant research fields inscience, which advanced also because of the technological progress in spectroscopy andmicroscopy. Simulating the interactions of atoms inside large system is computationallyvery challenging. Coarse grained (CG) models [34] were introduced to provide a compu-tational efficient concept. A CG-model is a simplified version of all atom representation,it simplifies according to the molecule structure and combines several atoms in one sin-gle interaction site. These simplified models are used for the rapid investigation of longtime- and length-scale processes in many important biological and soft matter processes.

To parameterize the interactions between different CG sites is an important researcharea in MD. These interactions (functions) are usually represented in term of theirdistance, bond angle or dihedral angle, etc. Assume a system with water molecules(H2O) at constant temperature and volume. The simplified CG-model handles everywater molecule as one CG site W . Hence, the non-bonded interactions of types O-O,O-H, H-H and bonded interactions of O-H, H-O-H in all atom water system are notneeded to be investigated. The only interaction type in the CG model which have to beconsidered are the non-bonded interaction W-W. (See figure 2a)

In our numerical example a cubic box containing 999 water molecules was simulatedto form atomistic trajectory using the GROMACS MD simulation package [4]. Thesimulation time step is set to 2 fs. The force matching method [34, 31] was applied tothe atomistic configurations to generate the CG potentials, and 10 ns of the trajectorywas used for analysis.

29

In the CG-model three atoms in one water molecule are combined to one singleCG-site W. For every frame of the trajectory, the position of each CG site W (ri =(xi, yi, zi))

999i=1 is obtained by the centers of geometry of three atoms in the corresponding

water molecule. In this way, each site Wi at position ri = (xi, yi, zi) is calculated by thetrajectory of the water atoms. The net force Fi = (F

xi , F

yi , F

zi ) which is acting on each

Wi is calculated based on Newton’s equations of motion, which should equal to the sumof the forces Wj(j 6= i) acting upon Wi. Given the net force Fi of each Wi, our aim isto derive the interaction force function between W and W in terms of their distance.

Let exi,j :=xj−xi|rj−ri| , e

yi,j :=

yj−yi|rj−ri| and e

zi,j :=

zj−zi|rj−ri| . The intermolecular radial dis-

tribution functions (RDF) (|rj − ri|)j 6=i are considered as our scattered data sites. Weapproximate the force function f between W and W in terms of Euclidean distance fromSh(φ,Ω) by solving

minu

∑i

∑ζ∈{x,y,z}

( N∑j=1,j 6=i

g(|rj − ri|)eζi,j − Fζi

)2+ ‖diag(λ)Wu‖`1 ,

in which g =∑

α u(α)φ(·h−α), φ = B4,W is the framelet transform with cubic framelets,

N = 999 and h = 0.005 nm. If u∗ is a minimizer, then f can be approximated by∑α u∗(α)φ( ·h − α).

The distribution of the RDF represents the probability of finding the relevant moleculesat a distance of (r, r + dr) as a function of the distance r. Most of the CG sites arelocated in a low energy state, whereas only a few CG sites are located in the high energystate, by the Boltzmann distribution in statistical mechanics [19]. Thus, the data sites(|rj − ri|)j 6=i in our model are very scattered. Moreover, due to inadequate sampling ofphase space and random fluctuations in the measurements, the data ri = (xi, yi, zi) andFi = (F

xi , F

yi , F

zi ) are usually very noisy.

In order to fit the force well while suppress noise, the parameters λ are choseninversely proportional to the RDF of experimental molecules, which is the cardinalityof #{|rj − ri| : |rj − ri| ∈ suppφ( ·h − α)}. The idea is that for the poor samplingregion, we put less trust, and large λ are chosen. On the contrary, for the relativelyadequate sampling region, we put more trust, and small λ are chosen. This trust regionregularization method can suppress noise over signal, provide appropriately smoothedresults, and fit the important features of the force curve. In figure 2b the curve offitting with efficient data can be seen as a benchmark, which was based on 10000 framesof the trajectory data. While for 30 frames of the trajectory data, it can be seen thatcompared with Laplacian regularization our approach preserves the minima better, whichis important for MD simulations.

30

(a) Water molecules in a box (b) Approximation of interaction force ofWater-Water by 999 water molecules

Figure 2: Approximation of interaction force of Water-Water molecules

Acknowledgement: The work of Jianbin Yang was partially supported by the re-search grant #11101120 from NSFC and the Fundamental Research Funds for the Cen-tral Universities, China; the work of Dominik Stahl was partially supported by the Ger-man Academic Exchange Service (DAAD); and the work of Zuowei Shen was partiallysupported by Singapore MOE AcRF Research Grant MOE2011-T2-1-116 and R-146-000-165-112.

References

[1] R. Arcangli, M. C. L. de. Silanes, J. J. Torrens. An extension of a bound for functionsin Sobolev spaces, with applications to (m, s)-spline interpolation and smoothing.Numer. Math., 107(2): 181-211, 2007.

[2] M. Arigovindan, M. Sühling, P. Hunziker and M. Unser. Variational image recon-struction from arbitrarily spaced samples: A fast multiresolution spline solution.IEEE Trans. Image Process., 14(4): 450-460, 2005.

[3] W. Beckner. Inequalities in Fourier analysis. Ann. of Math., 102(1): 159-182, 1975.

[4] H. J. C. Berendsen, D. van der Spoel and R. van Drunen. GROMACS: A message-passing parallel molecular dynamics implementation. Comput. Phys. Commun.,91(1): 43-56, 1995.

[5] L. Borup, R. Gribonval, M. Nielsen. Bi-framelet systems with few vanishing mo-ments characterize Besov spaces. Appl. Comput. Harmon. Anal., 17(1): 3-28, 2004.

[6] J. F. Cai, R. H. Chan, and Z. Shen. A framelet-based image inpainting algorithm.Appl. Comput. Harmon. Anal., 24 (2): 131-149, 2008

31

[7] J. F. Cai, B. Dong, S. Osher, and Z. Shen. Image restoration: total variation,wavelet frames, and beyond. J. Amer. Math. Soc., 25(4): 1033-1089, 2012.

[8] J. F. Cai, S. Osher, and Z. Shen. Split Bregman methods and frame based imagerestoration. Multiscale Model. Simul., 8(2): 337-369, 2009.

[9] J. F. Cai, Z. Shen, G. B. Ye. Approximation of frame based missing data recovery.Appl. Comput. Harmon. Anal., 31(2): 185-204, 2011.

[10] A. Chambolle, R. A. DeVore, N. Y. Lee, B. J. Lucier. Nonlinear wavelet imageprocessing: variational problems, compression, and noise removal through waveletshrinkage. IEEE Trans. Image Process., 7(3): 319-335, 1998.

[11] C. K. Chui. Multivariate splines. CBMS-NSF Regional Conference Series in AppliedMathematics. Society for Industrial and Applied Mathematics, 1988.

[12] I. Daubechies, B. Han, A. Ron and Z. Shen. Framelets: MRA-based constructionsof wavelet frames. Appl. Comput. Harmon. Anal., 14(1): 1-46, 2003.

[13] B. Dong, Q. T. Jiang, Z. Shen. Image restoration: wavelet frame shrinkage, nonlin-ear evolution PDEs, and beyond, manuscript.

[14] B. Dong, Z. Shen. Pseudo-splines, wavelets and framelets. Appl. Comput. Harmon.Anal., 22(1): 78-104, 2007.

[15] B. Dong, Z. Shen. MRA Based Wavelet Frames and Applications, IAS LectureNotes Series, Summer Program on The Mathematics of Image Processing, ParkCity Mathematics Institute, 2010.

[16] J. Duchon. Sur l’erreur d’interpolation des fonctions de plusieur variables par lesDm-splines. RAIRO Anal. Numer. 12(4): 325-334, 1978.

[17] T. Dupont, R. Scott. Polynomial approximation of functions in Sobolev spaces.Math. Comp., 34(150): 441-463, 1980.

[18] R. Franke. A critical comparison of some methods for interpolation of scattereddata. Naval Postgraduate School Monterey Ca, 1979.

[19] D. Frenkel, B. Smit. Understanding molecular simulation: from algorithms to ap-plications. Academic press, 2001.

[20] T. Goldstein, S. Osher. The split Bregman method for L1-regularized problems.SIAM J. Im. Sc., 2(2): 323-343, 2009.

[21] M. Gross, F. Pfister. Point-based Graphics. Morgan Kaufmann Series in ComputerGraphics, 2007.

[22] B. Han, Z. Shen. Wavelets with short support. SIAM J. Math. Anal., 38(2): 530-556,2006.

32

[23] B. Han, Z. Shen. Dual wavelet frames and Riesz bases in Sobolev spaces. Constr.Approx., 29(3), 369-406, 2009.

[24] H. Ji, Z. Shen, and Y. H. Xu. Wavelet frame based scene reconstruction from rangedata. J. Comput. Phys., 229(6): 2093-2108, 2010.

[25] R. Q. Jia. Approximation with scaled shift-invariant spaces by means of quasi-projection operators. J. Approx. Theory, 131(1): 30-46, 2004.

[26] R. Q. Jia, J. Lei. Approximation by multiinteger translates of functions havingglobal support. J. Approx. Theory, 72(1): 2-23, 1993.

[27] R. Q. Jia, C. A. Micchelli, Using the refinement equations for the construction ofpre-wavelets II: Powers of two, in Curves and Surfaces, P. J. Laurent, A. Laurent,A. Le Meh́auté, and L. L. Schumaker (Eds.), Academic Press, New York, 209-246,1991.

[28] M. J. Johnson. Scattered data interpolation from principal shift-invariant spaces.J. Approx. Theory, 113(2): 172-188, 2001.

[29] M. J. Johnson, Z. Shen, Y. H. Xu. Scattered data reconstruction by regularizationin B-spline and associated wavelet spaces. J. Approx. Theory, 159(2): 197-223, 2009.

[30] J. Li. Wavelet approximation and image restoration. Ph.D. Thesis. National Uni-versity of Singapore, 2013.

[31] L. Lu, J. F. Dama, G. A. Voth. Fitting coarse-grained distribution functions throughan iterative force-matching method. J. Chem. Phys., 139(12): 121906-1-10, 2013.

[32] L. Lu, Z. Shen, D. Tong, J. Yang. B-spline tight frames based force matchingmethod, manuscript, 2015.

[33] S. Mallat. A wavelet tour of signal processing: the sparse way. Academic press,2008.

[34] W. G. Noid, P. Liu, Y. Wang, J. W. Chu, G. S. Ayton, S. Izvekov, H. C. Andersen,and G. A. Voth. The multiscale coarse-graining method. II. Numerical implemen-tation for coarse-grained molecular models. J. Chem. Phys., 128(24): 244115-1-20,2008.

[35] A. Ron and Z. Shen. Affine systems in L2(Rd): The analysis of the analysis operator.J. Funct. Anal., 148(2): 408-447, 1997.

[36] Z. Shen. Wavelet frames and image restorations. Proceedings of the InternationalCongress of Mathematicians, Hyderabad, India, 2010.

[37] Z. Shen, Z. Xu. On B-spline framelets derived from the unitary extension principle.SIAM J. Math. Anal., 45(1): 127-151, 2013.

33

[38] G. Strang, G. Fix. A Fourier analysis of the finite-element variational method. In:Constructive Aspects of Functional Analysis. G. Geymonat (ed.), C.I.M.E., 793-840,1973.

[39] H. Wendland. Scattered data approximation. Cambridge monographs on appliedand computational mathematics. Cambridge University Press, 2005.

[40] J. Xian, S. Li. Sampling set conditions in weighted multiply generated shift-invariantspaces and their applications. Appl. Comput. Harmon. Anal. 23(2): 171-180, 2007.

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IntroductionAnalysis of the modelNotation and PreliminariesAsymptotic approximation analysisProof of Proposition 2.1 and 2.2

Applications to image restoration modelsImage inpaintingImage denoisingImage inpainting (Continued)

Algorithm and ExperimentsPiecewise continuous functionApplication to CG models in Molecular dynamics

an analysis of wavelet frame based scattered data reconstruction · 2019. 1. 28. · an analysis of...

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