# Compactly Supported Curvelet-Type Systems

Authors: Kenneth N. Rasmussen; Morten Nielsen
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

We study a flexible method for constructing curvelet-type frames. These curvelet-type systems have the same sparse representation properties as curvelets for appropriate classes of smooth functions, and the flexibility of the method allows us to give a constructive description of how to construct curvelet-type systems with a prescribed nature such as compact support in direct space. The method consists of using the machinery of almost diagonal matrices to show that a system of curvelet molecules which is sufficiently close to curvelets constitutes a frame for curvelet-type spaces. Such a system of curvelet molecules can then be constructed using finite linear combinations of shifts and dilates of a single function with sufficient smoothness and decay.


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## Body

## 1. Introduction

Second-generation curvelets were introduced by Candès and Donoho, who also proved that curvelets give an essentially optimal sparse representation of images (functions) that areC2 except for discontinuities along piecewise C2-curves [1]. It follows that efficient compression of such images can be archived by thresholding their curvelet expansions. Curvelets form a multiscale system with effective support that follows a parabolic scaling relation width≈length2. Moreover, they also provide an essentially optimal sparse representation of Fourier integral operators [2] and an optimal sparse and well organized solution operator for a wide class of linear hyperbolic differential equations [3]. However, curvelets are band-limited, and contrary to wavelets it is an open question whether compactly supported curvelet-type systems exist.In this paper we study a flexible method for generating curvelet-type systems with the same sparse representation properties as curvelets (when sparseness is measured in curvelet-type sequence spaces). The method uses a perturbation principle which was first introduced in [4], further generalized in [5], and refined for frames in [6]. We give a constructive description of how to construct curvelet-type system consisting of finite linear combinations of shifts and dilates of a single function with sufficient smoothness and decay. This gives the flexibility to construct curvelet-type systems with a prescribed nature (see Section 6) such as compact support in direct space. For the sake of convenience the construction will only be done in ℝ2, but it can easily be extended to ℝd. The main results can be found in Sections 4 and 5.The curvelet-type sequence spaces we use are associated with curvelet-type spacesGp,qs which were introduced in [7]. Here Gp,qs was constructed by applying a curvelet-type splitting of the frequency space to a general construction of decomposition spaces, thereby obtaining a natural family of smoothness spaces for which curvelets constitute frames (see Section 2). Originally, this construction of decomposition spaces based on structured coverings of the frequency space was introduced by Feichtinger and Gröbner [8] and Feichtinger [9]. For example, the classical Triebel-Lizorkin and Besov spaces correspond to dyadic coverings of the frequency space (see [10]).The outline of the paper is as follows. In Section2 we define second-generation curvelets and curvelet-type spaces. Furthermore, we introduce curvelet molecules which will be the building blocks for our compactly supported curvelet-type frames. Next, in Section 3 we use the properties of curvelet molecules to show that the “change of frame coefficient” matrix is almost diagonal if the curvelet molecules have sufficient regularity. With the machinery of almost diagonal matrices, we can then in Section 4 show that curvelet molecules which are close enough to curvelets constitute frames for the curvelet-type spaces. Finally, in Section 5 we give a constructive description of how to construct these curvelet molecules from finite linear combinations of shifts and dilates of a single function with sufficient smoothness and decay. We conclude the paper with a short discussion in Section 6 of the possible functions which can be used to construct the curvelet molecules.


## 2. Second-Generation Curvelets

We begin this section with a brief definition of curvelets and curvelet molecules which will later be used to construct curvelet-type frames. Furthermore, we define the curvelet-type spaces for which curvelets constitute frames. For a much more detailed discussion of the curvelet construction, we refer the reader to [1, 3], and for decomposition spaces, of which the curvelet-type spaces are a subclass, we refer to [7, 8].Letν be an even C∞(ℝ) window that is supported on [-π,π] such that its 2π-periodic extension obeys
(2.1)|ν(θ)|2+|ν(θ-π)|2=1,θ∈[0,2π).
Define νj,l(θ):=ν(2⌊j/2⌋θ-πl) for j≥2 and l=0,1,…,2⌊j/2⌋-1. Next, with the angular window in place, let w∈Cc∞(ℝ) obey
(2.2)|w0(t)|2+∑j≥2|w(2-jt)|2=1,t∈R,
with w0∈Cc∞(ℝ) supported in a neighborhood of the origin. We then define
(2.3)ϕj,l(ξ):=w(2-j|ξ|)(νj,l(θ)+νj,l(θ+π)),ξ=|ξ|(cos⁡⁡θ,sin⁡⁡θ)∈R2.
Notice that the support of w(2-j|ξ|)νj,0(θ) is contained in a rectangle Rj=I1j×I2j given by
(2.4)I1j:={ξ1,tj≤ξ1≤tj+Lj},I2j:={ξ2,2|ξ2|≤lj},
where tj is determined uniquely for a minimal Lj,Lj:=δ1π2j and lj:=δ22π2j/2 (δ1 depends weakly on j, see [1, Section 2.2]). With Ĩ1j:=±I1j and R̃j=Ĩ1j×I2j the system
(2.5)ej,k(ξ):=2-3j/42πδ1δ2ei((k1+1/2)2-jξ1/δ1)ei(k22-j/2ξ2/δ2),k∈Z2,
is an orthonormal basis for L2(R̃j).We letf̂(ξ):=ℱ(f)(ξ):=(2π)-1∫ℝ2f(x)e-ix·ξdx,f∈L1(ℝ2), and by duality extend it uniquely from 𝒮(ℝ2) to 𝒮′(ℝ2). Finally, we define(2.6)η̂μ(ξ):=ϕj,l(ξ)ej,k(Rθμ⊤ξ),μ=(j,l,k),
where Rθμ is rotation by the angle θμ:=π2⌊-j/2⌋l, and as coarse-scale elements we define η̂1,0,k(ξ):=δ0-1ϕ1,0(ξ)eik·ξ/δ0, where ϕ1,0(ξ):=ω0(|ξ|) and δ0>0 is sufficiently small. The system {ημ}μ∈𝒥×ℤ2 is called curvelets, 𝒥:={(j,l)∣j≥1,l=0,1,…,2⌊j/2⌋-1}. It can be shown that curvelets constitute a tight frame for L2(ℝ2) (see [1, Section 2.2]).To later construct curvelet-type frames, we need a system of functions which share the essential properties of curvelets. As we will see, curvelet molecules, which were introduced in [3] and used there to study hyperbolic differential equations, have all the properties we need. For κ∈ℕ02, we define |κ|:=κ1+κ2, and for suitably differentiable functions we define f(κ):=∂|κ|f/(∂ξ1κ1∂ξ2κ2).Definition 2.1.
A family of functions{ψμ}μ∈𝒥×ℤ2 is said to be a family of curvelet molecules with regularityR, R∈ℕ, if for j≥2 they may be expressed as
(2.7)ψμ(x)=23j/4aμ(D2-jRθμx-(k1δ1,k2δ2)),
where D2-jx=(2jx1,2j/2x2),δ1,δ2>0 and all functions aμ satisfy the following.(i)
For|κ|≤R there exist constants C>0 independent of μ such that
(2.8)|aμ(κ)(x)|≤C(1+|x|)-2R.(ii)
There exist constantsC>0 independent of μ such that
(2.9)|âμ(ξ)|≤Cmin⁡(1,2-j+|ξ1|+2-j/2|ξ2|)R.The coarse-scale molecules, j=1, must take the form ψμ(x)=aμ(x-k/δ0),δ0>0, where aμ satisfies (2.8).It can be shown that curvelets constitute a family of curvelet molecules with regularityR for any R∈ℕ (see [3, page 1489]).To define the curvelet-type spaces which together with the associated sequence spaces will characterize the sparse representation properties of curvelets, we need a suitable partition of unity.Definition 2.2.
LetQj,l:=supp(ϕj,l) for (j,l)∈𝒥, where ϕj,l was defined in (2.3). A bounded admissible partition of unity (BAPU) is a family of functions {Ψj,l}(j,l)∈𝒥⊂𝒮:=𝒮(ℝ2) satisfying:(i)
supp(Ψj,l)⊆Qj,l,(j,l)∈𝒥;(ii)
∑(j,l)∈𝒥Ψj,l(ξ)=1,ξ∈ℝ2;(iii)
sup(j,l)∈𝒥|Qj,l|1/p-1∥ℱ-1Ψj,l∥Lp(ℝ2)<∞,p∈(0,1].An example of a BAPU is{|ϕj,l|2}(j,l)∈𝒥 which follows from the construction of ϕj,l (see (2.1) and (2.2)) and curvelets being curvelet molecules with regularity R for any R∈ℕ. We are now ready to define curvelet-type spaces.Definition 2.3.
Let{Ψj,l}(j,l)∈𝒥 be a BAPU and Ψj,l(D)f:=ℱ-1(Ψj,lℱf). For s∈ℝ,0<q<∞ and 0<p≤∞, we define Gp,qs:=Gp,qs(ℝ2) as the set of distributions f∈𝒮′:=𝒮′(ℝ2) satisfying
(2.10)‖f‖Gp,qs:=(∑(j,l)∈J‖2jsΨj,l(D)f‖Lpq)1/q<∞.It can be shown thatGp,qs is a quasi-Banach space (Banach space for p,q≥1), and 𝒮 is dense in Gp,qs (see [7, 8]). Furthermore, Gp,qs is independent of the choice of BAPU.We also need the sequence spaces associated with the curvelet-type spaces. For the sake of convenience, we write∥fk∥ instead of ∥{fk}k∈K∥ when the index set is clear from the context.Definition 2.4.
Fors∈ℝ,0<q<∞ and 0<p≤∞, we define the sequence space gp,qs as the set of sequences {zμ}μ∈𝒥×ℤ2⊂ℂ satisfying
(2.11)‖zμ‖gp,qs:=‖2j(s+(3/2)(1/2-1/p))(∑k∈Z2|zμ|p)1/p‖lq<∞,
where the lp-norm is replaced with the l∞-norm if p=∞.Notice that the sequence spaceslq are special cases of gp,qs as we have gq,q-(3/2)(1/2-1/q)=lq.Next, we introduce frames forGp,qs and use the notation F≍G when there exist two constants 0<C1≤C2<∞, depending only on “allowable” parameters, such that C1F≤G≤C2F.Definition 2.5.
We say that a family of functions{ψμ}μ∈𝒥×ℤ2 in the dual of Gp,qs is a frame for Gp,qs if for all f∈Gp,qs we have
(2.12)‖f‖Gp,qs≍‖〈f,ψμ〉‖gp,qs.
The following is called the frame expansion of {ψμ}μ∈𝒥×ℤ2 when it exists:
(2.13)f=∑μ∈J×Z2〈f,S-1ψμ〉ψμ
in the sense of 𝒮′, where S is the frame operator Sf=∑μ∈𝒥×ℤ2〈f,ψμ〉ψμ, f∈Gp,qs.From [7, Lemma 4 and Section 7.3] we have that curvelets (2.6) constitute a frame for the curvelet-type spaces with a frame operator S that is equal to the identity, S=I.Proposition 2.6.
Assume thats∈ℝ, 0<q<∞ and 0<p≤∞. For any finite sequence {zμ}μ∈𝒥×ℤ2⊂ℂ, one has
(2.14)‖∑μ∈J×Z2zμημ‖Gp,qs≤C‖zμ‖gp,qs.
Furthermore, {ημ}μ∈𝒥×ℤ2 is a frame for Gp,qs with frame operator S=I,
(2.15)‖f‖Gp,qs≍‖〈f,ημ〉‖gp,qs,f∈Gp,qs.Notice that frame expansions for two frames{ψμ}μ∈𝒥×ℤ2 and {ημ}μ∈𝒥×ℤ2 have the same sparseness when measured in the associated sequence space gp,qs if {S-1ψμ}μ∈𝒥×ℤ2 and {S-1ημ}μ∈𝒥×ℤ2 also constitute frames for Gp,qs,
(2.16)‖〈f,S-1ψμ〉‖gp,qs≍‖f‖Gp,qs(R2)≍‖〈f,S-1ημ〉‖gp,qs.
Hence, to get a curvelet-type system {ψμ}μ∈𝒥×ℤ2 with the same sparse representation properties as curvelets {ημ}μ∈𝒥×ℤ2, it suffices to prove that {S-1ψμ}μ∈𝒥×ℤ2 constitutes a frame for Gp,qs.


## 3. Almost Diagonal Matrices

To generate curvelet-type frames in the following sections we introduce the machinery of almost diagonal matrices in this section. Almost diagonal matrices where used in [11] on Besov spaces, and here we find an associated notion of almost diagonal matrices on gp,qs. The goal is to find a definition so that the composition of two almost diagonal matrices gives a new almost diagonal matrix and almost diagonal matrices are bounded on gp,qs.To help us define almost diagonal matrices we use a slight variation of the pseudodistance introduced in [12] which was constructed in [3]. For this we need the center of ημ in direct space, xμ:=Rθμ(k12-j/δ1,k22-j/2δ2), and the “direction” of ημ,ρμ:=(cosθμ,sinθμ).Definition 3.1.
Given a pair of indicesμ=(j,l,k) and μ′=(j′,l′,k′), we define the dyadic-parabolic pseudodistance as
(3.1)ω(μ,μ′):=2|j-j′|(1+min⁡(2j,2j′)d(μ,μ′)),
where
(3.2)d(μ,μ′):=|θμ-θμ′|2+|xμ-xμ′|2+|〈ρμ,xμ-xμ′〉|.The dyadic-parabolic distance was studied in detail in [3], and from there we can deduce the following properties.(i)
Forδ>0 there exists C>0 such that
(3.3)∑k∈Z2ω(μ,μ′)-3/2-δ≤C.(ii)
Forδ>0 there exists C>0 such that
(3.4)∑(j,l)∈Jω(μ,μ′)-1/2-δ≤C.(iii)
ForN≥2 and δ>0 there exists C>0 such that
(3.5)∑μ′′∈J×Z2ω(μ,μ′′)-N-δω(μ′′,μ′)-N-δ≤Cω(μ,μ′)-N-δ/2.(iv)
Let{ψμ}μ∈𝒥×ℤ2 and {ημ}μ∈𝒥×ℤ2 be two families of curvelet molecules with regularity 4R, R∈ℕ. Then there exists C>0 such that
(3.6)|〈ψμ,ημ′〉|≤Cω(μ,μ′)-R.These properties lead us to the following definition of almost diagonal matrices ongp,qs.Definition 3.2.
Assume thats∈ℝ, 0<q<∞ and 0<p≤∞. Let r:=min(1,p,q) and t:=s+(3/2)(1/2-1/p). A matrix A={aμμ′}μ,μ′∈𝒥×ℤ2 is called almost diagonal ongp,qs if there exists C,δ>0 such that
(3.7)|aμμ′|≤C2(j′-j)tω(μ,μ′)-2/r-δ.Remark 3.3.
Note that by using (3.5), we get that the composition of two almost diagonal matrices on gp,qs gives a new almost diagonal matrix on gp,qs.We are now ready to show the most important property of almost diagonal matrices; they act boundedly on the curvelet-type spaces.Proposition 3.4.
IfA is almost diagonal on gp,qs, then A is bounded on gp,qs.Proof.
We only prove the result forp<∞ as the result for p=∞ follows in a similar way with lp replaced by l∞. Let ω0(μ,μ′):=ω(j,l,0,j′,l′,0),z:={zμ}μ∈𝒥×ℤ2∈gp,qs, and assume for now that p≥1. We begin with looking at the lp-norm of ∥Az∥gp,qs. By using Minkowski’s inequality, Hölder’s inequality and (3.3) we get
(3.8)(∑k∈Z2|(Az)μ|p)1/p≤C(∑k∈Z2(∑(j′,l′)∈J2(j′-j)tω0(μ,μ′)-1/2r-δ/2∑k′∈Z2|zμ′|ω(μ,μ′)-3/2r-δ/2)p)1/p≤C∑(j′,l′)∈J2(j′-j)tω0(μ,μ′)-1/2r-δ/2(∑k∈Z2(∑k′∈Z2|zμ′|ω(μ,μ′)-3/2r-δ/2)p)1/p≤C∑(j′,l′)∈J2(j′-j)tω0(μ,μ′)-1/2r-δ/2×(∑k∈Z2∑k′∈Z2|zμ′|pω(μ,μ′)-3/2r-δ/2(∑k′∈Z2ω(μ,μ′)-3/2r-δ/2)p-1)1/p≤C∑(j′,l′)∈J2(j′-j)tω0(μ,μ′)-1/2r-δ/2(∑k′∈Z2|zμ′|p)1/p.
We then have
(3.9)‖Az‖gp,qs≤C(∑(j,l)∈J(∑(j′,l′)∈J2j′tω0(μ,μ′)-1/2r-δ/2(∑k′∈Z2|zμ′|p)1/p)q)1/q.
For q≥1 we use Hölder’s inequality and (3.4) to get
(3.10)‖Az‖gp,qs≤C(∑(j,l)∈J∑(j′,l′)∈J2j′qtω0(μ,μ′)-1/2r-δ/2×(∑k′∈Z2|zμ′|p)q/p(∑(j′,l′)∈Jω0(μ,μ′)-1/2r-δ/2)q-1)1/q≤C‖z‖gp,qs.
For q<1 the result follows by a direct estimate. The case p<1 remains, and here we first observe that
(3.11)Ã:={ãμμ′}μ,μ′∈J×Z2={|aμμ′|p2(j′-j)(t-tp)}μ,μ′∈J×Z2
is almost diagonal on g1,q/ps. Furthermore, if we let v:={vμ}μ∈𝒥×ℤ2:={|zμ|p2-j(t-tp)}μ∈𝒥×ℤ2 we have
(3.12)‖v‖g1,q/ps1/p=(∑(j,l)∈J(∑k∈Z22jtp|zμ|p)q/p)1/q=‖z‖gp,qs.
Before we can put these two observations into use, we need that
(3.13)|(Az)μ|p≤∑(j′,l′)∈J∑k′∈Z2|aμμ′|p|zμ′|p=2j(t-tp)∑(j′,l′)∈J∑k′∈Z2ãμμ′vμ.
We then have
(3.14)‖Az‖gp,qs≤‖Ãv‖g1,q/ps1/p≤C‖v‖g1,q/ps1/p=C‖z‖gp,qs.


## 4. Curvelet-Type Frames

In this section we study a family of curvelet molecules{ψμ}μ∈𝒥×ℤ2 which is a small perturbation of curvelets {ημ}μ∈𝒥×ℤ2. The goal is first to show that if {ψμ}μ∈𝒥×ℤ2 is close enough to {ημ}μ∈𝒥×ℤ2, then {ψμ}μ∈𝒥×ℤ2 is a frame for Gp,qs. Next to get a frame expansion, we show that {S-1ψμ}μ∈𝒥×ℤ2 is also a frame. The results are inspired by [6] where perturbations of frames were studied in Triebel-Lizorkin and Besov spaces.Let{ψμ}μ∈𝒥×ℤ2⊂L2(ℝ2) be a system that is close to {ημ}μ∈𝒥×ℤ2 in the sense that there exists ɛ,δ>0 such that for j≥2(4.1)ημ(x)-ψμ(x)=23j/4cμ(D2-jRθμx-(k1δ1,k2δ2)),
where D2-jx=(2jx1,2j/2x2), δ1,δ2>0 and all functions cμ satisfy the following.(i)
For|κ|≤4⌈|t|+2/r⌉+δ we need
(4.2)|cμ(κ)(x)|≤ɛ(1+|x|)-8⌈|t|+2/r+δ⌉.(ii)
Furthermore we need(4.3)|ĉμ(ξ)|≤ɛmin⁡⁡(1,2-j+|ξ1|+2-j/2|ξ2|)4⌈|t|+2/r+δ⌉.We have used the notation from Definition3.2. The coarse-scale molecules, j=1, must take the form ημ(x)-ψμ(x)=cμ(x-k/δ0), δ0>0, where cμ satisfies (4.2).Then{ψμ}μ∈𝒥×ℤ2 is a family of curvelet molecules with regularity 4⌈|t|+2/r+δ⌉ and motivated by {ημ}μ∈𝒥×ℤ2 being a tight frame for L2(ℝ2), we formally define 〈f,ψμ′〉 as
(4.4)〈f,ψμ′〉:=∑μ∈J×Z2〈ημ,ψμ′〉〈f,ημ〉,f∈Gp,qs.
It follows from (3.6) and Proposition 3.4 that 〈·,ψμ′〉 is a bounded linear functional on Gp,qs; in fact we have
(4.5)∑μ∈J×Z2|〈ημ,ψμ′〉||〈f,ημ〉|≤‖{∑μ∈J×Z2|〈ημ,ψμ′〉||〈f,ημ〉|}μ′∈J×Z2‖gp,qs≤C‖〈f,ημ〉‖gp,qs≤C‖f‖Gp,qs.
Furthermore, {ψμ}μ∈𝒥×ℤ2 is a norming family for Gp,qs as it satisfies ∥〈f,ψμ〉∥gp,qs≤C∥f∥Gp,qs. This can be used to show that S is a bounded operator on Gp,qs, and for small enough ɛ this will be the key to showing that {ψμ} is a frame for Gp,qs.Theorem 4.1.
There existsɛ0,C1,C2>0 such that if {ψμ}μ∈𝒥×ℤ2 satisfies (4.1) for some 0<ɛ≤ɛ0, then one has
(4.6)C1‖f‖Gp,qs≤‖〈f,ψμ〉‖gp,qs≤C2‖f‖Gp,qsf∈Gp,qs.Proof.
That{ψμ}μ∈𝒥×ℤ2 is a norming family gives the upper bound; thus we only need to establish the lower bound. For this we use that {ɛ-1(ημ-ψμ)}μ∈𝒥×ℤ2 is also a norming family so we have
(4.7)‖〈f,ημ-ψμ〉‖gp,qs≤Cɛ‖f‖Gp,qs.
It then follows that
(4.8)‖f‖Gp,qs≤C‖〈f,ημ〉‖gp,qs≤C(‖〈f,ψμ〉‖gμs+‖〈f,ημ-ψμ〉‖gp,qs)≤C(‖〈f,ψμ〉‖gp,qs+ɛ‖f‖Gp,qs).
By choosing ɛ<1/C we get the lower bound.As one might guess from Theorem4.1, the boundedness of the matrix {〈ημ,S-1ψμ′〉}μ,μ′∈𝒥×ℤ2 on gp,qs is the key to showing that {S-1ψμ}μ∈𝒥×ℤ2 is also a frame for Gp,qs.Proposition 4.2.
There existsɛ0>0 such that if {ψμ}μ∈𝒥×ℤ2 satisfies (4.1) for some 0<ɛ≤ɛ0 and furthermore is a frame for G220=L2(ℝ2), then {〈ημ,S-1ψμ′〉}μμ′∈𝒥×ℤ2 is bounded on gp,qs.Proof.
The fact that{ψμ}μ∈𝒥×ℤ2 is a frame for L2(ℝ2) ensures that S-1 is a bounded operator on L2(ℝ2). We first show that S-1 is bounded on Gp,qs. This will follow from showing that
(4.9)‖(I-S)f‖Gp,qs≤Cɛ‖f‖Gp,qs,f∈Gp,qs,
choosing ɛ small enough and using the Neumann series. Assume for a moment that 𝒟:={dμ′μ}μ′,μ∈𝒥×ℤ2:={〈(I-S)ημ,ημ′〉}μ′,μ∈𝒥×ℤ2 satisfies
(4.10)‖Dz‖gp,qs≤Cɛ‖z‖gp,qs.
By using that S is self-adjoint, we then have
(4.11)‖(I-S)f‖Gp,qs≤C‖{〈(I-S)f,ημ′〉}‖gp,qs=C‖D{〈f,ημ〉}‖gp,qs≤Cɛ‖{〈f,ημ〉}‖gp,qs≤Cɛ‖f‖Gp,qs.
So to show (4.9) it suffices to prove (4.10). Note that
(4.12)〈(I-S)ημ,ημ′〉=∑μ′′∈J×Z2〈ημ,ημ′′〉〈ημ′′,ημ′〉-∑μ′∈J×Z2〈ημ,ψμ′′〉〈ψμ′′,ημ′〉=∑μ′′∈J×Z2〈ημ,ημ′′〉〈ημ′′-ψμ′′,ημ′〉+∑μ′′∈J×Z2〈ημ,ημ′′-ψμ′′〉〈ψμ′′,ημ′〉.
By setting
(4.13)D1:={d1(μ′)(μ′′)}:={〈ημ′′-ψμ′′,ημ′〉},D2:={d2(μ′′)(μ)}:={〈ημ,ημ′′〉},D3:={d3(μ′)(μ′′)}:={〈ψμ′′,ημ′〉},D4:={d4(μ′′)(μ)}:={〈ημ,ημ′′-ψμ′′〉},
we have the decomposition
(4.14)D=D1D2+D3D4.
Since {ɛ-1(ημ-ψμ)}μ∈𝒥×ℤ2 is a family of curvelet molecules with regularity 4⌈|t|+2/r+δ⌉, we have from (3.6) that ɛ-1𝒟1,𝒟2,𝒟3,ɛ-1𝒟4 are almost diagonal on gp,qs. Next, we use Remark 3.3, and by Proposition 3.4,
(4.15)‖Dz‖gp,qs≤Cɛ‖z‖gp,qs.
Consequently, (4.9) holds, and for sufficiently small ɛ the operator S-1 is bounded on Gp,qs. Finally, let z:={zμ}μ∈𝒥×ℤ2∈gp,qs and h=:∑μzμημ. By using (2.9) we have that h∈Gp,qs, and as {ψμ}μ∈𝒥×ℤ2 is a frame for L2(ℝ2), we have that S-1 is self-adjoint which gives
(4.16)∑μ∈J×Z2〈ημ,S-1ψμ′〉zμ=∑μ∈J×Z2〈S-1ημ,ψμ′〉zμ=〈S-1h,ψμ′〉.
If we combine this with {ψμ}μ∈𝒥×ℤ2 being a norming family (4.5), we get
(4.17)‖∑μ∈J×Z2〈ημ,S-1ψμ′〉zμ‖gp,qs=‖〈S-1h,ψμ′〉‖gp,qs≤C‖S-1h‖Gp,qs≤C‖h‖Gp,qs≤C‖z‖gp,qs
which proves that {〈ημ,S-1ψμ′〉}μ,μ′∈𝒥×ℤ2 is bounded on gp,qs.That{S-1ψμ}μ∈𝒥×ℤ2 is a frame for Gp,qs now follows as a consequence of {〈ημ,S-1ψμ′〉}μ,μ′∈𝒥×ℤ2 being bounded on gp,qs. We state the following results without proofs as they follow directly in the same way as in the Besov space case. The proofs can be found in [6]. First, we have the frame expansion.Lemma 4.3.
Assume that{ψμ}μ∈𝒥×ℤ2 is a family of curvelet molecules with regularity 4⌈|t|+2/r+δ⌉ and a frame for L2(ℝ2). If {〈ημ,S-1ψμ′〉}μ,μ′∈𝒥×ℤ2 is bounded on gp,qs, then for f∈Gp,qs one has
(4.18)f=∑μ∈J×Z2〈f,S-1ψμ〉ψμ
in the sense of 𝒮′.Next, we have that{S-1ψμ}μ∈𝒥×ℤ2 is a frame for Gp,qs.Theorem 4.4.
Assume that{ψμ}μ∈𝒥×ℤ2 is a family of curvelet molecules with regularity 4⌈|t|+2/r+δ⌉ and a frame for L2(ℝ2). Then {S-1ψμ}μ∈𝒥×ℤ2 is a frame for Gp,qs if and only if {〈ημ,S-1ψμ′〉}μ,μ′∈𝒥×ℤ2 is bounded on gp,qs.It follows from Proposition4.2, Lemma 4.3, and Theorem 4.4 that if {ψμ}μ∈𝒥×ℤ2 is a family of curvelet molecules which is close enough to curvelets, then the representation ∑μ∈𝒥×ℤ2〈f,S-1ψμ〉ψμ,f∈Gp,qs, has the same sparse representation properties as curvelets when measured in gp,qs. As a final result we also have a frame expansion with {S-1ψμ}μ∈𝒥×ℤ2.Lemma 4.5.
Assume that{ψμ}μ∈𝒥×ℤ2 is a family of curvelet molecules with regularity 4⌈|t|+2/r+δ⌉ and a frame for L2(ℝ2). If the transpose of {〈ημ,S-1ψμ′〉}μ,μ′∈𝒥×ℤ2 is bounded on gp,qs, then for f∈Gp,qs one has
(4.19)f=∑μ∈J×Z2〈f,ψμ〉S-1ψμ
in the sense of 𝒮′.All that remains now is to construct a flexible family of curvelet molecules which is close enough to curvelets in the sense of (4.1).


## 5.  Construction of Curvelet-Type Systems

In this section we construct a flexible curvelet-type system. We do this by showing that finite linear combinations of shifts and dilates of a functiong with sufficient smoothness and decay can be used to construct a system {ψμ}μ∈𝒥×ℤ2 that satisfies (4.1). From the previous section, we then have that the representation ∑μ∈𝒥×ℤ2〈f,S-1ψμ〉ψμ, f∈Gp,qs, has the same sparse representation properties as curvelets when measured in gp,qs.First we takeg∈CM+1(ℝ2), ĝ(0)≠0, which for fixed N′>2,M>0 satisfies(5.1)|g(κ)(x)|≤C(1+|x|)-N′,|κ|≤M+1.
Next, for m≥1 we define gm(x):=Cgm2g(mx), where Cg=:ĝ(0)-1. It then follows that(5.2)|gm(κ)(x)|≤Cm2+|κ|(1+m|x|)-N′,|κ|≤M+1,∫R2gm(x)dx=1.
We recall that curvelets (2.6) are a family of curvelet molecules for any regularity R∈ℕ. From the definition of a family of curvelet molecules (Definition 2.1), we have that for j≥2 curvelet molecules can be expressed as
(5.3)ημ(x)=23j/4aμ(D2-jRθμx-(k1δ1,k2δ2)),
where aμ must satisfy (2.8) and (2.9). So to construct a family of curvelet molecules {ψμ}μ∈𝒥×ℤ2 which satisfy (4.1), we need to construct a family of functions {bμ}μ∈𝒥×ℤ2 such that aμ-bμ satisfy (4.2) and (4.3). We define {ψμ}μ∈𝒥×ℤ2 as
(5.4)ψμ(x):=23j/4bμ(D2-jRθμx-(k1δ1,k2δ2))
for j≥2 and to construct {bμ}μ∈𝒥×ℤ2 we also need the following set of finite linear combinations:(5.5)ΘK,m:={bμ:bμ(⋅)=∑i=1Kcigm(⋅+di),ci∈C,di∈R2}.
We have omitted the construction of ψμ for j=1 as it follow in a similar way.Proposition 5.1.
LetN′>N>2,M>0 and j>0. If g∈CM+1(ℝ2),ĝ(0)≠0, fulfills (5.1) and aμ∈L2(ℝ2)∩CM+1(ℝ2) fulfills
(5.6)|aμ(κ)(x)|≤C(1+|x|)-N′,|κ|≤M+1,|âμ(ξ)|≤Cmin⁡(1,2-j+|ξ1|+2-j/2|ξ2|)M+1,
then for any ɛ>0 there exist K,m≥1 (m independent of j) and bμ∈ΘK,m such that
(5.7)|aμ(κ)(x)-bμ(κ)(x)|≤ɛ(1+|x|)-N,|κ|≤M,|âμ(ξ)-b̂μ(ξ)|≤ɛmin⁡⁡(1,2-j+|ξ1|+2-j/2|ξ2|)M.Proof.
Letɛ>0 and κ,|κ|≤M, be given. We construct the approximation of aμ in direct space in three steps. First by a convolution operator ωm=aμ*gm, then by θq,m which is the integral in ωm taken over a dyadic cube Q, and finally by a discretization over smaller dyadic cubes bl,q,m. From (5.2) we have
(5.8)aμ(κ)(x)-ωm(κ)(x)=∫R2(aμ(κ)(x)-aμ(κ)(x-y))gm(y)dy.
Define U:=mλ/2N, where λ:=min(1,N′-N). For |x|≤U, we use the mean value theorem to get
(5.9)|aμ(κ)(x)-aμ(κ)(x-y)|≤Cmin⁡(1,|y|).
Inserting this in (5.8) we have
(5.10)|aμ(κ)(x)-ωm(κ)(x)|≤C∫R2min⁡(1,|y|)m2(1+m|y|)N′dy≤Cm-λ≤Cm-λ/2UN≤Cm-λ/2(1+|x|)N.
For |x|>U, we split the integral over Ω:={y:|y|≤|x|/2} and Ωc. If y∈Ω, then |x-y|≥|x|/2, and we have
(5.11)∫Ω|aμ(κ)(x)-aμ(κ)(x-y)||gm(y)|dy≤C(1+|x|)-N′≤C(1+U)λ(1+|x|)N≤Cm-λ2/2N(1+|x|)N.
Integrating over Ωc with |x|>U gives
(5.12)∫Ωc|aμ(κ)(x)-aμ(κ)(x-y)||gm(y)|dy≤C(1+|x|)N′+∫ΩcCm2(1+|x-y|)N′(1+m|y|)N′dy≤C(1+|x|)N′+Cm-λ(1+|x|)N≤C(m-λ2/2N+m-λ)(1+|x|)N.
So by choosing m sufficiently large in (5.10)–(5.12), we get
(5.13)|aμ(κ)(x)-ωm(κ)(x)|≤ɛ3(1+|x|)-N.
For the next step we fix m and choose q∈ℕ. Let Q denote the dyadic cube with sidelength 2q+1, sides parallel with the axes and centered at the origin. We then approximate ωm with θq,m defined as
(5.14)θq,m(⋅)=∫Qaμ(y)gm(⋅-y)dy.
In which case we have
(5.15)ωm(κ)(x)-θq,m(κ)(x)=∫Qcaμ(y)gm(κ)(x-y)dy,
and it follows that
(5.16)|ωm(κ)(x)-θq,m(κ)(x)|≤∫|y|≥2qCm2+|κ|(1+|y|)N′(1+m|x-y|B)N′dy:=L.
We first estimate the integral for |x|≤2q-1 which gives |y|>|x| and |x-y|≥2q-1. Hence we obtain
(5.17)L≤Cm2+|κ|(1+|x|)N′∫|u|≥2q-11(1+m|u|)N′du≤Cm|κ|-λ2-λq(1+|x|)N′.
For |x|>2q-1, we split the integral over Ω:={y:|y|≥2q}∩{y:|y|≤|x|/2} and Ω′:={y:|y|≥2q}∩{y:|y|>|x|/2}. If y∈Ω, then |x-y|≥|x|/2, and we get
(5.18)∫Ωm2+|κ|(1+|y|)N′(1+m|x-y|)N′dy≤Cm2+|κ|(1+m|x|)N′∫|y|≥2q1(1+|y|)N′dy≤Cm|κ|-λ2-λq(1+|x|)N.
Similar for Ω′ we have
(5.19)∫Ω′m2+|κ|(1+|y|)N′(1+m|x-y|)N′dy≤C(1+|x|)N′∫R2m2+|κ|(1+m|x-y|)N′dy≤Cm|κ|(1+|x|)N′≤m|κ|2-λq(1+|x|)N.
By choosing q sufficiently large in (5.17)–(5.19), we obtain
(5.20)|ωm(κ)(x)-θq,m(κ)(x)|≤ɛ3(1+|x|)-N.
For the final step we fix q, choose l∈ℕ, and approximate θq,m by a discretization
(5.21)bl,q,m(⋅)=∑I∈Hl,q|I|aμ(xI)gm(⋅-xI),
where xI is the center of the dyadic cube I and Hl,q is the set of dyadic cubes with sidelength 2-l which together give Q. Note that bl,q,m∈ΘK,m, K=2q+l+1. We introduce F(·):=aμ(·)gm(κ)(x-·) which gives
(5.22)|θq,m(κ)(x)-bl,q,m(κ)(x)|≤∑I∈Hl,q∫I|aμ(y)gm(κ)(x-y)-aμ(xI)gm(κ)(x-xI)|dy≤∑I∈Hl,q∫I|F(y)-F(xI)|dy.
By using the mean value theorem, we then get
(5.23)|θq,m(κ)(x)-bl,q,m(κ)(x)|≤∑I∈Hl,q∫I|y-xI|max⁡|κ′|≤1z∈l(xI,y)|F(κ′)(z)|dy≤C22q-lmax⁡|κ′|≤|κ|+1|z|≤2q+1|gm(κ′)(x-z)|,
where l(xI,y) is the line segment between xI and y. If |x|≤2q+2 and |κ′|≤|κ|+1, then we have
(5.24)|gm(κ′)(x-z)|≤Cm3+|κ|≤Cm3+|κ|2qN(1+|x|)N.
For |x|>2q+2 and |z|≤2q+1, we have |x-z|≥|x|/2, and hence for |κ′|≤|κ|+1, it follows that
(5.25)|gm(κ′)(x-z)|≤Cm3+|κ|(1+m|x|)N′≤Cm3+|κ|(1+|x|)N′.
By choosing l sufficiently large, we obtain by combining (5.23)–(5.25) that
(5.26)|θq,m(κ)(x)-bl,q,m(κ)(x)|≤ɛ3(1+|x|)-N.
Finally by combining (5.13), (5.20) and (5.26), we get
(5.27)|aμ(κ)(x)-bl,q,m(κ)(x)|≤ɛ(1+|x|)-N.
To approximate aμ in frequency space we use three steps similar to the approximation in direct space. Note that bl,q,m still fulfills (5.27) if we choose l,q,m even larger. First we use ω̂m to approximate âμ in which case we have
(5.28)|âμ(ξ)-ω̂m(ξ)|=|âμ(ξ)M/(1+M)âμ(ξ)1/(1+M)(1-Cgĝ(ξm))|≤Cmin⁡(1,2-j+|ξ1|+2-j/2|ξ2|)M(1+|ξ|)-1|1-Cgĝ(ξm)|.
By choosing ξg>0 such that C(1+ξg)-1|1-Cgĝ(ξ/m)|≤ɛ/3 and m such that C|1-Cgĝ(ξ/m)|≤ɛ/3 for |ξ|<ξg, we get
(5.29)|âμ(ξ)-ω̂m(ξ)|≤ɛ3min⁡(1,2-j+|ξ1|+2-j/2|ξ2|)M.
Next, we fix m, choose q and limit the Fourier integral of aμ to Q from the approximation in direct space,
(5.30)θq,m′(ξ)=ĝm(ξ)∫Qaμ(x)eix⋅ξdx.
This gives
(5.31)|ω̂m(ξ)-θq,m′(ξ)|≤|ĝm(ξ)|∫|x|>2q|aμ(x)eix⋅ξ|dx≤C2-λq.
In the last step, we fix q and approximate θq,m′ by b̂l,q,m. We introduce G(x):=aμ(x)eix·ξ which gives
(5.32)|θq,m′(ξ)-b̂l,q,m(ξ)|≤|ĝm(ξ)||∫Qaμ(x)eix⋅ξdx-∑I∈Hl,q|I|aμ(xI)eixI⋅ξ|≤|ĝm(ξ)|∑I∈Hl,q∫I|G(x)-G(xI)|dx≤C22q-l1+|ξ/m|max⁡|κ′|≤1x∈R2|G(κ′)(x)|≤Cm22q-l.
By combining (5.29)–(5.32) for sufficiently large l,q,m, we get
(5.33)|âμ(ξ)-b̂l,q,m(ξ)|≤ɛmin⁡⁡(1,2-j+|ξ1|+2-j/2|ξ2|)M.
It follows that by choosing l,q,m large enough bl,q,m fulfills both (5.7) and (5.4). Furthermore, we have bl,q,m∈ΘK,m, K=2q+l+1.


## 6. Discussion

In this paper we studied a flexible method for generation curvelet-type systems with the same sparse representation properties as curvelets when measured ingp,qs. With Proposition 4.2, Lemma 4.3, and Theorem 4.4 we proved that a system of curvelet molecules which is close enough to curvelets has these sparse representation properties. Furthermore, with Proposition 5.1 we gave a constructive description of how such a system of curvelet molecules can be constructed from finite linear combinations of shifts and dilates for a single function with sufficient smoothness and decay.Examples of functions with sufficient smoothness and decay are the exponential functione-|·|2 and the rational functions (1+|·|2)-N with N sufficiently large. An example with compact support can be constructed by using a spline with compact support. Furthermore as the system is constructed using finite linear combinations of splines, we get a system consisting of modulated compactly supported splines.



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*Source: 876315-2012-02-06.xml*